CN107016656B

CN107016656B - Wavelet sparse basis optimization method in image reconstruction based on compressed sensing

Info

Publication number: CN107016656B
Application number: CN201710212715.7A
Authority: CN
Inventors: 魏子然; 徐智勇; 张健林; 吴润泽; 唐惜
Original assignee: Institute of Optics and Electronics of CAS
Current assignee: Institute of Optics and Electronics of CAS
Priority date: 2017-04-01
Filing date: 2017-04-01
Publication date: 2020-06-19
Anticipated expiration: 2037-04-01
Also published as: CN107016656A

Abstract

The invention relates to a wavelet sparse basis optimization method in image reconstruction based on compressed sensing, namely, a suppression matrix is used for optimizing a wavelet sparse basis. Firstly, multi-layer wavelet transformation is carried out on original signal data, the coefficient in a wavelet domain is observed, the size of the coefficient is gradually reduced, and therefore a suppression matrix is designed to suppress the small coefficient and serves as a part of a wavelet transformation base, and the purpose of enabling the coefficient to be more sparse is achieved. Simulation results show that when the sampling rate is within the range of 0.15 to 0.45, the quality improvement effect of the reconstructed image is the best, the peak signal-to-noise ratio is improved by about 0.5dB to 1dB, the method has a good effect on reconstruction of fingerprint texture images, and the defect that the reconstruction precision of the traditional compression sensing image reconstruction technology based on wavelet transformation is low is overcome to a certain extent.

Description

Wavelet sparse basis optimization method in image reconstruction based on compressed sensing

Technical Field

The invention relates to a wavelet sparse basis optimization method in image reconstruction based on compressed sensing, which is characterized in that original signal data of a signal is restored and reconstructed to original signal data with higher precision at a lower sampling rate, the method is applied to compression and restoration of the signal, image processing, computer vision and the like, and belongs to the field of signal compression transmission and restoration reconstruction in signal and information processing.

Background

The core of the compressive sensing is a linear measurement process, wherein x (N) is used as an original signal, the length is N, y (M) is obtained by multiplying a measurement matrix phi by left, and the length is M (M < N). If x (N) is not a sparse signal, orthogonal sparse transform is performed to obtain s (k), which is denoted as x ═ Ψ s, and the measurement process is rewritten as y ═ Θ s, where Θ ═ Φ Ψ (M × N) is referred to as a sensing matrix, and the process is shown in fig. 2.

The signal reconstruction algorithm means that the length of the reconstructed vector y is N (M) by M times of measurement<N) of the sparse signal x. The number N of unknown numbers in the equation set exceeds the number M of equations, x (N) cannot be directly recovered from y (M), and the minimum l can be solved₀The norm problem (1) is solved.

But minimum l₀The norm problem is an NP-hard problem that is exhaustive of all non-zero values in x

This permutation is possible and therefore cannot be solved. Whereby the solution is performed with a suboptimal solution algorithm, mainly involving a minimum of l₁Norm method, matching pursuit series algorithm, iterative threshold value method and minimum total variation method specially processing two-dimensional image problem. Meanwhile, the measurement matrix phi meets the constraint equidistant condition (RIP condition, 2 formula), and the original signal can be recovered through the reconstruction algorithm.

And simulating sparse data with the length of 256 by using MATLAB, and exploring the relation between data reconstruction precision and sparsity. Regarding the difference between the reconstructed data and the original data below a certain threshold as successful data reconstruction, the relationship (signal length d is 256) between the reconstruction success rate (input 1000 sets of test data) and the sparsity (m) is obtained in the case of different numbers of measured values (N) as shown in fig. 3. As can be seen from fig. 3, sufficient sparsity is crucial to improving data reconstruction accuracy, so the improvement of data reconstruction herein is to create a sparse change basis that makes the data more sparse.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: aiming at the problems that the sparsity of the coefficient in the wavelet domain is not enough after the existing wavelet transform in the compressed sensing signal reconstruction and the precision is not high after the signal reconstruction, a suppression matrix aiming at the wavelet transform is constructed, so that the wavelet coefficient is more sparse. Under the same sampling rate and the same reconstruction condition, the method can improve the accuracy and the signal-to-noise ratio of signal reconstruction to a certain extent.

The technical scheme adopted by the invention for solving the technical problems is as follows: a wavelet sparse basis optimization method in image reconstruction based on compressed sensing constructs an easily-realized wavelet coefficient suppression matrix, so that sparse coefficients of original signal data after wavelet transformation become more sparse; and reconstructing a sparse signal by a reconstruction algorithm according to a measured value obtained by the measuring matrix, and finally reconstructing an original signal by the sparse signal through wavelet inverse transformation.

Firstly, the original data signal is thinned through a wavelet sparse basis, the distribution condition of coefficients in a wavelet domain is observed, the overall coefficients show a gradually decreasing trend, and then the sparsity of the signal can be increased by considering the mode of restraining small coefficients.

After the distribution characteristics of the wavelet coefficients are analyzed, a diagonal suppression matrix W which is in the same dimension (n dimension) as the sparse signal is constructed, wherein the diagonal element is an arithmetic series with the initial term of 1 and the tolerance of-1/n, and then the wavelet coefficients are arranged backwards, the smaller the wavelet coefficients are, and the higher the suppression degree is. Such a suppression matrix W is used to suppress small coefficients in the wavelet domain to improve the sparsity of the wavelet coefficients, i.e., s '═ Ws, where s and s' are the wavelet coefficients before and after optimization, respectively, and the wavelet after optimization is changed to Ψ^-1＝WΨ₀ ^-1W is reversible, Ψ₀Is the original wavelet transform basis. Since the diagonal elements of the suppression matrix W are all nonzero values, the suppression matrix W is an invertible matrix

The initial wavelet coefficient and the suppression matrix are subjected to product to obtain a more sparse wavelet coefficient, and a measurement value obtained by the measurement matrix is approximated by a reconstruction algorithm to obtain the more sparse wavelet coefficient.

