CN114089334B - Double-base-circle track SAR imaging method based on space diversity idea - Google Patents
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Abstract
The invention relates to a double-base circular track SAR imaging method based on a space diversity idea, and belongs to the technical field of SAR imaging. Constructing a reference signal, compensating a received echo signal by using the reference signal, and FFT transforming the signal to a distance frequency domain along a distance dimension; compensating for RVP terms in the signal; after constructing a compensation function and multiplying the signal, performing inverse FFT; after the signals are subjected to IFFT conversion along the distance dimension, linear interpolation conversion is carried out, and the signals are multiplied by a distance difference correction function to obtain a final image. The invention can realize the effect of space time exchange under the condition of limited bandwidth, thereby obviously increasing the Fourier space sampling area of the signal and effectively inhibiting the high side lobe and ringing effect in the double-base circular track SAR imaging result.
Description
Technical Field
The invention belongs to the technical field of SAR imaging, and particularly relates to a double-base circular track SAR imaging method based on a space diversity idea.
Background
The bistatic circular track synthetic aperture radar (BCSAR) has the characteristics of circular observation tracks and transceiving separation, and can acquire high-resolution images and fine observation information of an observation scene. In addition, BCSAR has more flexible imaging configuration, and the configuration can be adjusted according to actual observation requirements, so that the system sensitivity of BCSAR is further improved, and further, the imaging performance is improved.
In recent years, the attention and research heat of BCSAR by domestic and foreign scientific institutions are continuously improved. The Wang et al design proposes a parallel motion double base circular track synthetic aperture radar (PM-BCSAR) imaging model under which the angular velocity and radius of rotation of the radar receiving and transmitting platforms are the same, the only difference being the working heights of the two radar platforms. Another exemplary BCSAR imaging model is a single fixed platform double base circular track synthetic aperture radar (OS-BCSAR) imaging model proposed by Xie et al, which is composed of a transmitter fixed at a specific location and a receiver doing circular track motion, and performs continuous observation for the same area for a long time. Although the various BCSAR imaging models have great application potential and advantages of high resolution and omnibearing observation, in practical treatment, when a radar performs 360-degree full-view observation imaging on a scene, serious ringing effect and high side lobe problems exist in the images, so that the imaging performance is obviously reduced. In conventional SAR imaging processing, when there are high level side lobes, the most common processing is windowing suppression, but for BCSAR imaging, this approach has little gain. In order to solve the problem, yu et al propose a circular SAR one-dimensional spectrum data extrapolation method based on an autoregressive model to inhibit high side lobe, but the method cannot improve the integral side lobe ratio of an imaging result, the inhibition effect of the method depends on the estimation precision of spectrum data, and when the spectrum estimation precision is poor, the side lobe inhibition effect is obviously reduced. Zhang Xiangkun, and the like, find that when the transmission bandwidth is increased, the ringing effect in the image is weakened and the sidelobe level is reduced, and this finding provides a certain thought for BCSAR image sidelobe suppression, but a large number of experiments show that the mode of directly increasing the bandwidth is inefficient in that a large bandwidth increment can only be replaced by a small quantity of sidelobe level reduction. Therefore, effective methods are needed to improve the ringing effect and high side lobe level problems in BCSAR imaging, and improve the final image quality.
Disclosure of Invention
Technical problem to be solved
In order to avoid the defects of the prior art, the invention provides an improved BCSAR (GD-BCSAR) imaging geometric configuration and a corresponding algorithm based on space diversity, which reduce the discontinuity of signals in the space sampling area of Fourier, and effectively inhibit the ringing effect and high-level side lobes in the imaging result.
