CN104574315B - Optical system imaging recovering method based on light intensity transmission matrix - Google Patents

Abstract

The invention relates to an optical system imaging recovering method based on a light intensity transmission matrix. The optical system imaging recovering method includes the steps that S1, each row of elements in a two-dimensional image matrix are sequentially connected end to end and arrayed to be one-dimensional image vectors; S2, according to a PSF matrix of an optical system and the one-dimensional image vectors, the two-dimensional light intensity transmission matrix of the optical system is solved, and inversion mathematical calculation is carried out on the two-dimensional light intensity transmission matrix to obtain a two-dimensional light intensity transmission inverse matrix; S3, the one-dimensional image vectors are multiplied with the two-dimensional light intensity transmission inverse matrix to obtain one-dimensional recovering object vectors, and two-dimensional processing is carried out on the one-dimensional recovering object vectors to obtain a two-dimensional recovering object matrix. Imaging is recovered through matrix operation, the ill-conditioned problem of a deconvolution method and the high frequency missing problem of a frequency domain recovering method in the previous imaging recovering technology are avoided, the error between the recovering object matrix and an original object matrix is very small, and therefore the recovering accuracy is high.

Description

Optical system imaging restored method based on light intensity transmission matrix

Technical field

The invention belongs to optical system imaging restores and image quality lift technique field, it is related to a kind of using two-dimentional light intensity Transmission matrix is by optical system as matrix reverts to the new method with picture matrix data amount identical reconstruction matrix.

Background technology

During optical system imaging, due to diffraction and the presence of aberration, each point on object plane can be imaged in image planes For a disc of confusion, so as to cause image blur, image quality to be degenerated, as inconsistent with thing.By being imaged recovery technique to picture Matter is lifted, and can improve the definition of picture, so that as being lifted with the similarity degree of thing.But in conventional image restoration Cheng Zhong, generally using deconvolution restored method and frequency spectrum restored method, both approaches all have the shortcomings that certain.First, exist The point spread function (PSF) that the light distribution function of picture can be with the light distribution function of thing with optical system in spatial domain makees convolution algorithm Try to achieve, the operation expression of the convolution is the integral equation of Fredholm types.In the case of known picture with PSF, from this The computing that reconstruction is solved in Fredholm type integral equations is ill-conditioning problem, it is impossible to obtain accurate analytic solutions, so, warp The reset error of product restored method is larger.Secondly, in a frequency domain as frequency spectrum the optics of optical system can be multiplied by with thing frequency spectrum Transmission function (OTF) is tried to achieve, and in the case of known picture with OTF, the principle of frequency domain restored method is by as making Fourier transformation Picture frequency spectrum is obtained, then picture frequency is composed divided by OTF acquisition reconstruction frequency spectrums, finally making inverse Fourier transform to reconstruction frequency spectrum can obtain To reconstruction.However, OTF has cut-off frequency, generally, the cut-off frequency of OTF will be less than thing frequency spectrum highest frequency. In optical system imaging, the spectrum component in thing frequency spectrum higher than OTF cut-off frequencies is truncated, and picture frequency spectrum is only ended comprising OTF Spectrum component within frequency.So, frequency domain restored method can only restore the thing frequency spectrum within OTF cut-off frequencies, cause to restore The radio-frequency component disappearance of thing, reset error are larger.

The content of the invention

It is an object of the invention to provide a kind of optical system imaging restored method based on light intensity transmission matrix, the method profit Imaging is realized with matrix operation to restore, it is to avoid the ill-conditioning problem and frequency domain of the Deconvolution Method in conventional imaging recovery technique The high frequency disappearance problem of restored method, reconstruction matrix are minimum with the error of original thing matrix, and recovery accuracy is high.

The purpose of the present invention is achieved through the following technical solutions：

A kind of optical system imaging restored method based on light intensity transmission matrix, comprises the steps：

The first step, the one-dimensional of two-dimensional image matrix are processed：

One-dimensional picture vector is arranged as by each row element in two-dimensional image matrix end to end successively.

Second step, two-dimentional light intensity transmission matrix and two-dimentional light intensity transmit the solution of inverse matrix：

For a certain known optical system, optical system is solved according to the PSF matrixes and one-dimensional picture vector of the optical system The two-dimentional light intensity transmission matrix of system；The mathematical computations inverted by two-dimentional light intensity transmission matrix are just obtained into two-dimentional light intensity transmission Inverse matrix.

3rd step, the solution of reconstruction matrix and reset error are calculated：

One-dimensional picture vector is multiplied by into two-dimentional light intensity transmission inverse matrix and tries to achieve one-dimensional reconstruction vector, by one-dimensional reconstruction vector Two-dimentional reconstruction matrix is obtained after two dimensionization.Known two-dimentional original thing matrix is subtracted each other with two-dimentional reconstruction matrix and is obtained Reset error matrix, and then it is equal with the error of corresponding element in two-dimentional reconstruction matrix to obtain known two-dimentional original thing matrix Root value RMS_E, RMS_ESize be used for evaluate imaging restored method recovery accuracy.

