CN104574315A - 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

Based on the optical system imaging restored method of light intensity transmission matrix

Technical field

The invention belongs to optical system imaging to restore and image quality lift technique field, relate to and a kind ofly utilize two-dimentional light intensity transmission matrix by the new method reverting to the reconstruction matrix identical with picture matrix data amount as matrix of optical system.

Background technology

In optical system imaging process, due to the existence of diffraction and aberration, each point on object plane all can be imaged as a disc of confusion in image planes, thus causes image blur, image quality to be degenerated, as inconsistent with thing.By imaging recovery technique, picture element is promoted, the sharpness of picture can be improved, thus picture is promoted with the similarity degree of thing.But in image restoration process in the past, generally adopting deconvolution restored method and frequency spectrum restored method, all there is certain shortcoming in these two kinds of methods.First, in spatial domain, the Light distribation function of picture can be tried to achieve as convolution algorithm with the point spread function (PSF) of the Light distribation function of thing and optical system, and the operation expression of this convolution is the integral equation of Fredholm type.When known picture and PSF, the computing solving reconstruction from this Fredholm type integral equation is ill-conditioning problem, cannot obtain accurate analytic solution, so the reset error of deconvolution restored method is larger.Secondly, in a frequency domain as frequency spectrum can try to achieve with the optical transfer function (OTF) that thing frequency spectrum is multiplied by optical system, when known picture and OTF, the principle of frequency domain restored method is that picture is obtained picture frequency spectrum as Fourier transform, again picture frequency spectrum is obtained reconstruction frequency spectrum divided by OTF, finally inverse Fourier transform is done to reconstruction frequency spectrum and can obtain reconstruction.But OTF has cutoff frequency, generally, the cutoff frequency of OTF will lower than thing frequency spectrum highest frequency.When optical system imaging, be truncated in thing frequency spectrum higher than the spectrum component of OTF cutoff frequency, picture frequency composes the spectrum component only comprised within OTF cutoff frequency.So frequency domain restored method can only restore the thing frequency spectrum within OTF cutoff frequency, cause the radio-frequency component of reconstruction to lack, reset error is larger.

Summary of the invention

The object of this invention is to provide a kind of optical system imaging restored method based on light intensity transmission matrix, the method utilizes matrix operation to achieve imaging recovery, avoid the ill-conditioning problem of the Deconvolution Method in imaging recovery technique in the past and the high frequency disappearance problem of frequency domain restored method, the error of reconstruction matrix and original thing matrix is minimum, and recovery accuracy is high.

The object of the invention is to be achieved through the following technical solutions:

Based on an optical system imaging restored method for light intensity transmission matrix, comprise the steps:

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

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

Solving of second step, two-dimentional light intensity transmission matrix and two-dimentional light intensity transmission inverse matrix:

For a certain known optical system, according to PSF matrix and the vectorial two-dimentional light intensity transmission matrix solving optical system of one dimension picture of this optical system; The mathematical computations of being carried out inverting by two-dimentional light intensity transmission matrix just can obtain two-dimentional light intensity transmission inverse matrix.

Solving and reset error calculating of 3rd step, reconstruction matrix:

One dimension picture vector is multiplied by two-dimentional light intensity transmission inverse matrix and tries to achieve one dimension reconstruction vector, obtain two-dimentional reconstruction matrix by after one dimension reconstruction vector two dimensionization.Original for known two dimension thing matrix and two-dimentional reconstruction matrix are subtracted each other and can obtain reset error matrix, and then the error mean square root RMS of corresponding element in known two dimension original thing matrix and two-dimentional reconstruction matrix can be obtained _e, RMS _esize for evaluating the recovery accuracy of imaging restored method.

The optical system imaging restored method based on light intensity transmission matrix that the present invention proposes is when the two-dimentional reconstruction matrix of known two-dimensional image matrix and two-dimentional PSF Matrix Solving, one dimension picture vector can be multiplied by two-dimentional light intensity transmission inverse matrix and obtain one dimension reconstruction vector, one dimension reconstruction vector two dimension be changed into two-dimentional reconstruction matrix and just can try to achieve accurate restoration result.The error of the two-dimentional reconstruction matrix that this method obtains and known two dimension original thing matrix is minimum, and recovery accuracy is high.

