CN103230283B - Method for optimizing ultrasonic probe imaging plane space position calibration - Google Patents

Abstract

An optimization method for calibrating the spatial position of an imaging plane of an ultrasonic probe, which belongs to the field of medical ultrasonic three-dimensional imaging and ultrasonic image fusion, is characterized in that it is a method for calibrating the transformation relationship between the spatial position of the imaging plane of the ultrasonic probe and the spatial position of a spatial position sensor. It is an optimization method for the space transformation matrix, which is realized by the following steps with the assistance of the 3D positioning system: 1. Construct the 3D positioning system; 2. Make the N-line model; 3. Establish the coordinate system set of the positioning system; 4. Calibrate the placement of the water tank model 5. Collect ultrasound images at different positions; 6. Extract the N-line markers and calculate the position of the ultrasound plane; 7. Map the N-line marker points on the ultrasound to the three-dimensional space plane; 8. Compare the three-dimensional coordinate points of the imaging plane with N on the water tank model for registration; 9. Optimal calculation; 10. Calculate the optimal transformation between the imaging plane and the position sensor. Its advantage is to reduce the deviation of the spatial position calibration of the imaging plane.

Description

An Optimal Method for Spatial Position Calibration of Ultrasonic Probe Imaging Plane

Technical field:

The invention belongs to the field of medical ultrasonic three-dimensional imaging and ultrasonic image fusion, and its specific applications include three-dimensional medical ultrasonic imaging, and spatial registration fusion imaging of medical ultrasonic images and other modes of images (such as CT and MRI).

Background technique:

As a nondestructive, real-time, and cheap imaging examination, medical ultrasound images have been widely used in medicine. At present, the ultrasound examination widely used in clinical practice adopts the B-mode imaging mode, commonly known as "B-ultrasound". B-type imaging obtains two-dimensional tomographic images that change in real time over time. Doctors rely on their understanding of human anatomy to form a judgment in their minds about whether the internal three-dimensional structure of the human body is normal or not. With the development of medical imaging technology, 3D ultrasound imaging has attracted more and more attention. At present, there are two ways to achieve 3D imaging. One is to directly use a 3D ultrasound probe to obtain ultrasonic detection in a rectangular pyramid-shaped sampling volume. The ultrasonic echoes in the three-dimensional sampling volume are used to reconstruct the three-dimensional image volume data; another way is to directly adopt the traditional B-mode imaging mode, through the acquisition of two-dimensional tomographic ultrasonic images of multiple known positions and the spatial grayscale The splicing of information reconstructs three-dimensional volume data and completes three-dimensional imaging. The above two methods have their own advantages and disadvantages. The former requires a specially designed and manufactured three-dimensional ultrasound probe, which is expensive, bulky, inconvenient to use, and requires specially matched ultrasound equipment; the latter can be directly used on existing ultrasound equipment through computer Processing to increase the three-dimensional imaging function, good compatibility with the ultrasound equipment in use, easy to use, and low cost.

When the traditional B-mode imaging mode is used for splicing and reconstruction of three-dimensional images, the core problem is to accurately know the relative positional relationship between each two-dimensional tomographic ultrasound images. At present, the commonly used method is to fix a position tracking sensor on the ultrasound probe to accurately track the spatial position of the ultrasound probe when collecting each two-dimensional ultrasound image, so as to perform three-dimensional splicing and reconstruction. In this mode, it is necessary to know the relative spatial position relationship between the spatial position sensor fixed on the ultrasonic probe and the imaging plane of the ultrasonic probe in advance , in order to carry out the three-dimensional splicing and reconstruction of the data. Therefore, it is necessary to design a method that can measure and calibrate the spatial transformation relationship between the imaging plane of the probe and the spatial position sensor. This process is called the spatial position calibration of the imaging plane of the probe .

In clinical medicine, it is a common application of ultrasound images to guide in vivo detection such as puncture by using ultrasound images. However, due to the relatively low resolution of ultrasound images, it is often necessary to use the fusion of other modes of images and ultrasound images to give full play to the real-time and safe ultrasound images, and the high resolution and high precision of radiological images (CT, MRI). Advantages, provide clinicians with more abundant and accurate anatomical structure information, and help them better complete interventional operations such as needle biopsy. In this case, in order to achieve the fusion of two or more images, the core technical problem is to achieve accurate alignment of the spatial positions of the two images, so that the image gray information of the two modalities complements each other. In this case, it is also necessary to track the spatial position of the real-time ultrasound image, so as to know where the ultrasound probe is in the three-dimensional space, and perform fusion display with the three-dimensional image data of CT or MRI at the corresponding position. Therefore, high-precision calibration of the spatial position of the imaging plane of the probe is also required.

During the development of the above technologies, a variety of methods for calibrating the spatial position of the imaging plane of the probe have been developed, mainly including point models, three-line models, plane models, and two-dimensional alignment models. For doctoral thesis at Hsu PW see Hsu PW. Freehandthree-dimensional ultrasound calibration, University of Cambridge; 2007 and Mercier see Mercier L, T, Lindseth F, et al. A review of calibration techniques for freehand3-D ultrasound systems. Ultrasound in medicine & biology. 2005, 31(2): 143-165 and others summarized the work in this field.

N line (also known as Z line) model see Comeau RM, Fenster A, Peters TM. Integrated MR and ultrasound imaging for improved image guidance in neurosurgery. Paper presented at: Medical Imaging1998: Image Processing1998; Comeau RM, Sadikot AF, Fenster A, et al al. Intraoperative ultrasound for guidance and tissue shift correction in image-guided neurosurgery. Medical Physics. 2000, 27:787-800 is the most commonly used two-dimensional alignment model. This model uses thin line-shaped targets with pre-designed positions, and every three thin lines form the shape of an English letter "N". When several N-shaped wire groups are placed in a water tank used for ultrasonic probe calibration, the spatial position relationship between these N-shaped wires is fixed (see Figure 2), and we use the water tank and the above fixed as The thin line of the ultrasound imaging target, as a whole model, is called the "water model". In practice, the precise spatial position of each group of N lines can be guaranteed through the design and manufacture of the water mold, or the precise spatial position of each group of N lines in the water mold can be known through measurement. When the ultrasonic probe is used to image the thin lines in the water phantom filled with clear water or ultrasonic coupling fluid, three N-shaped thin lines in groups of three will appear on the ultrasonic image as three bright spot targets E, F, and G (see Fig. 3a, Figure 3b). It can be seen from Figure 2 and Figure 3 that by identifying which group of N-lines on the water model a certain group of bright spot targets on the ultrasonic imaging plane presents, the location of the N-line can be known according to the structural design of the water model. The coordinate values corresponding to the two straight lines AB and CD in the coordinate system of the water tank model (see Figure 2). At the same time, as shown in Figure 3, when the three-dimensional space position of point ABCD on the water model is known, and the positions of E, F, and G are obtained through ultrasonic image measurement, the ratio of sides EF to GF can be obtained from similar triangles ΔEBF and ΔGCF , get the ratio of edge BF to CF. Through the principle of similar triangles, the ratio of the distances between the three bright spots E, F, and G on the ultrasound image can be used to obtain the Z coordinate of the oblique point in the three-dimensional space, thereby obtaining the three-dimensional space position of point F, In this way, an " N-line marker point " on an ultrasound image is obtained. A plane in the three-dimensional space of the water tank model can be fitted by multiple sets of "N-line marker points" simultaneously imaged on an ultrasound image plane (from the principle of determining the plane by three points, more than 3 N-line marker points are required), That is, the plane of ultrasound imaging. Then, according to the three-dimensional coordinates of these N-line marker points in the water model, the spatial position of the ultrasonic imaging plane relative to the water model coordinate system can be calculated. Then by querying the position of the spatial position sensor consolidated on the ultrasonic probe in the reference coordinate system when collecting the ultrasonic image, combined with the spatial position of the water model coordinate system in the reference coordinate system, the required calibration ultrasonic _See Comeau RM, Fenster A, Peters TM. Integrated MR and ultrasound imaging for improved image guidance in neurosurgery. Paper presented at: Medical Imaging1998: Image Processing 1998; Comeau RM, Sadikot AF, Fenster A, et al. Intraoperative ultrasound for guidance and tissue shift correction in image-guided neurosurgery. Medical Physics. 2000, 27:787-800.

However, by searching the literature and analyzing the existing calibration methods, it can be seen that in the actual calibration process, the effective imaging sound field of ultrasound imaging has a certain thickness (as shown in Figure 3a), and at the same time, the lateral resolution of ultrasound imaging is poor In the obtained ultrasound image, the marked thin line diffuses into a long elliptical bright spot in the transverse direction (X-axis direction) (as shown in Figure 3b), which makes it impossible to accurately determine the location of the thin line on the ultrasound image. Location. This not only affects the accuracy of selecting the positions of marker points E, F, and G in the ultrasonic image, but also, since the calculation of the three-dimensional coordinates of point F in the water model coordinate system is determined by the ratio of the lengths of EF to FG, the above E, F, and G Errors in the positions of F and G will lead to inaccurate calculation of the three-dimensional coordinates of point F. Moreover, in the actual design of the water model, in order to ensure that enough N-shaped line groups can be covered in one ultrasound image, the distance between the AB line and the CD line cannot be too wide, which leads to the comparison of the θ angle in Figure 3 Small, so that the position error of the marker point selection on the image is amplified in the Z direction. For multiple independently extracted N-line marker points in imaging, the accuracy of their three-dimensional positions in the water model coordinate system and the coplanarity of these marker points are lost, resulting in the calculated position of the imaging plane Inaccurate, so there is a large uncertainty error in the spatial transformation relationship between the imaging plane of the obtained ultrasonic image and the spatial position sensor.

