CN111669183B - Compressed sensing sampling and reconstruction method, equipment and storage medium - Google Patents

Compressed sensing sampling and reconstruction method, equipment and storage medium Download PDF

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CN111669183B
CN111669183B CN202010615006.5A CN202010615006A CN111669183B CN 111669183 B CN111669183 B CN 111669183B CN 202010615006 A CN202010615006 A CN 202010615006A CN 111669183 B CN111669183 B CN 111669183B
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CN111669183A (en
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刘少强
张熠铭
陈泽宇
樊晓平
李勇周
吴天功
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Central South University
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Abstract

The invention discloses a compressed sensing sampling and reconstruction method, a device and a storage medium, wherein the method comprises the following steps: determining the measurement number M of X according to the dimension and sampling rate of the sparse signal X to be measured, constructing two block diagonal matrixes which are respectively arranged along a main diagonal and a secondary diagonal and have the same dimension of row, column and X based on a Hadamard matrix, and respectively taking the front M of the two block diagonal matrixes1Line and front M2Row, construct two corresponding measurement matrices, and let M1＝2n‑j≥M2And M1+M2M; observing X by the two measurement matrixes to respectively obtain corresponding measurement values Y1And the measured value Y2(ii) a Finally, i.e. according to Y1、Y2And performing signal reconstruction with the corresponding measurement matrix to obtain a reconstructed signal
Figure DDA0002563443590000011
And combining the two parts to obtain a reconstructed signal of the sparse signal to be detected. The invention can realize compressed sensing of a resource-limited system and improve the quality of a reconstructed signal and the recognition rate of a reconstructed image.

Description

Compressed sensing sampling and reconstruction method, equipment and storage medium
Technical Field
The invention belongs to a signal compression sampling and reconstruction method in the technical field of signal processing, and particularly relates to a compressed sensing sampling and reconstruction method, equipment and a storage medium based on a block Hadamard measurement matrix with main diagonal and secondary diagonal arrangement.
Background
Compressed Sensing (CS) is an efficient signal processing theory proposed in 2004 by the scholars of Donoho, canddes, and Tao. The theory fully excavates the inherent sparse or compressible characteristic of a natural signal, projects an original high-dimensional signal to a low-dimensional space by using a non-correlation measurement matrix under the condition of being far lower than the Nyquist sampling frequency to obtain a low-dimensional measurement value, and reconstructs the original signal without distortion by solving a nonlinear optimization problem. Different from the traditional signal processing method based on the Nyquist sampling theorem, the compressed sensing can be properly compressed while signal sampling, the compressed measurement value of the signal can be directly obtained, a new thought is provided for realizing high-resolution signal sampling, and the method has wide application prospects in various fields such as image processing, biological sensing, wireless communication, analog information conversion and the like.
The measurement matrix is one of the core problems of the compressed sensing theory. The essence of compression measurement is that dimension reduction processing on an original signal is realized by using a measuring matrix of M × N (M < N) dimensions, and the compression measurement is a key step for distinguishing CS theory from the traditional sampling theorem. Common measurement matrices are three major classes, random measurement matrices, deterministic measurement matrices, and structural random matrices. The existing research results of the measurement matrix show that the block Hadamard measurement matrix arranged along the main diagonal has performance advantages compared with other measurement matrices, such as simple calculation, convenient hardware realization, high signal reconstruction quality and the like.
Although the block Hadamard measurement matrix arranged along the main diagonal has the advantages, the block Hadamard measurement matrix cannot observe a high-frequency component part in a sparse signal to be detected, so that a reconstructed signal at a low sampling rate is low in recognition rate, especially on image recognition of a two-dimensional image signal with relatively rich detail characteristics (or relatively high-frequency components after sparse transform processing), and a compressed sensing image which can be successfully recognized is practical. Therefore, an improvement to this drawback is needed to improve the utility of the compressed sensing method based on the block hadamard measurement matrix, and is a hot spot of application research.
Disclosure of Invention
The technical purpose of the invention is to provide a compressed sensing sampling and reconstruction method, equipment and storage medium based on the block Hadamard measurement matrix arranged along the main diagonal and the sub diagonal, aiming at the structural characteristics of the block Hadamard measurement matrix arranged along the main diagonal and the defects of the corresponding efficient reconstruction algorithm in performance in the prior art, and the quality of the reconstructed signal and the recognition rate of the reconstructed image can be improved by performing combined compressed sampling and reconstruction on different frequency spectrum regions of the signal and recovering the reconstruction.
