CN113822822A - Identification method of image fuzzy matrix structure - Google Patents

Identification method of image fuzzy matrix structure Download PDF

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CN113822822A
CN113822822A CN202111310670.XA CN202111310670A CN113822822A CN 113822822 A CN113822822 A CN 113822822A CN 202111310670 A CN202111310670 A CN 202111310670A CN 113822822 A CN113822822 A CN 113822822A
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袁小华
王令群
丁悦
王振华
郑宗生
陈明
张天蛟
潘海燕
马振玲
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Shanghai Ocean University
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Abstract

The patent discloses a method for identifying an image fuzzy matrix structure under certain 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 blurring matrix and the boundary matrix to obtain the structure of the image blurring matrix. The identification method 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 a basis for the calculation realization of image restoration.

Description

Identification method of image fuzzy matrix structure
Technical Field
The invention relates to image processing, in particular to a method for identifying a fuzzy matrix structure of an image in the recovery of linear degraded images under different boundary and corner conditions.
Background
On the basis of a linear degradation model g-Kf + eta pair image
Figure BDA0003341188030000011
In the image restoration, a matrix is calculated based on certain Boundary Conditions (BCs) and corner types
Figure BDA0003341188030000012
Line-first rearrangement vector with F
Figure BDA0003341188030000013
Kf, and the product of the transpose of K 'and f, K' f, where K is the function of implementing the point spread
Figure BDA0003341188030000014
H is PSF for short, and f is the image
Figure BDA0003341188030000015
Is represented by a line-prioritized vector. Generally, since K and K 'are severely sparse ultra-large matrices, the products Kf and K' f either 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 fuzzy matrix diagonalization, and specifically which calculation method can be adoptedThis is related to the structure of K, which is determined by the symmetry of the PSF, together with the boundary condition type and the corner type on which the image restoration is based, and thus, in image restoration based on linear degradation of a specific boundary condition and corner type, it is a prerequisite for the corresponding image restoration calculation to judge the matrix structure of the image blur matrix K.
The present identification method of image fuzzy matrix structure is to simply deduce the block structure of image fuzzy matrix in two-dimensional degradation process from the fuzzy matrix structure in one-dimensional degradation process, the method is: assuming a one-dimensional degradation process g1＝K1f1+η1Fuzzy matrix K in (1)1Is the sum of a plurality of structural matrices, wherein the vector g1、f1And η 1 has a size of n × 1,
Figure BDA0003341188030000016
the structure of the fuzzy matrix K in the two-dimensional degradation process is K1Each structure in (1) is taken as the sum of all possible combinations of intra-block and inter-block structures, respectively, without loss of generality, if K1＝Kx+KyIf K is equal to Kxx+Kxy+KyyIn which K isxyAnd a block matrix with an inter-block structure of x and an intra-block structure of y is represented. This simple method of pushing from one dimension to two dimensions only applies if the corner type is chosen as b in the boundary conditions.
The current image boundary conditions include Zero BCs (ZBC), Periodic BCs (PBC) and Reflective BCs (RFBC), and 6 newly proposed Anti-Reflective BCs (ARBC), Mean BCs (MBC), Repeated BCs (RPBC) boundary conditions and the like, wherein only a corner type b exists under the first three boundary conditions and the RPBC, and under the ARBC and MBC conditions, a corner type a or b needs to be further selected to uniquely determine the structure of the fuzzy matrix. Of the 6 boundary conditions, the first three boundary conditions, and the K corresponding to the b-corner type of ARBC (ARBC-b) and the b-corner type of MBC (MBC-b) have been identified, and the method for deriving the one-dimensional degradation to the two-dimensional degradation process is adopted. The structures of the remaining K corresponding to the a corner type of the ARBC (ARBC-a), the a corner type of the MBC (MBC-a), and the RPBC are not identified, wherein the structures of the K corresponding to the former two are 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 problem to be solved by the invention is 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 really applied to image restoration.
Because of the convolution effect realized by the product Kf in the linear degradation process of the image, the convolution effect can be decomposed into an image vector f which is firstly expanded according to boundary and corner conditions, and then the convolution calculation is carried out on the expanded image vector according to ZBC, namely Kf can be expressed as Kf
Figure BDA0003341188030000017
Wherein
Figure BDA0003341188030000018
Is an extended fuzzy matrix under the ZBC,
Figure BDA0003341188030000021
the effect of the boundary condition matrix for a particular boundary condition and corner type is to expand the boundary pixels with definite fingers in f so that the image degradation model can be solved. Here, the
Figure BDA0003341188030000022
The structure of B is known as a Teoplitz-like partition matrix both inter and intra, and can be derived from the definition of boundary conditions and corner types.