Technical proposal
A double-base circle locus SAR imaging method based on a space diversity idea is characterized by comprising the following steps:
step 1: constructing a reference signal;
Assuming that there is a target point p in the observed scene, its cartesian coordinates are (x p,yp,zp), and the coordinates of the transmitting end and the receiving end are (x t(ta),yt(ta),zt(ta)) and (x r(ta),yr(ta),zr(ta)), where t a represents slow time; the two instantaneous skew distances from the antenna phase center to the point target p are expressed as
Assuming that the radar transmits a chirp signal, the received echo signal expression is:
Wherein sigma, T p, gamma and f c respectively represent a target scattering coefficient, a transmission pulse width, a tone frequency and a center carrier frequency, rect [. Cndot ] represents a rectangular window function, and has
Wherein t r denotes a fast distance time, τ (t a) denotes a delay, and there is
Wherein c represents the speed of light; aiming at the problem of two-dimensional imaging of an x-y plane, the target points are all positioned in the x-y plane, namely z p =0;
selecting the center of an imaging scene as a reference point, and setting the distances from the phase centers of the receiver and the transmitter antenna to the reference point as R t_ref and R r_ref respectively; at this time, the time delay of the reference distance echo is τ ref, and the expression of the reference distance echo signal s ref(tr,ta) is:
step 2: compensating the received echo signals by using reference signals, and FFT transforming the signals to a distance frequency domain along a distance dimension;
Assuming that the scattering coefficient of the target is a constant value, performing declivity processing on the echo signal:
Wherein A represents the amplitude of the signal, represents the conjugate operation, and the conjugate reference signal The rectangular window function width of the echo signal s (t r,ta) should be slightly wider than the rectangular window function width of the echo signal s; ΔR (t a) represents the difference in distance between the point target instantaneous slope and the reference point slope, and has
ΔR(ta)＝Rt(ta)-Rt_ref+Rr(ta)-Rr_ref (11)
Performing FFT on the signal in the formula (10) along the distance dimension to obtain a signal s d(fr,ta) expression in the distance frequency domain as
Where sinc (·) represents a sinc function, with sinc (x) =sin (pi x)/(pi x);
Step 3: compensating for RVP terms in the signal;
ignoring the change of the signal envelope, wherein the second term phase term in the above formula is an RVP term, and the RVP term needs to be compensated; the compensation function is constructed according to (12) as follows
Multiplying the formula with formula (12) to realize RVP compensation; assuming that the number of samples of one intra-pulse distance frequency f r is K, a new distance frequency is defined as f (K) =f c+fr (K), which is represented by the sequence { f (K) |k=1, 2, …, K }; then the distance frequency domain signal S (f (k), t a) of GD-BCSAR is rewritten as
Response function writing of mth discrete distance unit in the above formula by using matched filtering mode
Considering that the distance frequency f (k) has the following relation
f(k)＝(k-1)Δf+f(1) (15)
Wherein Δf represents the frequency difference; after substituting the above formula into formula (14), the signal s (m, t a) is rewritten as
Step 4: after constructing a compensation function and multiplying the signal, performing inverse FFT;
to back-project equation (16), it is necessary to rewrite it to the form of the inverse fourier transform; giving the definition of the discrete fourier transform of the signal sequences S (n) and S (K)
Where the expression of the transformation factor w K is w K =exp { -j2 pi/K }, then the discrete inverse fourier transform of S (K) is
In order to rewrite the formula (16) into the form of Fourier transform, it is necessary to process it so that its phase contains a phase term similar to the above formula, and the rewritten signal s (m, t a) has the expression of
According to the fourier transform definition in equation (17), the signal s (m, t a) is finally written as follows
Defining a first term phase term in the above formula as a compensation function of the inverse Fourier transform, and a second term as a distance difference correction function; k represents a scaling factor of Fourier transform, and the scaling factor is 1/K in the case of inverse Fourier transform; in the actual processing, it is necessary to calculate the distance difference between each pulse and the pixel point of the projection plane and insert the pixel point coordinates (x, y, z) into (5);
step 5: after the signals are subjected to IFFT conversion along the distance dimension, linear interpolation conversion is carried out, and the signals are multiplied by a distance difference correction function to obtain a final image;
after zero padding operation of equation (20), the signal is expressed as
When K > K, S (f (K), t a)＝0;Nfft is defined as the number of points of the inverse Fourier transform, and N fft is the power of 2;
To obtain a given impulse response function for a pixel located at projection plane position P, it is necessary to calculate a distance difference Δr (t a) and interpolate the signal s (m, t a) to obtain s int(P,ta (n)); defining a full time sequence as { t a (N) |n=1, 2, …, N }, so that the final image response function I (P) is the sum of the response functions corresponding to all pulses:
A computer system, comprising: one or more processors, a computer-readable storage medium storing one or more programs, wherein the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the methods described above.
