CN112802135A - Ultrathin lens-free separable compression imaging system and calibration and reconstruction method thereof - Google Patents

Ultrathin lens-free separable compression imaging system and calibration and reconstruction method thereof Download PDF

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CN112802135A
CN112802135A CN202110055760.2A CN202110055760A CN112802135A CN 112802135 A CN112802135 A CN 112802135A CN 202110055760 A CN202110055760 A CN 202110055760A CN 112802135 A CN112802135 A CN 112802135A
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CN112802135B (en
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张�成
潘敏
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Anhui University
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Abstract

The invention provides an ultrathin lens-free separable compression imaging system and a calibration and reconstruction method thereof. The invention realizes imaging by adopting the random coding aperture, not only improves the luminous flux to the utmost extent, but also greatly reduces the thickness, the volume and the weight of an imaging system, reduces the cost, and has the advantages of compact structure, light weight and low cost; in addition, the invention applies the separable matrix to the measurement matrix of the imaging system according to the separable compressive sensing theory, obviously reduces the difficulty of the calibration and reconstruction of the imaging system, and has the advantages of feasible optical realization and calculation.

Description

Ultrathin lens-free separable compression imaging system and calibration and reconstruction method thereof
Technical Field
The invention relates to the technical field of coded aperture imaging, in particular to an ultrathin lens-free separable compression imaging system and a calibration and reconstruction method thereof.
Background
The existing imaging system is generally based on a lens, and is influenced by factors such as the number, thickness and focusing space of the lens, so that the problems of large volume, high price, complex assembly, limited installation space and the like exist, for example, a lens of the imaging system based on the lens for visible light can be made of cheap materials such as glass and plastic, but the lens for infrared and ultraviolet spectrums is very expensive; lens-based imaging systems always need to be assembled, reducing manufacturing efficiency.
Disclosure of Invention
The invention aims to solve the technical problem of providing an ultrathin lens-free separable compression imaging system and a calibration and reconstruction method thereof.
The technical scheme of the invention is as follows:
the utility model provides an ultra-thin separable compression imaging system that does not have lens, this imaging system includes random code aperture mask, image sensor, front and back both ends open-ended cavity box and opaque fixed plate, random code aperture mask is sealed to be covered and is established the front end opening part of cavity box, image sensor and cavity box are all fixed on opaque fixed plate, just image sensor is located in the rear end opening of cavity box, image sensor with random code aperture mask central point collineation, random code aperture mask comprises opaque element and transparent element, opaque element is used for hindering and is in the light, transparent element is used for transmitting light.
The ultrathin lens-free separable compression imaging system is characterized in that the opaque fixing plate is a black plate.
The ultrathin lens-free separable compression imaging system is characterized in that the hollow box body is of a cuboid structure consisting of opaque partition boards.
A calibration method for an ultrathin lens-free separable compression imaging system adopts a separable matrix as a measurement matrix of the imaging system, and comprises the following steps:
(1) calibrating the left separation matrix of the measurement matrix, which specifically comprises the following steps:
(11) selecting N calibration images Ck＝hk1TK is 1,2, …, N, wherein hkRepresenting the kth column of the Hadamard matrix H of order NxN, 1 representing an N-dimensional all-1-column vector, 1TRepresents a transposition of 1;
(12) will mark the image CkN is divided into two images, k is 1,2, …
Figure BDA0002900552230000021
And
Figure BDA0002900552230000022
and projected onto a display, respectively, wherein
Figure BDA0002900552230000023
Represents that C iskAll +1 elements in the-1 element set to 0,
Figure BDA0002900552230000024
is represented bykAll +1 elements in the image are reserved, and-1 element is set to be 0;
(13) two images are combined
Figure BDA0002900552230000025
And
Figure BDA0002900552230000026
MxM order measurement values obtained on an image sensor
Figure BDA0002900552230000027
And
Figure BDA0002900552230000028
subtracting to obtain a calibration image CkMeasured value Y ofk；
(14) For measured value YkK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure BDA0002900552230000029
(15) For matrix
Figure BDA0002900552230000031
Performing singular value decomposition, and comparing the orthogonal matrix containing left singular vector and diagonal matrix containing singular valueMultiplying, and recording the 1 st column of the matrix obtained by multiplying as ukK is 1,2, …, N, then ukIs an M-dimensional column vector;
