CN109143237B - PFA wavefront curvature correction method applicable to bistatic bunching SAR (synthetic aperture radar) with any platform track - Google Patents

PFA wavefront curvature correction method applicable to bistatic bunching SAR (synthetic aperture radar) with any platform track Download PDF

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CN109143237B
CN109143237B CN201811008004.9A CN201811008004A CN109143237B CN 109143237 B CN109143237 B CN 109143237B CN 201811008004 A CN201811008004 A CN 201811008004A CN 109143237 B CN109143237 B CN 109143237B
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武俊杰
缪昱宣
陈天夫
王雯璟
李易
李中余
杨建宇
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a PFA wave-front bending correction method of a double-base bunching SAR suitable for any platform track, which comprises the steps of initializing imaging system parameters, processing echo data by adopting a polar coordinate format algorithm and performing high-order space-variant phase compensation imaging. The invention adopts a method of constructing the bending error compensation filter to realize the correction of PFA wave front bending error of the double-base bunching SAR, constructs the space-variant wave front bending error compensation filter by an interpolation mapping method, can obtain a consistent focusing image of the bunching double-base SAR with higher efficiency, can realize high-resolution imaging processing in a larger imaging scene range, and enables an aircraft carrying the double-base SAR system to play better performances in the fields of battlefield reconnaissance and monitoring, material airdrop, earthquake disaster rescue and the like.

Description

PFA wavefront curvature correction method applicable to bistatic bunching SAR (synthetic aperture radar) with any platform track
Technical Field
The invention belongs to the technical field of radar signal processing, and particularly relates to a PFA wave front bending correction method of a double-base bunching SAR (synthetic aperture radar) suitable for any platform track.
Background
Since the birth of Synthetic Aperture Radar (SAR) technology, the Synthetic Aperture Radar has the characteristic of being free from the influence of weather and time, and can efficiently image ground objects, and the Synthetic Aperture Radar has been greatly developed in the aspects of high resolution, multi-polarization, multi-mode, multi-waveband and the like. And in order to improve the viability of SAR systems, researchers have proposed a new SAR system mechanism that places the transmitter system and the receiver system separately on two different platforms, such a system is called bistatic SAR.
The traditional SAR system works in a front side view mode or a squint mode, namely, a certain included angle exists between the flight direction and the beam direction. However, in practical situations, the target dynamics directly in front is more important than the lateral target dynamics. For the monostatic SAR, if the radar irradiates right ahead, the doppler direction is consistent with the range direction, and imaging cannot be performed. However, forward looking imaging is possible due to the special transceiving mode of bistatic SAR.
Disclosure of Invention
The invention aims to: in order to overcome the application defect of the conventional PFA wave-front bending correction method in actual bistatic SAR imaging, the invention provides the PFA wave-front bending correction method of the bistatic bunching SAR suitable for any platform track.
The technical scheme of the invention is as follows: a PFA wavefront curvature correction method suitable for a bistatic bunching SAR of any platform track comprises the following steps:
A. imaging system parameter initialization
The imaging system parameters comprise a reference point position and a three-dimensional track [ x ] of a transmitting stationT(t),yT(t),zT(t)]Receiving station three-dimensional locus [ x ]R(t),yR(t),zR(t)]Carrier frequency f of the transmitted signalcPulse width TpDistance-wise sampling frequency FsDistance direction sampling point number NrPulse repetition interval PRI, azimuth time vector t, range time vector tau, range frequency vector fτPoint object in scene (x)p,yp) Distance history R ofp(t；xp,yp)；
B. Processing echo data by adopting polar coordinate format algorithm
Constructing a point target reference function according to echo data of a reference point target, performing consistent compression on the echo data by using a distance frequency domain and an azimuth time domain, and performing polar coordinate format conversion on the consistently compressed echo data to obtain a PFA (Perfluoro fluoro ethylene) rough image;
C. high order space-variant phase compensation
Dividing the PFA rough image into a plurality of sub-images, constructing a space-variant wavefront bending error compensation filter, calculating a phase compensation factor corresponding to each sub-image, performing phase compensation, and splicing all the sub-images after phase compensation to obtain an imaging result.