Optical system imaging restored method based on light intensity transmission matrix proposed by the present invention known two-dimensional image matrix with During two-dimentional PSF Matrix Solvings two dimension reconstruction matrix, one-dimensional picture vector can be multiplied by two-dimentional light intensity transmission inverse matrix and obtain one-dimensional The vector two dimension chemical conversion of one-dimensional reconstruction two-dimentional reconstruction matrix just can be tried to achieve accurate restoration result by reconstruction vector.It is this The two-dimentional reconstruction matrix that method is obtained is minimum with the error of known two-dimentional original thing matrix, and recovery accuracy is high.

Description of the drawings

Fig. 1 is optical system spatial domain imaging schematic diagram；

Fig. 2 is the imaging schematic diagram that image planes are rotated 180 degree around optical axis；

Fig. 3 is the one-dimensional Principle of Process figure that the unknown original thing matrix of two dimension is converted into one-dimensional unknown original thing vector；

Fig. 4 is the one-dimensional Principle of Process figure that two-dimensional image matrix is converted into one-dimensional picture vector；

Fig. 5 is the graph of a relation of vectorial one-dimensional unknown material, one-dimensional picture vector and two-dimentional light intensity transmission matrix；

Fig. 6 is the one-dimensional picture vector after the two-dimensional image matrix and one-dimensional being made up of 5 row, 5 column element；

Fig. 7 be according in Fig. 6 as the two-dimentional unknown original thing matrix being made up of 5 row, 5 column element and one-dimensional that matrix is set up One-dimensional unknown material vector after change；

Fig. 8 is the PSF matrixes being made up of 3 row, 3 column element；

Element as of the Fig. 9 (a) for the unknown original thing matrix of two dimension in Fig. 7_1,1(corresponding in one-dimensional unknown original thing vector First element α₁) light distribution data that Jing optical systems are formed in image planes, it is (b) element a_1,1To the light as matrix each element Strong transmission coefficient；

Element as of the Figure 10 (a) for the unknown original thing matrix of two dimension in Fig. 7_1,2(corresponding in one-dimensional unknown original thing vector Second element α₂) light distribution data that Jing optical systems are formed in image planes, it is (b) element a_1,2To as matrix each element Light intensity transmission coefficient；

Element as of the Figure 11 (a) for the unknown original thing matrix of two dimension in Fig. 7_2,2(corresponding in one-dimensional unknown original thing vector The 7th element α₇) light distribution data that Jing optical systems are formed in image planes, it is (b) element a_2,2To as matrix each element Light intensity transmission coefficient；

Element as of the Figure 12 (a) for the unknown original thing matrix of two dimension in Fig. 7_5,5(corresponding in one-dimensional unknown original thing vector Last element α₂₅) light distribution data that Jing optical systems are formed in image planes, it is (b) element a_5,5To as matrix each element Light intensity transmission coefficient；

Figure 13 is the two-dimentional light intensity transmission matrix tried to achieve with picture vector in Fig. 6 by PSF matrixes in Fig. 8；

Figure 14 (a) is the known original thing of alphabetical F patterns, (b) is 7 row, 7 column matrix of the original thing of alphabetical F patterns

Figure 15 is optical system PSF matrix；

Figure 16 (a) be alphabetical F things Jing optical systems into picture, (b) be picture 7 row, 7 column matrix；

Figure 17 is the two-dimentional light intensity transmission matrix data obtained by the picture Matrix Calculating in the PSF matrixes and Figure 16 (b) in Figure 15 Graphics；

Figure 18 is the two-dimentional light intensity transmission inverse matrix tried to achieve by two-dimentional light intensity transmission matrix in Figure 17；

Figure 19 (a) is the reconstruction after restoring to the picture in Figure 16, is (b) two-dimentional reconstruction matrix；

Figure 20 is the error moments that the reconstruction matrix for deducting Figure 19 (b) by the known original thing matrix of Figure 14 (b) is obtained Battle array；

Figure 21 is containing original thing known to 400 row, 400 column element；

Figure 22 is the optical system PSF matrix containing 5 row, 5 column element；

Figure 23 is the picture containing 400 row, 400 column element；

Figure 24 be the picture in Figure 23 is restored after the reconstruction that obtains；

Figure 25 is the error matrix data three-dimensional figure containing 400 row, 400 column element.

Specific embodiment

Below in conjunction with the accompanying drawings technical scheme is further described, but is not limited thereto, it is every to this Inventive technique scheme is modified or equivalent, without deviating from the spirit and scope of technical solution of the present invention, all should cover In protection scope of the present invention.