(16) the M-dimensional column vector ukK is 1,2, …, N together, forming a matrix of M × N order [ u1；u2；…；uN]；
(17) The left separation matrix is calibrated using the following formula:
Figure BDA0002900552230000032
wherein,
Figure BDA0002900552230000033
representing the left separation matrix phiLCalibration matrix of H-1＝HT/N，HTRepresents the transpose of H;
(2) calibrating the right separation matrix of the measurement matrix, which specifically comprises the following steps:
(21) selecting N calibration images C'k＝1hk TK is 1,2, …, N, where 1 denotes an N-dimensional all-1-column vector, hkThe kth column, H, of the Hadamard matrix H representing an NxN orderk TRepresents hkTransposing;
(22) calibrating image C'kN is divided into two images, k is 1,2, …
Figure BDA0002900552230000034
And
Figure BDA0002900552230000035
and projected onto a display, respectively, wherein
Figure BDA0002900552230000036
Is represented by C'kAll +1 elements in the-1 element set to 0,
Figure BDA0002900552230000037
is represented by'kAll +1 elements inReserving an image with the-1 element set to 0;
(23) two images are combined
Figure BDA0002900552230000038
And
Figure BDA0002900552230000039
MxM order measurement values obtained on an image sensor
Figure BDA00029005522300000310
And
Figure BDA00029005522300000311
subtracting to obtain a calibration image C'kMeasured value of Y'k；
(24) To measured value Y'kK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure BDA00029005522300000312
(25) For matrix
Figure BDA00029005522300000313
Performing singular value decomposition, multiplying the diagonal matrix containing singular values obtained by decomposition and the transposition of the orthogonal matrix containing right singular vectors, and recording the 1 st column of the matrix obtained by multiplication as vkK is 1,2, …, N, then vkIs an M-dimensional column vector;
(26) the M-dimensional column vector vkK is 1,2, …, N together, forming a matrix of M × N orders [ v × ]1；v2；…；vN]；
(27) The right separation matrix is calibrated using the following formula:
Figure BDA0002900552230000041
wherein,
Figure BDA0002900552230000042
representing the right separation matrix phiRCalibration matrix of H-1＝HT/N，HTRepresenting the transpose of H.
A reconstruction method of an ultrathin lens-free separable compression imaging system, wherein a measurement matrix of the imaging system adopts a separable matrix, and the method comprises the following steps:
(1) performing singular value decomposition on calibration matrixes of a left separation matrix and a right separation matrix of the measurement matrix, and calculating the pseudo-inverse of the calibration matrixes by adopting the following formula:
Figure BDA0002900552230000043
Figure BDA0002900552230000044
wherein,
Figure BDA0002900552230000045
representing the left separation matrix phiLCalibration matrix of
Figure BDA0002900552230000046
The pseudo-inverse of (a) is,
Figure BDA0002900552230000047
representing the right separation matrix phiRCalibration matrix of
Figure BDA0002900552230000048
Pseudo-inverse of (U)LPresentation pair
Figure BDA0002900552230000049
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionLPresentation pair
Figure BDA00029005522300000410
Diagonal matrix containing singular values, V, obtained by singular value decompositionLPresentation pair
Figure BDA00029005522300000411
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure BDA00029005522300000412
represents ULThe transpose of (a) is performed,
Figure BDA00029005522300000413
representation sigmaLInverse matrix of, URPresentation pair
Figure BDA00029005522300000414
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionRPresentation pair
Figure BDA00029005522300000415
Diagonal matrix containing singular values, V, obtained by singular value decompositionRPresentation pair
Figure BDA00029005522300000416
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure BDA00029005522300000417
represents URThe transpose of (a) is performed,
Figure BDA00029005522300000418
representation sigmaRThe inverse matrix of (d);
(2) judging whether the left separation matrix and the right separation matrix are calibrated well, if so, skipping to the step (3), and if not, skipping to the step (4);
(3) according to the measured value of the target image, the reconstructed target image is calculated by adopting the following formula:
Figure BDA0002900552230000051
wherein,
Figure BDA0002900552230000052
representing a reconstructed target image, Y representing a measurement of the target image;
(4) according to the measured value of the target image, the reconstructed target image is calculated by adopting the following formula:
Figure BDA0002900552230000053
wherein,
Figure BDA0002900552230000054
representing the reconstructed target image, Y representing a measurement of the target image, σLAnd σRIs an intermediate variable, σL＝(ΣL)2，σR＝(ΣR)2,
Figure BDA0002900552230000055
Is expressed as sigmaRτ denotes the regularization parameter, 1 denotes the N-dimensional full 1-column vector, 1TThe transpose of the 1 is shown,
Figure BDA0002900552230000056
represents VRDenotes dot division.