Further, in the step B, the point target reference function is expressed as
Figure GDA0002562462970000021
Wherein S iso(fτT) represents a point target reference function, KrThe frequency is adjusted for the transmitted signal and c is the speed of light.
Further, in the step B, the echo data is uniformly compressed by conjugate multiplication of the distance frequency domain and the azimuth time domain, and the compressed distance frequency domain data is expressed as
Figure GDA0002562462970000022
Wherein, FFTr{. represents the distance direction FFT, sr(τ,t；xp,yp) Representing target echo data, So *(fτAnd t) represents the reference signal S for coherent compressiono(fτT) conjugation, Δ Rp(t；xp,yp) Representing a history of differential distances, wr[·]Representing a distance-to-time window function, wa[·]Representing an azimuthal time-domain window function, TaThe synthetic aperture time is indicated.
Further, in the step B, polar coordinate format conversion is performed on the echo data after the uniform compression, and two-dimensional low-pass interpolation is performed to convert the echo data into a polar coordinate format
Figure GDA0002562462970000023
And
Figure GDA0002562462970000024
transforming to two-dimensional wave number domain to obtain
Figure GDA0002562462970000025
And
Figure GDA0002562462970000026
further, in the step C, the effective scene size before compensation is calculated first, and the calculation formula is
Figure GDA0002562462970000027
Wherein M isiAnd NiAll the parameters are radar platform motion track correlation coefficients, and lambda represents carrier wave wavelength;
dividing the PFA coarse image h (x, y) into a plurality of sub-images h according to the effective scene sizei(x,y)。
Further, in the step C, the constructed space-variant wavefront curvature error compensation filter is represented as
Figure GDA0002562462970000028
Wherein k isxAnd kyThe wave numbers in the x and y directions respectively,
Figure GDA0002562462970000029
representing the total wavefront curvature phase of the wavenumber domain.
Further, in the step C, for each sub-image hi(x, y), the geometric center position (x) is determinedi,yi) Performing two-dimensional fast Fourier transform on the subimage to obtain a wave number spectrum H thereofi(kx,ky) (ii) a Then calculates the corresponding phase compensation factor Hc(kx,ky；xi,yi) Two-dimensional multiplication with sub-image wave number spectrum and solving product resultAnd performing inverse fast Fourier transform to obtain the compensated sub-image.
The invention has the beneficial effects that: the invention adopts a method of constructing the bending error compensation filter to realize the correction of PFA wave front bending error of the double-base bunching SAR, constructs the space-variant wave front bending error compensation filter by an interpolation mapping method, can obtain a consistent focusing image of the bunching double-base SAR with higher efficiency, can realize high-resolution imaging processing in a larger imaging scene range, and enables an aircraft carrying the double-base SAR system to play better performances in the fields of battlefield reconnaissance and monitoring, material airdrop, earthquake disaster rescue and the like.
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FIG. 1 is a schematic flow chart of a PFA wavefront curvature correction method of the double-base bunching SAR applicable to any platform trajectory according to the present invention;
FIG. 2 is a schematic geometric diagram of a bistatic SAR imaging system employed in an embodiment of the present invention;
FIG. 3 is a diagram of a target scene layout employed in an embodiment of the present invention;
fig. 4 is a schematic diagram of a simulation result of imaging a target scene in the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
For the convenience of describing the contents of the present invention, the following terms are first explained:
the term 1: bistatic Synthetic Aperture Radar (Bistatic Synthetic Aperture Radar, BiSAR)
The bistatic synthetic aperture radar is characterized in that in the motion process of a radar platform, a transmitting station antenna irradiates an imaging area, and a receiving station antenna receives a target scattered echo in the imaging area; the large bandwidth of the transmitted signal is utilized to form distance direction high resolution, the Doppler phase of the azimuth signal is compensated through an imaging processing algorithm to realize azimuth aperture synthesis so as to form azimuth direction high resolution, and therefore two-dimensional high resolution imaging in an imaging area is realized.