Based on the above 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 identification of the fuzzy matrix K is decomposed into the structure identification of the extended matrix B, and then the product identification is carried out
Figure BDA0003341188030000023
The structure of (1); when the corner type is b, the structure of the boundary condition matrix in the one-dimensional degradation process is given first, and then the fuzzy moment is extendedDeducing the structure of an image fuzzy matrix in the two-dimensional degradation process according to the structure of the product of the matrix and the boundary condition matrix; when the corner type is a, the corner condition matrix structure is directly analyzed in a two-dimensional degradation process, and then a structure of a product of a corresponding extended fuzzy matrix and a 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 demonstrate the correctness of the identified structure through experimentation.
The identification method provided by the invention can be applied to image restoration of a linear degradation model and has application value.
Drawings
FIG. 1 shows a process of the present invention;
FIG. 2 shows images used in the experiments of the present invention;
FIG. 3 shows the point spread function used in the experiments of the present invention;
FIG. 4 shows the mean square error of the product of the blur 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 this summary are set forth in connection with the accompanying drawings. For clarity of presentation and ease of programming, some calculation formulas employ the function of Matlab.
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 on which to construct K;
2) giving a structure identification method of B according to the corner type;
3) and according to the corner type, giving a structure identification method of K.
Example 2
When the corner type is a, the corresponding image fuzzy matrix is not the block matrix corresponding to the fuzzy matrix in the one-dimensional degradation process, and the identification method of K provided by the invention comprises the following steps: analyzing the structure of the two-dimensional boundary condition matrix B, and then analyzing
Figure BDA0003341188030000031
Finally based on
Figure BDA0003341188030000032
Will be provided with
Figure BDA00033411880300000310
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 process1The invention provides a K identification method, which comprises the following steps: first, analyze the one-dimensional boundary condition matrix B1Based on
Figure BDA0003341188030000034
Analysis K1Structure of (1), finally K1The block structure of (2) is determined as the structure of K.
Example 4
The invention specifically provides that the lower boundary condition matrix B of the ARBC-a is B
Figure BDA0003341188030000035
Blocks, each block having a size of
Figure BDA0003341188030000036
The block matrix has a specific structure of
Figure BDA0003341188030000037
Wherein, I12＝I1+I2，I13＝I1+I3，I123＝I1-I2+2I3To do so
Figure BDA0003341188030000038
And
Figure BDA0003341188030000039
example 5
The invention provides a structure of a boundary condition matrix B under MBC-a, which specifically comprises
Figure BDA0003341188030000041
Here, each size is
Figure BDA0003341188030000042
Are respectively a full rank matrix of
Figure BDA0003341188030000043
And
Figure BDA0003341188030000044
wherein i is more than or equal to 1 and less than or equal to p1,1≤j≤q1；s1,1,i,j＝(abs(d0)+1)(d1+1)，s1,2,i,j＝-d1abs(d0)+v2 d0,s2,1,i,j＝-d1abs(d0)-v1 d0,s2,2,i,j＝d1(abs(d0)-1),d0＝p1-q1-i + j, and d1＝(p1-i+1)v2+(q1-j+1)v1(ii) a Coefficient v1And v2Is set if d00, then v1＝1,v20, otherwise v1＝0,v2The function abs (x) represents the return of the absolute value of x, 1.
Example 6
The invention provides B in the one-dimensional degradation process under the RPBC condition1Has the structure of B1＝I13Correspondingly, B is B1A block matrix of, i.e.