Accompanying drawing explanation

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

Fig. 2 is imaging schematic diagram image planes being revolved turnback around optical axis;

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

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

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

Fig. 6 is the one dimension picture vector after the two-dimensional image matrix and of one-dimensional be made up of 5 row 5 column elements;

Fig. 7 is according to vectorial as the one dimension unknown material after the unknown original thing matrix of the two dimension be made up of 5 row 5 column elements of matrix foundation and of one-dimensional in Fig. 6;

Fig. 8 is the PSF matrix be made up of 3 row 3 column elements;

Fig. 9 (a) is the unknown original thing entry of a matrix element a of two dimension in Fig. 7 _1,1(corresponding to first element α in the unknown original thing vector of one dimension ₁) the Light distribation data that formed in image planes through optical system, (b) is element a _1,1to the light intensity transmission coefficient of each element of picture matrix;

Figure 10 (a) is the unknown original thing entry of a matrix element a of two dimension in Fig. 7 _1,2(corresponding to second element α in the unknown original thing vector of one dimension ₂) the Light distribation data that formed in image planes through optical system, (b) is element a _1,2to the light intensity transmission coefficient of each element of picture matrix;

Figure 11 (a) is the unknown original thing entry of a matrix element a of two dimension in Fig. 7 _2,2(corresponding to the 7th element α in the unknown original thing vector of one dimension ₇) the Light distribation data that formed in image planes through optical system, (b) is element a _2,2to the light intensity transmission coefficient of each element of picture matrix;

Figure 12 (a) is the unknown original thing entry of a matrix element a of two dimension in Fig. 7 _5,5(corresponding to last element α in the unknown original thing vector of one dimension ₂₅) the Light distribation data that formed in image planes through optical system, (b) is element a _5,5to the light intensity transmission coefficient of each element of picture matrix;

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

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

Figure 15 is optical system PSF matrix;

The picture that Figure 16 (a) becomes through optical system for alphabetical F thing, 7 row 7 column matrix that (b) is picture;

Figure 17 is by the three-dimensional plot of the two-dimentional light intensity transmission matrix data obtained as Matrix Calculating in the PSF matrix in Figure 15 and Figure 16 (b);

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

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

The error matrix that Figure 20 obtains for the reconstruction matrix being deducted Figure 19 (b) by the known original thing matrix of Figure 14 (b);

Figure 21 is the known original thing containing 400 row 400 column elements;

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

Figure 23 is the picture containing 400 row 400 column elements;

Figure 24 is the reconstruction obtained after restoring the picture in Figure 23;

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

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is further described; but be not limited thereto; everyly technical solution of the present invention modified or equivalent to replace, and not departing from the spirit and scope of technical solution of the present invention, all should be encompassed in protection scope of the present invention.

The invention provides a kind of new method utilizing two-dimentional light intensity transmission matrix the two-dimensional image matrix of optical system to be reverted to the two-dimentional reconstruction matrix identical with its data volume.Solving of two dimension reconstruction matrix carries out under the condition of known two-dimensional image matrix and two-dimentional PSF matrix, and in the whole process solving two-dimentional reconstruction matrix, two-dimentional original thing matrix is unknown condition.In order to set forth the principle of the imaging restored method that the present invention proposes, need to set up one with the identical two-dimensional matrix of two-dimensional image matrix data amount as the unknown original thing matrix of two dimension.The unknown original thing entry of a matrix element symbol a of two dimension _{m, n}represent, wherein, m, n are the position number of certain element in the unknown original thing matrix of two dimension.

The meaning of the unknown original thing matrix of two dimension and two dimension original thing matrix is on all four, and only the former all elements is all unknown, and all elements of the latter is all known, and the dimension of the two is all identical with the dimension of two-dimensional image matrix.The two-dimentional light intensity transmission matrix solved in this restored method and the unknown original thing entry of a matrix element a of two dimension _{m, n}irrelevant, only relevant with two-dimensional image matrix and PSF matrix.As needed to evaluate the recovery accuracy of this method, then need to provide the known original thing matrix of two dimension to analyze reset error.Concrete steps are as follows:

The first step: the of one-dimensional process of two-dimensional image matrix.

