CN110232697B - A method for fitting edge of light spot - Google Patents

Abstract

The invention provides a light spot edge fitting method. The invention extracts the laser spot edge image; passes through the edge pixel points to form a concentric circle with the center of the fitted circle; finds two tangent lines drawn from a point outside the circle to the concentric circle and makes the secant line ;Find the residual sum of squares from the point outside the circle to the secant line and convert it into polar coordinates; find the value of the corresponding parameter when the residual squared sum is the smallest, which is the circle center value. This method is a spot edge fitting method based on the improved least squares method, which combines algebraic and geometric methods, and uses geometric principles to construct the residual square term in the least squares circle fitting method, thereby improving the fitting accuracy of the center position of the spot. It solves the problem of transforming the solution of fitting parameters into linear algebra by the traditional least squares circle fitting method, and lacks consideration of the spot edge fitting process from a geometric point of view. The consideration is not comprehensive, the fitting effect of the spot is poor, and the accuracy and certainty are difficult to evaluate The problem.

Description

A method for fitting the edge of a light spot

Technical Field

The invention belongs to the field of machine vision measurement, and in particular relates to a light spot edge fitting method.

Background Art

Some visual measurement systems collect laser spot images, identify the movement of the spot position, and measure physical quantities such as displacement and length. The main process is to segment the spot image, extract and fit the spot edge, obtain the spot center position, and realize spot position detection. Among them, the accuracy of spot edge fitting is the key factor to ensure the accuracy of the visual measurement system.

At present, the representative studies on spot fitting algorithms are mainly:

First, the spot edge fitting algorithm based on geometric principles: In 2011, Zhang Haizhuang et al. proposed "Far-field laser spot center measurement based on circle positioning algorithm", taking three non-collinear points to determine the boundary circle equation, and then obtaining the boundary circle center coordinates. This method can adapt to the field measurement environment, but because the far-field light beam propagates over a long distance, the spot is prone to fragmentation and deformation. The actual collected spot image is irregular, and the uncertainty of taking three points to determine the spot center is large. In addition, the influence of image noise is not considered, and the anti-interference ability is low. When the spot image does not present a standard circle, the fitting error is large; in 2017, Guo Yujing et al. proposed "A laser spot center positioning algorithm based on brightness threshold", determined the brightness threshold and binarized the spot image to obtain an image with only the laser spot. According to the principle that the midpoint line of the parallel chord on the circle passes through the center of the circle, the center of the circle is determined by finding the midpoint of the chord. This method has a simple principle, small calculation amount, fast calculation speed, and is suitable for images with uniform distribution of spot images. When the spot area accounts for too small a proportion of the image area, it is difficult to select a suitable threshold to achieve better segmentation.

Second, the spot edge fitting algorithm based on algebraic principles: In 2013, Song et al. proposed "Research on sub-pixel location of the laser spot center", which continuously collected spot images and used the average background model to separate the image foreground and background. First, the spot center position was calculated in the image foreground, and then the grayscale distribution of the image was fitted with a quadratic curve in the horizontal and vertical directions of the grayscale center. The fitting result was used to obtain the corrected spot center position. This method accurately fits the spot position, but is sensitive to noise. It is suitable for occasions with high quality spot images and noise-free spot edges. In 2016, Wu Zekai et al. proposed "Laser spot center detection based on improved circle fitting algorithm", which uses the least squares circle fitting method and the least squares principle to fit the laser spot contour from an algebraic perspective to obtain the spot center position and its radius. It is fast, has high positioning accuracy, and can be used for real-time optical measurement. This method only uses algebraic methods to find the fitting circle equation, and the fitting result lacks certainty. Since the square term of the least squares fitting is very sensitive to outliers, this method is only suitable for occasions with dense distribution of edge pixels.

In summary, the traditional spot edge fitting algorithm only fits from an algebraic or geometric perspective, and its applicability has certain limitations. Least squares circle fitting is the most widely used fitting method. This method directly subtracts the distance from the edge pixel to the center of the circle from the square of the radius of the fitting circle to construct the residual sum of squares. However, spot edge circle fitting is a geometric process. Considering the geometric process only from an algebraic perspective may lead to deviations in the residual sum of squares, and the accuracy and certainty of the fitting results are difficult to evaluate.