According to the technical scheme, the random coding aperture is adopted to realize imaging, so that the luminous flux is improved to the maximum extent, the thickness, the volume and the weight of an imaging system are greatly reduced, the cost is reduced, and the random coding aperture has the advantages of compact structure, light weight and low cost; in addition, the invention applies the separable matrix to the measurement matrix of the imaging system according to the separable compressive sensing theory, obviously reduces the difficulty of the calibration and reconstruction of the imaging system, and has the advantages of feasible optical realization and calculation.
Drawings
Fig. 1 is a schematic diagram of the imaging system of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
As shown in fig. 1, the ultrathin lens-free separable compression imaging system comprises a random coded aperture mask 1, an image sensor 2, a hollow box 3 with openings at the front end and the rear end, and an opaque fixing plate 4. The random coding aperture mask 1 is hermetically covered on the front end opening of the hollow box body 3, the image sensor 2 and the hollow box body 3 are both fixed on the opaque fixing plate 4, the image sensor 2 is positioned in the rear end opening of the hollow box body 3, and the image sensor 2 and the central point of the random coding aperture mask 1 are collinear.
The imaging system mainly comprises a random coding aperture mask 1 and an image sensor 2, wherein the random coding aperture mask 1 randomly modulates object light field information, and the image sensor 2 records a measured value of the random coding aperture mask 1. The hollow box body 3 and the opaque fixing plate 4 are introduced, so that the distance between the random coding aperture mask 1 and the image sensor 2 can be kept fixed, stray light can be blocked, light rays can not bypass the random coding aperture mask 1 and can be irradiated onto the image sensor 2 from two sides, and noise of a measured value is enabled to be as small as possible. The hollow box body 3 is of a cuboid structure and is composed of opaque partition boards, and the opaque fixing board 4 is a black board.
The random coded aperture mask 1 and the image sensor 2 are considered to be planar and parallel to each other. The random coded aperture mask 1 is placed in front of the image sensor at a distance d (typical measurements are in the order of microns). The random coded aperture mask 1 is binary and is composed of opaque elements for blocking light and transparent elements for transmitting light, and the ideal random coded aperture mask 1 can maximally improve luminous flux. The invention can image under incoherent light, and an object can image under natural light.
The design of the coded aperture plays an important role in imaging. An ideal design would maximize the luminous flux while providing a well conditioned scene-image sensor transfer function to facilitate inversion. The primary purpose of the random coded aperture is to provide a more randomized modulation of the information, preserving as much information as possible.
In the imaging system of the present invention, the random coded aperture functions to measure and map a scene in the real world onto the image sensor 2, and the mathematical model thereof can be expressed as a projection matrix of M × N order by using a measurement matrix (i.e., a scene-image sensor transfer function matrix) Φ. Before a real experiment using the imaging system of the invention, the measurement matrix Φ has to be calibrated to determine the mapping between the scene and the measurements of the image sensor 2, so that a recovery of the scene from the measurements of the image sensor 2 can be achieved.
In order to reduce the dimension of the storage of the measurement matrix phi, the measurement matrix phi of the imaging system is improved by adopting a separable design concept, and the difficulty of the calibration and reconstruction of the imaging system is reduced. The measurement matrix Φ can be expressed as:
Figure BDA0002900552230000071
wherein phiL、ΦRA left separation matrix and a right separation matrix representing the measurement matrix phi respectively,
Figure BDA0002900552230000072
representing the Kronecker product, which may be a direct product or a tensor product.