By analyzing the research results reported by other researchers, it can be seen that during the calibration process of the traditional N-line model, the data of the translation component in the thickness direction of the imaging plane (the direction perpendicular to the ultrasound image plane) has a large dispersion. It shows that the estimation accuracy of the traditional N-line method is low. See Pagoulatos N, Haynor DR, Kim YA fastcalibration method for 3-D tracking of ultrasound images using a spatial localizer.Ultrasound in medicine\&biology.2001,27(9):1219-1229 in the research paper of Pagoulatos et al. error, but its data clearly show the existence of this problem. Hsu PW See Hsu PW, Prager RW, Gee AH, et al. Real-time freehand3D ultrasound calibration. Ultrasound in medicine & biology. 2008, 34(2):239-251 and Chen et al See Chen T, Thurston A, Moghari M , et al.Areal-time ultrasound calibration system with automatic accuracy control and incorporation of ultrasound sectionthickness.Proceedings of SPIE.2008,6918:2A1-11 and Chen TK,Thurston AD,Ellis RE,et al.A real-time freehand ultrasound calibration system with automatic accuracy feedback and control. Ultrasound in medicine & biology. 2009, 35(1): 79-93 also discussed this issue, and called it "elevational error ". The main source of this error is that the thickness and resolution of the ultrasonic beam cause the detected thin line to diffuse into a relatively large bright spot on the ultrasonic image, resulting in an error in its position estimation. This error is ubiquitous.

In order to overcome this defect, see Zhu Liren, Li Wenjun, Ding Hui, Wang Guangzhi, a solution method to improve the calibration accuracy of N-line model probes in the literature, Chinese Journal of Biomedical Engineering, 31(3):337-343, 2012, A method has been proposed to improve the accuracy of N-line marker point extraction by "coplanar constraints", thereby improving the calibration accuracy of the imaging plane of the probe . Experimental data show that this method can improve the accuracy performance. However, this method relies on rigidly projecting the extracted N-line marker points onto an artificially fitted virtual plane. Since there are errors in the fitted plane itself, each N-line can only be partially improved through projection processing. The impact caused by the non-coplanar marker points, but the accuracy of the plane cannot be guaranteed.

The research of Hsu P-W et al. proposes to introduce images of multiple imaging planes. The processing method is to add a separate step after the routine calibration process to refine the estimation of the translation in the thickness direction. See Hsu PW, Prager RW, Gee AH, et al. Real-time freehand3D ultrasound calibration. Ultrasound in medicine & biology. 2008,34(2):239-251. Their research proves that under the traditional solution framework, the introduction of information from multiple slices can help improve the repeatability of the calibration transformation. However, such methods require additional processing steps and require the use of a modified N-line model. See Hsu PW, Prager RW, Gee AH, et al. Real-time freehand3D ultrasound calibration. Ultrasound in medicine & biology.2008, 34(2):239-251; In addition, this method first assumes that all other position components except the imaging thickness direction are accurate after conventional calibration, but this assumption may not be true in most practical applications.

Considering the shortcomings of the above-mentioned calibration method, the present invention improves the spatial position calibration method of the imaging plane of the probe based on the N-line model . In view of the problem that the traditional N-line calibration method completely relies on a single two-dimensional ultrasonic image to extract the position of the target point and calculate the three-dimensional position , resulting in a large error in the thickness direction of the imaging, we propose to use the spatial position sensor consolidated on the ultrasonic probe to obtain multiple The relative positions of the imaging planes are fixed and known. By extracting the target points on these imaging planes, the three-dimensional spatial positions of the two-dimensional marker points extracted on each plane are reconstructed with the help of the spatial position sensor information, and then through these three-dimensional target points. The registration and optimization calculation with the thin lines in the imaged water model can achieve accurate calibration of the spatial position of the imaging plane of the probe. Because this method can effectively use the high-precision spatial position sensor to reconstruct the three-dimensional relative position information of the two-dimensional marker points, the thickness error of the marker points extracted in the traditional ultrasonic image is dispersed to each ultrasonic imaging plane, and the spatial orientation has randomness. It can reduce the deviation of the spatial position calibration of the imaging plane caused by the loss of coplanarity in the traditional method.

Invention content:

The purpose of the present invention is to provide a new calibration method and processing flow for the imaging plane position of an ultrasonic probe, thereby reducing the calibration error caused by the inherent effective sound field thickness and low imaging resolution of ultrasound imaging, and improving the accuracy of calibration. precision . The designed method and process can be applied to a three-dimensional ultrasound imaging system that uses optical or electromagnetic spatial position sensors to track the spatial position of the ultrasonic probe, or a system that uses these spatial position sensors to track the ultrasonic probe for multi-mode three-dimensional medical image fusion. A method and device for accurately and effectively calibrating the spatial position of the ultrasonic image plane of a system probe in the manufacture and use of such ultrasonic instruments. The present invention can overcome the disadvantage of "thickness error" caused by simply relying on the position extraction of diffused and blurred target points on the two-dimensional ultrasonic image when the existing N-line model is calibrated, so that the target points are out of coplanarity , making the measurement and calibration of the ultrasonic imaging plane more scientific, reasonable, simple and easy to implement, and the designed method and process are highly practical and have clear use value in ultrasonic instruments.

In the research, we noticed that when the traditional N-line model is used for calibration processing, since the target points on each ultrasound image are extracted and estimated separately, the direction with the largest error of these marker points appears in the vertical ultrasound imaging plane Therefore, it is difficult to reduce this error by averaging the errors of multiple images. Even if it is calibrated by collecting multiple two-dimensional ultrasound images as done in some literatures, the error is always in the direction perpendicular to the imaging plane of the probe. The direction is more prominent. Therefore, we hope to overcome the orientation of this error and reduce the error of calibration through the processing method and strategy of the present invention.

At the same time, we noticed that the process of using the N-line model to calibrate the spatial position of the ultrasonic imaging plane is actually to reconstruct and estimate the position of the target point on one or several two-dimensional slice images of a specific posture of the N-line model. The process of slicing position and pose information in 3D space. Therefore, the essence of the calibration problem is to register the two-dimensional target points extracted from the two-dimensional ultrasonic tomographic image with a batch of thin line objects in the three-dimensional space. The core of this technique is the registration of target points on a 2D image to slices of a 3D model. Theoretically speaking, like all registration techniques of single-plane 2D images to 3D space, the error is always the largest in the direction perpendicular to the 2D image, and by extending the registration problem to 3D space, it is possible Reduce the large error in a specific direction, so that the overall accuracy of the calibration is improved. We consider that a group of target points on two -dimensional ultrasound images with multiple orientations whose relative positions are known are mapped to several imaging planes distributed in space non-parallel with the help of the information of the spatial position sensor on the probe. The spatial distribution of two-dimensional marker points is restored to a set of three-dimensional spatial distribution, and then through the registration calculation from three-dimensional space to three-dimensional space, the marker points are registered to the known three-dimensional thin line model, thereby improving the accuracy of calibration . Compared with the traditional calibration method, the method of the present invention may obtain more accurate position calibration results through the averaging and redistribution of errors in all directions.

According to this idea, we established a processing strategy for registration optimization of landmark points on multiple 2D ultrasound images and a 3D N-line model, and designed a feasible processing method and process. Under the framework of this two-dimensional and three-dimensional registration problem, the accuracy and precision of the final calibration result can be significantly improved by introducing information of different angles and positions through optical or electromagnetic sensors on the ultrasound probe. According to our literature research, we have not found any researchers who have expounded the calibration of ultrasound imaging planes from the perspective of first performing 2D to 3D position reconstruction, and then performing 3D space registration and optimization. Therefore, our proposed processing strategy and The processing method is novel and original.

In terms of system configuration and data acquisition, the invented method is exactly the same as the hardware device required by traditional N-line model calibration, and maintains the advantages of N-line model calibration. When acquiring each ultrasonic imaging plane, the spatial coordinates and directions of the spatial position sensors consolidated on the ultrasonic probe should be acquired synchronously, so that the relative positional relationship between the acquired ultrasonic imaging planes can be known. After collecting the required set of ultrasonic images, instead of following the traditional method, by extracting the two-dimensional marker points on the ultrasonic image and using the three-dimensional position information of the thin lines designed by the water model, the N-line marker points are calculated one by one in the water model. The three-dimensional position in the coordinate system is then calculated by fitting the spatial position of the imaging plane, thereby calculating the transformation matrix between the position sensor and the imaging plane, but entering the following core processing flow designed by the present invention.

