CN113822822B - Identification method of image fuzzy matrix structure - Google Patents
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Abstract
The patent discloses an identification method of an image blurring matrix structure under some boundary conditions, which comprises the following steps of: 1) Determining the structure of a boundary condition matrix; 2) And analyzing the product of the extended blur matrix and the boundary matrix to obtain the structure of the image blur matrix. The identification method disclosed can be used for identifying the image blurring matrix structure under some boundary conditions and corner types, and the determined structure of the image blurring matrix can provide basis for the calculation implementation of image restoration.
Description
Technical Field
The invention relates to image processing, in particular to a method for identifying an image blur matrix structure in the recovery of linear degraded images under different boundary and corner conditions.
Background
On the basis of a linear degradation model g=kf+η, the image is subjected toIn the image restoration of (2), a matrix is calculated based on certain boundary conditions (Boundary Conditions, BCs) and corner types>Line-by-line reorder vector with FAnd the transpose of K is multiplied by f, K' f, where K is the implementation point spread function +.>Is the image blur matrix of the blurring effect of (h) PSF for short, f is the image +.>Is represented by a row-wise prioritized vector. Generally, since K and K 'are very large matrices with severe sparsity, products Kf and K' f need to be calculated in an accelerated manner based on a preset matrix or need to be calculated in an alternative manner based on convolution calculation or blurring matrix diagonalization, and in particular, what kind of calculation method can be adopted, which is related to the structure of K, and the latter is determined by the symmetry of PSF and the boundary condition type and corner type based on which the image restoration is based, therefore, in the image restoration based on the linear degradation of a specific boundary condition and corner type, the matrix structure of the image blurring matrix K is judged as a precondition for the corresponding image restoration calculation.
The current identification method of the image fuzzy matrix structure is to simply deduce the block structure of the image fuzzy matrix in the two-dimensional degradation process from the fuzzy matrix structure in the one-dimensional degradation process, and the method is as follows: let us assume a one-dimensional degradation process g 1 ＝K 1 f 1 +η 1 In (a) a fuzzy matrix K 1 Is the sum of a plurality of structural matrices, wherein the vector g 1 、f 1 And eta 1 is n x 1,the structure of the fuzzy matrix K in the two-dimensional degradation process is K 1 All possible combinations of structures in and between blocks, respectively, without loss of generality, if K 1 ＝K x +K y Then k=k xx +K xy +K yy Wherein K is xy Representing a blocking matrix with an inter-block structure x and an intra-block structure y. This simple one-dimensional to two-dimensional approach is only applicable when the corner type is selected as b in the boundary condition.
The current image boundary conditions include 6 kinds of Zero BCs (ZBC), periodic BCs (PBC) and Reflective BCs (RFBC), and recently proposed Anti-Reflective BCs (ARBC), mean BCs (MBC), repeated BCs (RPBC) boundary conditions, wherein only corner type b exists under the first three boundary conditions and RPBC, and only corner type a or b needs to be further selected under ARBC and MBC conditions to uniquely determine the structure of the fuzzy matrix. Among the 6 boundary conditions, the first three boundary conditions, and K corresponding to the b-corner type of ARBC (ARBC-b) and the b-corner type of MBC (MBC-b) have been identified, and a method of deriving one-dimensional degradation to two-dimensional degradation is adopted. The structure of K corresponding to the left corner type (ARBC-a), the corner type (MBC-a) of MBC, RPBC and the like is not identified, wherein the structure of K corresponding to the former two is very complex, the existing structure identification method is not suitable, and a new structure identification method needs to be found.
Disclosure of Invention
The technical problems to be solved by the invention are as follows: when the selected corner type is a, the structure of K cannot be identified by the existing method, so that the corner type a cannot be truly applied to image restoration.
The convolution effect realized by the product Kf in the linear degradation process of the image can be decomposed into the expansion of the image vector f according to boundary and corner conditions, and then the expanded image is subjected to the convolution effectThe image vector is convolved in ZBC, i.e. Kf itself can be expressed asWherein->Is an extended fuzzy matrix under ZBC, +.>For a boundary condition matrix under a specific boundary condition and corner type, the effect is to expand the boundary pixels with the determined fingers in f so that the image degradation model can be solved. Here->The structure of B is known as a blocking matrix of teopltz-like type, both inter and intra, and can be derived from the definition of boundary conditions and corner types.