Thus, scaling the measurement matrix Φ translates into its left-hand separation matrix ΦLAnd right separation matrix phiRAnd (4) calibrating.
A calibration method of an ultrathin lens-free separable compression imaging system comprises the following steps:
s1 left separation matrix phi for measurement matrix phiLAnd calibrating, specifically comprising:
s11, selecting N calibration images Ck＝hk1TK is 1,2, …, N, wherein hkRepresenting the kth column of the Hadamard matrix H of order NxN, 1 representing an N-dimensional all-1-column vector, 1TRepresenting the transpose of 1.
S12, calibrating the image CkN is divided into two images, k is 1,2, …
Figure BDA0002900552230000081
And
Figure BDA0002900552230000082
and projected onto a display, respectively, wherein
Figure BDA0002900552230000083
Represents that C iskAll +1 elements in the-1 element set to 0,
Figure BDA0002900552230000084
is represented bykAll +1 elements in the-1 element set to 0.
The Hadamard matrix H consists of elements +1 and-1, resulting in each calibration image C generated by a Hadamard patternkK is 1,2, …, N needs to be divided into two images
Figure BDA0002900552230000085
And
Figure BDA0002900552230000086
Figure BDA0002900552230000087
s13, combining the two images
Figure BDA0002900552230000088
And
Figure BDA0002900552230000089
MxM order measurement values obtained on an image sensor
Figure BDA00029005522300000810
And
Figure BDA00029005522300000811
subtracting to obtain a calibration image CkMeasured value Y ofk：
Figure BDA00029005522300000812
Figure BDA00029005522300000813
Figure BDA00029005522300000814
S14, for the measured value ykK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure BDA00029005522300000815
S15, pairing matrix
Figure BDA00029005522300000816
Performing singular value decomposition, multiplying an orthogonal matrix U containing left singular vectors obtained by decomposition by a diagonal matrix sigma containing singular values (only the first element of the diagonal matrix is not 0, and other elements are all 0), and recording the 1 st column of the matrix obtained by multiplication as UkK is 1,2, …, N, then ukFor an M-dimensional column vector, let:
Figure BDA00029005522300000817
due to Yk＝ΦLCk(ΦR)T＝(ΦLhk)(ΦR1)TThen, obtaining:
uk＝ΦLhk
s16, dividing the M-dimensional column vector ukK is 1,2, …, N together, forming a matrix of M × N order [ u1；u2；…；uN]Then, there are:
[u1；u2；…；uN]＝ΦL[h1；h2；…；hN]＝ΦLH
s17, adopting the following formula to separate the matrix phi on the leftLAnd (3) calibrating:
Figure BDA0002900552230000091
wherein,
Figure BDA0002900552230000092
representing the left separation matrix phiLCalibration matrix of H-1＝HT/N，HTRepresenting the transpose of H.
S2 right separation matrix phi for measurement matrix phiRAnd calibrating, specifically comprising:
s21, selecting N calibration images C'k＝1hk TK is 1,2, …, N, where 1 denotes an N-dimensional all-1-column vector, hkThe kth column, H, of the Hadamard matrix H representing an NxN orderk TRepresents hkThe transposing of (1).
S22, calibrating the image C'kN is divided into two images, k is 1,2, …
Figure BDA0002900552230000093
And
Figure BDA0002900552230000094
and projected onto a display, respectively, wherein
Figure BDA0002900552230000095
Is represented by C'kAll +1 elements in the-1 element set to 0,
Figure BDA0002900552230000096
is represented by'kAll +1 elements in the-1 element set to 0.