Current image is all stored as the form of e-file, and a gray level image two-dimensional matrix represents, in matrix each element positional representation image planes on the relative position of a point, the light intensity of numeric representation this point of each element.A color document image three-dimensional matrice represents, this three-dimensional matrice is equivalent to three two-dimensional matrixs, and each two-dimensional matrix represents red, green, blue three kinds of colors of image each point respectively.During process coloured image, data to red, green, blue three kinds of colors can be processed respectively according to the method for gray level image process, therefore the present invention only sets forth to the restored method of gray level image.

For setting forth our ratio juris, first the imaging process of optical system is described.As shown in Figure 1, the light intensity of a certain object point 2 on two dimension object plane 1 is imaged as a disc of confusion 5 through optical system 3 in two-dimentional image planes 4, and the light distribution of disc of confusion 5 accurately can be described by the data of the point spread function (PSF) 6 centered by corresponding picture point.Do not adding in the optical system rotating light path, image planes are stand upside down relative to object plane, and namely image planes have rotated 180 degree of angles relative to object plane.For the ease for the treatment of and analysis, image planes can be revolved turnback centered by optical axis, as shown in Figure 2, this disposal route does not change the imaging law of optical system, in actual optical system, rotates light path realization by adding.As shown in Figure 2, on two-dimentional object plane 1, the light intensity data of each point can be represented by the unknown original thing matrix 7 of the two dimension in Fig. 3, and equally, in two-dimentional image planes 4, the light intensity data of each point can be represented by the two-dimensional image matrix 9 in Fig. 4.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}light intensity transmission coefficient d _{m, n, q, w}must be four-dimensional parameter, then by d _{m, n, q, w}the matrix necessarily four-matrix formed, and four-matrix is not easy to treatment and analysis, is also not easy to draw represent.And for the element α in one dimension thing vector _ielement β in one dimension picture vector _jlight intensity transmission coefficient p _{i, j}two-dimensional parameter, by p _{i, j}the light intensity transmission matrix formed is two-dimensional matrix, and two-dimensional matrix is convenient to analyze, process, and being also convenient to draw represents.

In this restored method, solving of reconstruction matrix is carried out under the condition of known two-dimensional image matrix and two-dimentional PSF matrix, and the two dimension original thing matrix solved in two-dimentional reconstruction matrix process is unknown.Need to set up one with the identical two-dimensional matrix of two-dimensional image matrix data amount as the unknown original thing matrix of two dimension.

As shown in Figure 3, the of one-dimensional process of thing is exactly be a row vector by every a line in each row element of unknown for two dimension original thing matrix 7 successively end to end being arranged in order.As shown in Figure 4, as of one-dimensional process be exactly be a row vector by every a line in each for two-dimensional image matrix 9 row element successively end to end being arranged in order.Again press after tandem number to thing vector and the element in picture vector, just define the unknown original thing vectorial 8 of one dimension and one dimension picture vector 10.

As shown in Figure 3, Figure 4, original thing matrix unknown to two dimension, carry out of one-dimensional as matrix after, two dimension unknown original thing matrix certain element a _{m, n}change the respective element α in the unknown original thing vector of one dimension into _i, certain element b of two-dimensional image matrix _{q, w}change the respective element β in one dimension picture vector into _j, following relation should be met:

$\left\{\begin{array}{c}{\mathrm{\α}}_j=a_{q,w}\\ i=(m-1)\×N+n\end{array}\right.i\∈(1,M\×N),m\∈(1,M),n\∈(1,N)---\left(1\right);$

$\left\{\begin{array}{c}{\mathrm{\β}}_j=b_{q,w}\\ j=(q-1)\×W+w\end{array}\right.j\∈(1,Q\×W),q\∈(1,Q),w\∈(1,W)---\left(2\right).$

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

Second step: two-dimentional light intensity transmission matrix and two-dimentional light intensity transmit solving of inverse matrix.