Based on the analysis, the invention provides a new image blur matrix structure identification method aiming at the problem that some image blur matrices K are difficult to identify, and the thought is as follows: the structure of the fuzzy matrix K is identified and decomposed into the structure of the expansion matrix B which is identified first, and then the product is identifiedIs of a structure of (2); when the corner type is b, firstly giving out the structure of a boundary condition matrix in the one-dimensional degradation process, and then deducing the structure of an image fuzzy matrix in the two-dimensional degradation process from the structure of the product of the expanded fuzzy matrix and the boundary condition matrix; when the corner type is a, the corner condition matrix structure is directly analyzed from the two-dimensional degradation process, and then the product structure of the corresponding expansion fuzzy matrix and the boundary condition matrix is given. To support the above ideas:
some embodiments present the structure of the boundary condition matrix under ARBC-a, MBC-a and RPBC conditions.
Some implementations give the structure of the image blur matrix under ARBC-a, MBC-a and RPBC conditions.
Some embodiments experimentally illustrate the correctness of the identified structure.
The identification method provided by the invention can be applied to the image recovery of the linear degradation model and has application value.
Drawings
FIG. 1 shows a flow of the present invention;
FIG. 2 shows images used in the experiments of the present invention;
FIG. 3 shows the point spread function employed in the experiments of the present invention;
fig. 4 shows the mean square error of the product of the fuzzy matrix and the vector identified by the present invention.
Detailed Description
In the following description, for purposes of explanation, specific details of the embodiments described in the summary are set forth in connection with the accompanying drawings. For clarity of expression and ease of programming, matlab's function is applied to some calculation formulas.
Example 1
As shown in fig. 1, the present invention provides a method for identifying an image blur matrix structure under different boundary conditions, comprising the following steps:
1) Selecting boundary conditions and corner types based on which K is constructed;
2) According to the corner type, a structure identification method of the B is given;
3) And according to the corner type, a structure identification method of K is given.
Example 2
When the corner type is a, the corresponding image fuzzy matrix is not a blocking matrix corresponding to the fuzzy matrix in the one-dimensional degradation process, and the identification method of K is provided by the invention as follows: analyzing the structure of a two-dimensional boundary condition matrix BFinally based on->Will->The structure of (2) is determined as the structure of K.
Example 3
When the corner type is b, K is a fuzzy matrix K in the one-dimensional degradation process 1 The identification method of K provided by the invention is as follows: first analyzing one-dimensional boundary condition matrix B 1 Based on the structure of (2)Analysis K 1 And finally K is 1 The block structure of (2) is defined as the structure of K.
Example 4
The invention specifically provides that the matrix B of the ARBC-a lower boundary condition is BA plurality of blocks, each block having a size of +.>The specific structure of the block matrix of (2) is +.>
Wherein I is 12 ＝I 1 +I 2 ，I 13 ＝I 1 +I 3 ，I 123 ＝I 1 -I 2 +2I 3 While
And->
Example 5
The invention provides the structure of boundary condition matrix B under MBC-a, in particular
Here, the respective sizes are +.>Is respectively of the full order matrix of
And->Wherein i is more than or equal to 1 and p is more than or equal to p 1 ,1≤j≤q 1 ；s 1,1,i,j ＝(abs(d0)+1)(d 1 +1)，s 1,2,i,j ＝-d 1 abs(d0)+v 2 d 0 ,s 2,1,i,j ＝-d 1 abs(d 0 )-v 1 d 0 ,s 2,2,i,j ＝d 1 (abs(d0)-1),d 0 ＝p 1 -q 1 -i+j, and d 1 ＝(p 1 -i+1)v 2 +(q 1 -j+1)v 1 The method comprises the steps of carrying out a first treatment on the surface of the Coefficient v 1 And v 2 The setting method of (1) is if d 0 > =0, then v 1 ＝1,v 2 =0, otherwise v 1 ＝0,v 2 =1, the function abs (x) represents the return x absolute value.