The Hadamard matrix H consists of elements +1 and-1, resulting in each calibration image generated by a Hadamard patternC′kK is 1,2, …, N needs to be divided into two images
Figure BDA0002900552230000097
And
Figure BDA0002900552230000098
Figure BDA0002900552230000099
s23, combining the two images
Figure BDA00029005522300000910
And
Figure BDA00029005522300000911
MxM order measurement values obtained on an image sensor
Figure BDA00029005522300000912
And
Figure BDA00029005522300000913
subtracting to obtain a calibration image C'kMeasured value of Y'k：
Figure BDA00029005522300000914
Figure BDA00029005522300000915
Figure BDA00029005522300000916
S24, for measured value Y'kK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure BDA0002900552230000101
S25, pairing matrix
Figure BDA0002900552230000102
Performing singular value decomposition, and transposing (V ') a diagonal matrix sigma ' containing singular values (only the first element of the diagonal matrix is not 0 and the other elements are 0) obtained by decomposition and an orthogonal matrix V ' containing right singular vectorsTMultiplying, and recording the 1 st column of the matrix obtained by multiplying as vkK is 1,2, …, N, then vkFor an M-dimensional column vector, let:
Figure BDA0002900552230000103
due to Y'k＝ΦLC′k(ΦR)T＝(ΦL1)(ΦRhk)TThen, obtaining:
vk＝ΦRhk
s26, dividing the M-dimensional column vector vkK is 1,2, …, N together, forming a matrix of M × N orders [ v × ]1；v2；…；vN]Then, there are:
[v1；v2；…；vN]＝ΦR[h1；h2；…；hN]＝ΦRH
s27, adopting the following formula to separate the matrix phi on the rightRAnd (3) calibrating:
Figure BDA0002900552230000104
wherein,
Figure BDA0002900552230000105
representing the right separation matrix phiRCalibration matrix of H-1＝HT/N，HTRepresenting the transpose of H.
A reconstruction method of an ultrathin lens-free separable compression imaging system comprises the following steps:
s1 left separation matrix phi for measurement matrix phiLAnd right separation matrix phiRCalibration matrix of
Figure BDA0002900552230000106
And
Figure BDA0002900552230000107
singular value decomposition is carried out, and the pseudo-inverse is calculated by adopting the following formula:
Figure BDA0002900552230000108
Figure BDA0002900552230000109
wherein,
Figure BDA00029005522300001010
representing the left separation matrix phiLCalibration matrix of
Figure BDA00029005522300001011
The pseudo-inverse of (a) is,
Figure BDA00029005522300001012
representing the right separation matrix phiRCalibration matrix of
Figure BDA00029005522300001013
Pseudo-inverse of (U)LPresentation pair
Figure BDA00029005522300001014
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionLPresentation pair
Figure BDA0002900552230000111
Diagonal matrix containing singular values, V, obtained by singular value decompositionLPresentation pair
Figure BDA0002900552230000112
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure BDA0002900552230000113
represents ULThe transpose of (a) is performed,
Figure BDA0002900552230000114
representation sigmaLInverse matrix of, URPresentation pair
Figure BDA0002900552230000115
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionRPresentation pair
Figure BDA0002900552230000116
Diagonal matrix containing singular values, V, obtained by singular value decompositionRPresentation pair
Figure BDA0002900552230000117
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure BDA0002900552230000118
represents URThe transpose of (a) is performed,
Figure BDA0002900552230000119
representation sigmaRThe inverse matrix of (c).
s2, determining the left separation matrix phiLAnd right separation matrix phiRAnd if the calibration is good, jumping to step s3 if the calibration is good, and jumping to step s4 if the calibration is not good.
s3, calculating the reconstructed target image according to the measured value of the target image by adopting the following formula:
Figure BDA00029005522300001110
wherein,
Figure BDA00029005522300001111
representing the reconstructed target image and Y represents the measured value of the target image.
s4, calculating the reconstructed target image according to the measured value of the target image by adopting the following formula:
Figure BDA00029005522300001112
wherein,
Figure BDA00029005522300001113
representing the reconstructed target image, Y representing a measurement of the target image, σLAnd σRIs an intermediate variable, σL＝(ΣL)2，σR＝(ΣR)2,
Figure BDA00029005522300001114
Is expressed as sigmaRτ denotes the regularization parameter, 1 denotes the N-dimensional full 1-column vector, 1TThe transpose of the 1 is shown,
Figure BDA00029005522300001115
represents VRDenotes dot division.