(1) First extract the position of the thin line marker points on the ultrasound imaging plane (two-dimensional coordinates in each ultrasound imaging plane), corresponding to each ultrasound image, extract all N-shaped marker thin lines to present bright spots on the ultrasound image The two-dimensional position coordinates of ;

(2) With the help of the relative positional relationship between the ultrasonic imaging planes obtained by the spatial position sensor during ultrasonic image acquisition, the two-dimensional ultrasonic imaging planes are spatially combined to form several two-dimensional imaging planes distributed in a certain three-dimensional space, And bind the positions between these planes to form a set of imaging planes distributed in three-dimensional space. Then, according to the imaging plane corresponding to the two-dimensional position of each marker point, with a certain ultrasound image plane as a reference, with the help of the spatial position sensor information when each image is collected, the two-dimensional marker points extracted on each imaging plane are successively extracted Mapped to the corresponding three-dimensional ultrasound imaging plane, thus forming a batch of marker points distributed in three-dimensional space, which correspond to the intersection points of the imaging plane and the thin line when cutting the thin line model from different directions;

(3) Initialize the matching position between the ultrasonic imaging plane and the water model position. Using a certain ultrasonic image plane collected, according to the traditional N-line calibration method, calculate the initial three-dimensional space position of the imaging plane relative to the water model coordinate system, and get The initial space transformation matrix from the imaging plane to the water model coordinate system, and then map the three-dimensional marker points on all imaging planes bound to the position in the previous step to the corresponding three-dimensional space position of the water model according to this space transformation, forming a three-dimensional spatial distribution A batch of coarse registration points (initial registration points);

(4) In an ideal situation without errors, the above-mentioned three-dimensional marker points should accurately fall on the corresponding thin lines. Since there is a certain position error between the two-dimensional ultrasonic image plane calculated by using one of the above ultrasonic images and the three-dimensional water model, there will inevitably be a gap between the marker points on the other ultrasonic image planes and the thin lines distributed in three dimensions. If there is a certain deviation, our goal is to find a space transformation that minimizes the distance between the above-mentioned three-dimensional spatial distribution of the registration mark points and the corresponding thin lines on the three-dimensional water model, so as to obtain an optimal transformation relationship. Since these imaging planes are not parallel to each other, the "thickness error" inherent in conventional methods due to the maximum error of marker points having similar orientations can be overcome. In the processing, we use the optimization calculation method for iterative processing, and find the optimal transformation by changing the spatial position and orientation of the entire 3D marker point group to be registered;

(5) According to the above considerations, the objective function of setting optimization is to minimize the distance between the points to be registered on each ultrasonic imaging plane and the corresponding thin lines on the three-dimensional water model; the initial position of the space is calculated in the third step above The initial transformation matrix; the adjustable variable is the translation and rotation of the imaging plane position in the water tank model in the 6-degree-of-freedom space; therefore, the optimal iterative calculation can be performed based on the above conditions. By setting the error threshold or the maximum number of iteration steps of the optimization calculation, and controlling the termination of the iteration, an optimized space transformation matrix will finally be obtained, which represents the best match between the collected multiple imaging planes and the water model. The resulting transformation between the imaging plane and the sensor is optimal;

(6) According to the spatial position of the spatial position sensor solidified on the probe when collecting each ultrasonic image, and the corresponding position of a group of slices obtained above for the best match, calculate the imaging plane of the ultrasonic image and the position of the sensor solidified on the probe The optimal spatial transformation relationship between the spatial position sensors of That is the goal that the present invention is looking for.

The invention provides a method and a complete processing flow for improving the calibration accuracy of the ultrasonic probe imaging plane space position, which can reduce the calibration error caused by the inherently large sound field thickness and low resolution of the ultrasonic imaging. The method and process designed by the present invention can become a method for accurately and effectively calibrating the spatial position of the ultrasonic image plane of the system probe in the manufacture and use of this type of ultrasonic instrument. The present invention overcomes the "thickness error" caused by extracting the position of the diffused target point on the ultrasonic image when using the commonly used N-line model for calibration, and makes the processing of measurement and calibration more scientific, reasonable, simple and easy, and the designed method and flow The utility model has strong practicability and has good practical value.

Description of drawings

Figure 1 is a schematic diagram of the composition of the entire measurement and calibration system and the relationship between the coordinate systems of each subsystem;

1 personal computer

2 Three-dimensional positioning measuring instrument

3 positioning transmitter

4 3D Probing Pen

5 Ultrasonic probe and positioning sensor

6 water tank model

7 Ultrasound Imager

Figure 2 is the geometric representation of the N-line model and the definition of the model coordinate system;

1 positioning pit

Figure 3 is a schematic diagram of the geometric principle of N-line marker point extraction and the extraction error caused by sound field thickness and image blur;

Figure 3a shows the intersection of the imaging sound field with thickness and the N line

Figure 3b Diffuse spots of thin lines on the actual ultrasound image

Figure 3c Viewed from the top of the water tank, the geometric principle for solving the three-dimensional coordinates of point F

Fig. 4 adopts the schematic diagram of the geometric principle of multi-imaging plane synchronous calibration by the method of the present invention;

Figure 5 is a schematic diagram of the geometric relationship between multiple imaging planes and the initial registration of the water model;

Fig. 6 is a flow chart of optimizing calculations using multiple ultrasound image planes;

The traditional method that Fig. 7 actually calculates obtains and the comparative figure of calibration error of the present invention's method;

Fig. 8 is the program flow chart of the present invention.

Detailed ways

In order to realize the solution method proposed by the present invention and verify its effectiveness, we have actually built a test system and conducted a performance test comparison experiment, comparing the traditional N-line The calibration method and the solution accuracy of the method of the present invention and the repeatability error of calibration, and the situation that the error decreases with the number of ultrasonic images participating in the calibration, experiments have confirmed that the method of the present invention can be faster when there are multiple ultrasonic images. The calibration error is reduced, and the calibration accuracy is better when there is a larger "thickness error", so that the calibration process of the probe can be easier and more reliable. Provide the program flowchart of the present invention among Fig. 8, the example of the test comparison of concrete realization is as described in the following steps:

In the first step, a three-dimensional positioning measurement system for ultrasonic probe calibration is constructed. Our three-dimensional positioning measurement system for ultrasonic probe calibration includes four main devices: three-dimensional positioning measuring instrument, water tank model for calibration imaging, medical ultrasound imaging system and personal computer. Among them, the three-dimensional positioning measuring instrument, the water tank model, and the ultrasonic imaging system are used to measure the calibrated ultrasonic probe, and the personal computer is used for the acquisition and data processing of the measurement data. The three-dimensional positioning measuring instrument includes the following four main components: a 6-degree-of-freedom spatial position sensor fixed on the ultrasonic probe, a high-precision three-dimensional positioning probe, a wireless transmitter for positioning as a spatial reference coordinate system, and a measuring system control And the control box for data acquisition and transmission. The ultrasonic imaging system includes two parts, the calibrated ultrasonic probe and the imaging system host, which are used to generate the ultrasonic image of the water tank model to be collected. The above-mentioned 6-DOF spatial position sensor is consolidated on the calibrated ultrasonic probe. An image capture card is installed in the personal computer to collect the ultrasonic image output from the ultrasonic imaging system. At the same time, the personal computer collects the positioning of the 6-degree-of-freedom spatial position sensor output from the control box of the three-dimensional positioning measuring instrument through the USB interface. data and positioning data of the 3D positioning probe tip. The composition of the three-dimensional positioning measurement system for ultrasonic probe calibration is shown in Figure 1;

Step 2: Make the N line model. According to the following steps, a N-wire calibration model was made using a plexiglass plate and a thin nylon wire:

(1) First, make a rectangular water tank 1 with a plexiglass plate, the size of which is 240*180*120mm ³ , and the thickness of the tank wall is 5mm. According to the designed number of N wires and the spatial layout, a batch of small threading holes with a diameter of 0.2mm are coaxially drilled at the corresponding precise positions on the two opposite walls of the box with a size of 240*180mm for threading. Pass the thin nylon wire with a diameter of 0.08mm to form a batch of N-shaped wire pairs in the water tank. At the same time, at the four corners of the front and rear box walls, 5mm away from the outer edge of the box, four precisely positioned conical positioning pits are drilled. The depth of the conical positioning pit on the water tank we made is 0.2mm, and the apex angle of the cone is 90°. The four positioning pits are distributed in a rectangular shape on the plane of the box wall, the distance in the horizontal direction is 230mm, and the distance in the vertical direction is 170mm. The position of the positioning pit and the relative position of the above-mentioned small threading hole are processed by numerical control to ensure its accuracy, which is used to accurately measure the position of the water tank and the thin line in the space positioning system,

(2) After the water tank is completed, as shown in the partial enlarged view of Figure 2 and Figure 3a, a thin nylon wire with a diameter of 0.08mm is passed through a threading hole A on the tank wall and led to the opposite side The corresponding coaxial small hole B on the box wall, tighten the nylon wire and then fold it back from the small hole B to the adjacent small hole C with the same horizontal height on the same side wall to form an N-shaped oblique line; pull the nylon wire Then turn back from C to the small threading hole D on the opposite side wall plate, tighten and fix the nylon wire so that it forms an N-shaped wire pair in the water tank. By repeating this, several groups of horizontally arranged N-shaped wire pairs can be built in the water tank to form the target object for ultrasonic imaging calibration. Figure 1 shows three layers of horizontally arranged N lines, the highest layer contains 3 N shapes, and the remaining two layers each contain 5 N shapes. The heights of the three-layer N lines in the coordinate system of the water tank model are 65mm, 95mm and 125mm respectively, and the distance between the two parallel sides of the N line in the horizontal direction is 30mm, so the θ angle in Figure 3c is about 15°. The positions of the threading holes of each N wire in the water tank are precisely guaranteed by numerical control machining. In actual use, in order to improve the accuracy and ensure that there are enough N-line groups in the imaging field of view, you can choose to arrange 6-8 layers of N-lines according to the size of the field of view of the ultrasound probe and the resolution of the ultrasound image. 3 to 6 logarithms can also be arranged, so that the ultrasonic image can collect more thin-line targets, and make the target fill the field of view of the ultrasonic imaging as much as possible. After the whole threading work of the water tank is completed, use silica gel to close all threading holes so that the water tank is watertight. Then pour pure water into the water tank, and make it exceed the thin line of the highest layer by more than 30mm, and let the water tank stand still for 1 day, so that the air bubbles in the water can escape. Since all the fine wires are submerged in water, they can be imaged by an ultrasound imaging probe that is lowered into the water. In this way, the production of the water tank model for calibration is completed;