In order to achieve the technical purpose, the invention adopts the following technical scheme:
a compressed sensing sampling and reconstruction method based on a block Hadamard measurement matrix comprises the following steps:
step 1, determining the measurement number M of X according to the dimension N and the sampling rate alpha of a sparse signal X to be measured:
Figure GDA0003535735620000021
wherein N is 2n(ii) a Representing the range of M as M > M1＝2n-jN is more than j and is more than or equal to 1; memory M2＝M-M1And M is2Is shown as 2n-k+1＞M2≥2n-kN > k.gtoreq.j, and M2≥25；
Figure GDA0003535735620000022
Represents rounding up;
step 2, constructing a first measurement matrix
Figure GDA0003535735620000023
If 2n-j+1＞M＞2n-jJ is 1, then two are 2n-1Order hadamard matrix block
Figure GDA0003535735620000024
Arranged along the main diagonal and normalized to obtain a block diagonal matrix phiN×N；
If 2n-j+1≥M＞2n-jJ ≠ 1, then two are 2n-jOrder hadamard matrix block
Figure GDA0003535735620000025
And from 2n-j+1To 2n-1One for each Hadamard matrix block of order
Figure GDA0003535735620000026
Arranging the blocks from top to bottom along the main diagonal in the order of dimension from small to large, and normalizing to obtain a block diagonal matrix phiN×N；
Take phiN×NFront M of1The rows being a first measurement matrix
Figure GDA0003535735620000027
Step 3, constructing a second measuring matrix
Figure GDA0003535735620000028
Two are 2n-kOrder hadamard matrix block
Figure GDA0003535735620000029
And from 2n-k+1To 2n-1One for each Hadamard matrix block of order
Figure GDA00035357356200000210
Arranging the blocks from small to large from top to bottom along the secondary diagonal and normalizing to obtain a block diagonal matrix psiN×N；
Take psiN×NFront M of2The row being a second measuring matrix
Figure GDA00035357356200000211
Step 4, utilizing the first measuring matrix
Figure GDA00035357356200000212
And a second measurement matrix
Figure GDA00035357356200000213
Observing the sparse signal X to be measured, i.e. for the sparse signal to be measuredCarrying out compression sampling on the number X to respectively obtain M of the sparse signals X to be detected1Dimension measurement value Y1And M2Dimension measurement value Y2；
Step 5, according to M of sparse signal X to be measured1Dimension measurement value Y1And M2Dimension measurement value Y2And a first measurement matrix
Figure GDA00035357356200000214
And a second measurement matrix
Figure GDA00035357356200000215
Reconstruction signal for solving sparse signal X to be measured
Figure GDA00035357356200000216
First part of
Figure GDA00035357356200000217
And a second part
Figure GDA00035357356200000218
Finally, the first part is combined
Figure GDA00035357356200000219
And a second part
Figure GDA00035357356200000220
Obtaining the complete reconstructed signal
Figure GDA00035357356200000221
In a more preferred solution, the block diagonal matrix ΦN×NIs an orthogonal matrix, expressed as:
when 2 is inn-j+1＞M＞2n-jWhen j is 1, the ratio of the total of the two,
Figure GDA00035357356200000222
when 2 is inn-j+1≥M＞2n-jWhen j is not equal to 1,
Figure GDA0003535735620000031
the block diagonal matrix ΨN×NIs an orthogonal matrix, expressed as:
Figure GDA0003535735620000032
wherein the content of the first and second substances,
Figure GDA0003535735620000033
respectively representing dimension 2n-j、2n-k、2n-j+1、2n -k+1、2n-1The hadamard matrix block.