Example 6
The invention provides B in the one-dimensional degradation process under the RPBC condition 1 Is of the structure B 1 ＝I 13 Correspondingly, B is B 1 Block matrix of (a), i.e
Example 7
The invention specifically provides the structure of K under ARBC-a as
K＝K TT -K TH +K TR -K HT -K HH +K HL +K RT +K LH -K LL +K U
The structure is as follows: subscripts T, H, R respectively represent a Toeplitz matrix, a Hankel matrix and a rank-2 correction matrix in an image blur matrix under the ARBC-b condition, L and U respectively represent other two rank-2 correction matrices provided by the invention, and the structures of the two rank-2 correction matrices are as follows:
note p=2p 1 +1,q＝2q 1 +1, L matrixThe non-zero element in i is more than or equal to 1 and less than or equal to p is formed by the ith row h of h i,: The matrix structure is given as
The L-shaped block matrix has the structure of
1.ltoreq.i.ltoreq.p, where the matrix +.>May be Hankel or L-type matrix, X i The non-zero elements of h i,: Is given;
u-type blocking matrix K U The structure of (1) is that
Wherein (1)>r epsilon {1, p }, t epsilon {1, q }, the range of corresponding i is: if r=1, 1.ltoreq.i.ltoreq.p 1 Otherwise p 1 +2.ltoreq.i.ltoreq.p; the range of j is that if t=1, 1.ltoreq.j.ltoreq.q1, otherwise, q 1 +2≤j≤q。
Example 8
The invention specifically provides the structure of K under MBC-a as follows
K＝K TT +K TE +K ET +K EE +K D
The structure is as follows: subscript E denotes rank-4 matrix, K, of the image blur matrices under MBC-b boundary conditions D Another correction moment proposed by the invention is represented by a block structure of
Wherein a block matrix S of rank-4 t1,i ,t 1 ∈{1,2},1≤i≤p 1 The structure of (2) is->t 1 ∈{1,2},1≤i≤p 1 The calculation expression of each element in the structure is +.>
Example 9
The invention specifically provides the structure of K under RPBC as follows
K＝K TT +K TS +K ST +K SS
The structure is as follows: the subscript S represents the rank-2 structure determined by the invention, wherein the S type matrix isHere->t is {1, q }, and if t=1, 1.ltoreq.j.ltoreq.q1, otherwise q1+2.ltoreq.j.ltoreq.q; s-type block matrix is +.>Here->r.epsilon. {1, p }, i ranging from 1.ltoreq.i.ltoreq.p1 if r=1, else p 1 +2≤i≤p，X i1 From h i1,: Which itself may be an S-type matrix or a Toeplitz matrix.
Embodiment 10
The invention verifies the correctness of the B and K structures under the ARBC-a, MBC-a and RPBC, and also gives experimental results under PBC, RFBC, ARBC-and MBC-B with known structures for comparison. Experiments were performed on an Intel (R) Core (TM) i5 (3.2 GHz) machine using Matlab7.0. In the experiment: the image F adopts Matlab image tissue. Png shown in figure 2, the size is 506 multiplied by 800, and F is obtained by row-first rearrangement; 5 h are used, where Psf 1 And Psf 2 Psf of Gaussian type, sigma of 0.5, and sizes of 7×7 and 31×31, respectively, for simulating the symmetric shape of PSF and regular value of element, such as Psf shown in FIG. 3 3 To Psf 5 The normalized random matrix is used for simulating the situation that the PSF is asymmetric in shape and the element value is arbitrary in image recovery.
B correctness by mean square errorIs measured by->Is to expand the resulting image according to boundary conditions and corner types +.>Bf is the product of B and f identified by the invention, norm (x) and length (x) calculate the 2-norm of vector x and vector length x, respectively, and sqrt (x) returns the evolution of scalar x.
The correctness of K adopts the mean square errorIs measured by->Representing a true blurred image, kf represents a blur image based on kf=sum (+/-K) XY ) f, obtaining a blurred image.