Figure BDA0003341188030000051
Example 7
The invention specifically provides a structure of K under ARBC-a
K＝KTT-KTH+KTR-KHT-KHH+KHL+KRT+KLH-KLL+KU
In the structure: subscripts T, H and 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 matrices are as follows:
let p be 2p1+1,q＝2q1+1, L matrix
Figure BDA0003341188030000052
The non-zero element in 1 ≤ i ≤ p is selected from ith row h of hi,:Given, the matrix structure is
Figure BDA0003341188030000053
The structure of the L-shaped block matrix is
Figure BDA0003341188030000054
I is more than or equal to 1 and less than or equal to p, wherein the matrix
Figure BDA0003341188030000055
Can be Hankel or L type matrix, XiIs composed of hi,:Giving out;
block matrix K of U typeUIs structured as
Figure BDA0003341188030000056
Wherein the content of the first and second substances,
Figure BDA0003341188030000057
r belongs to {1, p }, t belongs to {1, q }, and the range corresponding to i is: if it is notr is 1, i is not less than 1 and not more than p1Else p1I is more than or equal to +2 and less than or equal to p; the range for j is 1 ≦ j ≦ q1 if t ≦ 1, otherwise q1+2≤j≤q。
Example 8
The invention specifically provides a structure of K under MBC-a
K＝KTT+KTE+KET+KEE+KD
In the structure: the subscript E denotes the rank-4 matrix, K, in the image blur matrix under MBC-b boundary conditionsDShows another correction moment proposed by the invention, and the block structure is
Figure BDA0003341188030000061
Block matrix S of rank-4 thereint1,i,t1∈{1,2},1≤i≤p1Is structured as
Figure BDA0003341188030000062
t1∈{1,2},1≤i≤p1The calculation expression of each element in the structure is
Figure BDA0003341188030000063
Figure BDA0003341188030000064
Example 9
The invention specifically provides a structure of K under RPBC as
K＝KTT+KTS+KST+KSS
In the structure: the subscript S denotes the rank-2 structure determined by the present invention, wherein the matrix of S type is
Figure BDA0003341188030000065
Here, the
Figure BDA0003341188030000066
t belongs to {1, q }, and if t is 1, j is more than or equal to 1 and less than or equal to q1, otherwise, q1+2 is more than or equal to j and less than or equal to q; the S-type block matrix is
Figure BDA0003341188030000067
Here, the
Figure BDA0003341188030000068
r ∈ {1, p }, the range of i is 1 ≦ i ≦ p1 if r ≦ 1, otherwise p1+2≤i≤p，Xi1From hi1,:Is generated, which itself may be an S-type matrix or a Toeplitz matrix.
10 th embodiment
The invention checks the correctness of the B and K structures under the given ARBC-a, MBC-a and RPBC, and also gives the experimental results under the PBC, RFBC, ARBC-and MBC-B with known structures at present for comparison. The experiments were performed on an Intel (R) core (TM) i5(3.2GHz) machine using Matlab7.0. In the experiment: png, the size of the Matlab image tissue.png shown in fig. 2 is 506 × 800, and F is obtained by preferentially rearranging rows; 5 h are used, where Psf1And Psf2Is of Gaussian type Psf with sigma of 0.5 and sizes of 7 × 7 and 31 × 31 respectively, and is used for simulating the conditions of symmetrical PSF shape and regular element values, such as the Psf shown in FIG. 33To Psf5The three-dimensional PSF is 3 normalized random matrixes which are used for simulating the conditions that the shape of the PSF is asymmetric and the value of an element is random in image recovery.
B correctness adopts mean square error
Figure BDA0003341188030000071
Is measured, wherein
Figure BDA0003341188030000072
Is to expand the resulting image according to boundary conditions and corner types
Figure BDA0003341188030000073
B f is the product of B and f identified by the present invention, norm (x) and length (x) calculate the 2 norm and vector length x of vector x, respectively, sqrt (x) returns the square of scalar x.
The correctness of K adopts mean square error
Figure BDA0003341188030000074
Is measured, wherein
Figure BDA0003341188030000075
Representing a real blurred image, Kf representing based on Kf ═ sum (± K)XY) f the resulting blurred image.
In the course of the experiments, it was shown that,
Figure BDA0003341188030000076
both intra-and inter-block of (a) are Toeplitz-like matrices, sum (. + -. K)XY) K in (1)TTBoth within and between blocks are Toeplitz matrices, and the non-zero elements of the two matrices are relatively dense and difficult to store, and the corresponding ones in the experiment
Figure BDA0003341188030000077
And KTTf is calculated using the two-dimensional convolution function of Matlab, i.e.