In a preferred embodiment, a first measurement matrix is used
Figure GDA0003535735620000034
Carrying out compression sampling on sparse signal X to be detected to obtain M1Dimension measurement value Y1The method comprises the following steps:
Figure GDA0003535735620000035
using a second measurement matrix
Figure GDA0003535735620000036
Carrying out compression sampling on sparse signal X to be detected to obtain M2Dimension measurement value Y2The method comprises the following steps:
Figure GDA0003535735620000037
in a more preferred embodiment, the signal is reconstructed
Figure GDA0003535735620000038
First part of
Figure GDA0003535735620000039
Solving according to a reversible matrix equation to obtain:
Figure GDA00035357356200000310
reconstructing a signal
Figure GDA00035357356200000311
Second part of (2)
Figure GDA00035357356200000312
The solving method comprises the following steps:
if M is2＝2n-kThen the second part is solved according to the reversible matrix equation
Figure GDA00035357356200000313
If 2n-k+1＞M2＞2n-kThen the second part
Figure GDA00035357356200000314
Front 2 ofn-kDimensional part
Figure GDA00035357356200000315
Solving according to the reversible matrix equation, the rest M2-2n-kSolving the dimension according to a general matching pursuit type reconstruction algorithm;
wherein the content of the first and second substances,
Figure GDA00035357356200000316
in a more preferred embodiment, the general matching pursuit type reconstruction algorithm is an orthogonal matching pursuit algorithm.
In a more preferable technical scheme, the sparse signal to be measured may be an image signal subjected to sparsification.
In a more preferred technical scheme, the sampling rate α has a value range of: alpha is more than 0.1 and less than 0.8.
The invention also provides an apparatus comprising a processor and a memory; wherein: the memory is to store computer instructions; the processor is configured to execute the computer instructions stored in the memory, and specifically, to perform the method according to any of the above technical solutions.
The present invention also provides a computer storage medium for storing a program, which when executed, is configured to implement the method according to any of the above-mentioned technical solutions.
Advantageous effects
When compressed sensing sampling is carried out on the sparse signal to be detected, the compressed measurement value of the sparse signal to be detected is divided into two parts, wherein the first part is the low-frequency component measurement value Y of the sparse signal to be detected1The second part is the measured value Y of the relatively high frequency of the sparse signal to be measured2(ii) a Then, reconstructing signals of each part by adopting an efficient reconstruction algorithm based on a diagonal block Hadamard measurement matrix, and combining to obtain complete reconstructed signals; and extracting the characteristics of the reconstructed image by adopting a typical image processing method such as an LBP (local binary pattern) method, and identifying the reconstructed image. Compared with the effect of the prior art that the signals are compressed and sampled and reconstructed by independently adopting the block Hadamard measurement matrix arranged along the main diagonal, the combined sampling provided by the invention has the advantages that the measured value simultaneously contains the low-frequency component part and part of the relatively high-frequency component of the sparse signal to be measured (because M is the M component)2Is relatively small (M)2≤M/2<N/2), therefore, only part of high-frequency signals in the sparse signal to be detected are obtained by sampling with the hadamard matrix block on the right of the sub-diagonal matrix, and the corresponding position is in the lower left block of the sparse signal, i.e. the relatively high-frequency part. But the left lower block and the right upper block of the detected sparse signal belong to relatively high-frequency components relative to the left upper block), so that the compressed sensing of a resource-limited system can be realized, and the quality of the finally obtained reconstructed signal is higher, and therefore, the identification rate for identifying the reconstructed signal can be obviously improved at a low sampling rate.
Drawings
FIG. 1 is a flow chart of an implementation of a high-recognition rate compressed sensing sampling method for two block Hadamard measurement matrices arranged along a major diagonal and a minor diagonal according to the present invention;
FIG. 2 is a graph showing the comparison between the sampling method (method 1) using only the block Hadamard measurement matrix arranged along the main diagonal and the reconstructed peak SNR after the compression measurement of the Peppers images with the pixel number of 512 × 512 according to the method of the present invention for different measurement values;
FIG. 3 is a graph showing the effect of method 1 and the method of the present invention on reconstruction after compression measurement of 512 × 512 Peppers images when the number of measured values is 200; wherein, FIG. 3(a) is a Peppers original image; FIGS. 3(b), (c) are images reconstructed for Peppers using method 1 and the method of the present invention, respectively;
FIG. 4 is a graph showing the effect of the reconstruction after compression measurement of 512 × 512 Peppers images by the method 1 and the method of the present invention when the number of measured values is 300; wherein, FIG. 4(a) is a Peppers original image; FIGS. 4(b), (c) are images reconstructed for Peppers using method 1 and the method of the present invention, respectively.