According to the basic theories of Fourier optics, optical imaging system belongs to incoherent imaging system, the light intensity that each object point on two dimension object plane sends is focused in image planes by optical system and forms the disc of confusion that covers multiple picture point, disc of confusion is centrally located at the picture point (namely in Fig. 2 with object point two-dimensional position identical picture point) corresponding with object point, and the light distribution of this disc of confusion accurately can be described by the PSF centered by this picture point.It should be noted that, the PSF in linear space-variant optical system centered by each picture point is all different, and the PSF in linear empty constant optical system centered by each picture point is all identical.So, the PSF matrix centered by each pixel element in known picture matrix is needed in the imaging recuperation of linear space-variant optical system, and in the imaging recuperation of linear empty constant optical system, only need the PSF matrix centered by some pixel elements in known picture matrix.

Due to certain element a of the unknown original thing matrix of two dimension _{m, n}be transferred to certain element b of two-dimensional image matrix _{q, w}light intensity transmission coefficient d _{m, n, q, w}can from b _{m, n}centered by PSF extracting data, and d _{m, n, q, w}with the element α in one dimension unknown material vector _ielement β in picture vector _jlight intensity transmission coefficient p _{i, j}(element namely in two-dimentional light intensity transmission matrix 11) is equal, that is:

$\left\{\begin{array}{c}{p}_{i,j}=d_{m,n,q,w}\\ i=(m-1)\×N+n,i\∈(1,M\×N),m\∈(1,M),n\∈(1,N)\\ j=(q-1)\×W+w,j\∈(1,Q\×W),q\∈(1,Q),w\∈(1,W)\end{array}\right.---\left(3\right);$

Wherein, parameter i, j, m, n, q, w, M, N, Q, W meet relation and meaning identical with the relation in formula (1), (2) and meaning.Therefore, as long as record the unknown original thing matrix of good two dimension, as the four-dimensional light intensity transmission coefficient d between each element of matrix _{m, n, q, w}, just can obtain the two-dimentional light intensity transmission coefficient p of each element of the unknown original thing vector of one dimension to each element of one dimension picture vector _{i, j}, the respective element in this coefficient and the two-dimentional light intensity transmission matrix 11 shown in Fig. 5.According to the method, obtain each element of the unknown original thing vector of one dimension successively to the light intensity transmission coefficient of each element of one dimension picture vector, just can obtain two-dimentional light intensity transmission matrix 11.Due to the transmission coefficient p in two-dimentional light intensity transmission matrix _{i, j}be the data in PSF, so, two-dimentional light intensity transmission matrix and the unknown original thing entry of a matrix element a of two dimension _{m, n}irrelevant, two-dimentional light intensity transmission matrix completely can according to two-dimensional image matrix and two-dimentional PSF Matrix Solving.

After solving two-dimentional light intensity transmission matrix 11, the unknown original thing vector 8 of one dimension is multiplied by the relation that two-dimentional light intensity transmission matrix 11 equals one dimension picture vector 10 and just establishes, as shown in Figure 5.This product calculation relation can the form of formulate (4)

A·P＝B (4)；

Wherein, A is the unknown original thing vector of one dimension, and meaning is equal to the unknown original thing vector 8 of one dimension in Fig. 5; P is two-dimentional light intensity transmission matrix, and meaning is equal to the two-dimentional light intensity transmission matrix 11 in Fig. 5; B is one dimension picture vector, and meaning is equal to the picture vector 10 in Fig. 5.

The mathematical operation being carried out inverting by two-dimentional light intensity transmission matrix just can obtain two-dimentional light intensity transmission inverse matrix.Two dimension light intensity transmission matrix and two-dimentional light intensity transmit inverse matrix following relation:

P·P ^-1＝E (5)；

Wherein, matrix P is two-dimentional light intensity transmission matrix, matrix P ^-1for two-dimentional light intensity transmission inverse matrix, matrix E is the unit matrix identical with P exponent number.

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

$P^{-1}=\frac{P*}{\left|P\right|}---\left(6\right);$

Wherein, | P| is the determinant of matrix P, matrix P ^*for the adjoint matrix that the algebraic complement of each element of matrix P is formed.

Solving and reset error calculating of 3rd step, two-dimentional reconstruction matrix:

The one dimension picture vector first step obtained is multiplied by the two-dimentional light intensity transmission inverse matrix that second step obtains just has obtained one dimension reconstruction vector, namely

C＝B·P ^-1(7)，

B is one dimension picture vector, P ^-1for two-dimentional light intensity transmission inverse matrix, C is one dimension reconstruction vector.

