CN103780267A - Measurement matrix design method based on LDPC matrix - Google Patents
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Abstract
The invention discloses a measurement matrix design method based on an LDPC matrix. The method comprises the following steps: the step one in which the number L of continuous '1' at the beginning of the first row of the matrix is determined at first, and positions of '1' in the following row of the matrix are subjected to translation to the right for L positions successively according to positions of '1' in the previous row so as to ensure that the positions of '1' between every two rows or every two columns in a cycle submatrix are different to construct the cycle submatrix; the step two in which required rows or columns are randomly selected from the cycle submatrix to construct a measurement matrix based on the LDPC matrix; and the step three in which four-side loops are found in the whole matrix through searches of a limited number, and the four-side loops are eliminated to construct the measurement matrix based on the LDPC matrix. The measurement matrix inherits advantages, such as good sparsity, small column correlation values, etc., of the LDPC matrix, and at the same time, the shortage that optimal d values (the average number of '1' in each column) need to be determined for different dimensions of measurement matrixs in advance when the LDPC matrix is used as the compression sensing measurement matrix can be overcome. The matrix has advantages of good sparsity, simple structure, good orthogonality, easy hardware implementation and good reconstruction effect.
Description
Technical field
The present invention relates to image compression cognition technology field, particularly relate to the method for designing of measuring matrix in compressed sensing field.
Background technology
In recent years, compressive sensing theory (the Compressive Sensing being proposed by the people such as Candes and Donoho, CS) process theoretical support is provided for New Image collection and compression, first utilize random observation matrix, sparse on certain orthogonal basis or tight frame or compressible high dimensional signal are projected on lower dimensional space, then by solving-optimizing problem from a small amount of projection with high probability reconstruct primary signal or image.The core concept of CS theory is that compression is merged and carried out with sampling, and it has broken through the bottleneck of Shannon's sampling theorem, only need to be by just reconstituting initial image accurately of a small amount of sample point.
The design of measuring matrix is vital link in compressed sensing design process, it must guarantee that signal is in the case of being far smaller than primary signal through observation and the dimension of measuring matrix, can also comprise all information of primary signal, rear end can be recovered signal accurately.For the design of measuring matrix, constraint isometry (Restricted Isometry Property, RIP) condition becomes matrix as measuring an adequate condition that also has definite recovery solution after matrix, even designed measurement matrix meets RIP condition, utilizes so the primary signal that recovers that the data that collect just can very large probability.Be a NP-Hard problem but utilize RIP to judge matrix or design, be difficult to directly use it to the quality of prediction matrix.For reducing the complexity of matrix design, researcher has proposed the RIP condition of equivalence, even provable go out related measurement matrix irrelevant with sparse base, this measurement matrix there is a strong possibility sexual satisfaction RIP character.
Consider hard-wired complexity, conventional measurement matrix can be divided into two classes.The first kind is non-zero-1 matrix containing different real number elements in matrix.Comprise random Gaussian matrix, random Bernoulli Jacob's matrix and part orthogonal matrix etc., these matrixes are proved to be and meet RIP or RIP criterion of equal value.But these matrix ubiquity complex structures and random are difficult to adopt hardware to be realized, and to difficult problems such as memory capacity have higher requirements; Equations of The Second Kind matrix is that element is 0 or 1 0-1 matrix, common are the check matrix (being called for short LDPC matrix) of sparse random measurement matrix and LDPC code etc., these matrixes generally have higher sparse property, and be easy to hardware and realize, be particularly useful for compressing the design with printing opacity or the light tight encoding mask representing in imaging system.