The present invention is provided a kind of reverting to the two-dimensional image matrix of optical system using two-dimentional light intensity transmission matrix and is counted with which According to the new method of amount identical two dimension reconstruction matrix.The solution of two-dimentional reconstruction matrix is in known two-dimensional image matrix and two dimension Carry out under conditions of PSF matrixes, during the entire process of two-dimentional reconstruction matrix is solved, two-dimentional original thing matrix is unknown bar Part.In order to illustrate the principle of imaging restored method proposed by the present invention, need to set up one it is complete with two-dimensional image matrix data amount Identical two-dimensional matrix is used as the unknown original thing matrix of two dimension.The element symbol a of the unknown original thing matrix of two dimension_{M, n}Represent, its In, m, n are the position number of certain element in the unknown original thing matrix of two dimension.

The unknown original thing matrix of two dimension is on all four with the meaning of two-dimentional original thing matrix, and only the former is all Element is unknown, and all elements of the latter are known, and the dimension of the two is identical with the dimension of two-dimensional image matrix. The two-dimentional light intensity transmission matrix solved in this restored method and the element a of the unknown original thing matrix of two dimension_{M, n}It is unrelated, only with two Dimension is relevant as matrix and PSF matrixes.Evaluate if desired for the recovery accuracy to this method, then need to provide known two dimension original Beginning thing matrix is being analyzed to reset error.Comprise the following steps that：

The first step：The one-dimensional of two-dimensional image matrix is processed.

Current image stores into the form of e-file, and gray level image is represented with a two-dimensional matrix, in matrix The relative position of a point in the positional representation image planes of each element, the numerical value of each element represent the light intensity of the point. Color document image represents that with a three-dimensional matrice this three-dimensional matrice is equivalent to three two-dimensional matrixs, each two-dimensional matrix point Not Biao Shi image each point three kinds of colors of red, green, blue.When processing coloured image, can be according to the method difference of gray level image process The data to three kinds of colors of red, green, blue are processed, therefore the present invention is only illustrated to the restored method of gray level image.

To illustrate the principle of this method, first the imaging process of optical system is described.As shown in figure 1, two-dimentional thing The light intensity Jing optical system 3 of a certain object point 2 on face 1 is imaged as a disc of confusion 5, the light intensity of disc of confusion 5 in two-dimentional image planes 4 Distribution can be accurately described by the data of the point spread function (PSF) 6 centered on corresponding picture point.In the light that turns light path of not putting english In system, image planes are to stand upside down relative to object plane, i.e., image planes have rotated 180 degree angle relative to object plane.For the ease of analysis and Process, image planes can be rotated 180 degree centered on optical axis, as shown in Fig. 2 this processing method does not change the imaging of optical system Rule, in actual optical system, can turn light path realization by putting english.As shown in Fig. 2 on two-dimentional object plane 1 each point light intensity number According to can by Fig. 3 in two-dimentional unknown original thing matrix 7 represent that equally, the light intensity data of each point can be by Fig. 4 in two-dimentional image planes 4 Two-dimensional image matrix 9 represent.Certain element a of the unknown original thing matrix of two dimension_{M, n}To certain element b of two-dimensional image matrix_{Q, w}'s Light intensity transmission coefficient d_{M, n, q, w}Must be four-dimensional parameter, then by d_{M, n, q, w}The matrix of composition necessarily four-matrix, and four-matrix It is not easy to analyze and processes, being also not easy to draw represents.And for the element α in one-dimensional thing vector_iTo one-dimensional picture vector In element β_jLight intensity transmission coefficient p_{I, j}It is two-dimensional parameter, by p_{I, j}The light intensity transmission matrix of composition is two-dimensional matrix, and two Dimension matrix is easy to analyze, is processed, and also allowing for drawing represents.

The solution of reconstruction matrix in this restored method is entered under conditions of known two-dimensional image matrix with two dimension PSF matrixes Capable, the two-dimentional original thing matrix during the two-dimentional reconstruction matrix of solution is unknown.Need to set up one with two-dimensional image square The identical two-dimensional matrix of battle array data volume is used as the unknown original thing matrix of two dimension.

As shown in figure 3, the one-dimensional of thing process be exactly by the every a line in two-dimentional unknown 7 each row element of original thing matrix according to It is secondary end to end to be arranged in order as a row vector.As shown in figure 4, it is exactly by 9 each row of two-dimensional image matrix that the one-dimensional of picture is processed Every a line in element is end to end successively to be arranged in order as a row vector.Thing vector is pressed again with the element in picture vector After tandem numbering, one-dimensional unknown original thing vector 8 and one-dimensional picture vector 10 are just defined.

As shown in Figure 3, Figure 4, to the unknown original thing matrix of two dimension, after carrying out one-dimensional as matrix, two dimension is unknown original Certain element a of thing matrix_{M, n}Respective element α being changed in one-dimensional unknown original thing vector_i, certain unit of two-dimensional image matrix Plain b_{Q, w}Respective element β being changed in one-dimensional picture vector_j, following relation should be met：

Wherein, m, n are the sequence number of certain element in the unknown original thing matrix of two dimension, and i is in one-dimensional unknown original thing vector The sequence number of certain element, q, w are the sequence number of certain element in two-dimensional image matrix, and j is the sequence number of certain element in one-dimensional picture vector, M, N are the line number of the unknown original thing matrix of two dimension, columns, Q, W total line number, total columns for two-dimensional image matrix, and i, j, m, n, Q, w, M, N, Q, W are positive integer.Due to recovering and two-dimensional image matrix function in imaging recovery technique of the present invention According to etc. big two-dimentional reconstruction matrix, the element of two-dimentional reconstruction matrix and two-dimensional image matrix as many, so, M is equal with Q , N be equal with W.