Detailed Description
The following describes embodiments of the present invention in detail, which are developed based on the technical solutions of the present invention, and give detailed implementation manners and specific operation procedures to further explain the technical solutions of the present invention.
As shown in fig. 1, the compressed sensing sampling and reconstructing method based on the partitioned hadamard measurement matrix with primary and secondary diagonal arrangements provided by the present invention includes the following steps:
step 1, determining the measurement number M of X according to the dimension N and the sampling rate alpha of a sparse signal X to be measured:
Figure GDA0003535735620000051
wherein N is 2n(ii) a Representing the range of M as M > M1＝2n-jN is more than j and is more than or equal to 1; memory M2＝M-M1And M is2Is shown as 2n-k+1＞M2≥2n-kN > k.gtoreq.j, and M2≥25；
Figure GDA0003535735620000052
Indicating rounding up.
Step 2, constructing a first measurement matrix
Figure GDA0003535735620000053
If 2n-j+1＞M＞2n-jJ is 1, then two are 2n-1Order hadamard matrix block
Figure GDA0003535735620000054
Arranged along the main diagonal and normalized to obtain a block diagonal matrix phiN×N(ii) a Wherein the block diagonal matrix phiN×NIs an orthogonal matrix and can be expressed as:
Figure GDA0003535735620000055
if 2n-j+1≥M＞2n-jJ ≠ 1, then two are 2n-jOrder hadamard matrix block
Figure GDA0003535735620000056
And from 2n-j+1To 2n-1One for each Hadamard matrix block of order
Figure GDA0003535735620000057
Arranging the blocks from top to bottom along the main diagonal in the order of dimension from small to large, and normalizing to obtain a block diagonal matrix phiN×N(ii) a Wherein the block diagonal matrix is phiN×NIs an orthogonal matrix and can be expressed as:
Figure GDA0003535735620000058
obtaining a block diagonal matrix phiN×NThen, take phiN×NFront M of1The rows being a first measurement matrix
Figure GDA0003535735620000059
Step 3, constructing a second measuring matrix
Figure GDA00035357356200000510
Two are 2n-kOrder hadamard matrix block
Figure GDA0003535735620000061
And from 2n-k+1To 2n-1One for each Hadamard matrix block of order
Figure GDA0003535735620000062
Arranging the blocks from small to large from top to bottom along the secondary diagonal and normalizing to obtain a block diagonal matrix psiN×N(ii) a The resulting block diagonal matrix ΨN×NIs an orthogonal matrix, expressed as:
Figure GDA0003535735620000063
obtain the block diagonal matrix ΨN×NThen, take psiN×NFront M of2The row being a second measuring matrix
Figure GDA0003535735620000064
Wherein the content of the first and second substances,
Figure GDA0003535735620000065
respectively representing blocks of hadamard matrices of dimensions 2n-j, 2n-k, 2n-j +1, 2n-k +1, 2 n-1.
Step 4, utilizing the first measuring matrix
Figure GDA0003535735620000066
And a second measurement matrix
Figure GDA0003535735620000067
Observing the sparse signal X to be measured, namely performing compression sampling on the sparse signal X to be measured to respectively obtain M of the sparse signal X to be measured1Dimension measurement value Y1And M2Dimension measurement value Y2；
Wherein a first measurement matrix is used
Figure GDA0003535735620000068
Carrying out compression sampling on sparse signal X to be detected to obtain M1Dimension measurement value Y1The method comprises the following steps:
Figure GDA0003535735620000069
using a second measurement matrix
Figure GDA00035357356200000610
Carrying out compression sampling on sparse signal X to be detected to obtain M2Dimension measurement value Y2The method comprises the following steps:
Figure GDA00035357356200000611
the sparse signal to be measured in the present invention may be an image signal subjected to a thinning process (for example, DCT transform, wavelet transform). The frequency spectrum distribution characteristics of the image sparse signal are as follows: the upper left part has the lowest signal frequency and the lower right part has the highest signal frequency, but the data with higher frequency contains less effective information, so the data in the lower right part can be discarded. The signal frequencies of the upper right and lower left portions are higher overall relative to the signal frequency of the upper left portion. The signal spectrum of the lower left part is higher and lower than the signal spectrum of the upper right part, and the situation is relatively complex due to different specific sparse transform methods (the distribution characteristics of the signal spectrum after DCT transform and the distribution characteristics of the signal spectrum after wavelet transform are obviously different in the upper right part and the lower left part). Therefore, for the tested sparse signal containing relatively high frequency components, effective signal data of relatively high frequency can be obtained by sampling the Hadamard matrix block at the right upper part of the secondary diagonal matrix.