Step 3, establishing the transformation relationship between each relevant coordinate system and the coordinate system of the three-dimensional positioning measurement system used for ultrasonic probe calibration:

As shown in Figure 1, four coordinate systems are involved in the calibration experiment. The coordinate system includes the defined reference coordinate system of the high-precision three-dimensional positioning measurement system, represented by X _w Y _w Z _w , which is set on the wireless transmitter device for positioning, and the coordinate origin is at the center of the transmitter device; the water tank model coordinate system, Expressed by X _m Y _m Z _m , X _m is the length direction of the water tank, Z _m is the width direction of the water tank, Y _m is the height direction of the water tank, the origin of the coordinate system is set on the positioning pit at the upper left corner of the front wall of the water tank, each coordinate axis The direction follows the right-hand rule; the spatial position sensor coordinate system, represented by X _s Y _s Z _s , is set on the 6-DOF spatial position sensor, and the coordinate origin is at the center of the spatial position sensor; the ultrasonic imaging plane coordinate system, Denoted by X _u Y _u , it is set on the ultrasound image, the coordinate origin is set at the midpoint of the uppermost end of the ultrasound image, the X _u direction is the horizontal direction of the image, and the Y _u direction is the depth direction of the ultrasound imaging. In order to establish the transformation relationship between each coordinate system, the space transformation matrix T _wm of 4*4 is set to be the space from the water tank model coordinate system X _m Y _m Z _m to the reference coordinate system X _w Y _w Z _w Transformation matrix, T _ws is the space transformation matrix from the spatial position sensor coordinate system X _s Y _s Z _s to the reference coordinate system X _w Y _w Z _w , T _su is the coordinate system from the ultrasound imaging plane X _u Y _u to the space position sensor coordinate system X _s Y _s Z _s space transformation matrix used for positioning, T _um is from the water tank model coordinate system X _m Y _m Z _m to the ultrasonic imaging plane coordinate system X _u The space transformation matrix of Y _u ;

Step 4, calibrate the spatial position of the water model in the reference coordinate system

According to the position of the positioning pit, we determined the coordinate system of the water tank model X _m Y _m Z _m , that is, the positioning pit at the upper left corner of the front wall of the water tank is used as the origin of the coordinate system of the water tank model , and the straight line connecting the left and right upper corners of the positioning pit is defined as The X _m axis, the straight line connecting the two positioning pits on the left side and the downward direction is defined as the Y _m axis, and the direction connecting the positioning pits of the front and rear water tank walls is defined as the Z _m axis. In this way, in the water tank model coordinate system, each N-shaped thin line group is parallel to the X _m -Z _m plane, and each N-shaped straight line side is parallel to the Z _m axis, and its position coordinates in the water tank model coordinate system are known , and its specific arrangement is shown in Figure 2. In the calibration, the transformation matrix T _wm from the water tank model coordinate system to the reference coordinate system must be marked first by measurement. We use the high-precision three-dimensional electromagnetic positioning system of Canada NDI company for measurement. Positioning system transmitter used as a reference position. As shown in Figure 1, the launcher of the positioning system is selected as a fixed reference coordinate system X _w Y _w Z _w , according to the instructions of the positioning system, the water tank is placed in the most suitable space for the detection of the positioning launcher, and the positioning system is equipped with The tip of the high-precision three-dimensional positioning probe pen detects the three-dimensional space positions of the positioning pits arranged on the front and rear walls of the water tank one by one, and records their three-dimensional space positions in the reference coordinate system sequentially by the computer P _wi =(x _wi , y _wi , z _wi ), where i=1, 2, ..., 8 are the serial numbers of the positioning pits on the water tank model. From the definition of the water tank model coordinate system, the coordinates P _mi = (x _mi , y _mi , z _mi ) of these positioning pits in the water tank model coordinate system can be obtained as: P _m1 = (0,0,0), P _m2 =(230,0,0), P _m3 =(0,170,0), P _m4 =(230,170,0), P _m5 =(0,0,120), P _m6 =(230,0,120), P _m7 =( 0,170,120), P _m8 = (230,170,120), the unit is mm,

Note that the space transformation matrix T _wm from the water tank model coordinate system X _m Y _m Z _m to the reference coordinate system X _w Y _w Z _w is:

${\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; R</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ \mathrm{<span\; class="notranslate">\; t</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 00</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 01</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 02</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 10</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; 20</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; one</span>}}& {\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; twenty\; two</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}& {\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}& {\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; z</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

The 3*3 block R in the upper left corner is the rotation transformation matrix, and the 1*3 block t in the lower left corner is the translation transformation. Then the three-dimensional spatial positions P _wi =( x _wi ,y _wi ,z _wi ) of the eight positioning pits in the reference coordinate system and their corresponding coordinates P _mi =(x _mi ,y _mi ,z _mi ), satisfy the following space transformation equation:

$\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

Therefore, the space transformation matrix T _wm from the water tank model coordinate system to the reference coordinate system can be solved from equation group (2).

Step 5: Acquire multiple ultrasound images at different positions for the N line in the water model

As shown in Figure 1, a 6-DOF position sensor is fixed on the ultrasonic probe before the experiment, so that it is firmly connected with the ultrasonic probe and cannot be moved. Use an ultrasonic probe with a position sensor to image the thin lines in the water tank model, select an imaging position with clear imaging and no artifacts, stabilize the ultrasonic probe, and collect images and corresponding spatial positions. The ultrasonic image is output through the image output interface of the medical ultrasonic instrument, and is captured by the video capture card installed in the PC computer (the OSPREY100 video capture card is used in the experiment). The position and attitude matrix of the spatial position sensor in the reference coordinate system can be obtained by The high-precision three-dimensional positioning measurement system is directly read out and transmitted to the computer through the USB interface. After the collected ultrasonic images are input into the computer one by one, they are stored as Img(j), j=1, 2,..., J, J is the number of collected ultrasonic images, and the corresponding spatial position sensor to the reference coordinate system is stored at the same time Space transformation matrix T _ws (j). At different positions in the water model, with different probe attitudes, repeatedly acquire and store several ultrasonic images of different planes, and store the corresponding spatial transformation matrix T _ws (j) of the spatial position sensor. In our experiment, more than 7 ultrasound images and corresponding positioning data of different positions were collected each time.

Step 6, extract the position of the marker point on the ultrasound image and calculate the position of the ultrasound imaging plane

First define the ultrasound imaging plane coordinate system X _u Y _u , as shown in Figure 1. In this coordinate system, the two-dimensional coordinates of each marker bright spot in the ultrasound image coordinate system X _u Y _u are extracted by manually selecting the N-shaped thin-line marker spot on each ultrasound image. Specifically, on the ultrasound image Img(j), according to the order of increasing X _m first, and then Y _m increasing, manually select all the three ones formed by the intersection of the N lines in the field of view of the ultrasound image and the plane of the ultrasound image. The two-dimensional coordinate position of the center of the group of bright spots, and write down the sequence number k of each group of N lines in the order in which they were clicked. Let the bright spots on the ultrasonic image of the three sides of a group of selected N lines be E(j,k), F(j,k), G(j,k), where k=1,2,..., K _j , K _j is the total number of N-line marker points obtained on the ultrasound image Img(j). The two-dimensional coordinates of the selected bright spots are: $x_u^{\left(\mathrm{E.}\right)}(j,k)=[x_u^{\left(\mathrm{E.}\right)}(j,k),{\mathrm{the\; y}}_u^{\left(\mathrm{E.}\right)}(j,k)],$ $x_u^{\left(f\right)}(j,k)=[x_u^{\left(f\right)}(j,k),{\mathrm{the\; y}}_u^{\left(f\right)}(j,k)],$ $x_u^{\left(G\right)}(j,k)=[x_u^{\left(G\right)}(j,k),{\mathrm{the\; y}}_u^{\left(G\right)}(j,k)],$ The distance |EF| _jk and |EG| _jk between the three selected points can be obtained according to the image resolution:

${<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ;</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

${<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; .</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

According to the above method, the positions of the N-line marker points on all the collected ultrasound images are extracted and stored according to the serial numbers.