The term 2: polar Format Algorithm (Polar Format Algorithm, PFA)
The PFA algorithm is a mature algorithm which is provided for processing the beamforming SAR imaging earlier, has the advantages of simplicity, high efficiency, small calculation amount, easiness in combination with a self-focusing algorithm and the like, and is still an ideal choice in the ultrahigh resolution beamforming SAR imaging nowadays. However, the traditional polar format algorithm has two obvious defects, and the application and the development of the traditional polar format algorithm are limited. One is that when the conventional PFA is implemented, data distributed in a polar coordinate format needs to be converted into data uniformly arranged in a rectangular format through a two-dimensional interpolation operation, so as to facilitate subsequent two-dimensional FFT processing. Since the two-dimensional interpolation process is performed in the spatial frequency domain (wave number domain), local errors in the spatial frequency domain may cause degradation of the overall image quality. Therefore, high requirements are provided for the precision of the interpolation algorithm for ensuring the imaging quality; however, the high-precision interpolation algorithm will impose certain restrictions on the calculation efficiency, so there is a contradiction between the interpolation precision and the calculation efficiency. Meanwhile, the process of using programming to realize interpolation operation is also complex. PFA uses a plane wave front assumption, but the radar actually emits spherical waves, so that primary and secondary space-variant phase errors of a spatial frequency domain are generated, geometric distortion and defocusing effects are generated on an image, and the size of an effective scene of PFA is limited. Especially under the requirement of high resolution or near field scene, the effective imaging range of the traditional PFA can hardly meet the practical requirement.
The term 3: wavefront bending error
The wavefront bending error is caused by the fact that a PFA algorithm ignores a high-order space variant term of distance history on the premise of the assumption of a far-field plane wave. Wavefront curvature error can have two manifestations in the image domain: firstly, a target point far away from a scene central point in an imaging result can be defocused; secondly, geometric distortion occurs in the imaging result.
Fig. 1 is a schematic flow chart of a PFA wavefront curvature correction method suitable for bistatic bunching SAR of any platform trajectory according to the present invention. A PFA wavefront curvature correction method suitable for a bistatic bunching SAR of any platform track comprises the following steps:
A. imaging system parameter initialization
The imaging system parameters comprise a reference point position and a three-dimensional track [ x ] of a transmitting stationT(t),yT(t),zT(t)]Receiving station three-dimensional locus [ x ]R(t),yR(t),zR(t)]Carrier frequency f of the transmitted signalcPulse width TpDistance-wise sampling frequency FsDistance direction sampling point number NrPulse repetition interval PRI, azimuth time vector t, range time vector tau, range frequency vector fτPoint object in scene (x)p,yp) Distance history R ofp(t；xp,yp)；
B. Processing echo data by adopting polar coordinate format algorithm
Constructing a point target reference function according to echo data of a reference point target, performing consistent compression on the echo data by using a distance frequency domain and an azimuth time domain, and performing polar coordinate format conversion on the consistently compressed echo data to obtain a PFA (Perfluoro fluoro ethylene) rough image;
C. high order space-variant phase compensation
Dividing the PFA rough image into a plurality of sub-images, constructing a space-variant wavefront bending error compensation filter, calculating a phase compensation factor corresponding to each sub-image, performing phase compensation, and splicing all the sub-images after phase compensation to obtain an imaging result.
In an optional embodiment of the present invention, the step a initializes the parameters of the imaging system, including the reference point position, defined as (0,0, 0); three-dimensional trajectory of the transmitting station, denoted xT(t),yT(t),zT(t)](ii) a Three-dimensional trajectory of the receiving station, denoted xR(t),yR(t),zR(t)](ii) a Carrier frequency of the transmitted signal, denoted fc(ii) a Pulse width, noted as Tp(ii) a Distance-wise sampling frequency, denoted Fs(ii) a Number of sampling points in the distance direction, which is recorded as Nr(ii) a Pulse repetition interval, denoted PRI; the azimuth time vector, denoted t [ -PRI · N [ ]a/2,-PRI·(Na/2-1),…,PRI·(Na/2-1)]Distance-time vector, noted as τ [ -1/Fs·Nr/2,-1/Fs·(Nr/2-1),…,1/Fs·(Nr/2-1)](ii) a Distance frequency vector, noted as fτ＝[-Fs/2,-Fs·(Nr/2-1)/Nr,…,Fs·(Nr/2-1)/Nr]；
Point object in scene (x)p,yp) The distance history of (d) is:
Rp(t；xp,yp)＝RT(t；xp,yp)+RR(t；xp,yp)
wherein the content of the first and second substances,
Figure GDA0002562462970000051
RT(t；xp,yp) Representing the distance history of the transmitting station (T),
Figure GDA0002562462970000052
represents a trajectory vector of the transmitting station (T),
Figure GDA0002562462970000053
RR(t；xp,yp) Indicating the distance history of the receiving station (R),
Figure GDA0002562462970000054
represents the trajectory vector of the receiving station (R).