Figure BDA0003341188030000078
And conv2(F, h, 'same'). Bf and sum (. + -. K)XYf) Other of (1) KXYf, due to corresponding B and KXYSufficiently sparse, so these matrices were constructed directly in the experiment and Bf and each K were calculatedXYf。
Experimental results found that, under the various boundary conditions and corner types tested: the RMSEs of Bf are all 0, which shows that all the matrixes B including the three matrixes B provided by the invention have correct structures; the mean square error of K is shown in FIG. 4, where the values are 10 of the true error14It can be seen that the error of RMSEs of Kf under each boundary condition is small, which indicates that the structure of the K obtained by the invention is correct.

Claims (3)

1. A method for identifying fuzzy matrix structure of image under boundary conditions includes using point spread function under boundary conditions of ARBC, MBC and RPBC respectively
Figure FDA0003341188020000011
Figure FDA0003341188020000012
And noise
Figure FDA0003341188020000013
Induced image
Figure FDA0003341188020000014
In order to apply the blurring matrix involved in the corresponding image restoration to the linear degradation model g ═ Kf + η
Figure FDA0003341188020000015
And image vector
Figure FDA0003341188020000016
The product of (a) provides a calculation basis, and can be decomposed into a boundary condition matrix under corresponding boundary conditions based on the fuzzy action of K on f
Figure FDA0003341188020000017
Extension of the fuzzy matrix under ZBC conditions
Figure FDA0003341188020000018
In succession to the fuzzy action of, i.e. presence of
Figure FDA0003341188020000019
According to the mechanism, the structure of the fuzzy matrix K is identified by adopting the following steps according to boundary conditions and corner types:
1) analyzing the structure of a boundary condition matrix in a one-dimensional or two-dimensional degradation process;
2) and analyzing the structure of the product 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 comprises:
when the type of the corner is a type, namely an extended image, and the pixels outside the four corners are directly obtained by pixel extension in the anti-symmetric direction in the corner, the structure of the boundary condition matrix B of the two-dimensional degradation process in image recovery is not the structure of the boundary condition matrix B in the one-dimensional case
Figure FDA00033411880200000110
The block structure of (a), the structure of B in the two-dimensional degradation process needs to be directly analyzed; 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 expanding the pixels according to the column direction, and at this time, the structure of the boundary condition matrix B corresponding to the two-dimensional degradation process is the structure of the boundary condition matrix B under the condition of one-dimensional degradation1So that only B need be analyzed1The structure of (1) is as follows.
In the one-dimensional degeneration process, the point spread function is
Figure FDA00033411880200000111
The column vector to be recovered is
Figure FDA00033411880200000112
f1The corresponding spreading vector is
Figure FDA00033411880200000113
In the two-dimensional degradation process, the extended image of the image F to be restored is recorded as
Figure FDA00033411880200000114
h1H can be of any size, and p can be set in application without loss of generality1And q is1Are all less than or equal to 5. Then B is1The method for structural analysis of B is as follows:
1) when the corner type is a, the method for structural analysis of B is as follows:
first according to the cornerDefinition of conditions construction extension image FeThe method comprises the following steps: filling F with FeMiddle part of (1), i.e. Fe(p1+1:m+p1,q1+1:n+q1) F; then according to the boundary condition and corner type, calculating F according to the sequence from the boundary and corner from near to fareWherein the pixels located outside the F-contour range are linearly represented by the corresponding elements in the F-contour, i.e. have
Figure FDA00033411880200000115
Wherein constant coefficient ai,jDeducing from boundary conditions and corner types, determining the value range of i, j from the boundary conditions and the corner types, and comparing with i1,j1And (4) correlating. Line-wise prioritization of FeAnd F is the loss
Figure FDA00033411880200000116
And f, feAnd f have a relationship of
Figure FDA00033411880200000117
Then f and f areeThe relationship between (A) and (B) is arranged as feConstructing a boundary condition matrix B as Bf, wherein the method comprises the following steps: analysis by one feAnd f to obtain non-zero elements of the boundary condition matrix B