Second step：Two-dimentional light intensity transmission matrix and two-dimentional light intensity transmit the solution of inverse matrix.

According to the basic theories of Fourier Optics, optical imaging system belongs to incoherent imaging system, two-dimentional object plane On the light intensity that sends of each object point be focused in image planes by optical system and form a disc of confusion for covering multiple picture points, Disc of confusion is centrally located at the picture point (i.e. in Fig. 2 with object point two-dimensional position identical picture point) corresponding with object point, the disc of confusion Light distribution can be accurately described by the PSF centered on the picture point.It should be noted that with every in linear space-variant optical system PSF centered on individual picture point is different, and the PSF in linear empty constant optical system centered on each picture point is phase With.So, need in the imaging recuperation of linear space-variant optical system it is known as matrix in each pixel element be The PSF matrixes of the heart, and in the imaging recuperation of linear empty constant optical system, it is only necessary to some in known picture matrix PSF matrixes centered on pixel element.

Due to certain element a of the unknown original thing matrix of two dimension_{M, n}It is transferred to certain element b of two-dimensional image matrix_{Q, w}Light Strong transmission coefficient d_{M, n, q, w}Can be from b_{M, n}Centered on PSF extracting datas, and d_{M, n, q, w}With the unit in one-dimensional unknown material vector Plain α_iElement β in as vector_jLight intensity transmission coefficient p_{I, j}(element i.e. in two-dimentional light intensity transmission matrix 11) be it is equal, I.e.：

Wherein, the relation and meaning in parameter i, j, the relation of m, n, q, w, M, N, Q, W satisfaction and meaning and formula (1), (2) Justice is identical.Therefore, as long as the unknown original thing matrix of the good two dimension of record, as the four-dimensional light intensity transmission coefficient between matrix each element d_{M, n, q, w}, one-dimensional unknown original thing vector each element just can be obtained to the two-dimentional light intensity transmission coefficient of one-dimensional picture vector each element p_{I, j}, the coefficient is the respective element in the two-dimentional light intensity transmission matrix 11 shown in Fig. 5.According to the method, obtain successively one-dimensional Unknown original thing vector each element is just obtained two-dimentional light intensity transmission matrix to the light intensity transmission coefficient of one-dimensional picture vector each element 11.Due to the transmission coefficient p in two-dimentional light intensity transmission matrix_{I, j}The data being in PSF, so, two-dimentional light intensity transmission matrix with The element a of the unknown original thing matrix of two dimension_{M, n}Unrelated, two-dimentional light intensity transmission matrix completely can be according to two-dimensional image matrix and two dimension PSF Matrix Solvings.

After solving two-dimentional light intensity transmission matrix 11, one-dimensional unknown original thing vector 8 is multiplied by two-dimentional light intensity transmission matrix 11 Just establish equal to the relation of one-dimensional picture vector 10, as shown in Figure 5.This product calculation relation can be write as the form of formula (4)

A P=B (4)；

Wherein, A is one-dimensional unknown original thing vector, and meaning is equal to the one-dimensional unknown original thing vector 8 in Fig. 5；P is two Dimension light intensity transmission matrix, meaning are equal to the two-dimentional light intensity transmission matrix 11 in Fig. 5；B is that one-dimensional picture is vectorial, and meaning is equal to figure Picture vector 10 in 5.

The mathematical operation inverted by two-dimentional light intensity transmission matrix is just obtained into two-dimentional light intensity transmission inverse matrix.Two-dimentional light Strong transmission matrix has following relation with two-dimentional light intensity transmission inverse matrix：

P·P^-1=E (5)；

Wherein, matrix P is two-dimentional light intensity transmission matrix, matrix P^-1Inverse matrix is transmitted for two-dimentional light intensity, matrix E is and P ranks Number identical unit matrix.

For matrix P, its Methods of Finding Inverse Matrix is as follows：

Wherein, determinants of | the P | for matrix P, matrix P^*For the adjoint matrix that the algebraic complement of matrix P each element is constituted Battle array.

3rd step, the solution of two-dimentional reconstruction matrix and reset error are calculated：

The one-dimensional picture vector that the first step is obtained is multiplied by into the two-dimentional light intensity transmission inverse matrix that second step obtains and just obtains one Dimension reconstruction vector, i.e.,

C=B P^-1(7),

B is that one-dimensional picture is vectorial, P^-1Inverse matrix is transmitted for two-dimentional light intensity, C is one-dimensional reconstruction vector.