Fig. 2 is a schematic geometric diagram of a bistatic SAR imaging system used in the embodiment of the present invention. The system coordinate system takes a ground surface point pointed by the center of the radar double-station beam as a coordinate origin, and the platform moves along a curved track. The imaging system parameters are shown in table 1.
TABLE 1 imaging System parameters
Figure GDA0002562462970000055
As shown in fig. 3, it is a target scene layout diagram adopted in the embodiment of the present invention; the dots in the figure are 49 point targets arranged 7 × 7 on the ground. The 49 points are spaced apart by 400m in both the x and y directions. The position coordinates of any point in the scene are denoted as P (x, y).
Constructing an orientation time vector t [ -PRI · N [ ]a/2,-PRI·(Na/2-1),…,PRI·(Na/2-1)]Where PRI is the pulse repetition time, NaAnd counting the number of sampling points in the azimuth direction of the target echo. Distance-time vector, noted as τ [ -1/F [ ]s·Nr/2,-1/Fs·(Nr/2-1),…,1/Fs·(Nr/2-1)]Wherein F issIs the range-wise sampling rate, NrAnd counting the number of sampling points in the direction of the target echo distance.
In an optional embodiment of the present invention, in step B, the point target echo data s is simulated by using MATLAB according to the radar platform flight trajectory and the point target coordinate position initialized in step ar(τ,t；xp,yp) Is shown as
Figure GDA0002562462970000061
Wherein σp(xp,yp) Representing the target scattering cross-sectional area (RCS); w is ar[·]Representing a distance-to-time window function representing the envelope of the transmitted pulse signal, the window width being Tp；wa[·]Representing the time domain window function of azimuth direction, representing the modulation effect of antenna directional diagram, and the window width is the time T of synthetic aperturea；KrAdjusting the frequency of the transmitting signal; c is the speed of light; λ is the carrier wavelength.
Constructing a consistent compressed reference signal, expressed as
Figure GDA0002562462970000062
Wherein S iso(fτAnd t) represents a point target reference function.
And then conjugate point multiplication is carried out on the echo signal and the reference signal by utilizing a distance frequency domain and an azimuth time domain, the echo data are compressed uniformly, and the compressed distance frequency domain data are expressed as follows:
Figure GDA0002562462970000063
wherein, FFTr{. represents the distance direction FFT, So *(fτAnd t) represents the reference signal S for coherent compressiono(fτT) taking the conjugate of the compound,
Figure GDA0002562462970000064
wherein, the lower subscript 0 indicates that the point target is located at the central point, Δ indicates the difference, Δ RT(t；xp,yp) Representing the history of the differential distance of the transmitting station, Δ RR(t；xp,yp) Representing the differential distance history of the receiving station.
Will differentiate the distance history Δ Rp(t；xp,yp) For space coordinate (x)p,yp) Two-dimensional Taylor expansion and splitting the expansion into linear terms and higher order terms can result in:
Figure GDA0002562462970000071
wherein R isT(t)＝||rT(t)||2，RR(t)＝||rR(t)||2；ΔRcur(t,xp,yp) Generally remain to the third order, i.e.:
Figure GDA0002562462970000072
and calculating a data vector by using platform track data:
Figure GDA0002562462970000073
Figure GDA0002562462970000074
according to the mapping relation
Figure GDA0002562462970000075
Figure GDA0002562462970000076
The polar coordinate format conversion is carried out on the echo data after the consistent compression through two-dimensional low-pass interpolation, and the echo data are converted in the same conversion mode
Figure GDA0002562462970000077
And
Figure GDA0002562462970000078
transforming to two-dimensional wave number domain to obtain
Figure GDA0002562462970000079
And
Figure GDA00025624629700000710
wherein
Figure GDA00025624629700000711
The second-order taylor expansion coefficient is represented,
Figure GDA00025624629700000712
the third-order taylor expansion coefficient is represented,
Figure GDA00025624629700000713
representing the second order wavefront curvature phase of the wavenumber domain,
Figure GDA00025624629700000714
representing the third-order wavefront curvature phase in the wavenumber domain.