Figure FDA00033411880200000118
According to the above method, among the two existing boundary conditions that enable the selection of the a-corner type:
under ARBC, when the corner type is a, the boundary condition matrix B is obtained as containing
Figure FDA0003341188020000028
Each block is divided into blocks with the size of each block
Figure FDA0003341188020000029
The block matrix has a specific structure of
Figure FDA0003341188020000021
Wherein, I12＝I1+I2，I13＝I1+I3，I123＝I1-I2+2I3And is and
Figure FDA0003341188020000022
and
Figure FDA0003341188020000023
under MBC, when the corner type is a, the boundary condition matrix B is obtained as containing
Figure FDA0003341188020000024
Each block is divided into blocks with the size of each block
Figure FDA0003341188020000025
The block matrix has a specific structure of
Figure FDA0003341188020000026
Here, each size is
Figure FDA0003341188020000027
Are respectively a full rank matrix of
Figure FDA0003341188020000031
And
Figure FDA0003341188020000032
wherein i is more than or equal to 1 and less than or equal to p1,1≤j≤q1；s1,1,i,j＝(abs(d0)+1)(d1+1)，s1,2,i,j＝-d1abs(d0)+v2d0,s2,1,i,j＝-d1abs(d0)-v1d0,s2,2,i,j＝d1(abs(d0)-1),d0＝p1-q1-i + j, and d1＝(p1-i+1)v2+(q1-j+1)v1(ii) a Coefficient v1And v2Is set if d00, then v1＝1,v20, otherwise v1＝0,v2Function abs (x) denotes return of the absolute value of x;
2) when the corner type is B, the analysis method of B is as follows:
firstly, according to the definition structure f of boundary conditione1The method comprises the following steps: first using f1Filling fe1Middle part of (i), i.e. fe1(q1+1:n+q1，1)＝f1Then f is calculated according to the boundary condition and according to the sequence from the near to the far away from the boundarye1Middle excess f1The upper and lower boundary portions of the range, i.e. fue1＝fe1(1:q11) and fde1＝fe1(n+q1+1：n+2q11) of the elements of (1), the resulting element being f1Linear representation of internal elements, e.g. in the form of
Figure FDA0003341188020000033
1≤i≤q1And f andde1(i,1)＝∑ajfj，n+q1+1≤i≤n+2q1wherein f isjIs f1Pixels close to the boundary, j is determined by the boundary condition and i, and constant coefficient ajDeducing from the boundary condition;
then according to f1And its extension vector fe1Relation f betweene1＝B1f1Constructing an extended matrix B1The specific method comprises the following steps: b is1Middle row B of1(q1+1:n+q1Is a unit array of nxn; in B1Each of (a) and (b) is q1The value of the non-zero element in the ith row can be uniquely defined by fue1Or fde1Linear representation of the ith pixel in (A) is derived, i.e. B1(i,j)＝aj,1≤j is less than or equal to n; the remaining elements are 0 values;
in the boundary condition capable of selecting the corner type B, only the boundary condition matrix structure under RPBC is unknown, and B in the one-dimensional degradation process can be obtained according to the method1Has the structure of B1＝I13Correspondingly, B is B1By a block matrix, i.e.
Figure FDA0003341188020000041
3. The method for identifying the structure of the image blur matrix according to claim 2, wherein the method for analyzing the block structure of the blur matrix K comprises:
when the corner type is a, the image fuzzy matrix K in the two-dimensional degradation process is not in the one-dimensional degradation process
Figure FDA0003341188020000042
The block matrix of (a), the structure of K needs to be directly analyzed at this time; 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 process1Block matrix of, in this case, only K has to be analyzed1The structure of (1) is as follows. K1The specific method for structural analysis 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 fuzzy matrix
Figure FDA0003341188020000043
Figure FDA0003341188020000044
Is a non-square and sparse block matrix, and is a Toeplitz-like structure with 2p on the ith row block1+1 successive non-zero matrices are made up of elements of h, i.e. having
Figure FDA0003341188020000045
Wherein
Figure FDA0003341188020000046
Is constructed by the method of
Figure FDA0003341188020000047
Figure FDA0003341188020000048
Then a Toeplitz-like matrix constructed from row i of h (i,: h);
then according to
Figure FDA0003341188020000049
Calculating linear expressions of non-zero elements in K, e.g. in the form of
Figure FDA00033411880200000410
Wherein 1 is less than or equal to i1≤mn，1≤j1≤mn，1≤i≤2p1+1，1≤j≤2q1The actual value ranges of +1, i and j are related to i1 and j 1;