Wherein, LDPC matrix is a kind of check matrix of linear block codes, it not only has sparse 0-1 structure, and there is very little row coherence, for the LDPC matrix of different sizes, by selecting optimum d value (number of average each row 1 in matrix) can make the reconstruct effect optimum of signal.And in the construction process of LDPC matrix, generally need to avoid the existence of " four side rings " in matrix as far as possible.So-called " four side rings " is embodied in matrix is the quadrangle being made up of four elements " 1 ".The square array that four " 1* " forms is as shown in Equation (6) exactly one " four side rings ".The position of existence explanation two row " 1 " of " four side rings " has at least two to overlap, and this will inevitably increase the correlation between two row.In theory, the ring of formation is larger, and the node of process is more, just represents that the correlation of variable node used and other variable node is less, is reflected in check matrix, just represents that the correlation between each row is less.Correlation is less, is embodied in matrix and just represents that it has better orthogonality.Good orthogonality and the RIP character of matrix are closely related, and in compressed sensing, a well behaved measurement matrix often has good orthogonality.This is also the very important reason that LDPC matrix is applicable to compressed sensing.
But in the existing measurement matrix design method based on LDPC code, for the different measurement matrix of dimension, all need to determine in advance the d value that can make its best performance before each structural matrix, design process is loaded down with trivial details, is unfavorable for practical application.And LDPC matrix requires the number of each row " 1 " at least will have 3, and its sparse property need further raising.
Summary of the invention
In order to overcome the problem of prior art existence, the present invention proposes a kind of measurement matrix of half random mode that partly circulates based on LDPC matrix.
The present invention is achieved by following technical solution.
A kind of measurement matrix design method based on LDPC matrix that the present invention proposes, the method comprises the following steps:
The structure of step 1, simple cycle submatrix.First determine the matrix the first row beginning number L of " 1 " continuously, matrix go subsequently the position at " 1 " place according to the position of lastrow " 1 " successively to a right translation L position, with in this submatrix that guarantees to circulate between every two row or row the position of " 1 " not identical, construct circulation submatrix;
Step 2, select required row or column at random from described circulation submatrix, the random model of selecting according to a random number generator herein, this model is defined by following formula (1):
Num=（CONS
1×Seed+CONS
2）mod(n) （1）
In formula (1), Num represent selected go out random row or the position of random column in circulation submatrix, line number or row number.And CONS
1and CONS
2represent two very large prime number constants, span is 2
10～2
16, Seed represents a variable constantly changing with system time, its span is 1～2
16, n represents the dimension size of circulation submatrix entirety, the i.e. numerical value of line number or columns.Mod(n) represent mould n, can obtain a numerical value between 1～n; Through make numerical value that random generator obtains be no more than the dimension size of circulation submatrix to n delivery, make got random row or be listed in circulation submatrix to produce;
The elimination of step 3, four side rings.The whole matrix of search by limited number of times is found four side rings, and by four side ring cancellations, the measurement matrix of structure based on LDPC matrix;
The measurement matrix based on LDPC matrix being obtained by above step, according to varying in size of needed compression sampling rate R in compressed sensing process, is generally even number for a M × 1(M) one-dimensional signal x, measure the building method of matrix and be divided into three kinds of situations:
(1), in the time of sample rate R<0.5, need the measurement matrix A of structure
1dimension be N × M, wherein N is the numerical value after M × R rounds.Structure A
1time, adopt and first construct circulation submatrix B
1, and then the required row of random selection form the method for supplementing Matrix C 1 from circulation submatrix, shown in (2).
A
1＝[B
1 C
1] （2）
Circulation submatrix B
1in every a line have L " 1 ", L is the maximum integer that is not more than 1/R.Structure B
1time, first stator matrix B
1the position of middle the first row " 1 " is front L, go subsequently the position of " 1 " successively to a right translation L position, forms thus the subcycle matrix B that the capable L of N shown in formula (3) × N is listed as
1.Supplement Matrix C
1generation method is: utilize the defined randomizer of formula (1) to produce M-L × N 1 random number arriving between L × N, select B according to the random number generating
1in accordingly row, as measure matrix A
1supplementary Matrix C
1.