However, in the prior art, for example, in the method 1 described herein, when compressed sensing sampling is performed based on a block hadamard measurement matrix, a block hadamard diagonal matrix is constructed only along a main diagonal, which is equivalent to only the upper left part, i.e., the low frequency part, of an image sparse signal is retained, and the information at the lower part, i.e., the high frequency information, is completely discarded, so that the recognition rate of the reconstructed signal at a low sampling rate is not high, especially for image recognition of a two-dimensional image signal with relatively rich detail features (or relatively high frequency components after sparse transform processing).
Compared with the prior art, the block Hadamard diagonal matrix is constructed along the main diagonal line and the secondary diagonal line, namely, on the basis of reserving the low-frequency signal in the sparse signal to be detected, partial signals with relatively high frequency in the sparse signal to be detected are further reserved, so that the quality of a reconstructed signal at a low sampling rate, particularly a two-dimensional image signal with relatively rich detail characteristics, is higher, the practicability of the reconstructed signal is further higher, and the recognition rate of the reconstructed two-dimensional image signal is higher.
Step 5, according to M of sparse signal X to be measured1Dimension measurement value Y1And M2Dimension measurement value Y2And a first measurement matrix
Figure GDA0003535735620000071
And a second measurement matrix
Figure GDA0003535735620000072
Reconstruction signal for solving sparse signal X to be measured
Figure GDA0003535735620000073
First part of
Figure GDA0003535735620000074
And a second part
Figure GDA0003535735620000075
Finally, the first part is combined
Figure GDA0003535735620000076
And a second part
Figure GDA0003535735620000077
Obtaining the complete reconstructed signal
Figure GDA0003535735620000078
Wherein the signal is reconstructed
Figure GDA0003535735620000079
First part of
Figure GDA00035357356200000710
Solving according to a reversible matrix equation to obtain:
Figure GDA00035357356200000711
reconstructing a signal
Figure GDA00035357356200000712
Second part of (2)
Figure GDA00035357356200000713
The solving method of (2) is divided into two cases:
if M is2＝2n-kThen the second part is solved according to the reversible matrix equation
Figure GDA00035357356200000714
If 2n-k+1＞M2＞2n-kThen the second part
Figure GDA00035357356200000715
Front 2 ofn-kDimensional part
Figure GDA00035357356200000716
Solving according to the reversible matrix equation, the rest M2-2n-kSolving the dimension according to a general matching pursuit type reconstruction algorithm;
wherein the content of the first and second substances,
Figure GDA00035357356200000717
corresponding to the compressed sensing sampling and reconstruction method, the invention also provides a device, which comprises a processor and a memory; wherein: the memory is to store computer instructions; the processor is used for executing the computer instructions stored by the memory, and particularly executing the method.
In response to the compressed sensing sampling and reconstruction method provided above, the present invention also provides a computer storage medium for storing a program, which when executed, implements the method described above.
The following embodiment takes the sparse signal to be measured as an image signal to verify the excellent performance of the algorithm provided by the invention. It should be noted that the embodiment is only exemplary and is not intended to limit the applicable scope of the present invention.
Example 1:
the method comprises the following steps of carrying out compression reconstruction on Peppers, Baboon, Goldhill and Bridge gray images with the pixel number of 512 multiplied by 512 and 5011 gray images with the size of 256 multiplied by 256:
firstly, carrying out sparsification processing on the image signal by adopting a Discrete Cosine Transform (DCT) basis to obtain a sparse signal X.