One-dimensional reconstruction vector C two dimensionizations are just obtained into two-dimentional reconstruction matrix, such as shown in formula (8)

Wherein, a_{K, l}' it is element in two-dimentional reconstruction matrix, α_c' it is element in one-dimensional reconstruction vector C, k, l are The sequence number of certain element in two-dimentional reconstruction matrix, c are the sequence number of certain element in one-dimensional reconstruction vector, and K, L are multiple for two dimension Total line number of original matrix, total columns, K are equal to the W that Q, L in formula (2) are equal in formula (2), and c, k, l, K, L are just Integer.

In order to evaluate the recovery accuracy of the imaging restored method, needs are imaged using known two-dimentional original thing, Imaging is obtained two-dimentional reconstruction matrix according to the method described above, two-dimentional original thing matrix is deducted two-dimentional reconstruction matrix can Two-dimentional reset error matrix E is obtained, such as shown in formula (9), reset error root-mean-square value RMS_ECan be calculated by formula (10)：

Wherein, e_{K, l}For the element of two-dimentional error matrix,For the element of known two-dimentional original thing matrix, a_{K, l}' be The element of two-dimentional reconstruction matrix, k, l are element in known two-dimentional original thing matrix, two-dimentional reconstruction matrix and error matrix Sequence number, K, L are total line number, total columns of known two-dimentional original thing matrix, two-dimentional reconstruction matrix and error matrix, and k, L, K, L are positive integer, and K is equal to the Q in formula (2), and L is equal to the W in formula (2), and c, k, l, K, L are positive integer.

So far, the optical system imaging restored method based on light intensity transmission matrix of the present invention has all been illustrated Finish, below with three examples illustrating principle, recuperation, restoration result and the reset error of methods described.

Example one：

In order to absolutely prove the method for solving of two-dimentional light intensity transmission matrix of the present invention, Fig. 6 gives and is arranged by 5 rows 5 The example of the one-dimensional picture vector after the two-dimensional image matrix and one-dimensional of element composition.Fig. 7 is given according to two-dimensional image matrix in Fig. 5 It is unknown original after the two-dimentional unknown original thing matrix containing 5 row, 5 column element big with two-dimensional image matrix etc. set up and one-dimensional Thing vector, in solution procedure, the unknown original thing matrix of two dimension is not with all elements in one-dimensional unknown original thing vector Know.Fig. 8 gives the PSF matrix examples being made up of 3 row, 3 column element, and the optical system is linear sky invariant system.Two dimension is not Know the effect of each element Jing optical systems PSF in original thing matrix, disc of confusion can be formed in image planes, but in actual optical system In system, due to the restriction of diaphragm and image planes size, exceeding because of the diffusion of PSF as the part at edge cannot be recorded in detector On, so, cannot record in as matrix beyond the part as matrix in disc of confusion.Fig. 9 (a) is given two in Fig. 6 Tie up the element a of unknown original thing matrix_1,1(corresponding to first element α in one-dimensional unknown original thing vector₁) Jing optical systems In the light distribution data that image planes are formed, Fig. 9 (b) gives a_1,1To the light intensity transmission coefficient of two-dimensional image matrix each element.Figure 10 A () gives the element a of the unknown original thing matrix of two dimension_1,2(corresponding to second element in one-dimensional unknown original thing vector α₂) the light distribution data that formed in image planes of Jing optical systems, Figure 10 (b) gives a_1,2To the light intensity of two-dimensional image matrix each element Transmission coefficient.The like, Figure 11 (a) gives the element a2 of the unknown original thing matrix of two dimension, and 2 (correspond to one-dimensional unknown original The 7th element α in beginning thing vector₇) the light distribution data that formed in image planes of Jing optical systems, Figure 11 (b) gives a_2,2Arrive The light intensity transmission coefficient of two-dimensional image matrix each element.Figure 12 (a) gives the element a of the unknown original thing matrix of two dimension_5,5(correspondence Last element α in one-dimensional unknown original thing vector₂₅) the light distribution data that formed in image planes of Jing optical systems, Figure 12 B () gives a_5,5To the light intensity transmission coefficient of two-dimensional image matrix each element.On the whole, the matrix in Fig. 9 (a)～Figure 12 (a) For the light intensity diffusion matrix that the element of the unknown original thing matrix of two dimension is formed in image planes, Fig. 9 (b)～Figure 12 (b) is two-dimentional unknown Light intensity diffusion coefficient matrix of the element of original thing matrix to two-dimensional image matrix each element.Two-dimensional image matrix is equal to all two dimensions not Know the light intensity diffusion matrix sum of form that the element of original thing matrix formed in image planes as shown in Fig. 9 (a)～Figure 12 (a).Can To find out, light intensity diffusion coefficient matrix shown in Fig. 9 (b)～Figure 12 (b) and the two-dimentional unknown original thing matrix element shown in Fig. 6 Unrelated, the PSF matrixes that light intensity diffusion matrix completely can be as shown in Figure 8 are asked by formula (3) with the two-dimensional image matrix shown in Fig. 6 .The light intensity diffusion coefficient matrix that Fig. 9 (b)～Figure 12 (b) is given is the unknown original thing matrix element a of two dimension_{M, n}To two-dimensional image Matrix element b_{Q, w}Four-dimensional light intensity transmission coefficient d_{M, n, q, w}Set, as the unknown original thing matrix of two dimension has 25 units Element, the number of light intensity diffusion coefficient matrix is also 25.For example, what the light intensity diffusion coefficient matrix shown in Fig. 9 (b) was represented is a_1,1To b_1,1、b_1,2、b_2,1、b_2,2The four-dimensional strong transmission coefficient d of light_1,1,1,1、d_1,1,1,2、d_1,1,2,1、d_1,1,2,2, according to formula (1), (2) understand, a_1,1Corresponding to the element α in one-dimensional unknown original thing vector₁, and b_1,1、b_1,2、b_2,1、b_2,2Correspond respectively to one-dimensional As the element β in vector₁、β₂、β₆、β₇, understood according to formula (3), element α in one-dimensional unknown original thing vector₁To one-dimensional picture to Element β in amount₁、β₂、β₆、β₇Between two-dimentional light intensity transmission coefficient p_1,1、p_1,2、p_1,6、p_1,7Respectively equal to d_1,1,1,1、d_1,1,1,2、 d_1,1,2,1、d_1,1,2,2, and α₁To except β₁、β₂、β₆、β₇Other one-dimensional two-dimentional light intensity diffusion coefficients as vector element in addition are 0, so far, the first row element p of two-dimentional light intensity transmission coefficient_{1, j}(j=1～25) all solve out.For α₂～α₂₅, press The all elements p of the 2nd～25 row in two-dimentional light intensity transmission matrix can be obtained according to same method_{2, j}～p_{25, j}(j=1～25). By all light intensity transmission coefficient p_{I, j}Write as the form of matrix by subscript position number, just can be tried to achieve the two-dimentional light shown in Figure 13 Strong transmission matrix.As can be seen that the two-dimentional light intensity transmission matrix shown in Figure 13 has 25 rows, 25 row, its all elements is Fig. 8 The element a of the two-dimentional unknown original thing matrix in the data in shown PSF matrixes, with Fig. 7_{M, n}It is unrelated.