From the above reconstruction method, if ΦLAnd phiRAre well-calibrated, the unknown scene X can be estimated by solving a least squares problem:
Figure BDA0002900552230000121
the solution is in closed form:
Figure BDA0002900552230000122
if phiLAnd phiRCalibrationWhen the conditions are not good or not sufficient, least square method estimation needs to be considered
Figure BDA0002900552230000123
The influence of noise amplification, one simple method to reduce noise is to add a regularization term to the least squares problem:
Figure BDA0002900552230000124
where τ > 0, solutions of the above formula may also be used
Figure BDA0002900552230000125
And
Figure BDA0002900552230000126
the SVD of (1) explicitly writes out:
Figure BDA0002900552230000127
the above-mentioned embodiments are merely illustrative of the preferred embodiments of the present invention, and do not limit the scope of the present invention, and various modifications and improvements made to the technical solution of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims (5)

1. An ultra-thin lens-free separable compression imaging system, characterized in that: the imaging system comprises a random coding aperture mask, an image sensor, a hollow box body with front and back openings and an opaque fixed plate, wherein the random coding aperture mask is covered at the opening of the front end of the hollow box body in a sealing manner, the image sensor and the hollow box body are fixed on the opaque fixed plate, the image sensor is positioned in the opening of the back end of the hollow box body, the image sensor and the central point of the random coding aperture mask are collinear, the random coding aperture mask is composed of opaque elements and transparent elements, the opaque elements are used for blocking light, and the transparent elements are used for transmitting light.
2. The ultra-thin lens-less separable compressed imaging system of claim 1, wherein: the non-transparent fixing plate is a black plate.
3. The ultra-thin lens-less separable compressed imaging system of claim 1, wherein: the hollow box body is of a cuboid structure consisting of opaque partition boards.
4. A calibration method for an ultrathin lens-free separable compression imaging system, wherein a measurement matrix of the imaging system adopts a separable matrix, is characterized by comprising the following steps:
(1) calibrating the left separation matrix of the measurement matrix, which specifically comprises the following steps:
(11) selecting N calibration images Ck＝hk1TK is 1,2, …, N, wherein hkRepresenting the kth column of the Hadamard matrix H of order NxN, 1 representing an N-dimensional all-1-column vector, 1TRepresents a transposition of 1;
(12) will mark the image CkN is divided into two images, k is 1,2, …
Figure FDA0002900552220000011
And
Figure FDA0002900552220000012
and projected onto a display, respectively, wherein
Figure FDA0002900552220000013
Represents that C iskAll +1 elements in the-1 element set to 0,
Figure FDA0002900552220000014
is represented bykAll +1 elements in the-1 element are reserved and the-1 element is set to 0 to obtainThe image of (a);
(13) two images are combined
Figure FDA0002900552220000021
And
Figure FDA0002900552220000022
MxM order measurement values obtained on an image sensor
Figure FDA0002900552220000023
And
Figure FDA0002900552220000024
subtracting to obtain a calibration image CkMeasured value Y ofk；
(14) For measured value YkK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure FDA0002900552220000025
(15) For matrix
Figure FDA0002900552220000026
Performing singular value decomposition, multiplying the orthogonal matrix containing the left singular vector obtained by decomposition and the diagonal matrix containing the singular value, and recording the 1 st column of the matrix obtained by multiplication as ukK is 1,2, …, N, then ukIs an M-dimensional column vector;
(16) the M-dimensional column vector ukK is 1,2, …, N together, forming a matrix of M × N order [ u1；u2；…；uN]；
(17) The left separation matrix is calibrated using the following formula:
Figure FDA0002900552220000027
wherein,
Figure FDA0002900552220000028
representing the left separation matrix phiLCalibration matrix of H-1＝HT/N，HTRepresents the transpose of H;
(2) calibrating the right separation matrix of the measurement matrix, which specifically comprises the following steps:
(21) selecting N calibration images C'k＝1hk TK is 1,2, …, N, where 1 denotes an N-dimensional all-1-column vector, hkThe kth column, H, of the Hadamard matrix H representing an NxN orderk TRepresents hkTransposing;