Summary of the invention

In view of the above problems, the present invention provides a spot edge fitting method, which is a spot edge fitting method based on the improved least squares method, combines algebraic and geometric methods, and utilizes geometric principles to construct the residual square term in the least squares circle fitting method, thereby improving the edge fitting accuracy. It solves the problem that the traditional least squares circle fitting method converts the solution of fitting parameters into a linear algebra problem, lacks consideration of the spot edge fitting process from a geometric perspective, is incomplete, has poor spot fitting effect, and is difficult to evaluate in terms of accuracy and certainty.

The technical solution adopted by the present invention to solve the technical problem is: a light spot edge fitting method, comprising the following steps:

Spot edge extraction: extract the laser spot edge image;

Draw concentric circles and secants of the fitting circle: Assume that O is the center of the fitting circle, and its coordinates are (a, b). The set of pixel points at the edge of the light spot is {x ₁ ,x ₂ ,…, _xi }, and _Ri is the distance from each edge pixel point to the center of the circle, that is, the radius, where i is 1, 2, 3…n, n is the total number of pixels, and the coordinates of the edge pixel _pointxi are ( _xi , _yi ); draw concentric circles _C1 with O as the center of x1, and draw two tangent lines about the concentric circles _C1 at a point outside the circle in the rectangular coordinate system. The coordinates of the point outside the _circle are ( _xV , _yV ), and the secant line _L1 of the concentric circles _C1 is obtained by connecting the two tangent points. Draw the secant line of the corresponding circle of each pixel point in the edge of the light spot, and record it as _Li ;

Transform the residual sum of squares into coordinates: Calculate the residual sum of squares from the point outside the circle to the secant line and transform it into polar coordinates;

Solve to get the center value when the residual square sum is the smallest: find the center coordinates and radius of the fitting circle corresponding to the minimum residual square sum, which is the center value, to achieve the center positioning of the light spot.

In the above solution, the Canny operator is used to extract the edge of the light spot in the light spot edge extraction step.

In the above scheme, the equation of the secant line _Li is:

In the above scheme, the step of coordinate transformation of the residual square sum Q is specifically as follows:

Obtain the distance f ₁ from a point outside the circle to the secant line, then make concentric circles of the fitting circle and the secant lines of the concentric circles for each pixel point on the edge of the light spot, and respectively obtain the distance _fi from the point outside the circle to the corresponding secant line _Li ;

Assume that the distance from a point outside the circle to the corresponding secant line L of the fitting circle is f, and construct the residual sum of squares Q between _fi and f according to the least squares principle;

Convert the residual sum of squares form to polar coordinate form.

Furthermore, in a rectangular coordinate system, the _fi equation is:

Furthermore, in a rectangular coordinate system, the residual sum of squares Q is in the form of:

Furthermore, the residual sum of squares is converted into polar coordinate form as follows:

Convert the rectangular coordinate system parameters into polar coordinates:

In the polar coordinate system, the _fi equation is transformed into:

In the polar coordinate system, the residual sum of squares of _fi is transformed into:

Furthermore, the step of solving the center value of the circle when the residual square sum Q is minimized is specifically as follows:

To find the partial derivatives of the three variables ρ, θ, and f of the residual sum of squares, set the partial derivatives to zero, and solve the system of equations to obtain the center coordinates (a, b) and radius R of the fitting circle; the solutions of a, b, and R are:

Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention is a spot edge fitting method based on the improved least squares method, which constructs concentric circles and parallel secants passing through the pixel points of the spot edge, and constructs the residual sum of squares using the distance from the outer point of the circle to the parallel secants, improves the traditional least squares circle fitting method based on geometric principles, increases the certainty and accuracy of the least squares circle fitting results, and improves the fitting accuracy of the center position of the spot.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and/or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:

FIG1 is a flow chart of the spot edge fitting method of the present invention;

FIG2 is a schematic diagram of the operation of the light spot edge fitting method of the present invention;

FIG3 is a binary segmentation image of a laser spot in the implementation of the spot edge fitting method of the present invention;

FIG4 is a Canny operator edge extraction image in the implementation of the spot edge fitting method of the present invention;

FIG5 is an image after edge fitting of a laser spot in the implementation of the spot edge fitting method of the present invention;

FIG. 6 is an image showing the fitting effect based on the circle positioning algorithm in the prior art.