A computer readable storage medium, characterized by storing computer executable instructions that when executed are configured to implement the method described above.
A computer program comprising computer executable instructions which when executed are adapted to implement the method described above.
Advantageous effects
The novel BCSAR imaging configuration and the processing method provided by the invention inhibit high side lobe and ringing effects existing in the traditional BCSAR imaging, so that the image resolution is improved. Analysis by combining with a Fourier space sampling theory reveals the reason that the imaging of the traditional BCSAR system can generate ringing effect and high side lobe: the signal has serious discontinuity in the sampling area in fourier space due to the limitation of transmission bandwidth, and the discontinuity causes high side lobe and ringing effect. Based on the analysis, the thought of space diversity is introduced, the GD-BCSAR imaging geometric configuration and the corresponding algorithm are designed, and the sampling characteristic is deeply analyzed. According to calculation SFAR, GD-BCSAR can achieve the effect of space exchange time under the condition of limited bandwidth, further the Fourier space sampling area of signals is remarkably increased, and high side lobe and ringing effects in the double-base circular track SAR imaging result are effectively restrained.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, like reference numerals being used to refer to like parts throughout the several views.
FIG. 1 is a chart of the Fourier space sampling area of a GD-BCSAR single-frequency signal and a bandwidth signal;
FIG. 2 is a graph showing the variation of the effective Fourier sampling area with signal bandwidth according to the present invention compared with the conventional BCSAR;
FIG. 3 is a diagram of an imaging geometry employed in the present invention;
fig. 4 is a flowchart of an imaging process of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
The invention utilizes the Fourier sampling theorem to analyze and explain the reason of ringing effect and high side lobe generation in BCSAR images from the angles of theory and physical significance, and indicates that the high side lobe and the ringing effect are substantially determined by the Fourier space sampling area (SFA). Because of the limitation of the transmission bandwidth, the existing BCSAR imaging mode can only sample a very small area in the fourier space, and the phenomenon is the reason for the ringing effect and high side lobe of the final imaging result. Based on Fourier space sampling analysis, a concept of space diversity is introduced, and on the basis, a brand new BCSAR imaging geometric configuration, namely a space diversity double-base circular track synthetic aperture radar (GD-BCSAR), is designed, a corresponding processing algorithm is provided, and the performance of GD-BCSAR is deeply analyzed. The GD-BCSAR can break through the limitation of the transmitting bandwidth by increasing the space diversity order, effectively increase the sampling area of the signal in the Fourier space, and effectively inhibit the ringing effect and high-level side lobes in the final imaging result.
The method for realizing the invention comprises the following steps:
(1) Carrying out qualitative analysis on sampling areas of corresponding imaging geometric configurations in a Fourier space by combining with a Fourier space sampling theorem, and determining the reasons of image ringing effect and high side lobe generation;
(2) According to the required diversity degree, designing a system working mode and corresponding working parameters of the space diversity double-base circular SAR, and deducing a signal echo model thereof;
(3) Imaging processing is carried out by utilizing a back projection algorithm;
Wherein:
the step (1) mainly comprises the following steps:
1a) Establishing a GD-BCSAR Fourier space sampling area schematic diagram;
1b) Calculating the corresponding geometric parameters of the sampling area;
1c) And calculating the Fourier effective sampling area ratio corresponding to the corresponding geometric configuration.