Example two：

In order to further illustrate the meter of the method for solving and reset error of two-dimentional light intensity transmission matrix in imaging recovery technique Calculate, Figure 14 (a) gives the known two-dimentional original thing of the alphabetical F patterns containing 7 rows, 7 column elements, and Figure 14 (b) gives alphabetical F Two-dimentional original thing matrix known to pattern, in recuperation, it is believed that the original thing matrix is unknown, answering after reconstruction In the calculating process of former error, the original thing matrix is known.Figure 15 gives certain optical system PSF matrix data, the light System is linear sky invariant system.Figure 16 (a) gives the picture of alphabetical F things, and Figure 16 (b) gives two-dimensional image matrix, can be with Find out, due to optical system imaging quality it is poor, as obscuring very much, and two-dimensional image matrix data and known two-dimentional original thing square Battle array data differ greatly.Two-dimentional light intensity transmission matrix is built according to the method described in formula (3) and example one, be obtained 49 rows, The two-dimentional light intensity transmission matrix of 49 row, as the element number of the matrix is excessive, inconvenience is listed as Figure 13 one by one, and Figure 17 gives The graphics of the two-dimentional light intensity transmission matrix data drawn with matlab softwares is gone out, wherein x coordinate represents the row sequence of the matrix Number, the y coordinate representation matrix column sequence number, z coordinate represents the numerical value of the matrix element.As can be seen that two-dimentional light intensity transmission square Battle array has 3 data strips, and intermediate strap data value is maximum, and both sides Rearrangments, band number are equal to the line number of PSF data. According to formula (6) to Figure 17 in the mathematical operation that carries out finding the inverse matrix of two-dimentional light intensity transmission matrix two-dimentional light intensity transmission is obtained Inverse matrix, Figure 18 give the graphics that inverse matrix data are transmitted with the two-dimentional light intensity that matlab softwares draw, wherein x coordinate table Show row sequence number, the y coordinate representation matrix column sequence number of the matrix, z coordinate represents the numerical value of the matrix element.According to formula (2) by the two-dimensional image matrix shown in Figure 16 (b) one-dimensional chemical conversion one-dimensional picture vector, one-dimensional picture vector is multiplied by by figure according to formula (7) Two-dimentional light intensity transmission inverse matrix shown in 18 is just obtained one-dimensional reconstruction vector, one-dimensional reconstruction is vectorial according to formula (8) Two dimensionization is just obtained two-dimentional reconstruction matrix, shown in reconstruction and two-dimentional reconstruction matrix such as Figure 19 (a), (b).According to formula (9) known two-dimentional original thing matrix in Figure 14 (b) is deducted into the two-dimentional reconstruction matrix in Figure 19 (b) and two-dimentional recovery is obtained Error matrix, the data of two-dimentional reset error matrix are as shown in figure 20.Two-dimentional reset error can be calculated according to formula (10) Root-mean-square value RMS_EFor 8.6787 × 10^-14, it is seen then that the imaging restored method described in this patent has minimum reset error, In the case of known optical systems PSF, accurately can restore two-dimentional original thing matrix by two-dimensional image matrix.