One dimension reconstruction vector C two dimensionization just can be obtained two-dimentional reconstruction matrix, as shown in formula (8)

$\left\{\begin{array}{c}{a_{k,l}}^{\′}={{\mathrm{\α}}_c}^{\′}\\ c=(k-1)\×L+l\end{array}\right.c\∈(1,K\×L),k\∈(1,K),l\∈(1,L)---\left(8\right),$

Wherein, a _{k, l}' be the element in two-dimentional reconstruction matrix, α _c' be the element in one dimension reconstruction vector C, k, l are the sequence number of certain element in two-dimentional reconstruction matrix, c is the sequence number of certain element in one dimension reconstruction vector, K, L are total line number of two-dimentional reconstruction matrix, total columns, K Q, L equaled in formula (2) equals the W in formula (2), and c, k, l, K, L are positive integer.

In order to evaluate the recovery accuracy of described imaging restored method, need to utilize the original thing of known two dimension to carry out imaging, imaging is obtained according to the method described above two-dimentional reconstruction matrix, original for two dimension thing matrix is deducted two-dimentional reconstruction matrix and can obtain two-dimentional reset error matrix E, as shown in formula (9), reset error root-mean-square value RMS _ecan be calculated by formula (10):

${\mathrm{RMS}}_E=\sqrt{\frac{\underset{k=1}{\overset{K}{\mathrm{\Σ}}}\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{({\hat{a}}_{k,l}-{a_{k,l}}^{\′})}^2}{K\×L}}=\sqrt{\frac{\underset{k=1}{\overset{K}{\mathrm{\Σ}}}\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{\left(e_{k,l}\right)}^2}{K\×L}}---\left(10\right).$

Wherein, e _{k, l}for the element of two-dimentional error matrix, for the original thing entry of a matrix element of known two dimension, a _{k, l}' be two-dimentional reconstruction entry of a matrix element, k, l are the sequence number of element in known two dimension original thing matrix, two-dimentional reconstruction matrix and error matrix, K, L are total line number of known two dimension original thing matrix, two-dimentional reconstruction matrix and error matrix, total columns, and k, l, K, L are positive integer, K equals the Q in formula (2), L equals 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 is all set forth complete, sets forth described side's ratio juris, recuperation, restoration result and reset error below with three examples.