In the course of the experiments described herein,is similar to Toeplitz matrix, sum (+ -K) XY ) K in (B) TT The inside and the between of the blocks are Toeplitz matrixes, the non-zero elements of the Toeplitz matrixes are relatively dense and difficult to store, and the Toeplitz matrixes correspond to each other in experiments>And K TT f is calculated using a two-dimensional convolution function of Matlab, i.e. +.>And conv2 (F, h, 'same'). Bf and sum (+ -K) XY f) Other K in (a) XY f, due to corresponding B and K XY Sufficiently sparse, so these matrices are constructed directly in the experiment, and Bf and each K are calculated XY f。
Experimental results found that under the various boundary conditions and corner types of the experiment: the RMSEs of Bf is 0, which indicates that the structures of all the matrixes B including the three matrixes B provided by the invention are correct; the mean square error of K is shown in FIG. 4, where the values are 10 of the true error 14 From this, it can be seen that the RMSEs error of Kf under each boundary condition is small, which indicates that the structure of K obtained by the present invention is correct.
Claims (3)
1. An identification method for image fuzzy matrix structure under boundary conditions includes such steps as under the boundary conditions as ARBC, MBC and RPBC, and point spread functionAnd noise->Image of the cause->In order to give the corresponding image a blur matrix involved in restoration of the image in the linear degradation model g=kf+ηAnd image vector->Provides a calculation basis based on the fuzzy action of K to f, and can be decomposed into a boundary condition matrix under the corresponding boundary condition +.>Spreading function of (a) and spreading fuzzy matrix under ZBC conditionIn succession of ambiguous actions, i.e. there is +.>The structure of the fuzzy matrix K is identified by the mechanism according to boundary conditions and corner types by adopting the following steps:
1) Analyzing the structure of a boundary condition matrix in a one-dimensional or two-dimensional degradation process;
2) And analyzing the product structure of the boundary condition matrix and the extended fuzzy matrix in the one-dimensional or two-dimensional degradation process to obtain a fuzzy matrix structure.
2. The method for identifying an image blur matrix structure according to claim 1, wherein the method for analyzing the boundary condition matrix structure in the one-dimensional or two-dimensional degradation process according to the boundary condition and the corner type is as follows:
when the type of the corner is a type, namely, when the image is expanded, the pixels outside the four corners are directly obtained by expanding the pixels in the antisymmetric direction in the corner, the structure of the boundary condition matrix B of the two-dimensional degradation process in the image recovery is not the boundary condition matrix in the one-dimensional caseIn the block structure of (a), the B in the two-dimensional degradation process is directly analyzedA structure; when the corner type is B type, i.e. the image is expanded, the pixels outside the four corners are obtained by expanding the pixels in the corresponding frame according to the row direction and then the column direction, and at this time, the structure of the boundary condition matrix B corresponding to the two-dimensional degradation process is the boundary condition matrix B under the one-dimensional degradation condition 1 Such that only analysis B is required 1 The structure of (2) is just needed;
in the one-dimensional degradation process, the point spread function is recorded asColumn vector to be restored is +.>f 1 The corresponding extension vector is +.>In the two-dimensional degradation process, the expanded image of the image F to be restored is recorded as +.>h 1 And h can be any size, without losing generality, p can be set in application 1 And q 1 All are less than or equal to 5; then B is 1 And B specifically the following steps:
1) When the corner type is a, the structural analysis method of B is as follows:
the extended image F is constructed according to the definition of the corner condition e The method comprises the following steps: f is filled with F e In the middle part of (F) e (p 1 +1:m+p 1 ,q 1 +1:n+q 1 ) =f; then according to the boundary condition and the corner type, calculating F in the order from the boundary and the corner from near to far e Pixels lying outside the F-profile range, the resulting pixels being represented linearly by corresponding elements within the F-profile, i.e. havingWherein the constant coefficient a i,j Derived from the boundary condition and the corner type, the value range of i, j is determined by the boundary condition and the corner type and is equal to i 1 ,j 1 Correlation; rearrangement by row first F e And F is the loss->And f, f e And f has the relation of