Example three：

In order to verify recovery effect of this restored method to the picture matrix of big data quantity, Figure 21 give containing 400 rows, The known original thing of 400 column elements, Figure 22 gives the PSF matrixes of optical system, and (optical system is linear sky invariant system System), Figure 23 gives the original thing Jing optical system imagings shown in Figure 21.As can be seen that due to optical system imaging matter Amount is very poor, and original thing imaging is obscured very much.The two-dimensional matrix of the picture in Figure 23 is carried out one-dimensional first to be contained The one-dimensional picture vector of 160000 elements, then according to formula (3) and the method described in example one build two-dimentional light intensity transmission square Battle array, is obtained 160000 rows, the two-dimentional light intensity transmission matrixs of 160000 row, due to matrix element number excessively, matlab numbers Value analysis software cannot draw the datagraphic of the matrix.According to formula (6) to 160000 rows, the two-dimentional light intensity of 160000 row Transmission matrix finding the inverse matrix is obtained the two-dimentional light intensity transmission inverse matrix of 160000 rows, 160000 row, similarly, since the matrix Element number is excessive, and matlab numerical analysis softwares cannot draw the datagraphic that two-dimentional light intensity transmits inverse matrix.According to formula (7) one-dimensional picture vector is multiplied by into two-dimentional light intensity transmission inverse matrix, one-dimensional reconstruction vector is obtained.According to formula (8), will be one-dimensional Reconstruction vector two dimensionization is just obtained two-dimentional reconstruction matrix.Two-dimentional reconstruction is as shown in figure 24, it can be seen that in Figure 24 Reconstruction there is no difference with the original thing in Figure 21.According to formula (9) by two-dimentional original thing matrix known in Figure 21 The two-dimentional reconstruction matrix for deducting Figure 24 is obtained two-dimentional reset error matrix, is drawn using the data of two-dimentional reset error matrix Reset error distribution map as shown in figure 25, wherein x coordinate represents the row sequence number of the matrix, the y coordinate representation matrix column sequence Number, z coordinate represents the numerical value of the matrix element.As can be seen that each element and two dimension in known two-dimentional original thing matrix The difference of each element in reconstruction matrix is less than 6 × 10^-12.Reset error root-mean-square value can be calculated according to formula (10) RMS_EFor 2.6293 × 10^-13.It can be seen that, the recovery technique described in this patent still has minimum recovery for the picture of big data quantity Error, in the case of known optical systems PSF, the big data quantity such as accurately can restore by the two-dimensional image matrix of big data quantity Two-dimentional original thing matrix.

Claims (6)

1. a kind of optical system imaging restored method based on light intensity transmission matrix, it is characterised in that the step of methods described such as Under：

The first step, the one-dimensional of two-dimensional image matrix are processed：

When the one-dimensional of two-dimensional image matrix is processed, one is set up with the identical matrix of two-dimensional image matrix data amount as two Unknown original thing matrix is tieed up, to the unknown original thing matrix of two dimension, after carrying out one-dimensional as matrix, the unknown original thing matrix of two dimension Certain element a_{M, n}Respective element α being changed in one-dimensional unknown original thing vector_i, certain element b of two-dimensional image matrix_{Q, w}Turn Respective element β being changed in one-dimensional picture vector_j, following relation should be met：

$\left\{\begin{array}{cccc}\begin{array}{c}{\mathrm{\α}}_i=\frac{a_{m,n}}{\underset{m=1}{\overset{M}{\Σ}}\underset{n=1}{\overset{N}{\Σ}}{a}_{m,n}}\\ i=(m-1)\×N+n\end{array},& i\∈(1,M\×N)& m\∈(1,M)& n\∈(1,N)\end{array}\right.;$

$\left\{\begin{array}{cccc}\begin{array}{c}{\mathrm{\β}}_j=\frac{b_{q,w}}{\underset{q=1}{\overset{Q}{\Σ}}\underset{w=1}{\overset{W}{\Σ}}{b}_{q,w}}\\ j=(q-1)\×W+w\end{array},& j\∈(1,Q\×W)& q\∈(1,Q)& w\∈(1,W)\end{array}\right.;$

Wherein, m, n are the sequence number of certain element in the unknown original thing matrix of two dimension, and i is certain in one-dimensional unknown original thing vector The sequence number of element, q, w are the sequence number of certain element in two-dimensional image matrix, and j is the sequence number of certain element in one-dimensional picture vector, M, N For the line number of the unknown original thing matrix of two dimension, columns, Q, W are total line number, the total columns as matrix, and i, j, m, n, q, w, M, N, Q, W are positive integer, and M is equal with Q, N is equal with W；