And secondly, constructing block Hadamard measurement matrixes arranged along the main diagonal and the secondary diagonal respectively. Since N is 512, for a sampling rate of 0.5 < α < 0.8, the corresponding M > 256, M1＝256，M2＝M-M1E.g. when the number of measurements M is 300, M1＝256，M244, two Hadamard matrix sub-blocks H with 256 dimensions256Arranged along the main diagonal and normalized to form a block diagonal matrix phi512×512(ii) a Two Hadamard matrix subblocks H with dimension 3232A sub-block H of a Hadamard matrix of dimension 6464A sub-block H of a Hadamard matrix of dimension 128128And a sub-block H of Hadamard matrix with dimension 256256Arranging the two dimensions from small to large along a secondary diagonal and normalizing to obtain the matrix psi512×512：
Figure GDA0003535735620000081
Figure GDA0003535735620000082
Then select phi512×512The first 256 lines ofAs a first measurement matrix phi256×512Selecting Ψ512×512First 44 rows of as a second measurement matrix Ψ44×256。
For a sampling rate of 0.25 < alpha < 0.5, the corresponding 128 < M<256，M1＝128，M2＝M-M1E.g. when the number of measurements M is 200, M1＝128，M272, two hadamard matrix sub-blocks H with dimension 128128And a sub-block H of Hadamard matrix with dimension 256256Arranged along the main diagonal and normalized to obtain a matrix phi512×512(ii) a Two sub-blocks H of the Hadamard matrix with dimension 6464A sub-block H of a Hadamard matrix of dimension 128128And a sub-block H of Hadamard matrix with dimension 256256Arranging the two dimensions from small to large along a secondary diagonal and normalizing to obtain the matrix psi512×512。
Figure GDA0003535735620000083
Figure GDA0003535735620000091
Then select phi512×512As the first measurement matrix phi128×512Selecting Ψ512×512First 72 rows as the second measurement matrix Ψ72×512。
Thirdly, using phiM×NCompressing and measuring the signal X after the image sparsification processing to obtain a measured value Y, and compressing and measuring the signal X after the image sparsification processing by using the method of the invention, namely respectively using
Figure GDA0003535735620000092
And
Figure GDA0003535735620000093
carrying out compression measurement on the signal X after image sparsification processing to obtain a measurement value Y1And Y2。
Fourthly, utilizing compressed sensing efficient reconstruction algorithm based on block Hadamard measurement matrix and measurement value Y1Reconstructing the compressed image signal to obtain a reconstructed signal
Figure GDA0003535735620000094
Compressed sensing efficient reconstruction algorithm based on block Hadamard measurement matrix and measurement value Y2Reconstructing the compressed image signal to obtain a reconstructed signal
Figure GDA0003535735620000095
FIG. 2 shows a block Hadamard measurement matrix using primary and secondary diagonal permutations for different measurement values
Figure GDA0003535735620000096
And
Figure GDA0003535735620000097
the combined sampling strategy provided by the invention is used for compressing and measuring a reconstructed peak signal-to-noise ratio contrast curve of a 512 x 512 Peppers image. For N-2n(512＝29) M is determined according to the magnitude of M or the sampling rate alpha1And M2And then determining the matrix phi according to the size of M1N×NSize and combination of sub-blocks of Hadamard matrix, if M>2n-1Then M is1＝2n-1，M2＝M-M1Orthogonal matrix phiN×NFrom two identical 2n-1The Victoria Hadamard matrix subblocks are arranged along the main diagonal and are normalized to form the Victoria Hadamard matrix subblocks; if 2n-2<M≤2n-1Then M is1＝2n-2，M2＝M-M1In the previous case, 2 in the upper left cornern-1Decomposition of a sub-block of the Vihadamard matrix into two 2n-2Dimension blocks, i.e. matrices ΦN×NIs composed of two units 2n-2And a 2n-1The Victoria Hadamard matrix subblocks are arranged along the main diagonal and are normalized to form the Victoria Hadamard matrix subblocks; if M is smaller, the minimum subblock at the upper left corner in the last case is continuously decomposed into two subblocks one level lower. Then according to M2Determine the orthogonal momentMatrix psiN×NEach Hadamard matrix sub-block size and combination, if M2＝2n-1，ΨN×NFrom two identical 2n-1The sub-blocks of the Victoria Hadamard matrix are arranged along the secondary diagonal and are normalized to form the sub-blocks; if 2n-2≤M2<2n-1Then 2 of the upper left corner in the last casen -1Decomposition of a sub-block of the Vihadamard matrix into two 2n-2Dimension blocks, i.e. measuring matrices phiN×NIs composed of two units 2n-2And a 2n-1The Victoria Hadamard matrix subblocks are arranged along the primary and secondary angular lines and are formed in a normalized mode; if M is smaller, the minimum subblock at the upper left corner in the last case is continuously decomposed into two subblocks one level lower. Finally using phiN×NAnd ΨN×NAnd sampling the image signal after the thinning processing in a combined mode. In practice, if the sparse signal to be measured is an image signal, the dimension of the minimum subblock corresponding to N ═ 512 is not suitable to be smaller than 32, because the sampling rate at this time is lower than 0.1, and an excessively low sampling rate cannot guarantee whether the signal can be successfully reconstructed, and such a sampling rate is meaningless. It can be seen from fig. 2 that the method of the present invention has performance advantages regardless of the value of M.