The coordinate axis of the mapped two-dimensional wave number is expressed as
Figure GDA00025624629700000715
Figure GDA00025624629700000716
Wherein k isx min,ky minThe minimum values of the wave number in the x direction and the wave number in the y direction, kx max,ky maxThe maximum values of the wave number in the x direction and the wave number in the y direction are respectively calculated according to the wave number mapping relation given above; dx,DyThe x-direction dimension and the y-direction dimension of the imaged scene, respectively.
After the data mapping is finished, a wave number spectrum H is obtainedp(kx,ky) Then to Hp(kx,ky) And performing two-dimensional inverse fast Fourier transform to obtain a PFA coarse image h (x, y).
In an alternative embodiment of the present invention, the step C first calculates the effective scene size before compensation, and the calculation formula is
Figure GDA0002562462970000081
Wherein the coefficient MiAnd NiThe expression of (a) is as follows:
Figure GDA0002562462970000082
Figure GDA0002562462970000083
Figure GDA0002562462970000084
Figure GDA0002562462970000085
Figure GDA0002562462970000086
Figure GDA0002562462970000087
Figure GDA0002562462970000088
finding a relation to xpAnd ypAfter the value range of (a), dividing the PFA coarse image h (x, y) into a plurality of sub-images h with the size determined by the value rangei(x,y)。
Then, based on the center position (x) of each sub-imagei,yi) Constructing a space-variant wavefront curvature error compensation filter represented as:
Figure GDA0002562462970000091
wherein the content of the first and second substances,
Figure GDA0002562462970000092
by passing
Figure GDA0002562462970000093
And obtaining the interpolation by mapping the interpolation to a two-dimensional wave number domain.
To construct the filter, the filter can be constructed by dividing Δ Rcur(t；xp,yp) Factorization is carried out:
Figure GDA0002562462970000094
it can be seen that Δ Rcur(t；xp,yp) Is divided into two parts, the first part is xpypCoefficient of irrelevance
Figure GDA0002562462970000095
And
Figure GDA0002562462970000096
the second part being xp iyp j；
Then can obtain
Figure GDA0002562462970000097
Combining the products obtained in step B
Figure GDA0002562462970000098
And
Figure GDA0002562462970000099
to obtain
Figure GDA00025624629700000910
Where N represents a set of natural numbers.
Thereby obtaining a compensated phase matrix
Figure GDA00025624629700000911
For each sub-image hi(x, y), the geometric center position (x) is determinedi,yi) Performing two-dimensional fast Fourier transform on the subimage to obtain a wave number spectrum H thereofi(kx,ky) (ii) a Then calculates the corresponding phase compensation factor Hc(kx,ky；xi,yi) Multiplying the sub-image wave number spectrum and solving the two-dimensional inverse fast Fourier transform of the product result to obtain a compensated sub-image expressed as
h′i(x,y)＝IFFT2{FFT2{hi(x,y)}·Hc(kx,ky；xi,yi)}。
And after all the sub-images are subjected to phase compensation, image splicing is carried out on all the sub-images subjected to phase compensation according to a geometric relation, and an imaging result h' (x, y) of large scene uniform focusing is obtained.