In the figure, 1. point outside the circle; 2. tangent point; 3. secant line; 4. secant line; 5. secant line; 6. tangent point; 7. fitted circle center point; 8. edge pixel point; 9. edge pixel point; 10. circle; 11. fitted circle; 12. concentric circles.

DETAILED DESCRIPTION

Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, and should not be construed as limiting the present invention.

FIG1 is a preferred embodiment of the light spot edge fitting method of the present invention, and the light spot edge fitting method comprises the following steps:

Spot edge extraction: extract the laser spot edge image, preferably, use the Canny operator to extract the spot edge.

Draw concentric circles and secant lines of the fitting circle: Assume the coordinates of the center of the fitting circle to be (a, b), draw two tangent lines of the circle for any pixel point on the edge of the light spot, and connect the two tangent points to form a secant line L ₁ . Specifically:

Assume that O is the center of the fitting circle, and its coordinates are (a, b). The set of pixels at the edge of the spot is {x ₁ ,x ₂ ,…, _xi }, and _Ri is the distance from each edge pixel to the center of the circle, i.e., the radius, where i is 1, 2, 3…n, and n is the total number of pixels. Draw a circle C ₁ with O as the center for x ₁ , and draw two tangent lines about the circle C ₁ at a point outside the circle in the rectangular coordinate system. The coordinates of the point outside the circle are (x _v ,y _v ), and connect the two tangent points to get the secant line L ₁ of C _1. Repeat the above steps to draw the secant line of the circle corresponding to each pixel point at the edge of the spot, denoted by _Li .

The equation of the secant line _Li is:

The residual sum of squares is transformed into coordinates: the distance f ₁ from a point outside the circle to the secant line is obtained, and then the above operation is repeated for each pixel point on the edge of the light spot, that is, concentric circles and secants of the concentric circles of the fitting circle are made for each pixel point, and the distance _fi from the point outside the circle to the corresponding secant line _Li is obtained respectively; let the distance from a point outside the circle to the corresponding secant line L of the fitting circle be f, and according to the least squares principle, the residual sum of squares Q of _fi and f is constructed; the rectangular coordinate form of the residual sum of squares Q is converted into the polar coordinate form. Specifically:

For the convenience of calculation, move the point outside the circle to the origin, and record the distance from the origin to L ₁ as f _1. Construct a concentric circle C ₂ with O as the center of C ₁ , and repeat the above operation to obtain f ₂ ,f ₃ ,…, _fi .

The equation for _fi is:

Assuming that the distance between the secant line L of the fitting circle C and the outer point of the circle is f, according to the least squares principle, the residual sum of squares is taken:

The secant line L in the above process has been transformed into:

(x _i -a)a+(y _i -b)b+R _i ² =0 Formula 4

Repeat the above steps to find Q ₁ ,Q ₂ ,…,Q _i .

Convert the rectangular coordinate system parameters into polar coordinates:

The rectangular coordinate form of the residual sum of squares is also converted into a polar coordinate equation. In the polar coordinate system, the _fi equation is converted into:

The residual sum of squares of _fi in the polar coordinate system is obtained as follows:

Solve to get the center value of the circle when the residual square sum is the smallest: Let Q be the smallest, and you can solve for the center coordinates (a, b) and radius R of the fitting circle. Specifically:

The method to find the minimum value of Q is to find the partial derivatives of the three variables ρ, θ, and f of the residual sum of squares, set the partial derivatives to zero, and solve the system of equations to obtain the center and radius of the fitting circle.