and then extracting the structure of the fuzzy matrix K, wherein the method comprises the following steps: according to the common block matrix in image processing, elements of the known block matrix are detected from the expression of each element of K, and then the residual content is further decomposed into block structures to be utilized, wherein the known matrix structure comprises possible inter-block and intra-block structures formed by circulation, Teoplitz, Hankel, rank-2 correction matrix, rank-4 correction matrix and the like, so that K is decomposed into
Figure FDA00033411880200000411
Wherein the indices x, y denote a certain matrix structure, Kx,yA block matrix which represents that the inter-block structure is not x and the intra-block structure is y;
according to the above method, in two boundary conditions where the corner type a can be selected:
under ARBC, when the corner type is a, the structure of the obtained fuzzy matrix K is
K＝KTT-KTH+KTR-KHT-KHH+KHL+KRT+KLH-KLL+KU
In the structure: subscripts T, H and R respectively represent a Toeplitz matrix, a Hankel matrix and a rank-2 correction matrix in an image fuzzy matrix under an ARBC-b boundary condition, L and U respectively represent other two rank-2 correction matrices provided by the invention, and the structures of the matrices are as follows:
let p be 2p1+1,q＝2q1+1, L matrix
Figure FDA00033411880200000412
The non-zero element in 1 ≤ i ≤ p is selected from ith row h of hi,:Given, the matrix structure is
Figure FDA0003341188020000051
The structure of the L-shaped block matrix is
Figure FDA0003341188020000052
Wherein the matrix
Figure FDA0003341188020000053
Can be Hankel or L type matrix, XiIs composed of hi,:Giving out; block matrix K of U typeUIs structured as
Figure FDA0003341188020000054
The range corresponding to i is: if r is 1, then 1 ≦ i ≦ p1Else p1I is more than or equal to +2 and less than or equal to p; the range for j is 1 ≦ j ≦ q if t is 11Otherwise, q1+2≤j≤q；
Under MBC, when the corner type is a, the structure of the fuzzy matrix K can be obtained as
K＝KTT+KTE+KET+KEE+KD
In the structure: the subscript E denotes the rank-4 matrix, K, in the image blur matrix under MBC-b boundary conditionsDShows another correction moment proposed by the invention, and the block structure is
Figure FDA0003341188020000055
Block matrix S of rank-4 thereint1,i,t1∈{1,2},1≤i≤p1Is structured as
Figure FDA0003341188020000056
The computational expression of each element in the structure is
Figure FDA0003341188020000057
Figure FDA0003341188020000058
2) The structural analysis method of K when the corner type is b is as follows:
first structure K1The method comprises the following steps: firstly, constructing an extended fuzzy matrix in the one-dimensional degradation process
Figure FDA0003341188020000061
Figure FDA0003341188020000062
Non-square and sparse, Toelpliz-like matrix, i.e., with non-zero elements on row i
Figure FDA0003341188020000063
Then according to
Figure FDA0003341188020000064
Calculating K1Linear expression of non-zero elements, e.g. in form of
Figure FDA0003341188020000065
Wherein 1 is less than or equal to i1≤n，j1Value range and boundary condition of (1) and (i)1Is related to the value of j is more than or equal to 1 and less than or equal to 2q1+1, j value range and i1And j1Is related to the value of;
and analyzing the structure of the fuzzy matrix K, wherein the method comprises the following steps: first according to the imageThe structure of the matrix is common in the process, from K1In the expression of each element, a matrix element with a known structure is detected, and then the rest part in the expression of each element is further decomposed into a matrix with an available structure, wherein the known matrix structure comprises circulation, Teoplitz, Hanke, a correction matrix and the like, so that K is obtained1Is shown as
Figure FDA0003341188020000066
Where subscript x represents a certain matrix type; then determining the structure of the fuzzy matrix K as K1The sum of the possible block structures of the respective decomposition matrices, in the form of
Figure FDA0003341188020000067
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 fuzzy matrix under the boundary condition is obtained as
K＝KTT+KTS+KST+KSS
In the structure: the subscript S denotes the rank-2 structure determined by the present invention, wherein the matrix of S type is
Figure FDA0003341188020000068
Here, the
Figure FDA0003341188020000069
And if t is 1, then 1 ≦ j ≦ q1Otherwise q1J is more than or equal to +2 and less than or equal to q; the S-type block matrix is
Figure FDA00033411880200000610
Here, the
Figure FDA00033411880200000611
i is in the range of 1 ≦ i ≦ p if r ≦ 11Else p1+2≤i≤p，Xi1From hi1,:Is generated, which itself may be an S-type matrix or a Toeplitz matrix.
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