The step (2) mainly comprises the following steps:
2a) Setting the moving speed and the moving speed direction of a receiver platform and a transmitter platform;
2b) Setting the working heights of a receiver platform and a transmitter platform;
2c) Setting the rotation radius of a receiver platform and a transmitter platform;
2d) And calculating and deducing corresponding pitch histories according to the motion characteristics of the receiver and the transmitter.
The step (3) mainly comprises the following steps:
3a) Constructing a reference signal;
3b) Compensating the received echo signals by using reference signals, and FFT transforming the signals to a distance frequency domain along a distance dimension;
3c) Compensating for RVP terms in the signal;
3d) After constructing a compensation function and multiplying the signal, performing inverse FFT;
3e) After the signals are subjected to IFFT conversion along the distance dimension, linear interpolation conversion is carried out, and the signals are multiplied by a distance difference correction function to obtain a final image.
The steps are as follows:
(1) Fourier space effective sampling area ratio calculation
From a priori knowledge, when the transmitter and the receiver move, the sampling area of the transmitted signal in the fourier space is a hollow circle with the center deviating from the origin. Unlike the conventional BCSAR receiver and transmitter simultaneous motion, GD-BCSAR forms a "fixed" receiver/transmitter at multiple locations throughout the observation, which also indicates that the signal is offset and distorted in the fourier space sampling area. Thus, the sampling area in the fourier space when the GD-BCSAR system transmits a single frequency signal can be obtained as shown in fig. 1 (a).
In combination with the above analysis, when the transmitted signal is a bandwidth signal, the fourier space sampling area of GD-BCSAR can be obtained as shown in fig. 1 (b). As can be seen, the Fourier space sampling area of GD-BCSAR can be seen as the rotation of the sampling region of OS-BCSAR about the origin of coordinates, and in an ideal case, GD-BCSAR would result in a nearly complete circular sampling region. To verify the advantages of GD-BCSAR in terms of fourier space sampling, we introduce here a fourier space sampling area Ratio (SFA Ratio, SFAR) to calculate the actual effective sampling area rate of the signal.
Using wavenumber sampling vectors to define the radius of a sampling circular region, a corresponding expression can be given as
From the above, it can be seen that when the bandwidth is fixed to the center carrier frequency, the maximum sampling area of the signal in the Fourier space isSFAR can be defined as the ratio of the actual sampling area to the maximum sampling area, i.e. there is
For the traditional BCSAR, the sampling area is regular in shape, so that the corresponding SFAR can be calculated by using the formula (2). However, when the GD-BCSAR system samples, since each sampling area overlaps with each other, it is difficult to give an accurate calculation formula of the entire sampling area. Considering that the sampling area of GD-BCSAR can be regarded as consisting of an overlapping of sampling areas of several OS-BCSAR, we can thus approximate GD-BCSAR area using a sampling area combination like that in fig. 1 (b), i.e. eight sets of OS-BCSAR sampling areas offset from each other by 45 °. According to the geometric relationship, the approximate sampling area S GD_BCSAR of GD-BCSAR in the Fourier space can be calculated as
SGD_BCSAR＝4·SOS_BCSAR-4.ΔS1+4·ΔS2 (3)
Wherein the method comprises the steps of
Since the actual sampling area of BCSAR system is mainly determined by the bandwidth of the transmitted signal and the central carrier frequency, we choose the X-band as the working band of the radar system, the bandwidth of the transmitted signal varies in the range from 100MHz to 1GHz, and fig. 2 shows the SFAR variation graph of three BCSAR configurations with respect to the bandwidth of the signal.