Find the partial derivative of f and set it to zero:

The solution is:

Find the partial derivative of ρ and set it to zero:

The solution is:

Take the partial derivative of θ and set it to zero, that is:

The solution is:

Where _K1 is:

_K2 is:

Finally, combine the above equations to obtain a, b, and R respectively:

As shown in FIG2 , the spot edge fitting method of the present invention includes the following specific steps:

First, assume that the fitting center point 7 (a, b) is the center of the spot fitting circle, and draw a circle 10 with the fitting center point 7 (a, b) as the center through the edge pixel point 8 (x ₁ , y ₁ ), whose radius is R ₁ ; point 1 (x _a , y _a ) is a point outside the circle in the rectangular coordinate system, and two tangent lines about the circle 10 are drawn through the point 1 outside the circle, and the tangent points are 2 and 6 respectively; connect the tangent points 2 and 6 to form a secant line 5, and the equation of the secant line 5 is obtained according to the above formula 1, and the equation of the secant line 5 is:

Repeat the above steps to draw a concentric circle 12 of the circle 10 through the edge pixel point 9 (x ₂ , y ₂ ), with a radius of R ₂ . Similarly, draw a tangent point from the outer point 1 of the circle to form a secant line 3. The equation of the secant line 3 is:

Draw concentric circles for each pixel point on the edge of the spot image, and fit the edge pixel points of the circle to (x _i , y _i ). The equation of the secant line 4 of the fitted circle 11 is:

Next, move the outer point 1 (x _a , y _a ) to the origin of the coordinate system, and record the distance from the outer point 1 (0,0) to the secant line 5 as f ₁ ; f ₁ is calculated using the above formula 2.

Then the distance from the outer point 1 (0,0) to the secant line 3 is f ₂ .

Repeat the above operation to obtain f ₃ , f ₄ …, _fi , and obtain the distance _fi from the outer point 1 of the circle to the fitting circle secant line 4.

With the change of corresponding points, each secant equation in the above process has been transformed into:

(x _i -a)a+(y _i -b)b+R _i ² =0

According to the least squares principle, let the distance between the secant line L of the fitting circle C and the outer point of the circle be f, and the square of the residual of f ₁ is obtained in the form of Q ₁ ,Q ₂ ,…,Q _i according to the above formula 3.

Q ₁ =(f ₁ -f ) ²

_Q2 ＝( _f2 -f) ²

_Qi ＝( _fi -f) ²

Integration gives the residual sum of squares in the form:

Then, polar coordinate transformation is performed to convert the parameters of the fitting center point 7 into polar coordinate form, and the parameters are converted into the form of formula 5:

Then the distance formula becomes the form of the above formula 6:

Further, in the polar coordinate system, the residual sum of squares of _fi is converted into:

Using algebraic methods, we can find the minimum value of Q, that is, let the partial derivatives of the three variables in equation 7 be 0. By solving equation 8 for f, we can get:

By using formula 10 to solve ρ for deviation to 0, we can obtain:

By using formula 12 to find the partial derivative of θ to 0, we can get:

k ₁ and k ₂ are shown in the above formulas 14 and 15 respectively.

Finally, combining the above equations, we can solve a, b, and R as follows:

The above a, b, and R are the parameters of the fitting center point 7 (a, b).

As shown in Figure 3, in the implementation of the spot edge fitting method of the present invention, the binary segmentation image is implemented. The threshold of the laser spot area and the image background area is calculated to maximize the pixel differentiation contained in the two types of areas, that is, the variance reaches the maximum value. At this time, the threshold is the optimal threshold of the image, thereby achieving the effect of binary segmentation.

As shown in FIG4 , the Canny operator edge extraction image in the implementation of the spot edge fitting method of the present invention. After the spot is binarized, the edge area is still not smooth, and there are certain burr noises or isolated pixels. After edge extraction, the Canny operator extracts the spot edge most accurately and smoothly, has a good noise suppression effect, and the edge contour is closed.

As shown in Figure 5, the edge fitting result of the laser spot image in the implementation of the spot edge fitting method of the present invention. The fitting circle basically fits the spot edge and reflects the spot edge shape more completely. There are some irregular areas on the right side of the spot edge. Although some edge pixels do not completely overlap with the fitting circle, the edge points are distributed both outside and inside the fitting circle. The fitting error has a certain compensation, and the spot edge fitting result is reliable and accurate.