Second step, two-dimentional light intensity transmission matrix and two-dimentional light intensity transmit the solution of inverse matrix：

The two-dimentional light intensity transmission matrix of optical system is solved according to the PSF matrixes and one-dimensional picture vector of optical system；By two-dimentional light The mathematical computations inverted by strong transmission matrix, obtain two-dimentional light intensity transmission inverse matrix；

The solution procedure of described two-dimentional light intensity transmission matrix is as follows：

From the PSF extracting datas of optical system by the unknown original thing matrix of two dimension certain element representation light intensity a_{M, n}Transmission To light intensity b of certain element representation of two-dimensional image matrix_{Q, w}Four-dimensional light intensity transmission coefficient d_{M, n, q, w}, d_{M, n, q, w}With one-dimensional unknown original Element α in beginning thing vector_iTo the element β in one-dimensional picture vector_jTwo-dimentional light intensity transmission coefficient p_{I, j}It is equal, p_{I, j}For two Element in dimension light intensity transmission matrix, i.e.,：

$\left\{\begin{array}{cccc}{p}_{i,j}=d_{m,n,q,w}& & & \\ i=(m-1)\×N+n,& i\∈(1,M\×N),& m\∈(1,M),& n\∈(1,N)\\ j=(q-1)\×W+w,& j\∈(1,Q\×W),& q\∈(1,Q),& w\∈(1,W)\end{array}\right.;$

Wherein, m, n are the sequence number of certain element in the unknown original thing matrix of two dimension, and i is certain in one-dimensional unknown original thing vector The sequence number of element, q, w are the sequence number of certain element in two-dimensional image matrix, and j is the sequence number of certain element in one-dimensional picture vector, M, N For the line number of the unknown original thing matrix of two dimension, columns, Q, W total line number, total columns for two-dimensional image matrix, and i, j, m, n, q, w, M, N, Q, W are positive integer, and M is equal with Q, N is equal with W；Obtain one-dimensional unknown original thing vector according to the method described above successively each Element is just obtained two-dimentional light intensity transmission matrix to the light intensity transmission coefficient of one-dimensional picture vector each element；

3rd step, the solution of reconstruction matrix：

One-dimensional picture vector is multiplied by into two-dimentional light intensity transmission inverse matrix and tries to achieve one-dimensional reconstruction vector, by one-dimensional reconstruction vector two dimension Two-dimentional reconstruction matrix is obtained after change.

2. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterised in that institute State in step 2, two-dimentional light intensity transmission matrix has following relation with two-dimentional light intensity transmission inverse matrix：

P·P^-1=E；

Wherein, matrix P is two-dimentional light intensity transmission matrix, matrix P^-1Inverse matrix is transmitted for two-dimentional light intensity, matrix E is and P exponent number phases Same unit matrix.

3. the optical system imaging restored method based on light intensity transmission matrix according to claim 2, it is characterised in that two Dimension light intensity transmission matrix P of matrix^-1Ask method as follows：

$P^{-1}=\frac{P*}{\left|P\right|};$

Wherein, determinants of | the P | for matrix P, matrix P^*For the adjoint matrix that the algebraic complement of matrix P each element is constituted.

4. the optical system imaging restored method based on light intensity transmission matrix according to claim 1,2 or 3, its feature exist In the step 3, the one-dimensional picture vector that the first step is obtained is multiplied by into the two-dimentional light intensity transmission inverse matrix that second step obtains and just may be used One-dimensional reconstruction vector is obtained, i.e.,

B·P^-1=C；

B is that one-dimensional picture is vectorial, P^-1Inverse matrix is transmitted for two-dimentional light intensity, C is one-dimensional reconstruction vector.

5. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterised in that institute State in step 3, one-dimensional reconstruction vector is just obtained into two-dimentional reconstruction matrix according to equation below two dimensionization：

$\left\{\begin{array}{cccc}\begin{array}{c}{a_{k,l}}^{\′}={{\mathrm{\α}}_c}^{\′}\\ c=(k-1)\×L+l\end{array}& c\∈(1,K\×L)& k\∈(1,K)& L\∈(1,L)\end{array}\right.;$

Wherein, a_{K, l}' it is element in two-dimentional reconstruction matrix, α_c' it is element in one-dimensional reconstruction vector C, k, l are two dimension The sequence number of certain element in reconstruction matrix, c are the sequence number of certain element in one-dimensional reconstruction vector, and K, L are two-dimentional reconstruction Total line number of matrix, total columns, K are equal to Q, and L is equal to W, Q, W total line number, total columns for two-dimensional image matrix, and c, k, l, K, L It is positive integer.

6. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterised in that institute State in step 3, known two-dimentional original thing matrix is subtracted each other with two-dimentional reconstruction matrix and be obtained error matrix, Jin Erqiu Go out error mean square root RMS of known two-dimentional original thing matrix and corresponding element in two-dimentional reconstruction matrix_E, RMS_ESize For evaluating the recovery accuracy of imaging restored method.