Fig. 4 is a schematic diagram illustrating an imaging simulation result of a target scene according to an embodiment of the present invention; it can be seen that the invention realizes the compensation of PFA wavefront bending error and the imaging processing of high resolution consistent focusing within a larger imaging scene range.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (7)

1. A PFA wavefront curvature correction method suitable for a bistatic bunching SAR of any platform track is characterized by comprising the following steps:
A. imaging system parameter initialization
The imaging system parameters comprise a reference point position and a three-dimensional track [ x ] of a transmitting stationT(t),yT(t),zT(t)]Receiving station three-dimensional locus [ x ]R(t),yR(t),zR(t)]Carrier frequency f of the transmitted signalcPulse width TpDistance-wise sampling frequency FsDistance direction sampling point number NrPulse repetition interval PRI, azimuth time vector t, range time vector tau, range frequency vector fτPoint object in scene (x)p,yp) Distance history R ofp(t；xp,yp)；
B. Processing echo data by adopting polar coordinate format algorithm
Constructing a consistent compression reference signal according to echo data of a reference point target, performing consistent compression on the echo data by using a distance frequency domain and a direction time domain, and performing polar coordinate format conversion on the consistently compressed echo data to obtain a PFA (Perfluoro fluoro ethylene) rough image;
C. high order space-variant phase compensation
Dividing the PFA rough image into a plurality of sub-images, constructing a space-variant wavefront bending error compensation filter, calculating a phase compensation factor corresponding to each sub-image, performing phase compensation, and splicing all the sub-images after phase compensation to obtain an imaging result.
2. The PFA wavefront curvature correction method for bistatic beamforming SAR with arbitrary platform trajectory as claimed in claim 1, wherein in said step B, the uniformly compressed reference signal is represented as
Figure FDA0002562462960000011
Wherein S iso(fτT) represents a point target reference function, KrThe frequency is adjusted for the transmitted signal and c is the speed of light.
3. The PFA wavefront curvature correction method for bistatic beamforming SAR with arbitrary platform trajectory as claimed in claim 2, wherein in said step B, the echo data is compressed uniformly by conjugate multiplication of distance frequency domain and azimuth time domain, and the compressed distance frequency domain data is represented as
Figure FDA0002562462960000012
Wherein, FFTr{. represents the distance direction FFT, sr(τ,t；xp,yp) Representing target echo data, So *(fτAnd t) represents the reference signal S for coherent compressiono(fτT) conjugation, Δ Rp(t；xp,yp) Representing a history of differential distances, wr[·]Representing a distance-to-time window function, wa[·]Representing an azimuthal time-domain window function, TaThe synthetic aperture time is indicated.
4. The PFA wavefront curvature correction method for bistatic beamforming SAR with arbitrary platform trajectory as claimed in claim 3, wherein in said step B, the polar coordinate format transformation is performed on the echo data after the uniform compression, and the two-dimensional low-pass interpolation is used to correct the data
Figure FDA0002562462960000021
And
Figure FDA0002562462960000022
transforming to two-dimensional wave number domain to obtain
Figure FDA0002562462960000023
And
Figure FDA0002562462960000024
wherein
Figure FDA0002562462960000025
The second-order taylor expansion coefficient is represented,
Figure FDA0002562462960000026
the third-order taylor expansion coefficient is represented,
Figure FDA0002562462960000027
representing the second order wavefront curvature phase of the wavenumber domain,
Figure FDA0002562462960000028
representing the third-order wavefront curvature phase in the wavenumber domain.
5. The PFA wavefront curvature correction method for bistatic bunching SAR with any platform trajectory as set forth in claim 4, wherein in the step C, the effective scene size before compensation is first calculated, and the calculation formula is
Figure FDA0002562462960000029
Wherein M isiAnd NiAll the parameters are radar platform motion track correlation coefficients, and lambda represents carrier wave wavelength;
dividing the PFA coarse image h (x, y) into a plurality of images according to the effective scene sizeSubimage hi(x,y)。
6. The PFA wavefront curvature correction method for bistatic beamforming SAR with arbitrary platform trajectory as claimed in claim 5, wherein in said step C, the constructed space-variant wavefront curvature error compensation filter is expressed as
Figure FDA00025624629600000210
Wherein k isxAnd kyThe wave numbers in the x and y directions respectively,
Figure FDA00025624629600000211
representing the total wavefront curvature phase of the wavenumber domain.
7. The PFA wavefront curvature correction method for bistatic beamforming SAR with arbitrary platform trajectory as claimed in claim 6, characterized in that in said step C, for each sub-image hi(x, y), the geometric center position (x) is determinedi,yi) Performing two-dimensional fast Fourier transform on the subimage to obtain a wave number spectrum H thereofi(kx,ky) (ii) a Then calculates the corresponding phase compensation factor Hc(kx,ky；xi,yi) And after the sub-image wave number spectrum is multiplied, the two-dimensional inverse fast Fourier transform of the product result is solved to obtain the compensated sub-image.
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