As can be seen from fig. 2, SFAR of GD-BCSAR is significantly higher than conventional BCSAR, and when the transmission bandwidth of GD-BCSAR is 100MHz, its corresponding SFAR is close to the sampling rate when PM-BCSAR transmits an 800MHz bandwidth signal, which also verifies the capability of the previously mentioned GD-BCSAR to realize space utilization for bandwidth. When the transmission signal bandwidth is increased to 1GHz, SFAR of PM-BCSAR can be increased only to about 18%, and OS-BCSAR is less than 5%, which indicates that the method of enlarging the fourier space sampling area by directly increasing the signal bandwidth is inefficient, which is also why the increase of bandwidth is not obvious for the side lobe suppression effect. In contrast, GD-BCSAR had a SFAR of more than 25% and a sampling area ratio significantly exceeding that of conventional BCSAR at a transmission bandwidth of 1 GHz. In addition, the space diversity degree can be further increased by increasing the difference between the rotation radius, the working height and the rotation angular velocity of the transmitter and the receiver, and the sampling area ratio of the GD-BCSAR is further improved at the moment, so that the effectiveness and the superiority of the GD-BCSAR are fully verified by the characteristics.
(2) GD-BCSAR imaging geometry and signal model
The main idea of GD-BCSAR is to change the geometrical configuration constituent elements such as the height, the rotation radius or the rotation angular velocity of the radar platform, so as to achieve the effect of increasing the spatial diversity. The GD-BCSAR imaging geometry is given in FIG. 3. As shown in the figure, the radar transmitter and the receiver on-board platform move at the same height H at uniform speeds and circular motions respectively with the rotating radiuses r T and r R, and the rotating angular speeds of the transmitter and the receiver are omega T,ωR respectively. The ground wiping angle between the center of the illumination scene and the transmitter is theta T_in, and the ground wiping angle between the center of the illumination scene and the receiver is theta R_in. During radar operation, the antenna beam always illuminates the center of the observation scene.
Assume that there is a target point p in the observed scene, its cartesian coordinates are (x p,yp,zp), and the coordinates of the transmitting end and the receiving end are (x t(ta),yt(ta),zt(ta)) and (x r(ta),yr(ta),zr(ta), respectively, where t a represents slow time. The two instantaneous skew of the antenna phase center to the point target p at this time can be expressed as
Assuming that the radar transmits a chirp signal, the received echo signal expression may be written as
Wherein sigma, T p, gamma and f c respectively represent a target scattering coefficient, a transmission pulse width, a tone frequency and a center carrier frequency, rect [. Cndot ] represents a rectangular window function, and has
Wherein t r denotes a fast distance time, τ (t a) denotes a delay, and there is
Where c represents the speed of light. The invention mainly aims at the problem of two-dimensional imaging of an x-y plane, so that target points are all located in the x-y plane, namely z p = 0.
(3) GD-BCSAR imaging processing algorithm
The center of the imaging scene is selected as a reference point, and the distances from the phase centers of the receiver and the transmitter antennas to the reference point are set as R t_ref and R r_ref respectively. At this time, the time delay of the reference distance echo is τ ref, and the expression of the reference distance echo signal s ref(tr,ta) can be written as
Assuming that the target scattering coefficient is a constant value, performing declining treatment on the echo signal, and multiplying the conjugated value by the formula (6) to obtain
Wherein A represents the amplitude of the signal, represents the conjugate operation, and the conjugate reference signalThe rectangular window function width of (c) should be slightly wider than the rectangular window function width of the echo signal s (t r,ta). ΔR (t a) represents the difference in distance between the point target instantaneous slope and the reference point slope, and has
ΔR(ta)＝Rt(ta)-Rt_ref+Rr(ta)-Rr_ref (11)
Performing FFT on the signal in the formula (10) along the distance dimension to obtain a signal s d(fr,ta) expression in the distance frequency domain as
Where sinc (·) represents a sinc function, there is sinc (x) =sin (pi x)/(pi x). Neglecting the variation of the signal envelope, the second term in the above equation is RVP term, which needs to be compensated for, assuming that the number of samples of the distance frequency f r in one pulse is K, a new distance frequency is defined as f (K) =f c+fr (K), which can be represented by the sequence { f (K) |k=1, 2, …, K }. Then the distance frequency domain signal S (f (k), t a) of GD-BCSAR can be rewritten as