Fig. 3 and 4 show the effect of method 1 and the inventive method on reconstruction after compression measurement of 512 × 512 Peppers images, when the number of measured values is 200 and 300, respectively. Since the method 1 has better effect, the difference is not obvious enough when the visual effect of the method is distinguished by human eyes compared with the method 1.
Without loss of generality, table 1 shows the peak snr contrast for the reconstructed images of 512 × 512 babon, Goldhill and Bridge gray scale images under method 1 and the proposed method of the present invention, when the number of measured values is 200, 300 and 400, respectively.
Table 1: method 1 and peak signal-to-noise ratio of the inventive method to the reconstruction of three exemplary 512 x 512 images
Figure GDA0003535735620000101
As can be seen from Table 1, the peak signal-to-noise ratio of the reconstructed image of the method of the present invention is improved under different numbers of the sampled data, compared with the method 1. This is consistent with previous conclusions drawn from Peppers image reconstruction.
Table 2 shows experimental test data for comparing feature extraction and recognition effects of images reconstructed by the method of the present invention and the method 1 by using the LBP method under the same sampling rate, where 5011 grayscale images with 256 × 256(N ═ 28) in the VoC2007 data set (http:// pjred.
Table 2: method 1 and the method of the invention recognition rate (%) -of image reconstruction at the same sampling rate and same processing algorithm
Figure GDA0003535735620000102
As can be seen from Table 2, under the same sampling rate and the same processing algorithm, the recognition rate of the reconstructed image by the method of the present invention is higher than that of the reconstructed image by the method 1, and the performance advantage of the method is more obvious at a low sampling rate. The data comparison in table 2 shows that, in the combined compression sampling provided by the present invention, when the sampling rate is the same, the compression measurement process of the sparsified image signal is divided into two parts, and the low frequency component part and the relatively high frequency component part of the sparsified image signal are respectively collected, whereas in the method 1, when the compression measurement is performed on the sparsified image signal, only the low frequency part of the signal is collected, but the relatively high frequency component part often plays an important role in image recognition (often provides detailed information or more local information), so that the reconstructed image of the method provided by the present invention has a higher recognition rate in recognition at a low sampling rate.
The method 1 described herein refers to the technical scheme described in patent No. ZL 2017108254705.
The above embodiments are preferred embodiments of the present application, and those skilled in the art can make various changes or modifications without departing from the general concept of the present application, and such changes or modifications should fall within the scope of the claims of the present application.