According to the number of columns and the number of layers where the processed N lines are located, the X _m and Y _m coordinate values of the threading hole A _k B _k C _k D _k of the kth group on the water model are obtained:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

Among them, hole A is coaxial with hole B, and hole C is coaxial with hole D, namely: $x_m^{\left(A\right)}(j,k)=x_m^{\left(B\right)}(j,k),$ ${\mathrm{the\; y}}_m^{\left(A\right)}(j,k)={\mathrm{the\; y}}_m^{\left(B\right)}(j,k),$ $x_m^{\left(C\right)}(j,k)=x_m^{\left(\mathrm{D.}\right)}(j,k),$ ${\mathrm{the\; y}}_m^{\left(C\right)}(j,k)={\mathrm{the\; y}}_m^{\left(\mathrm{D.}\right)}(j,k),$ And the holes in the N line group are at the same height in the water model coordinate system, that is: ${\mathrm{the\; y}}_m^{\left(A\right)}(j,k)={\mathrm{the\; y}}_m^{\left(B\right)}(j,k)={\mathrm{the\; y}}_m^{\left(C\right)}(j,k)={\mathrm{the\; y}}_m^{\left(\mathrm{D.}\right)}(j,k).$ Then measure the distances between E(j,k), F(j,k), and G(j,k) through ultrasound images (see Figure 3b, here for the sake of brevity, the subscripts of each symbol are omitted, the same below ), the ratio of _{| BF} | _jk to |CF| _jk can be obtained from the similar triangles EBF and GCF through the ratio of |EF| _jk to |GF| (See Figure 3c), thus obtaining point F In this way, for an N-line pair (bright spot in a group of three) with the serial number k in the ultrasonic image Img(j), the N-line marker point F(j,k) in a water tank model coordinate system can be obtained 3D coordinates. Assuming that in the water tank model coordinate system shown in Figure 2, the three-dimensional coordinates of point A are:

$x_m^{\left(A\right)}=[x_m^{\left(A\right)}(j,k),{\mathrm{the\; y}}_m^{\left(A\right)}(j,k),z_m^{\left(A\right)}(j,k)],$ Then the three-dimensional coordinates of point F in the tank model coordinate system

$x_m^{\left(f\right)}=[x_m^{\left(f\right)}(j,k),{\mathrm{the\; y}}_m^{\left(f\right)}(j,k),z_m^{\left(f\right)}(j,k)]$ It can be expressed as:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}<span\; class="notranslate">\; \&CenterDot;</span>{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; AB</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Wherein, j is the serial number of the ultrasonic image Img(j) used for calibration, k=1, 2,..., K _j , and K _j is the number of marker points extracted on the image. and: ${\left|\mathrm{BD}\right|}_{\mathrm{jk}}=\sqrt{{(x_m^{\left(\mathrm{D.}\right)}(j,k)-x_m^{\left(B\right)}(j,k))}^2},$ ${\left|\mathrm{AB}\right|}_{\mathrm{jk}}=\sqrt{{(z_m^{\left(B\right)}(j,k)-z_m^{\left(A\right)}(j,k))}^2}.$ In the above formula, the coordinate values in the X _m and Z _m directions are determined according to the similar triangle relationship between △BEF and △CGF, and the position of the reference point (here, point A) defined during the water mold manufacturing , and the coordinate value of point F in the Y _m direction is obtained from the position of the layer where the N line is located.

All the N-line groups that can be obtained in the image Img(j) are processed in this way, and the two-dimensional N-line marker point coordinates of each serial number k obtained by imaging There is a three-dimensional coordinate point in the model coordinate system The point pairs corresponding to the sequence are formed. Ideally, these N-line marker points will form a plane in the three-dimensional coordinate system of the water tank. For Img(j), we can use equation (4) to solve the coordinate mapping relationship T _um (j) between the water tank model coordinate system and the ultrasonic image plane coordinate system:

$\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

or written as:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

Where j=1,2,...,J,J is the number of ultrasonic images collected, k=1,2,...,K _j , K _j is the number of N-line marker points extracted on the image Img(j) .

Step 7: Map the N-line marker points on each two-dimensional ultrasound image to multiple imaging planes distributed in three-dimensional space

As mentioned above, when collecting each ultrasound image, the ultrasound probes are distributed in a certain way in the three-dimensional space, which forms the cutting of the thin lines on the water model at different spatial positions. According to the position T _ws (j) of the spatial position sensor when each ultrasonic image is collected, we can virtually combine multiple two-dimensional ultrasonic imaging planes in the three-dimensional space in the reference coordinate system to form several images with a certain three-dimensional spatial distribution. imaging plane. The position of any two-dimensional marker point extracted on each ultrasound imaging plane Img(j) is the intersection of the ultrasonic image plane at a specific spatial position and a specific thin line in the water model. Therefore, from the perspective of ultrasonic imaging, these intersection points are two-dimensional marker points falling in their respective virtual imaging planes. From the perspective of the water model, they are These thin lines are cut by the batch of spatially distributed virtual imaging planes to form a batch of intersection points of three-dimensionally distributed planes and lines in space. Our goal is to find: when these cutting surfaces are located, the marker points extracted from multiple 2D ultrasound images with known positions can best match the thin lines in the water model. In actual calculation, it is not necessary to actually find the spatial positions of these virtual imaging planes, but only need to extract the two images extracted on each imaging plane according to the position T _ws (j) of the position sensor when each ultrasonic image Img(j) is collected Dimensional landmark position According to the position and posture of the spatial position sensor when the image is collected, it is mapped to the three-dimensional space and combined together to form a batch of registration points in the three-dimensional space. These three-dimensional space points can be three-dimensionally registered with the thin lines in the water model. For the specific implementation algorithm, see step 8.

Taking the calibration of the sector-scanning ultrasound probe as an example, the geometric concept of combining each virtual imaging plane with the help of the position information T _ws (j) of the spatial position sensor is shown in Fig. 4. The five sectors in the figure are five virtual imaging planes, and each Several marker points can be extracted from a fan-shaped two-dimensional ultrasound image.

Therefore, the position of the marker point on each ultrasound image Img(j) extracted in step 6 With the help of the position transformation matrix T _ws (j) of the position sensor when each ultrasound image Img(j) is collected, the positions of the two-dimensional marker points on each imaging plane can be successively mapped to the jth corresponding virtual imaging plane in the three-dimensional space , thus forming a batch of marker points distributed in three-dimensional space, which correspond to the intersection points of multiple imaging planes and thin lines when the thin line model is imaged from different directions.

For the sake of standardization, we unified the marker points extracted on the ultrasound imaging plane Img(j) Two-dimensional coordinates are written as: Among them, l=1,2,...,3×K _j , K _j is the number of N-line mark points extracted on the image Img(j); The three-dimensional coordinates mapped to each imaging plane in three-dimensional space are marked as $x_u^{3\mathrm{D.}}(j,l)=(x_u^{3\mathrm{D.}}(j,l),{\mathrm{the\; y}}_u^{3\mathrm{D.}}(j,l),z_u^{3\mathrm{D.}}(j,l)),$ but:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

Wherein, T _ws (j) is the position transformation matrix of the position sensor corresponding to the Img (j) imaging plane obtained from the personal computer from the three-dimensional positioning measuring instrument, and j=1, 2, ..., J, J is the ultrasonic wave collected Number of images, l=1,2,...,3×K _j , K _j is the number of N-line marker points extracted on the image Img(j). The spatial points mapped to each virtual ultrasound imaging plane are shown as dots on each sector in FIG. 4 .

Step 8: Initialize the spatial matching position

Using the equation (10) in step 6, the three-dimensional coordinates of each N-line marker point position extracted from a certain ultrasound image plane Img(j) And the two-dimensional coordinates of the corresponding bright spot on the ultrasound image Solve the coordinate mapping relationship T _um (j) from the water tank model coordinate system to the ultrasonic image plane coordinate system. This transformation establishes the transformation relationship from the marker point on the ultrasonic imaging plane to the intersection point of the thin line in the water model and the imaging plane. Since all the extracted marker points have been reconstructed in 3D space according to their imaging orientations in step 7, a batch of 3D spatially distributed points to be registered is obtained When we select an ultrasound image to obtain the transformation relationship between the image and the water model, we can use this transformation to transform all the landmark points in the three-dimensional imaging space Mapped to the initial position in the water model that is relatively close to the corresponding thin line. Assuming that the nth ultrasound image Img(n) is selected for initial registration, the transformation matrix T _um (n) can be obtained according to equation (13), namely:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

Then, all the marker points on other virtual imaging planes obtained (j=1,2,...,n-1,n+1,...J,l=1,2,...,3×K _j ), perform spatial translation and rotation along with the plane Img(n), and transform to the water tank In the model coordinate system, a batch of 3D coordinate points to be registered in the model coordinate system are formed

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

So far, the three-dimensional space marker points on the virtual ultrasound imaging plane have been transformed into the coordinate system of the water tank model, and initially registered with the thin lines on the water model, that is, the marker points and the corresponding thin lines in the water model The lines will be close together, as shown in Diagram 5.