The response function of the mth discrete distance unit in the above formula can be written by using a matched filtering mode
Considering that the distance frequency f (k) has the following relation
f(k)＝(k-1)Δf+f(1) (15)
Where Δf represents the frequency difference. After substituting the above formula into formula (14), the signal s (m, t a) can be rewritten as
To back project the above equation, it is necessary to rewrite the above equation into an inverse fourier transform form. Here we give the definition of the discrete Fourier transform of the signal sequences S (n) and S (K)
Where the expression of the transformation factor w K is w K =exp { -j2 pi/K }, then the discrete inverse fourier transform of S (K) is
In order to rewrite the formula (16) into the form of Fourier transform, it is necessary to process it so that its phase contains a phase term similar to the above formula, and the rewritten signal s (m, t a) has the expression of
According to the fourier transform definition in equation (17), the signal s (m, t a) can be finally written as follows
The first term in the above equation is defined as the compensation function of the inverse fourier transform and the second term is the distance difference correction function. K represents the scaling factor of the Fourier transform, which takes a constant value and is 1/K in the inverse Fourier transform. In the actual processing, it is necessary to calculate the distance difference between each pulse and the projection plane pixel, and insert the pixel coordinates (x, y, z) into equation (5). It should be noted here that to obtain the final GD-BCSAR image by equation (20), an interpolation operation is also required, because the discretized distance difference will generate a certain deviation when projected directly onto the imaging plane.
Considering that the GD-BCSAR signal is a wideband signal, interpolation processing can be performed using a sinc kernel function. This operation can be approximately seen as zero padding when performing the inverse fourier transform and linear interpolation when finally outputting the discretized result. N fft is defined as the number of points of the inverse fourier transform, and N fft should generally be 10 times the length of the original signal, but N fft is generally 2 times in order to optimize the computational complexity. After zero padding operation of equation (20), the signal can be expressed as
When K > K, there is S (f (K), t a) =0.
To obtain a given impulse response function for a pixel located at projection plane position P, it is necessary to calculate a distance difference Δr (t a) and interpolate the signal s (m, t a) to obtain s int(P,ta (n)). The full time sequence is defined as { t a (N) |n=1, 2, …, N }, so the resulting image response function I (P) is the sum of the response functions corresponding to all pulses.
While the invention has been described with reference to certain preferred embodiments, it will be understood by those skilled in the art that various changes and substitutions of equivalents may be made without departing from the spirit and scope of the invention.
Claims (4)
1. A double-base circle locus SAR imaging method based on a space diversity idea is characterized by comprising the following steps:
step 1: constructing a reference signal;
Assuming that there is a target point p in the observed scene, its cartesian coordinates are (x p,yp,zp), and the coordinates of the transmitting end and the receiving end are (x t(ta),yt(ta),zt(ta)) and (x r(ta),yr(ta),zr(ta)), where t a represents slow time; the two instantaneous skew distances from the antenna phase center to the point target p are expressed as
Assuming that the radar transmits a chirp signal, the received echo signal expression is:
Wherein sigma, T p, gamma and f c respectively represent a target scattering coefficient, a transmission pulse width, a tone frequency and a center carrier frequency, rect [. Cndot ] represents a rectangular window function, and has
Wherein t r denotes a fast distance time, τ (t a) denotes a delay, and there is
Wherein c represents the speed of light; aiming at the problem of two-dimensional imaging of an x-y plane, the target points are all positioned in the x-y plane, namely z p =0;
Selecting the center of an imaging scene as a reference point, and setting the distances from the phase centers of the receiver and the transmitter antenna to the reference point as R t_ref and R r_ref respectively; at this time, the time delay of the reference distance echo is τ ref, and the expression of the reference distance echo signal s ref(tr,ta) is:
step 2: compensating the received echo signals by using reference signals, and FFT transforming the signals to a distance frequency domain along a distance dimension;
Assuming that the scattering coefficient of the target is a constant value, performing declivity processing on the echo signal:
Wherein A represents the amplitude of the signal, represents the conjugate operation, and the conjugate reference signal The rectangular window function width of the echo signal s (t r,ta) should be slightly wider than the rectangular window function width of the echo signal s; ΔR (t a) represents the difference in distance between the point target instantaneous slope and the reference point slope, and has
ΔR(ta)＝Rt(ta)-Rt_ref+Rr(ta)-Rr_ref (11)
Performing FFT on the signal in the formula (10) along the distance dimension to obtain a signal s d(fr,ta) expression in the distance frequency domain as
Where sinc (·) represents a sinc function, with sinc (x) =sin (pi x)/(pi x);
Step 3: compensating for RVP terms in the signal;
ignoring the change of the signal envelope, wherein the second term phase term in the above formula is an RVP term, and the RVP term needs to be compensated; the compensation function is constructed according to (12) as follows
Multiplying the formula with formula (12) to realize RVP compensation; assuming that the number of samples of one intra-pulse distance frequency f r is K, a new distance frequency is defined as f (K) =f c+fr (K), which is represented by the sequence { f (K) |k=1, 2, …, K }; then the distance frequency domain signal S (f (k), t a) of GD-BCSAR is rewritten as
Response function writing of mth discrete distance unit in the above formula by using matched filtering mode
Considering that the distance frequency f (k) has the following relation
f(k)＝(k-1)Δf+f(1) (15)
Wherein Δf represents the frequency difference; after substituting the above formula into formula (14), the signal s (m, t a) is rewritten as
Step 4: after constructing a compensation function and multiplying the signal, performing inverse FFT;
to back-project equation (16), it is necessary to rewrite it to the form of the inverse fourier transform; giving the definition of the discrete fourier transform of the signal sequences S (n) and S (K)
Where the expression of the transformation factor w K is w K =exp { -j2 pi/K }, then the discrete inverse fourier transform of S (K) is
In order to rewrite the formula (16) into the form of Fourier transform, it is necessary to process it so that its phase contains a phase term similar to the above formula, and the rewritten signal s (m, t a) has the expression of
According to the fourier transform definition in equation (17), the signal s (m, t a) is finally written as follows
Defining a first term phase term in the above formula as a compensation function of the inverse Fourier transform, and a second term as a distance difference correction function; k represents a scaling factor of Fourier transform, and the scaling factor is 1/K in the case of inverse Fourier transform; in the actual processing, it is necessary to calculate the distance difference between each pulse and the pixel point of the projection plane and insert the pixel point coordinates (x, y, z) into (5);
step 5: after the signals are subjected to IFFT conversion along the distance dimension, linear interpolation conversion is carried out, and the signals are multiplied by a distance difference correction function to obtain a final image;
after zero padding operation of equation (20), the signal is expressed as
When K > K, S (f (K), t a)＝0;Nfft is defined as the number of points of the inverse Fourier transform, and N fft is the power of 2;
To obtain a given impulse response function for a pixel located at projection plane position P, it is necessary to calculate a distance difference Δr (t a) and interpolate the signal s (m, t a) to obtain s int(P,ta (n)); defining a full time sequence as { t a (N) |n=1, 2, …, N }, so that the final image response function I (P) is the sum of the response functions corresponding to all pulses:
2. a computer system, comprising: one or more processors, a computer-readable storage medium storing one or more programs, wherein the one or more programs, when executed by the one or more processors, cause the one or more processors to implement the method of claim 1.
3. A computer readable storage medium, characterized by storing computer executable instructions that, when executed, are adapted to implement the method of claim 1.
4. A computer program product comprising computer executable instructions which, when executed, are adapted to implement the method of claim 1.
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