(22) calibrating image C'kN is divided into two images, k is 1,2, …
Figure FDA0002900552220000029
And
Figure FDA00029005522200000210
and projected onto a display, respectively, wherein
Figure FDA00029005522200000211
Is represented by C'kAll +1 elements in the-1 element set to 0,
Figure FDA00029005522200000212
is represented by'kAll +1 elements in the image are reserved, and-1 element is set to be 0;
(23) two images are combined
Figure FDA00029005522200000213
And
Figure FDA00029005522200000214
MxM order measurement values obtained on an image sensor
Figure FDA00029005522200000215
And
Figure FDA00029005522200000216
subtracting to obtain a calibration image C'kMeasured value of Y'k；
(24) To measured value Y'kK is 1,2, …, N is rank-first approximated to obtain a matrix
Figure FDA0002900552220000031
(25) For matrix
Figure FDA0002900552220000032
Performing singular value decomposition, multiplying the diagonal matrix containing singular values obtained by decomposition and the transposition of the orthogonal matrix containing right singular vectors, and recording the 1 st column of the matrix obtained by multiplication as vkK is 1,2, …, N, then vkIs an M-dimensional column vector;
(26) the M-dimensional column vector vkK is 1,2, …, N together, forming a matrix of M × N orders [ v × ]1；v2；…；vN]；
(27) The right separation matrix is calibrated using the following formula:
Figure FDA0002900552220000033
wherein,
Figure FDA0002900552220000034
representing the right separation matrix phiRCalibration matrix of H-1＝HT/N，HTRepresenting the transpose of H.
5. A reconstruction method for an ultra-thin lens-free separable compressed imaging system, the measurement matrix of which adopts a separable matrix, the method comprising the steps of:
(1) performing singular value decomposition on calibration matrixes of a left separation matrix and a right separation matrix of the measurement matrix, and calculating the pseudo-inverse of the calibration matrixes by adopting the following formula:
Figure FDA0002900552220000035
Figure FDA0002900552220000036
wherein,
Figure FDA0002900552220000037
representing the left separation matrix phiLCalibration matrix of
Figure FDA0002900552220000038
The pseudo-inverse of (a) is,
Figure FDA0002900552220000039
representing the right separation matrix phiRCalibration matrix of
Figure FDA00029005522200000310
Pseudo-inverse of (U)LPresentation pair
Figure FDA00029005522200000311
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionLPresentation pair
Figure FDA00029005522200000312
Diagonal matrix containing singular values, V, obtained by singular value decompositionLPresentation pair
Figure FDA00029005522200000313
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure FDA00029005522200000314
represents ULThe transpose of (a) is performed,
Figure FDA00029005522200000315
representation sigmaLInverse matrix of, URPresentation pair
Figure FDA00029005522200000316
An orthogonal matrix including left singular vectors, sigma, obtained by singular value decompositionRPresentation pair
Figure FDA00029005522200000317
Diagonal matrix containing singular values, V, obtained by singular value decompositionRPresentation pair
Figure FDA0002900552220000041
An orthogonal matrix containing right singular vectors obtained by singular value decomposition,
Figure FDA0002900552220000042
represents URThe transpose of (a) is performed,
Figure FDA0002900552220000043
representation sigmaRThe inverse matrix of (d);
(2) judging whether the left separation matrix and the right separation matrix are calibrated well, if so, skipping to the step (3), and if not, skipping to the step (4);
(3) according to the measured value of the target image, the reconstructed target image is calculated by adopting the following formula:
Figure FDA0002900552220000044
wherein,
Figure FDA0002900552220000045
representing a reconstructed target image, Y representing a measurement of the target image;
(4) according to the measured value of the target image, the reconstructed target image is calculated by adopting the following formula:
Figure FDA0002900552220000046
wherein,
Figure FDA0002900552220000047
representing the reconstructed target image, Y representing a measurement of the target image, σLAnd σRIs an intermediate variable, σL＝(ΣL)2，σR＝(ΣR)2,
Figure FDA0002900552220000048
Is expressed as sigmaRτ denotes the regularization parameter, 1 denotes the N-dimensional full 1-column vector, 1TThe transpose of the 1 is shown,
Figure FDA0002900552220000049
represents VRDenotes dot division.
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