Step 9: Set the optimized objective function and initial value, and perform optimization calculation

Through the processing of step 8, all the three-dimensional coordinate points to be registered have been mapped to the water model coordinate system. Ideally, these points should coincide with the corresponding thin lines on the water model (the distance is zero), but due to the There is a position error when extracting marker points on the image, so There may also be a certain distance to the corresponding thin line. We choose the optimization objective function to be the marker points of all three-dimensional spatial distributions The average distance to the nearest thin line on the 3D water model is the smallest. Note that the N-shaped lines in the design of the water model are distributed on the horizontal planes of different heights, so the y _m coordinates of each layer of thin lines are fixed, and for the parallel sides of the N-shaped lines, the x _m coordinates are also Fixed, assuming that there are R layers of N-shaped lines in the water model, and each layer has S parallel straight lines, these thin lines can be numbered sequentially, and the straight line equations of these parallel thin lines can be written as:

$\left\{\begin{array}{c}{x}_m\left(l\right)=L_x(r,\mathrm{the\; s});\\ {\mathrm{the\; y}}_m\left(l\right)=L_{\mathrm{the\; y}}(r,\mathrm{the\; s});\end{array}\right.$ in $\left\{\begin{array}{c}r=\mathrm{1,2},...,R\\ \mathrm{the\; s}=\mathrm{1,2},...,S\end{array}\right.---\left(15\right)$

Among them, L _x (r, s) and _Ly (r, s) are respectively the coordinates of the sth thin parallel line in the rth layer defined in the water model design in the coordinate system of the water tank model, which are given constants, l is the serial number of each thin line.

The hypotenuse of the N-shaped line is distributed in the horizontal plane of the water model coordinate system, and its three-dimensional space linear equation can be defined by the positions of the starting point and the ending point of the diagonal line (also the small hole passing through the parallel line) of the oblique line:

$\left\{\begin{array}{c}\frac{\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; the\; y</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

Therefore, the spatial positions of all thin lines in the water model are strictly defined. In order to obtain the spatial marker points The distance to its corresponding straight line, after the above initialization processing, can directly solve a certain marker point The distance to all straight lines, for parallel thin lines, only need to be Subtract the x and y components of the thin line from the corresponding components of the thin line, and calculate the distance according to the Pythagorean theorem. For oblique lines, because they are distributed horizontally, you can first solve the distance on the x and z components, and then combine them with the y component . in getting After the distance to all thin lines, select one of the smallest distances D(j,l) as the space point The distance from the nearest thin line on the 3D water model, so that the marker point can be obtained Average distance from the nearest thin line on the 3D water model:

${\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; avg</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

Among them, J is the number of ultrasound images involved in the calculation, and K _j is the number of marker points that can be extracted from the jth ultrasound image.

The initial spatial position T _um (n) of the above optimization calculation is the spatial position of the imaging plane in the water model coordinate system calculated by using a certain ultrasound image Img(n) in step 8 above. On this basis, the variable of T _um can be adjusted for optimal iterative calculation, and the average distance D _avg can be minimized through optimal calculation. The variable to be optimized is the space transformation matrix T _um (including three translations and three rotations, a total of 6 degrees of freedom) between the ultrasound imaging coordinate system and the water model coordinate system. By setting the error threshold of the average distance D _avg (taken as 2mm) and the maximum number of iteration steps (500 steps), the end of the optimization iterative process is controlled, and finally an optimized transformation matrix is obtained, which represents the acquired multi-imaging plane A Best Matching Transform to Thin Line Objects in Water Models

In this example, we use the optimization calculation tool of sequential quadratic programming algorithm (SQP) in the optimization toolbox of the Matlab software for optimization calculation. Of course, other mathematical calculation tools can also be used for optimization calculation.

Step 10: Calculate the optimal spatial transformation between the imaging plane and the sensor According to the spatial position T _ws (j) of the spatial position sensor fixed on the probe when each ultrasonic image Img(j) is collected, the spatial transformation of the position corresponding to the best slice obtained in step 9 With the help of the relationship between the various coordinate systems defined in Figure 1, it can be known that: for the point set in the ultrasound imaging coordinate system or can be transformed through space Transform to the water model coordinate system to get the corresponding point set In the actual calculation, we use the formula (12) in step 7 to extract the marker points on each ultrasound image Transform to the virtual imaging plane in three-dimensional space to obtain the point set Used to solve the optimal ultrasound image-to-sensor transformation The calculation flow is as follows. First, transform the marker points extracted from the ultrasound image to the virtual imaging plane to obtain the three-dimensional marker position:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

Then point set on the virtual imaging plane Transform to the water model coordinate system:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

Then, the point set in the water model coordinate system Transform into sensor coordinate system:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 20</span>}<span\; class="notranslate">\; )</span>$

Among them, T _ws (j) is the position of the position sensor in the reference coordinate system when the ultrasonic image Img (j) is collected, which can be read from the positioning system, and T _wm is the water model coordinate system measured and solved in the fourth step to Spatial transformation matrix between reference coordinate systems.

In this way, a batch of paired marker point data in the ultrasound image coordinate system and the corresponding position sensor coordinate system is obtained: and Where j=1,2,...,J, J is the number of ultrasonic images collected, l=1,2,...,3×K _J , K _J is the N-line mark that can be extracted from the corresponding image Img(j) the number of points. Using these data, the global optimal transformation between the ultrasonic image coordinate system to be solved and the spatial position sensor on the ultrasonic probe can be calculated according to equation (21)

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

In fact, another process can also be used to obtain the same result. We can use the formula (22) to calculate the optimal transformation between the ultrasound image and the sensor one by one according to the order of the acquired images For any image Img(j):

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$

The advantage of this is that from equations (11), (12) and (14), the optimal transformation between the ultrasonic image to be solved and the sensor can be directly deduced as:

${\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

Finally, the optimal spatial transformation matrix calibrated by multiple ultrasound images is averaged to obtain the optimal calibration result

${\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; J</span>}}{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; four</span>}<span\; class="notranslate">\; )</span>$

where J is the number of ultrasound images acquired. Calculated using Equation (21) or Equation (24) It is the optimal transformation between the coordinate system of the target ultrasound image and the coordinate system of the spatial position sensor on the ultrasound probe that the present invention seeks.

In summary, the entire processing flow of the present invention is shown in FIG. 6 .

In order to compare the method of the present invention with the traditional N-line method, we also extracted and calculated the three-dimensional spatial position of each "N-line marker point" in the water tank model coordinate system on each ultrasonic image acquired according to the N-line calibration method And the positions of the collected ultrasound imaging planes are obtained through these position fitting calculations. At the same time, the spatial transformation relationship T _su (j) between the ultrasonic imaging plane Img(j) and the position sensor is calculated according to the spatial position and attitude of the corresponding position sensor according to the traditional N-line calibration method.

In order to solve the spatial transformation relationship T _su (j) between the ultrasonic imaging plane Img(j) and the position sensor, it is only necessary to find a batch of paired spatial points in the sensor coordinate system and the ultrasonic imaging plane coordinate system for calibration. With the help of the relationship between the various coordinate systems defined in Figure 1, we record the three-dimensional coordinates of the intersection point F of the extracted N-line marker point and the Img(j) imaging plane in the water tank model coordinate system These spatial points can be transformed and mapped to the reference coordinate system of the position sensor by the following equation (25), and the three-dimensional coordinates in the reference coordinate system of the position sensor can be obtained $x_{\mathrm{the\; s}}^{\left(f\right)}(j,k)=$ $[x_{\mathrm{the\; s}}(j,k),{\mathrm{the\; y}}_{\mathrm{the\; s}}(j,k),z_{\mathrm{the\; s}}(j,k)]:$

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 25</span>}<span\; class="notranslate">\; )</span>$

Among them, T _ws (j) is the spatial position transformation matrix of the position sensor in the reference coordinate system during Img (j) imaging, which can be read from the positioning system in real time, and T _wm is the coordinate system of the water tank model solved in the fourth step to Spatial transformation matrix between reference coordinate systems.

Note the points in the above sensor space The paired points on the image coordinate system are the two-dimensional coordinates of the oblique line in the N-line pair extracted on the ultrasound image Img(j) In this way, a batch of point pairs corresponding to each other from the image coordinate system to the sensor reference coordinate system is formed. and By writing them as homogeneous coordinates, the transformation T _su between them can be calculated, as in equation (26):

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 26</span>}<span\; class="notranslate">\; )</span>$

Equation (26) obtains the spatial transformation matrix T _su (j) of an ultrasonic image Img(j) calibrated by the traditional N-line method. In order to improve its accuracy, the spatial transformation matrix calibrated by multiple ultrasonic images can be On average, the result of traditional N-line calibration using multiple ultrasound images is obtained, which is denoted as T ^T _su .

${\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; J</span>}}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 27</span>}<span\; class="notranslate">\; )</span>$

where J is the number of ultrasound images acquired.

Compared with the traditional processing flow, it can be seen that the traditional N-line model calibration method directly uses the water model design data and water model positioning information, and uses the two-dimensional marker points extracted from the ultrasonic image and the similar triangle relationship to solve the ultrasonic plane and water model thin line intersection point, and use the extracted paired point set to calculate and solve the space transformation T _su from the ultrasonic image to the sensor by using the equation group (26), and our invention, by using the relative position information of multiple ultrasonic images during imaging , using the spatial correlation between multiple images to form an optimized calculation of the distance between a family of 2D image targets and 3D thin lines in the water model coordinate system, which makes more effective use of the obtained information , so the calibration accuracy can be improved.