(2) in the time of sample rate R=0.5, the measurement matrix A of required structure
2dimension be 0.5M × M.Circulation submatrix B
2in every a line have 2 " 1 ", i.e. L=2, fixing B
2the position of middle the first row " 1 " is the first two, and back row is successively to two positions of right translation, the circulation submatrix B constructing
2shown in (4), B
2dimension be 0.5M × M, equate with the dimension of required measurement matrix, the submatrix that now circulates is final measurement matrix, i.e. A
2=B
2.
(3), in the time of sample rate R>0.5, need the measurement matrix A of structure
3dimension be N × M, wherein N is the value after M × R rounds.Structure A
3time, adopt and first construct circulation submatrix B
3, and then the required row of random selection forms supplementary Matrix C from circulation submatrix
3method, shown in (5).
Circulation submatrix B
3dimension be 0.5M × M dimension, its generation method is identical during with R=0.5.Supplement Matrix C
3generation method is: first incite somebody to action submatrix B in proper order
3upset at random generator matrix T by row
3, then utilize the defined randomizer of formula (1) to produce the random number between N-0.5M individual 1 to 0.5M, select T according to the random number generating
3in accordingly go, as measure matrix A
3supplementary Matrix C
3.
Due to supplementary Matrix C
3the position of middle random row two " 1 " is random, measures matrix A
3likely there will be " four side rings ", in matrix A
3after construction complete, need to search and eliminate four side rings that may exist by the whole matrix of traversal of limited number of times, to optimize the orthogonal performance of matrix.
Compared with prior art, the present invention has following good effect:
This measurement matrix has been inherited the advantages such as the sparse property of LDPC matrix is good, row correlation is little, has overcome the deficiency that need to pre-determine optimum d value (quantity of each column average " 1 ") when LDPC matrix is measured matrix as compressed sensing for the measurement matrix of different dimensions simultaneously.The measurement matrix of the designed matrix of this patent to arbitrary size, has fixing constituted mode, does not need the d value of determining that in advance it is optimum, thereby on the basis of LDPC matrix, further improved its sparse property and realizability in when design.This matrix has that sparse property is high, simple in structure, orthogonality is strong and be easy to hard-wired feature, and reconstruct is respond well.
Accompanying drawing explanation
Fig. 1 is that Lena image reconstructed image PSNR value under different sample rates compares;
Fig. 2 has provided lena figure in the situation that sample rate is 0.3, utilize that half random matrix that partly circulates, LDPC matrix, random Bernoulli Jacob's matrix and the sparse random matrix of this patent design obtain reconstructed image;
Fig. 3 is algorithm flow chart.
Embodiment
Below in conjunction with drawings and Examples, further describe the specific embodiment of the present invention.
The specific implementation of technical scheme of the present invention, is described below:
1, the structure of simple cycle submatrix
Under normal circumstances, the dimension of compressed sensing measurement matrix is closely-related with the size of sample rate and the length of signal.For the measurement matrix that makes to construct is when the number that meets the every a line of LDPC matrix and each row 1 equates this feature, in matrix, not there is not " four side rings ", first need to construct a simple circulation submatrix, determine the position of its every a line " 1 ".This process can be by determining the first row beginning number L of " 1 " continuously in matrix, matrix go subsequently the position at " 1 " place according to the position of lastrow " 1 " successively to a right translation L position, with in this submatrix that guarantees to circulate between every two row or row the position of " 1 " not identical.Like this, a preliminary circulation submatrix is constructed out.