Then f and f e The relation between them is organized as f e =bf, i.e. the boundary condition matrix B is constructed by: analysis of f one by one e And f to obtain non-zero elements of the boundary condition matrix B
According to the above method, among the two existing boundary conditions that can select the type of a corner:
under ARBC, when the corner type is a, the available boundary condition matrix B is inclusiveEach block is divided into blocks, and the size of each block is +.>The specific structure of the block matrix of (a) is that
Wherein I is 12 ＝I 1 +I 2 ，I 13 ＝I 1 +I 3 ，I 123 ＝I 1 -I 2 +2I 3 And (2) and
and
under MBC, when the corner type is a, the available boundary condition matrix B is containingEach block is divided into blocks, and the size of each block is +.>The specific structure of the block matrix of (a) is that
Here, the respective sizes are +.>Is respectively of the full order matrix of
Andwherein i is more than or equal to 1 and p is more than or equal to p 1 ,1≤j≤q 1 ；s 1,1,i,j ＝(abs(d 0 )+1)(d 1 +1)，s 1,2,i,j ＝-d 1 abs(d 0 )+v 2 d 0 ,s 2,1,i,j ＝-d 1 abs(d 0 )-v 1 d 0 ,s 2,2,i,j ＝d 1 (abs(d 0 )-1),d 0 ＝p 1 -q 1 -i+j, and d 1 ＝(p 1 -i+1)v 2 +(q 1 -j+1)v 1 The method comprises the steps of carrying out a first treatment on the surface of the Coefficient v 1 And v 2 The setting method of (1) is if d 0 > =0, then v 1 ＝1,v 2 =0, otherwise v 1 ＝0,v 2 =1, the function abs (x) represents the return x absolute value；
2) When the corner type is B, the analysis method of B is as follows:
first constructing f according to definition of boundary condition e1 The method comprises the following steps: first using f 1 Filling f e1 Of (f) e1 (q 1 +1:n+q 1 ，1)＝f 1 Then, according to the boundary conditions, f is calculated in the order from the near to the far from the boundary e1 Is beyond f 1 Upper and lower boundary portions of the range, i.e. f ue1 ＝f e1 (1:q 1 1) and f de1 ＝f e1 (n+q 1 +1：n+2q 1 Each element in 1), the obtained element is f 1 Linear representation of internal elements, e.g. shapeAnd f de1 (i,1)＝∑a j f j ，n+q 1 +1≤i≤n+2q 1 Wherein f j Is f 1 The value range of the pixel close to the boundary in the pixel is determined by the boundary condition and the value of i, and the constant coefficient a j Derived from boundary conditions;
then according to f 1 And an expansion vector f e1 Relationship f between e1 ＝B 1 f 1 Constructing an expansion matrix B 1 The specific method is as follows: b (B) 1 Middle row B of (2) 1 (q 1 +1:n+q 1 N is a unit array of n multiplied by n; at B 1 Q is respectively above and below 1 In the row, the value of the non-zero element in the i-th row can be uniquely represented by f ue1 Or f de1 Linear representation push-out of the ith pixel in (B) 1 (i,j)＝a j J is more than or equal to 1 and less than or equal to n; the remaining elements are 0 values;
in the boundary condition with the selectable corner type B, only the matrix structure of the boundary condition under RPBC is unknown, and the method can obtain B in the one-dimensional degradation process 1 Is of the structure B 1 ＝I 13 Correspondingly, B is B 1 Block matrix of (a), i.e
3. The image blur matrix structure recognition method according to claim 2, wherein the method of analyzing the block structure of the blur matrix K is:
when the corner type is a, the image blur matrix K in the two-dimensional degradation process is not in the one-dimensional degradation processDirectly analyzing the structure of K at the moment; when the corner type is b, the fuzzy matrix K in the two-dimensional degradation process is the fuzzy matrix K in the one-dimensional degradation process 1 Is only needed to analyze the block matrix of K at this time 1 The structure of (2) is just needed; k (K) 1 And the structural analysis specific method of K is as follows:
1) The structural analysis method of K when the corner type is a is as follows:
firstly, constructing an extended image blurring matrix Is a non-square sparse partitioned matrix, and has a Toeplitz-like structure between blocks and in the blocks, and the 2p of the ith row block is formed 1 +1 consecutive non-zero matrices are made up of elements of h, i.e. ofWherein->The construction method of (2) is that Then is the first of hi rows h (i:) constructed Toeplitz-like matrix;