As shown in FIG6 , it is an image of the fitting effect based on the circle positioning algorithm in the prior art. Compared with the light spot edge fitting method described in the present invention, this method has obviously poorer fitting effect, has a low fit with the light spot edge, and is greatly affected by noise.

The present invention performs binarization processing on the laser spot image; uses the Canny operator to extract the edge image of the laser spot; draws concentric circles with the center of the fitting circle through the edge pixel point; finds two tangent lines from a point outside the circle to the concentric circles and makes their secants; calculates the residual sum of squares from the point outside the circle to the secant and converts it into polar coordinate form; calculates the value of the corresponding parameter when the residual sum of squares is the minimum, which is the center value of the circle. The traditional least squares circle fitting method is improved by geometric principles, the certainty and accuracy of its fitting results are increased, and the fitting accuracy of the center position of the spot is improved, which is of great significance to ensure the reliability of the visual measurement system.

It should be understood that although this specification is described according to various embodiments, not every embodiment contains only one independent technical solution. This narrative method of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

The series of detailed descriptions listed above are only specific descriptions of feasible embodiments of the present invention. They are not intended to limit the scope of protection of the present invention. All equivalent embodiments or changes that do not deviate from the technical spirit of the present invention should be included in the scope of protection of the present invention.

Claims (5)

1. A method for fitting a light spot edge, characterized in that it comprises the following steps: Spot edge extraction: extract the laser spot edge image; Draw concentric circles and secants of the fitting circle: Assume that O is the center of the fitting circle, and its coordinates are (a, b). The set of pixel points at the edge of the light spot is {x ₁ ,x ₂ ,…, _xi }, and _Ri is the distance from each edge pixel point to the center of the circle, that is, the radius, where i is 1, 2, 3…n, n is the total number of pixels, and the coordinates of the edge pixel _pointxi are ( _xi , _yi ); draw concentric circles _C1 with O as the center of x1, and draw two tangent lines about the concentric circles _C1 at a point outside the circle in the rectangular coordinate system. The coordinates of the point outside the _circle are ( _xV , _yV ), and the secant line _L1 of the concentric circles _C1 is obtained by connecting the two tangent points. Draw the secant line of the corresponding circle of each pixel point in the edge of the light spot, and record it as _Li ; Transform the residual sum of squares into coordinates: Calculate the residual sum of squares from the point outside the circle to the secant line and transform it into polar coordinates; Solve to get the center value of the circle when the residual square sum is the smallest: find the center coordinates and radius of the fitting circle corresponding to the minimum residual square sum, which is the center value of the circle, to achieve the center positioning of the light spot; The steps of coordinate transformation of the residual square sum Q are specifically as follows: Obtain the distance f ₁ from a point outside the circle to the secant line, then make concentric circles of the fitting circle and the secant lines of the concentric circles for each pixel point on the edge of the light spot, and respectively obtain the distance _fi from the point outside the circle to the corresponding secant line _Li ; Assume that the distance from a point outside the circle to the corresponding secant line L of the fitting circle is f, and construct the residual sum of squares Q between _fi and f according to the least squares principle; Convert the residual sum of squares form into polar coordinate form; Converting the residual sum of squares into polar coordinates is as follows: Convert the rectangular coordinate system parameters into polar coordinates: a＝ρcosθ, b＝ρsinθ,

In the polar coordinate system, the _fi equation is transformed into:

In the polar coordinate system, the residual sum of squares of _fi is transformed into:

The step of solving the circle center value with the minimum residual square sum Q is specifically as follows: Take partial derivatives of the three variables ρ, θ, and f of the residual sum of squares, set the partial derivatives to zero, and solve the system of equations to obtain the center coordinates (a, b) and radius R of the fitting circle; the solutions for a, b, and R are:

2. The spot edge fitting method according to claim 1 is characterized in that the Canny operator is used to extract the spot edge in the spot edge extraction step. 3. The spot edge fitting method according to claim 1, characterized in that the equation of the secant line _Li is:

4. The spot edge fitting method according to claim 1 is characterized in that, in a rectangular coordinate system, the _fi equation is:

5. The spot edge fitting method according to claim 1, characterized in that, in a rectangular coordinate system, the residual square sum Q is in the form of:

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