2, the generation of random row or column
In the middle of practical application, the dimension of our needed measurement matrix is uncertain, so the dimension of the designed circulation submatrix out of the first step often can not reach the dimension of required matrix, thus, just need by adding corresponding row or being listed as the structure of matrix.This step can select required row or column to complete by random from constructed circulation submatrix, and the realization of random process needs the model of a random number generator, and this model can be defined by following formula (1):
Num=（CONS
1×Seed+CONS
2）mod(n) （1）
In formula (1), Num represent selected go out random row or the position of random column in circulation submatrix, line number or row number.And CONS
1and CONS
2represent two very large prime number constants, span is 2
10～2
16, Seed represents a variable constantly changing with system time, its span is 1～2
16, n represents the dimension size of circulation submatrix entirety, the i.e. numerical value of line number or columns.Mod(n) represent mould n, can obtain a numerical value between 1～n; Through make numerical value that random generator obtains be no more than the dimension size of circulation submatrix to n delivery, make got random row or be listed in circulation submatrix to produce;
3, the elimination of four side rings.The matrix being constructed by said method, owing to there being the existence of random number " 1 ", likely there will be the situation that is similar to four side rings in LDPC matrix, in order to make it meet the characteristic of LDPC matrix, find four side rings by the whole matrix of search of limited number of times herein, and by its cancellation, reduce the impact of four side rings on matrix as far as possible.Obviously, only there is 1 or 2 " 1 " in this matrix, be better than original LDPC matrix in sparse property in each row.
For correctness and the validity of half random measurement matrix in image compression perception is processed that partly circulates of verifying that this patent is designed, adopt lena, cameraman, butterfly and plane tetra-width international standard gray scale test patterns to carry out emulation experiment, the resolution of image is 256 × 256 pixels.Utilizing measurement matrix image to be compressed after measurement, utilize TVAL3 algorithm to carry out Image Reconstruction.In order to embody the size of matrix dimension to the impact of reconstruction result, image is carried out to piecemeal processing, image is divided into 16 × 16 piece.Table 1, for utilizing this patent designed partly circulate half random measurement matrix and LDPC matrix, random Bernoulli Jacob's matrix and sparse random matrix respectively above-mentioned test pattern compress after measurement, utilizes the comparative result of the Y-PSNR (PSNR) of TVAL3 algorithm reconstructed image.
Table 1 simulation result comparison (dB)
In order more intuitively simulation result to be compared, Fig. 1 has provided Lena image, and under different sample rates, each measures the PSNR value curve chart of matrix Recovery image compared with original image.
As can be seen from Table 1, what this patent was designed partly circulate, and its reconstructed image of half random matrix PSNR value is obviously better than other three kinds of matrixes, PSNR value than LDPC matrix, random Bernoulli Jacob's matrix and sparse random matrix reconstructed image on average exceeds respectively 0.89dB, 1.23dB and 1.65dB.For Lena and the more image of Butterfly details, the matrix of this patent design has more advantage qualitatively at reconstructed image, compare with three kinds of matrixes of sparse random matrix with LDPC matrix, random Bernoulli Jacob's matrix, its PSNR value on average exceeds respectively 1.25dB, 1.51dB and 1.91dB.Can more significantly find out from the PSNR curve chart shown in Fig. 1, along with the increase of sample rate, the quality of reconstructed image is also increasing thereupon.And under different sample rates, when half random mode that partly circulates based on LDPC matrix of this patent structure is measured matrix for the compressed sensing of image, its reconstruct effect is all better than all the other several conventional measurement matrixes.
Fig. 2 has provided lena figure in the situation that sample rate is 0.3, utilize that half random matrix that partly circulates, LDPC matrix, random Bernoulli Jacob's matrix and the sparse random matrix of this patent design obtain reconstructed image.Wherein Fig. 2 (a) is original image, 2 (b), 2 (c), 2 (d) and 2 (e) are respectively the utilization reconstructed image that half random matrix, LDPC matrix, random Bernoulli Jacob's matrix and sparse random matrix obtain that partly circulates.Fig. 2 can find out, in the more region of the details such as hair and eyes, utilizes the image of the half random matrix reconstruct that partly circulates more clear, and more approaching with original image, subjective effect is better.
Visible, no matter half random mode that partly circulates based on LDPC matrix that this patent is constructed measures matrix aspect objective PSNR value, or aspect subjective reconstructed image, effect is all better than its excess-three kind and measures matrix.