Claims (9)

1. A compressed sensing sampling and reconstruction method based on a block Hadamard measurement matrix is characterized by comprising the following steps:
step 1, determining the measurement number M of X according to the dimension N and the sampling rate alpha of a sparse signal X to be measured:
Figure FDA0003535735610000011
wherein N is 2n(ii) a Representing the range of M as M > M1＝2n-jN is more than j and is more than or equal to 1; memory M2＝M-M1And M is2Is shown as 2n-k+1＞M2≥2n -kN > k.gtoreq.j, and M2≥25；
Figure FDA0003535735610000012
Represents rounding up;
step 2, constructing a first measurement matrix phiM1×N：
If 2n-j+1＞M＞2n-jJ is 1, then two are 2n-1Order hadamard matrix block
Figure FDA0003535735610000013
Arranged along the main diagonal and normalized to obtain a block diagonal matrix phiN×N；
If 2n-j+1≥M＞2n-jJ ≠ 1, then two are 2n-jOrder hadamard matrix block
Figure FDA0003535735610000014
And from 2n-j+1To 2n-1One for each Hadamard matrix block of order
Figure FDA0003535735610000015
Arranging the blocks from top to bottom along the main diagonal in the order of dimension from small to large, and normalizing to obtain a block diagonal matrix phiN×N；
Take phiN×NFront M of1The rows being a first measurement matrix
Figure FDA0003535735610000016
Step 3, constructing a second measuring matrix
Figure FDA0003535735610000017
Two are 2n-kOrder hadamard matrix block
Figure FDA0003535735610000018
And from 2n-k+1To 2n-1One for each Hadamard matrix block of order
Figure FDA0003535735610000019
Arranging the blocks from small to large from top to bottom along the secondary diagonal and normalizing to obtain a block diagonal matrix psiN×N；
Take psiN×NFront M of2The row being a second measuring matrix
Figure FDA00035357356100000110
Step 4, utilizing the first measuring matrix
Figure FDA00035357356100000111
And a second measurement matrix
Figure FDA00035357356100000112
Observing the sparse signal X to be measured, namely performing compression sampling on the sparse signal X to be measured to respectively obtain M of the sparse signal X to be measured1Dimension measurement value Y1And M2Dimension measurement value Y2；
Step 5, according to M of sparse signal X to be measured1Dimension measurement value Y1And M2Dimension measurement value Y2And a first measurement matrix
Figure FDA00035357356100000113
And a second measurement matrix
Figure FDA00035357356100000114
Reconstruction signal for solving sparse signal X to be measured
Figure FDA00035357356100000115
First part of
Figure FDA00035357356100000116
And a second part
Figure FDA00035357356100000117
Finally, the first part is combined
Figure FDA00035357356100000118
And a second part
Figure FDA00035357356100000119
Obtaining the complete reconstructed signal
Figure FDA00035357356100000120
2. The method of claim 1, wherein the block diagonal matrix ΦN×NIs an orthogonal matrix, expressed as:
when 2 is inn-j+1＞M＞2n-jWhen j is 1, the ratio of the total of the two,
Figure FDA0003535735610000021
when 2 is inn-j+1≥M＞2n-jWhen j is not equal to 1,
Figure FDA0003535735610000022
the block diagonal matrix ΨN×NIs an orthogonal matrix, expressed as:
Figure FDA0003535735610000023
wherein the content of the first and second substances,
Figure FDA0003535735610000024
respectively representing dimension 2n-j、2n-k、2n-j+1、2n-k+1、2n-1The hadamard matrix block.
3. Method according to claim 1, characterized in that a first measurement matrix is used
Figure FDA0003535735610000025
Carrying out compression sampling on sparse signal X to be detected to obtain M1Dimension measurement value Y1The method comprises the following steps:
Figure FDA0003535735610000026
using a second measurement matrix
Figure FDA0003535735610000027
Carrying out compression sampling on sparse signal X to be detected to obtain M2Dimension measurement value Y2The method comprises the following steps:
Figure FDA0003535735610000028
4. the method of claim 1, wherein the signal is reconstructed
Figure FDA0003535735610000029
First part of
Figure FDA00035357356100000210
Solving according to a reversible matrix equation to obtain:
Figure FDA00035357356100000211
reconstructing a signal
Figure FDA00035357356100000212
Second part of (2)
Figure FDA00035357356100000213
The solving method comprises the following steps:
if M is2＝2n-kThen the second part is solved according to the reversible matrix equation
Figure FDA00035357356100000214
Figure FDA00035357356100000215
If 2n-k+1＞M2＞2n-kThen the second part
Figure FDA00035357356100000216
Front 2 ofn-kDimensional part
Figure FDA00035357356100000217
Solving according to the reversible matrix equation, the rest M2-2n -kSolving the dimension according to a general matching pursuit type reconstruction algorithm;
wherein the content of the first and second substances,
Figure FDA0003535735610000031
5. the method of claim 4, wherein the generic matching pursuit-like reconstruction algorithm is an orthogonal matching pursuit algorithm.
6. The method according to claim 1, wherein the sparse signal under test is an image signal subjected to sparsification.
7. The method of claim 1, wherein the sampling rate α is in a range of: alpha is more than 0.1 and less than 0.8.
8. An apparatus comprising a processor and a memory; wherein: the memory is to store computer instructions; the processor is configured to execute the computer instructions stored by the memory, in particular to perform the method according to any one of claims 1 to 7.
9. A computer storage medium storing a program which, when executed, performs the method of any one of claims 1 to 7.
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