We used the above processing flow to carry out practical experimental tests. A total of 8 sets of data were tested, and each set included 7 sets of water model ultrasonic images collected at different locations and corresponding spatial position sensor data. In the processing, we selected 5 sets of ultrasound images and positioning data in each set of data for the calibration of the imaging plane and the spatial position of the sensor, and used the other two sets of data for accuracy verification. Therefore, there are a total of C ₅ ⁷ =21 combinations for each set of data, and we take the average precision calibrated by these 21 sets of combinations as the precision of the overall calculation. In the experiment, the traditional N-line model calculation method and the method of the present invention are respectively used for processing, and the obtained calibration accuracy is compared. Figure 7 shows a typical set of data with thickness errors. During the calculation process, the calibration error decreases with the number of ultrasound images involved in the calculation. It can be seen that when an ultrasonic image is selected for calibration, the error can be reduced to a certain extent by using our method for optimization, but the calibration error is still relatively large, and after introducing multiple ultrasonic images, the calibration of the method of the present invention The error decreases more rapidly than the traditional method. When more than 3 images are used, a relatively small error can be achieved, and when 5 images are used for calibration, a good calibration effect is obtained. Therefore, the processing method of the present invention can obtain higher calibration accuracy on the basis of collecting a small amount of ultrasonic images and positioning data.

Inventor

Inventors: Wang Guangzhi, Ding Hui, Zhu Liren

Unit: Department of Biomedical Engineering, School of Medicine, Tsinghua University

Tel: (010)62783631

Contact person: Ding Hui, E-mail: dinghuitsinghua.edu.cn

Type of patent to be applied for: invention patent

Claims (1)