The specific implementation of algorithm flow chart is as shown in Figure 3 expressed as follows:
R in flow chart represents needed compression sampling rate in compressed sensing process, and according to varying in size of R, the building method of matrix can be divided into three kinds of situations.
(1), in the time of sample rate R<0.5, need the measurement matrix A of structure
1dimension be N × M, wherein N is the numerical value after M × R rounds.Structure A
1time, adopt and first construct circulation submatrix B
1, and then the required row of random selection form supplementary Matrix C from circulation submatrix
1method, shown in (2).
A
1＝[B
1 C
1] （2）
Circulation submatrix B
1in every a line have L " 1 ", L is the maximum integer that is not more than 1/R.Structure B
1time, first stator matrix B
1the position of middle the first row " 1 " is front L, go subsequently the position of " 1 " successively to a right translation L position, forms thus the subcycle matrix B that the capable L of N shown in formula (3) × N is listed as
1.Supplement Matrix C
1generation method is: utilize the defined randomizer of formula (1) to produce M-L × N 1 random number arriving between L × N, select B according to the random number generating
1in accordingly row, as measure matrix A
1supplementary Matrix C
1.
(2) in the time of sample rate R=0.5, the measurement matrix A of required structure
2dimension be 0.5M × M.Circulation submatrix B
2in every a line have 2 " 1 ", i.e. L=2, fixing B
2the position of middle the first row " 1 " is the first two, and back row is successively to two positions of right translation, the circulation submatrix B constructing
2shown in (4), B
2dimension be 0.5M × M, equate with the dimension of required measurement matrix, the submatrix that now circulates is final measurement matrix, i.e. A
2=B
2.
(3), in the time of sample rate R>0.5, need the measurement matrix A of structure
3dimension be N × M, wherein N is the value after M × R rounds.Structure A
3time, adopt and first construct circulation submatrix B
3, and then the required row of random selection forms supplementary Matrix C from circulation submatrix
3method, shown in (5).
Circulation submatrix B
3dimension be 0.5M × M dimension, its generation method is identical during with R=0.5.Supplement Matrix C
3generation method is: first incite somebody to action submatrix B in proper order
3upset at random generator matrix T by row
3, then utilize the defined randomizer of formula (1) to produce the random number between N-0.5M individual 1 to 0.5M, select T according to the random number generating
3in accordingly go, as measure matrix A
3supplementary Matrix C
3.
Due to supplementary Matrix C
3the position of middle random row two " 1 " is random, measures matrix A
3likely there will be " four side rings ", in matrix A
3after construction complete, need to search and eliminate four side rings that may exist by the whole matrix of traversal of limited number of times, to optimize the orthogonal performance of matrix.
Claims (1)
1. the measurement matrix design method based on LDPC matrix, is characterized in that, the method comprises the following steps:
The structure of step 1, simple cycle submatrix, first determine the matrix the first row beginning number L of " 1 " continuously, matrix go subsequently the position at " 1 " place according to the position of lastrow " 1 " successively to a right translation L position, with in this submatrix that guarantees to circulate between every two row or row the position of " 1 " not identical, construct circulation submatrix;
Step 2, select required row or column at random from described circulation submatrix, the random model of selecting according to a random number generator herein, this model is defined by following formula (1):
Num=（CONS
1×Seed+CONS
2）mod(n) （1）
In formula (1), Num represent selected go out random row or the position of random column in circulation submatrix, line number or row number; And CONS
1and CONS
2represent two very large prime number constants, span is 2
10～2
16, Seed represents a variable constantly changing with system time, its span is 1～2
16, n represents the dimension size of circulation submatrix entirety, the i.e. numerical value of line number or columns; Mod(n) represent mould n, can obtain a numerical value between 1～n; Through make numerical value that random generator obtains be no more than the dimension size of circulation submatrix to n delivery, make got random row or be listed in circulation submatrix to produce;