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 the example of the one dimension picture vector after the two-dimensional image matrix and of one-dimensional be made up of 5 row 5 column elements.Fig. 7 gives the original thing vector of the unknown after the unknown original thing matrix such as the two dimension containing 5 row 5 column elements large with two-dimensional image matrix etc. and of one-dimensional set up according to two-dimensional image matrix in Fig. 5, in solution procedure, all elements in the original thing vector of two dimension unknown original thing matrix and one dimension the unknown is all unknown.Fig. 8 gives the PSF matrix example be made up of 3 row 3 column elements, and this optical system is linear empty invariant system.In the unknown original thing matrix of two dimension, each element is through the effect of optical system PSF, all disc of confusion can be formed in image planes, but in actual optical system, due to the restriction of diaphragm and image planes size, the part exceeded because of the diffusion of PSF as edge cannot record on the detector, so the part exceeded in disc of confusion as matrix cannot be recorded in picture matrix.Fig. 9 (a) gives two dimension in Fig. 6 unknown original thing entry of a matrix element a _1,1(corresponding to first element α in the unknown original thing vector of one dimension ₁) the Light distribation data that formed in image planes through optical system, Fig. 9 (b) gives a _1,1to the light intensity transmission coefficient of each element of two-dimensional image matrix.Figure 10 (a) gives two dimension unknown original thing entry of a matrix element a _1,2(corresponding to second element α in the unknown original thing vector of one dimension ₂) the Light distribation data that formed in image planes through optical system, Figure 10 (b) gives a _1,2to the light intensity transmission coefficient of each element of two-dimensional image matrix.The like, Figure 11 (a) gives two dimension unknown original thing entry of a matrix element a2, and 2 (corresponding to the 7th element α in the unknown original thing vector of one dimension ₇) the Light distribation data that formed in image planes through optical system, Figure 11 (b) gives a _2,2to the light intensity transmission coefficient of each element of two-dimensional image matrix.Figure 12 (a) gives two dimension unknown original thing entry of a matrix element a _5,5(corresponding to last element α in the unknown original thing vector of one dimension ₂₅) the Light distribation data that formed in image planes through optical system, Figure 12 (b) gives a _5,5to the light intensity transmission coefficient of each element of two-dimensional image matrix.On the whole, matrix in Fig. 9 (a) ~ Figure 12 (a) is the light intensity diffusion matrix that the unknown original thing entry of a matrix element of two dimension is formed in image planes, and Fig. 9 (b) ~ Figure 12 (b) is the light intensity diffusion coefficient matrix of the unknown original thing entry of a matrix element of two dimension to each element of two-dimensional image matrix.Two-dimensional image matrix equals the light intensity diffusion matrix sum of form as shown in Fig. 9 (a) ~ Figure 12 (a) that the unknown original thing entry of a matrix element of all two dimensions is formed in image planes.Can find out, the unknown original thing matrix element of light intensity diffusion coefficient matrix shown in Fig. 9 (b) ~ Figure 12 (b) and the two dimension shown in Fig. 6 has nothing to do, and light intensity diffusion matrix can PSF matrix as shown in Figure 8 and the two-dimensional image matrix shown in Fig. 6 be tried to achieve by formula (3) completely.The light intensity diffusion coefficient matrix that Fig. 9 (b) ~ Figure 12 (b) provides 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, because the unknown original thing matrix of two dimension has 25 elements, the number of light intensity diffusion coefficient matrix is also 25.Such as, that the light intensity diffusion coefficient matrix shown in Fig. 9 (b) represents 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, known according to formula (1), (2), a _1,1corresponding to the element α in the unknown original thing vector of one dimension ₁, and b _1,1, b _1,2, b _2,1, b _2,2correspond respectively to the element β in one dimension picture vector ₁, β ₂, β ₆, β ₇, known according to formula (3), element α in the unknown original thing vector of one dimension ₁element β in one dimension picture vector ₁, β ₂, β ₆, β ₇between two-dimentional light intensity transmission coefficient p _1,1, p _1,2, p _1,6, p _1,7equal d respectively _1,1,1,1, d _1,1,1,2, d _1,1,2,1, d _1,1,2,2, and α ₁to removing β ₁, β ₂, β ₆, β ₇other one dimensions are in addition 0 as the two-dimentional light intensity coefficient of diffusion of vector element, so far, and the first row element p of two-dimentional light intensity transmission coefficient _{1, j}(j=1 ~ 25) all solve out.For α ₂~ α ₂₅, all elements p of 2nd ~ 25 row in two-dimentional light intensity transmission matrix can be obtained after the 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 intensity transmission matrix shown in Figure 13.Can find out, the two-dimentional light intensity transmission matrix shown in Figure 13 has 25 row, 25 row, and its all elements is the data in the PSF matrix shown in Fig. 8, original thing entry of a matrix element a unknown with the two dimension in Fig. 7 _{m, n}irrelevant.

Example two:

In order to further illustrate the two-dimentional method for solving of light intensity transmission matrix and the calculating of reset error in imaging recovery technique, Figure 14 (a) gives the original thing of known two dimension of the alphabetical F pattern containing 7 row, 7 column elements, Figure 14 (b) gives the known two dimension of alphabetical F pattern original thing matrix, in recuperation, can think that this original thing matrix is unknown, in the computation process of reset error after reconstruction, this original thing matrix is known.Figure 15 gives certain optical system PSF matrix data, and this optical system is linear empty invariant system.Figure 16 (a) gives the picture of alphabetical F thing, Figure 16 (b) gives two-dimensional image matrix, can find out, because optical system imaging quality is poor, picture is very fuzzy, and two-dimensional image matrix data and the original thing matrix data of known two dimension differ greatly.Two-dimentional light intensity transmission matrix is built according to the method described in formula (3) and example one, the two-dimentional light intensity transmission matrix of 49 row, 49 row can be obtained, because this entry of a matrix prime number order is too much, inconvenience is listed one by one as Figure 13, Figure 17 gives the three-dimensional plot of the two-dimentional light intensity transmission matrix data drawn with matlab software, wherein x coordinate represents the line order number of this matrix, y coordinate represents this matrix column sequence number, and z coordinate represents the numerical value of this matrix element.Can find out, two-dimentional light intensity transmission matrix has 3 data strip, and intermediate strap data value is maximum, both sides Rearrangments, and band number equals the line number of PSF data.Two-dimentional light intensity transmission inverse matrix can be obtained to the mathematical operation that the two-dimentional light intensity transmission matrix in Figure 17 carries out finding the inverse matrix according to formula (6), Figure 18 gives the three-dimensional plot of the two-dimentional light intensity transmission inverse matrix data drawn with matlab software, wherein x coordinate represents the line order number of this matrix, y coordinate represents this matrix column sequence number, and z coordinate represents the numerical value of this matrix element.According to formula (2), the two-dimensional image matrix one dimension shown in Figure 16 (b) is changed into one dimension picture vector, according to formula (7), one dimension picture vector is multiplied by the two-dimentional light intensity transmission inverse matrix shown in Figure 18 and just can obtains one dimension reconstruction vector, according to formula (8), one dimension reconstruction vector two dimensionization just can be obtained two-dimentional reconstruction matrix, reconstruction and two-dimentional reconstruction matrix are as shown in Figure 19 (a), (b).According to formula (9), by original for two dimension known in Figure 14 (b) thing matrix, the two-dimentional reconstruction matrix deducted in Figure 19 (b) can obtain two-dimentional reset error matrix, and the data of two-dimentional reset error matrix as shown in figure 20.The root-mean-square value RMS of two-dimentional reset error can be calculated according to formula (10) _ebe 8.6787 × 10 ^-14, visible, the imaging restored method described in this patent has minimum reset error, when known optical systems PSF, accurately can restore two-dimentional original thing matrix by two-dimensional image matrix.

Example three:

In order to verify the recovery effect of this restored method to the picture matrix of big data quantity, Figure 21 gives the known original thing containing 400 row, 400 column elements, Figure 22 gives the PSF matrix (this optical system is linear empty invariant system) of optical system, and Figure 23 gives the original thing shown in Figure 21 through optical system imaging.Can find out, because optical system imaging quality is very poor, original thing imaging is very fuzzy.First the two-dimensional matrix of the picture in Figure 23 is carried out the one dimension picture vector that of one-dimensional can obtain containing 160000 elements, then two-dimentional light intensity transmission matrix is built according to the method described in formula (3) and example one, the two-dimentional light intensity transmission matrix of 160000 row, 160000 row can be obtained, because entry of a matrix prime number order is too much, matlab numerical analysis software cannot draw the datagraphic of this matrix.The two-dimentional light intensity transmission inverse matrix of 160000 row, 160000 row can be obtained according to the two-dimentional light intensity transmission matrix finding the inverse matrix of formula (6) to 160000 row, 160000 row, equally, because this entry of a matrix prime number order is too much, matlab numerical analysis software cannot draw the datagraphic of two-dimentional light intensity transmission inverse matrix.According to formula (7), one dimension picture vector is multiplied by two-dimentional light intensity transmission inverse matrix and can obtains one dimension reconstruction vector.According to formula (8), one dimension reconstruction vector two dimensionization just can be obtained two-dimentional reconstruction matrix.Two dimension reconstruction as shown in figure 24, can be found out, the original thing in the reconstruction in Figure 24 and Figure 21 there is no difference.The two-dimentional reconstruction matrix according to formula (9) original for two dimension known in Figure 21 thing matrix being deducted Figure 24 can obtain two-dimentional reset error matrix, utilize reset error distribution plan that the data of two-dimentional reset error matrix are drawn as shown in figure 25, wherein x coordinate represents the line order number of this matrix, y coordinate represents this matrix column sequence number, and z coordinate represents the numerical value of this matrix element.Can find out, the difference of each element in known two dimension original thing matrix and each element in two-dimentional reconstruction matrix is no more than 6 × 10 ^-12.Reset error root-mean-square value RMS can be calculated according to formula (10) _ebe 2.6293 × 10 ^-13.Visible, the recovery technique described in this patent still has minimum reset error for the picture of big data quantity, when known optical systems PSF, accurately can be restored etc. by the two-dimensional image matrix of big data quantity the original thing matrix of two dimension of big data quantity.