1. An optimization method for ultrasonic probe imaging plane spatial position calibration, characterized in that, the optimization method is a method for demarcating the transformation relationship between the spatial position of the ultrasonic probe imaging plane and the spatial position sensor spatial position, i.e. the spatial transformation matrix The optimization method is realized in the following steps with the assistance of a three-dimensional positioning measurement system: Step 1. Construct a 3D positioning measurement system for ultrasonic probe calibration: The three-dimensional positioning measuring system used for ultrasonic probe calibration includes: a three-dimensional positioning measuring instrument, a water tank model, a medical ultrasonic imaging system and a personal computer, wherein the three-dimensional positioning measuring instrument, the water tank model, and the ultrasonic imaging system are jointly used to measure the calibrated The ultrasonic probe is used for measurement, and the personal computer is used for the acquisition and data processing of measurement data. The three-dimensional positioning measuring instrument includes: a 6-degree-of-freedom spatial position sensor fixed on the ultrasonic probe, and a three-dimensional positioning probe with a tip positioning accuracy of 0.2mm , as a wireless transmitting device for the positioning of the spatial reference coordinate system, the ultrasonic imaging system includes a calibrated ultrasonic probe and an imaging system host for generating ultrasonic images of the water tank model to be collected, wherein the calibrated ultrasonic probe is consolidated with the above-mentioned A 6-DOF space position sensor, an image capture card is installed in the personal computer to collect the ultrasonic image output from the ultrasonic imaging system, and at the same time, the personal computer collects the 6-freedom image output from the three-dimensional positioning measuring instrument through a USB interface. The positioning data of the three-dimensional position sensor and the positioning data of the tip of the three-dimensional positioning probe; Step 2. Make an N-line model: The N-line model is to arrange multiple groups of thin lines in the shape of "N", layer by layer and fix them at equal heights layer by layer in a pre-set dimension in three directions of length, width and height. In the cuboid water tank, each N shape forms an "N-line target". The N-line model is formed by making coaxial small holes through the front and rear walls of the cuboid water tank, and passing thin nylon wires through them to form the water tank. According to the field of view size and image resolution of the ultrasonic probe, the N-line model in the N-line model has three layers of N-lines arranged from top to bottom, the highest layer contains three N-lines, and the remaining two layers each contain 5 N-shaped, so that the ultrasonic image can collect more N-line targets, and make each N-line target fill the field of view of the ultrasonic imaging as much as possible. The height and position of each group of N-lines in the coordinate system of the water tank model have been Set in advance, and ensure its position accuracy through the manufacture of the water tank model. Four pit marks for positioning and measurement are arranged on the four corners of the front and rear walls of the water tank, each forming a rectangle, which is used to define the coordinate system of the water tank and accurately measure the water tank. placement; Step 3. Establish the set of coordinate systems of the three-dimensional positioning measurement system used for ultrasonic probe calibration: including reference coordinate system, water tank model coordinate system, spatial position sensor coordinate system, and ultrasonic imaging plane coordinate system. The reference coordinate system uses X _w Y _w Z _w represents that it is located on the wireless transmitter device for positioning, and the coordinate origin is at the center of the transmitter device; the water tank model coordinate system is represented by X _m Y _m Z _m , X _m is the length direction of the water tank, and Z _m is the width of the water tank direction, Y _m is the height direction of the water tank, the origin of the coordinate system is set on the positioning pit in the upper left corner of the front wall of the water tank, and the directions of each coordinate axis follow the right-hand rule; the spatial position sensor coordinate system is represented by X _s Y _s Z _s , set On the 6-DOF spatial position sensor, the coordinate origin is at the center of the spatial position sensor; the ultrasonic imaging plane coordinate system is represented by X _u Y _u , which is set on the ultrasonic image, and the coordinate origin is set at the uppermost midpoint of the ultrasonic image , X _u direction is the image horizontal direction, Y _u direction is the depth direction of ultrasound imaging, In order to establish the transformation relationship between each coordinate system, the space transformation matrix T _wm of 4*4 is set to be the space from the water tank model coordinate system X _m Y _m Z _m to the reference coordinate system X _w Y _w Z _w Transformation matrix, T _ws is the space transformation matrix from the spatial position sensor coordinate system X _s Y _s Z _s to the reference coordinate system X _w Y _w Z _w , T _su is the coordinate system from the ultrasound imaging plane X _u Y _u to the space position sensor coordinate system X _s Y _s Z _s space transformation matrix used for positioning, T _um is from the water tank model coordinate system X _m Y _m Z _m to the ultrasonic imaging plane coordinate system X _u The space transformation matrix of Y _u ; Step 4, calibrate the spatial position where the water tank model is placed, and obtain the space transformation matrix from the coordinate system X _m Y _m Z _m of the water tank model to the reference coordinate system X _w Y _w Z _w : Step 4.1: Use the tip of the three-dimensional positioning probe configured by the three-dimensional positioning measurement system to detect the four positioning recesses arranged at the four corners on the front and rear walls of the water tank model located in the effective measurement area of the wireless transmitting device for positioning one by one. The three-dimensional space position P _wi of the pit in the reference coordinate system X _w Y _w Z _w = (x _wi , y _wi , z _wi ), where i=1, 2, ..., 8 are the sequence numbers for positioning the pits , Step 4.2: Query or measure the coordinates P _mi =(x _mi ,y _mi ,z _mi ) of each positioning pit in the water tank model coordinate system in the design and manufacture of the water tank model in the same order, where i=1,2, ..., 8, is the serial number of the positioning pit, Step 4.3 solves the space transformation matrix T _wm of described water tank model coordinate system to described reference coordinate system by following equation group $\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 3</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 4</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 5</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 6</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 7</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; 8</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; ;</span>$ Step 5, collecting multiple ultrasonic images at different positions in the water tank model: use the calibrated ultrasonic probe with a 6-degree-of-freedom spatial position sensor to image multiple groups of N-line thin lines in the water tank, and image the ultrasonic The probe is arranged above the water tank model. By moving the ultrasonic probe, more than 7 ultrasonic images are collected in different imaging planes that are not overlapped or parallel, and the collected ultrasonic images are input into the Personal computer, stored as Img(j), j=1, 2,..., J, J is the number of ultrasonic images collected, and at the same time, through the USB interface, read the spatial position when collecting each ultrasonic image Img(j) The spatial position and attitude of the sensor in the reference coordinate system X _w Y _w Z _w , and calculate the spatial transformation matrix from the spatial position sensor coordinate system X _s Y _s Z _s to the reference coordinate system X _w Y _w Z _w T _ws (j), the input ultrasonic image can be displayed on the screen of the personal computer, and the cross-section of each N thin line is displayed as a bright spot on the ultrasonic image; Step 6, extracting the positions of the N-line marker points on each ultrasound image and calculating the position of the ultrasound plane, the steps are as follows: Step 6.1 On the ultrasonic image Img(j) displayed on the personal computer, according to the order of increasing X _m first, and then increasing Y _m , manually select all the N lines in the field of view of the ultrasonic image to intersect with the ultrasonic image plane The two-dimensional coordinate position of each bright spot in the formed group of three bright spots, and write down its serial number k according to the order in which each group of N lines is selected, and set the three sides of the selected group of N lines at The bright spots on the ultrasonic image are E(j,k), F(j,k), G(j,k) respectively, where k=1,2,...,K _j , K _j is the ultrasonic image Img (j) The total number of N-line mark points obtained above, and the two-dimensional coordinates of the bright spot selected by the point $x_u^{\left(\mathrm{E.}\right)}(j,k)=[x_u^{\left(\mathrm{E.}\right)}(j,k),{\mathrm{the\; y}}_u^{\left(\mathrm{E.}\right)}(j,k)],$ $x_u^{\left(f\right)}(j,k)=[x_u^{\left(f\right)}(j,k),{\mathrm{the\; y}}_u^{\left(f\right)}(j,k)],$ Get the distance |EF| _jk and |EG| _jk between the three selected points according to the image resolution: ${<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ;</span>$ ${<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ;</span>$ According to the method in step 6.1, the position of the N-line marker point on all the collected ultrasound images is extracted, and stored according to the serial number, Step 6.2 Calculate the space transformation matrix T _um (j) from the X _m Y _m Z _m coordinate system to the ultrasound imaging plane coordinate system X _u , Y _u for any ultrasound image Img(j) according to the following steps: Step 6.2.1 According to the sequence number k of the N-line selected on the ultrasonic image in step 6.1, select the N-line group corresponding to the position in the water tank model. The three-dimensional coordinates of the four threading holes A, B, C, and D in the coordinate system X _m Y _m Z _m of the water tank model are respectively ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ Among them, hole A is coaxial with hole B, and hole C is coaxial with hole D, namely: ${\mathrm{the\; y}}_m^{\left(A\right)}(j,k)={\mathrm{the\; y}}_m^{\left(B\right)}(j,k),$ $x_m^{\left(C\right)}(j,k)=x_m^{\left(\mathrm{D.}\right)}(j,k),$ ${\mathrm{the\; y}}_m^{\left(C\right)}(j,k)={\mathrm{the\; y}}_m^{\left(\mathrm{D.}\right)}(j,k),$ Therefore, the two thin lines connecting holes A and B and holes C and D constitute the straight line side of the N-line mark, and the thin lines connecting holes B and C constitute the hypotenuse of the N-line mark. The image Img(j) is cut with a group of N lines with the serial number k, the ultrasonic image intersects with the thin line AB in the N lines at E, intersects with the thin line BC at F, and intersects with the thin line CD at G, defined The three-dimensional coordinates of points E, F, and G in the water tank model are: ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$ Step 6.2.2 Calculate the three-dimensional coordinates of point F in the N line with serial number k in the water tank model coordinate system X _m Y _m Z _m according to the following formula $x_m^{\left(f\right)}(j,k)=[x_m^{\left(f\right)}(j,k),{\mathrm{the\; y}}_m^{\left(f\right)}(j,k),z_m^{\left(f\right)}(j,k)]:$ $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; BD</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EF</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}{{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; EG</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; AB</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; jk</span>}}\end{array}\right.$ in: ${\left|\mathrm{BD}\right|}_{\mathrm{jk}}=\sqrt{{(x_m^{\left(B\right)}(j,k)-x_m^{\left(\mathrm{D.}\right)}(j,k))}^2},$ ${\left|\mathrm{AB}\right|}_{\mathrm{jk}}=\sqrt{{(z_m^{\left(B\right)}(j,k)-z_m^{\left(\mathrm{D.}\right)}(j,k))}^2}$ Step 6.2.3 The personal computer obtains the coordinates of each group of N-line marker points F in the X _m Y _m Z _m coordinate system by measuring all the complete N-line bright spots obtained on the ultrasonic image Img(j). 3D coordinates k=1,2,...,K _j , K _j is the total number of N-line marker points obtained on the ultrasonic image Img(j), Step 6.2.4 According to step (6.2.3) and step (6.2.2) obtained according to a set of N-line marker points F corresponding to an ultrasonic image j, the three-dimensional coordinate representation under the X _m Y _m Z _m coordinate system for: , the coordinate value of this point has been solved through step 6.2.2, and the F mark points with different serial numbers k extracted on the same ultrasound image Img(j) In the three-dimensional coordinate system of the water tank, a plane will be formed, and the corresponding coordinates of these points in the ultrasound image coordinate system X _u Y _u are For Img(j), use the following equation to solve the coordinate mapping relationship T _um (j) from the water tank model coordinate system to the ultrasonic image plane coordinate system: $\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; =</span>\left[\begin{array}{cccc}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& <span\; class="notranslate">\; .</span>& \\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& {\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>$ or written as: $x_u^{2\mathrm{D.}}(j,k)=x_m(j,k)\&Center\; Dot;T_{u-m}\left(j\right)$ Where j=1,2,...,J,J is the number of ultrasonic images collected, k=1,2,...,K _j , K _j is the N-line marker point F that can be extracted on the image Img(j) Number of; Step 7. Map the N-line marker points on the two-dimensional ultrasound image to the multi-imaging planes distributed in three-dimensional space, and the multi-imaging planes refer to the X _w Y _w Z _w coordinate system by means of the spatial position sensor The resulting spatial position matrix corresponding to the ultrasound image Img(j) Combine each two-dimensional ultrasonic imaging plane in the X _w Y _w Z _w coordinate system to form multiple imaging planes with a certain three-dimensional distribution, and use the spatial position sensor information to extract the N-line marks extracted from each two-dimensional ultrasonic image Points are mapped to the imaging plane in three-dimensional space, and then the N-line marker points on the ultrasound image extracted in step 6.1 According to the combination of the position and posture of the spatial position sensor when the image is collected, a batch of registration points in the three-dimensional space is formed, and the two-dimensional coordinates of the N-line marker points extracted on the ultrasonic image Img(j) are uniformly written as: Where l=1,2,...,3*K _j , K _j is the number of N-line marker points extracted on the image Img(j); The three-dimensional coordinates mapped to each imaging plane in three-dimensional space are $x_u^{3\mathrm{D.}}(j,l)=(x_u^{3\mathrm{D.}}(j,l),{\mathrm{the\; y}}_u^{3\mathrm{D.}}(j,l),z_u^{3\mathrm{D.}}(j,l)),$ but: ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span>$ Wherein, T _ws (j) is the position transformation matrix of the position sensor corresponding to the Img (j) imaging plane obtained from the personal computer from the three-dimensional positioning measuring instrument, and j=1, 2, ..., J, J is the ultrasonic wave collected Number of images, l=1,2,...,3*K _j , K _j is the number of N-line marker points extracted on the image Img(j); Step 8. According to the following steps, a batch of three-dimensional coordinate points mapped to multiple imaging planes Initialize the registration with the N line on the tank model: Step 8.1, use the N-line marker points on the ultrasound image obtained in step 6.1 and step 6.2.3 and the corresponding coordinates in the tank model Choose an ultrasonic image whose serial number is n, and use the following equations to calculate the space transformation matrix T _um (n) from the X _m Y _m Z _m coordinate system to the X _u Y _u coordinate system: ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$ Step 8.2, using the initial space transformation matrix T _um (n) obtained in step 8.1, the three-dimensional coordinates of all marker points obtained in step 7 j=1,2,...,n-1,n+1,...J,l=1,2,...,3*K _j is mapped to the coordinate system of the water tank model, and a batch of waiting in the coordinate system of the water tank model is calculated Registered 3D coordinate points ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$ Therefore, initially match the N-line mark points on each ultrasonic plane with the N-line on the water tank model, Step 9. Set the objective function and initial value of the optimization calculation and perform the optimization calculation: Optimizing the objective function is to adjust the space transformation matrix T _um between the X _m Y _m Z _m coordinate system and the X _u Y _u coordinate system described in step 8.1, so that all the three-dimensional space distribution obtained in step 8.2 The landmark points to be registered The average distance from the corresponding thin line on the tank is the smallest, Therefore, the variable to be optimized is the variable sequence of the transformation matrix T _um , and the initial position of the optimization calculation is the transformation matrix T _um (n) calculated by using the nth ultrasound image. It is assumed that a total of J layers of images participate in the calculation, and the marker points of each layer The number is K _j , so there are a total of J*K _j three-dimensional points used to calculate the distance to the thin line in the water tank model, by directly solving a certain marker point The distance D to all straight lines in the tank, and one of the smallest distances D(j,l) is selected as the space point The distance from the nearest thin line on the 3D water tank model to get the marker point Average distance from the nearest thin line on the 3D tank model: ${\mathrm{<span\; class="notranslate">\; D.</span>}}_{\mathrm{<span\; class="notranslate">\; avg</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; J</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\underset{\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span>$ The error threshold of the average distance D _avg is selected as 2 mm, and the maximum number of iteration steps is 500 steps. The average distance D _avg is minimized through optimization calculation, and an optimal matching transformation between the multi-imaging plane and the thin line target in the water tank model is obtained. for The optimization tool used is the sequential quadratic programming function in the Matlab optimization toolbox, the English name is sequential quadratic programming algorithm, referred to as SQP; Step 10, calculate the optimal spatial transformation between the imaging plane and the spatial position sensor Step 10.1, the coordinates of the two-dimensional N-line marker points extracted on each ultrasound image With the help of the spatial position sensor transformation matrix T _ws (j) when the image is collected, it is transformed into a three-dimensional space to obtain the three-dimensional marker point positions on multiple imaging planes ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}$ Step 10.2, the three-dimensional point set With the help of the best matching transformation obtained in step 9 Mapped to the tank model coordinate system X _m Y _m Z _m : ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; ,</span>$ Step 10.3, the point set in the tank model coordinate system X _m Y _m Z _m Transform into the sensor coordinate system X _s Y _s Z _s : ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; ,</span>$ Step 10.4, use the corresponding point set obtained in step 10.3 and step 10.1 and Use the following equations to solve the global optimal transformation between the ultrasound image and the sensor ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; D.</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; u</span>}}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; l</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; ,</span>$ Calculated by solving It is the optimal transformation between the ultrasonic image coordinate system to be calibrated and the spatial position sensor coordinate system on the ultrasonic probe.