The elimination of step 3, four side rings, finds four side rings by the whole matrix of search of limited number of times, and by four side ring cancellations, the measurement matrix of structure based on LDPC matrix;
The measurement matrix based on LDPC matrix being obtained by above step, according to varying in size of needed compression sampling rate R in compressed sensing process, is generally even number for a M × 1(M) one-dimensional signal x, measure the building method of matrix and be divided into three kinds of situations:
(1), in the time of sample rate R<0.5, need the measurement matrix A of structure
1dimension be N × M, wherein N is the numerical value after M × R rounds; Structure A
1time, adopt and first construct circulation submatrix B
1, and then the required row of random selection form supplementary Matrix C from circulation submatrix
1method, shown in (2):
A
1＝[B
1 C
1] （2）
Circulation submatrix B
1in every a line have L " 1 ", L is the maximum integer that is not more than 1/R; Structure B
1time, first stator matrix B
1the position of middle the first row " 1 " is front L, go subsequently the position of " 1 " successively to a right translation L position, forms thus the subcycle matrix B that the capable L of N shown in formula (3) × N is listed as
1; Supplement Matrix C
1generation method is: utilize the defined randomizer of formula (1) to produce M-L × N 1 random number arriving between L × N, select B according to the random number generating
1in accordingly row, as measure matrix A
1supplementary Matrix C
1.
(2) in the time of sample rate R=0.5, the measurement matrix A of required structure
2dimension be 0.5M × M; Circulation submatrix B
2in every a line have 2 " 1 ", i.e. L=2, fixing B
2the position of middle the first row " 1 " is the first two, and back row is successively to two positions of right translation, the circulation submatrix B constructing
2shown in (4), B
2dimension be 0.5M × M, equate with the dimension of required measurement matrix, the submatrix that now circulates is final measurement matrix, i.e. A
2=B
2;
(3), in the time of sample rate R>0.5, need the measurement matrix A of structure
3dimension be N × M, wherein N is the value after M × R rounds; Structure A
3time, adopt and first construct circulation submatrix B
3, and then the required row of random selection forms supplementary Matrix C from circulation submatrix
3method, shown in (5):
Circulation submatrix B
3dimension be 0.5M × M dimension, its generation method is identical during with R=0.5; Supplement Matrix C
3generation method is: first incite somebody to action submatrix B in proper order
3upset at random generator matrix T by row
3, then utilize the defined randomizer of formula (1) to produce the random number between N-0.5M individual 1 to 0.5M, select T according to the random number generating
3in accordingly go, as measure matrix A
3supplementary Matrix C
3;
Due to supplementary Matrix C
3the position of middle random row two " 1 " is random, measures matrix A
3while there is " four side rings ", in matrix A
3after construction complete, need to search and eliminate by the whole matrix of traversal of limited number of times four side rings of existence, to optimize the orthogonal performance of matrix.
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| CN105931282A (en) * | 2016-04-14 | 2016-09-07 | 中南大学 | Construction method and signal processing method of random-dimension partial Hadamard measuring matrix |
| CN105931282B (en) * | 2016-04-14 | 2018-09-28 | 中南大学 | A kind of arbitrary part Hadamard calculation matrix construction of dimension and signal processing method |
| CN106027070A (en) * | 2016-05-16 | 2016-10-12 | 南京理工大学 | Diagonalizable LDPC (Low Density Parity Check) observation matrix generating method based on compressed sensing |
| CN106027070B (en) * | 2016-05-16 | 2019-10-11 | 南京理工大学 | The generation method of compressed sensing based diagonalization LDPC observing matrix |
| CN106452456A (en) * | 2016-09-29 | 2017-02-22 | 天津大学 | Compressed sensing measurement matrix establishment method based on LDPC matrix |
| CN106936440A (en) * | 2017-02-20 | 2017-07-07 | 东南大学 | A kind of compressed sensing observing matrix generation method and device |
| CN111492586A (en) * | 2017-12-15 | 2020-08-04 | 华为技术有限公司 | L DPC code original pattern diagram expanding method |
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