then according toCalculating a linear expression of the non-zero element in K, shaped as +.>Wherein i is 1 to or less 1 ≤mn，1≤j 1 ≤mn，1≤i≤2p 1 +1，1≤j≤2q 1 Actual range of values for +1, i and j and i 1 And j 1 Correlation;
the structure of the fuzzy matrix K is extracted again, and the method is as follows: according to the common blocking matrix in the image processing, elements of the known blocking matrix are detected from the expression of each element of K, and then the rest is further decomposed into block structures to be utilized, wherein the known matrix structures comprise possible inter-block and intra-block structures formed by Circulant, teoplitz, hankel, rank-2 correction, rank-4 correction matrix and the like, so that K is decomposedWherein the subscript x, y denotes a matrix structure, K x,y Representing a block matrix with an inter-block structure x and an intra-block structure y;
according to the above method, among two boundary conditions that can select the corner type a:
under ARBC, when the corner type is a, the structure of the blur matrix K is
K＝K TT -K TH +K TR -K HT -K HH +K HL +K RT +K LH -K LL +K U
The structure is as follows: subscripts T, H, R respectively represent Toeplitz matrix, hankel matrix and ARBC-b boundary condition image blur matrix rank-2 correction matrix, L and U respectively represent the other two rank-2 correction matrices provided by the invention, and their structures are:
note p=2p 1 +1,q＝2q 1 +1, L matrixNon-zero elements in (a) are represented by row i h of h i,: The matrix structure is +.>The L-shaped block matrix has the structure of +.>Wherein matrix->Is Hankel or L-type matrix, X i The non-zero elements of h i,: Is given; u-type blocking matrix K U The structure of (1) is that
Wherein (1)>The range of correspondence i is: if r=1, 1.ltoreq.i.ltoreq.p 1 Otherwise p 1 +2.ltoreq.i.ltoreq.p; the range of j is that if t=1, 1.ltoreq.j.ltoreq.q 1 Otherwise, q 1 +2≤j≤q；
Under MBC, when the corner type is a, the structure of the obtained fuzzy matrix K is that
K＝K TT +K TE +K ET +K EE +K D
The structure is as follows: subscript E denotes rank-4 matrix, K, of the image blur matrices under MBC-b boundary conditions D Another correction moment proposed by the invention is represented by a block structure of
Wherein a block matrix S of rank-4 t1,i ,t 1 ∈{1,2},1≤i≤p 1 The structure of (1) is that/>The calculation expression of each element in the structure is +.>
2) The structural analysis method of K when the corner type is b is as follows:
first construct K 1 The method comprises the following steps: first, an extended fuzzy matrix in a one-dimensional degradation process is constructed Non-square and sparse, toelpliz-like matrix, i.e. non-zero element on ith row of the matrix is +.>Then according to->Calculation of K 1 Linear expression of non-zero elements, shaped as +.>Wherein i is 1 to or less 1 ≤n，j 1 Value range and boundary condition of (a) and i 1 Is related to the value of 1.ltoreq.j.ltoreq.2q 1 Value range of +1, j and i 1 And j 1 Is related to the value of (a);
the structure of the fuzzy matrix K is analyzed by the following steps: first, according to the structure of a common matrix in image processing, K is used as a reference 1 In the expressions of each element, matrix elements with known structures are detected, and the rest of the expressions of each element are further dividedTo solve the matrix of the available structure, the known matrix structure includes Circulant, teoplitz, hankel, correction matrix, etc., to thereby divide K 1 Represented asWherein the subscript x represents a certain matrix type; the structure of the blur matrix K is then determined as K 1 The sum of the possible block structures of the respective decomposition matrices, shaped as +.>
Under the boundary condition that the corner type b can be selected, only the structure of the fuzzy matrix under the RPBC is unknown, and according to the method, the structure of the fuzzy matrix under the boundary condition can be obtained as
K＝K TT +K TS +K ST +K SS
The structure is as follows: the subscript S represents the rank-2 structure determined by the invention, wherein the S type matrix isHere->And if t=1, 1.ltoreq.j.ltoreq.q 1 Otherwise q 1 +2.ltoreq.j.ltoreq.q; s-type block matrix is +.>Here->The range of i is that if r=1, 1.ltoreq.i.ltoreq.p 1 Otherwise p 1 +2≤i≤p，X i1 From h i1,: Which itself is an S-type matrix or Toeplitz matrix.
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