Claims (9)

1., based on an optical system imaging restored method for light intensity transmission matrix, it is characterized in that the step of described method is as follows:

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

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

Solving of second step, two-dimentional light intensity transmission matrix and two-dimentional light intensity transmission inverse matrix:

According to PSF matrix and the vectorial two-dimentional light intensity transmission matrix solving optical system of one dimension picture of optical system; Two-dimentional light intensity transmission matrix is carried out the mathematical computations of inverting, obtain two-dimentional light intensity transmission inverse matrix;

Solving of 3rd step, reconstruction matrix:

One dimension picture vector is multiplied by two-dimentional light intensity transmission inverse matrix and tries to achieve one dimension reconstruction vector, obtain two-dimentional reconstruction matrix by after one dimension reconstruction vector two dimensionization.

2. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterized in that in described step one, when the of one-dimensional process of two-dimensional image matrix, set up one with the identical matrix of two-dimensional image matrix data amount as the unknown original thing matrix of two dimension, the unknown original thing entry of a matrix element of two dimension only uses symbol of element a _{m, n}represent, wherein, m, n are the position number of certain element in this matrix.

3. the optical system imaging restored method based on light intensity transmission matrix according to claim 2, is characterized in that: original thing matrix unknown to two dimension, carry out of one-dimensional as matrix after, two dimension unknown original thing matrix certain element a _{m, n}change the respective element α in the unknown original thing vector of one dimension into _i, certain element b of two-dimensional image matrix _{q, w}change the respective element β in one dimension picture vector into _j, following relation should be met:

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

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

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

4. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterized in that in described step 2, the solution procedure of two-dimentional light intensity transmission matrix is as follows: from light intensity a by certain element representation of the unknown original thing matrix of two dimension of the PSF extracting data of optical system _{m, n}be transferred to the 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 the element α in the unknown original thing vector of one dimension _ielement β in one dimension picture vector _itwo-dimentional light intensity transmission coefficient p _{i, j}equal, p _{i, j}for the element in two-dimentional light intensity transmission matrix, that is:

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

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

5. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, is characterized in that in described step 2, and two-dimentional light intensity transmission matrix and two-dimentional light intensity transmit inverse matrix following relation:

P·P ^-1＝E ；

Wherein, matrix P is two-dimentional light intensity transmission matrix, matrix P ^-1for two-dimentional light intensity transmission inverse matrix, matrix E is the unit matrix identical with P exponent number.

6. the optical system imaging restored method based on light intensity transmission matrix according to claim 5, is characterized in that two-dimentional light intensity transmission matrix P of matrix ^-1ask method as follows:

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

Wherein, | P| is the determinant of matrix P, matrix P ^*for the adjoint matrix that the algebraic complement of each element of matrix P is formed.

7. the optical system imaging restored method based on light intensity transmission matrix according to claim 1,5 or 6, it is characterized in that in described step 3, the one dimension picture vector first step obtained is multiplied by the two-dimentional light intensity transmission inverse matrix that second step obtains just can obtain one dimension reconstruction vector, namely

B·P ^-1＝C；

B is one dimension picture vector, P ^-1for two-dimentional light intensity transmission inverse matrix, C is one dimension reconstruction vector.

8. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, is characterized in that in described step 3, and one dimension reconstruction vector just can be obtained two-dimentional reconstruction matrix according to formula two dimensionization below:

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

Wherein, a _{k, l}' be the element in two-dimentional reconstruction matrix, α _c' be the element in one dimension reconstruction vector C, k, l are the sequence number of certain element in two-dimentional reconstruction matrix, c is the sequence number of certain element in one dimension reconstruction vector, K, L are total line number of two-dimentional reconstruction matrix, total columns, K equals Q, L equals W, and Q, W are total line number of two-dimensional image matrix, total columns, and c, k, l, K, L are positive integer.

9. the optical system imaging restored method based on light intensity transmission matrix according to claim 1, it is characterized in that in described step 3, original for known two dimension thing matrix and two-dimentional reconstruction matrix are subtracted each other and can obtain error matrix, and then obtains the error mean square root RMS of corresponding element in known two dimension original thing matrix and two-dimentional reconstruction matrix _e, RMS _esize for evaluating the recovery accuracy of imaging restored method.