After a wavelet coefficient suppression matrix with diagonal elements in an arithmetic series is added, the product of the suppression matrix and wavelet transformation jointly forms a sparse basis. Since the small coefficient values in the wavelet domain are shrunk after being suppressed by the suppression matrix, the original signal is also easily restored due to the reversibility of the suppression matrix. The product of the inverse matrix of the sparse basis and the more sparse wavelet coefficient can obtain the restored and reconstructed original signal, and the restored and reconstructed final signal is not influenced by adding the suppression matrix in the wavelet sparse transform.

The principle of the invention is as follows:

designing a suppression matrix capable of suppressing small coefficients to become a part of a sparse transformation base, wherein the technical scheme adopted by the invention is as follows:

the raw data is thinned out by wavelet transform, as shown in fig. 4 and 5. As can be seen from fig. 4 and 5, the wavelet transform performs a large degree of sparseness on the data, but the sparseness is still not ideal enough for reconstruction of the compressed sensing data. As can be seen by observing the coefficient sequence in the wavelet domain of the data signal, the coefficient in front of the wavelet coefficient sequence is large, and the coefficient behind the wavelet coefficient sequence is small, so that the trend of gradual decrease is basically presented, and the method of inhibiting the small coefficient can be considered to improve the sparsity of the wavelet coefficient. A wavelet coefficient suppression matrix as shown in fig. 6 was thus designed.

In fig. 6, the wavelet coefficient suppression matrix is an n-dimensional diagonal matrix, the diagonal element is an arithmetic sequence with a first term of 1 and a tolerance of-1/n, and a new coefficient vector is obtained by multiplying the matrix by the coefficients in the previous wavelet transform domain, as shown in fig. 7.

As can be seen from fig. 7, compared with the initial distribution of coefficients in the wavelet domain of fig. 4, the sparsity of the wavelet coefficients is greatly improved, and the purpose of suppressing small coefficients is achieved.

Compared with the prior art, the invention has the advantages that:

(1) the invention designs a wavelet coefficient inhibiting matrix which is easy to realize aiming at the coefficient distribution arrangement characteristic of the original data in the wavelet domain, and can directly multiply the matrix and the wavelet transformation matrix to become a part of a sparse change base, so that the original wavelet domain coefficient becomes more sparse, the reconstruction of data is facilitated, and the theoretical analysis and the concrete implementation are easy.

(2) The sampling rate interval which has great improvement on the image reconstruction precision is between 0.15 and 0.45, the reconstruction peak signal-to-noise ratio can be improved by about 0.5dB to 1dB, and the sampling rate interval is also the most common sampling rate interval for compressed sensing image reconstruction engineering application, so that the improved technology can be conveniently put into engineering practice and application and has more practical significance.

(3) The invention generally has poor image reconstruction effect on texture details based on the compressed sensing image reconstruction of wavelet transform, and the improved method greatly improves the reconstruction precision of the high-texture image such as fingerprint and enhances the detail reconstruction capability of the image, thereby making up the problem of low reconstruction precision of the texture image by the traditional compressed sensing image reconstruction technology based on wavelet transform to a certain extent.

Drawings

FIG. 1 is a flow chart of an implementation of the method of the present invention for compressed sensing data signal reconstruction;

FIG. 2 is a basic block diagram of the compressive sensing linearity measurement process of the present invention;

FIG. 3 is a diagram showing the relationship between the success rate of reconstruction of compressed sensing signals and the number of measured values and sparsity;

FIG. 4 is raw signal data;

FIG. 5 is the thinned data after wavelet transform of the original data;

FIG. 6 is a wavelet coefficient suppression matrix constructed in the present invention;

FIG. 7 is the wavelet coefficient distribution after the wavelet transform is added with the suppression matrix in the present invention;

FIG. 8 is a graph showing the relationship between the signal-to-noise ratio and the sampling rate of a Lena image reconstructed before and after the improvement of the present invention;

FIG. 9 is a graph showing the relationship between the signal-to-noise ratio and the sampling rate of a reconstructed Fingerprint image before and after improvement;

FIG. 10 is a comparison of the Lena image reconstruction effects before and after the improvement of the present invention;

FIG. 11 is a comparison of reconstruction effects of finger images before and after improvement in accordance with the present invention;

FIG. 12 is a comparison of the Lena partial image reconstruction effect before and after the improvement of the present invention;

fig. 13 is a comparison of reconstruction effects of Fingerprint local images before and after improvement of the present invention.

Detailed Description

The following detailed description of the invention is provided in connection with the accompanying drawings.

As can be seen from the schematic block diagram in FIG. 2, the original smallThe wave sparse transform basis is Ψ₀' if the suppression matrix is w, the last sparse transform is denoted by w Ψ₀' x ═ s, and let w Ψ₀'as an improved wavelet sparse transform basis, i.e. Ψ' in the schematic block diagram. The MATLAB is used for respectively using the gray-scale images of Lena (512 x 512) and Fingerrint (512 x 512) to carry out simulation experiments of image reconstruction, and due to the fact that the size of the image is too large, the image is reconstructed in rows and then spliced into a reconstructed full-width image. Since the single data length is 512, the size of the suppression matrix is 512 x 512, the head of the diagonal element is 1, and the tolerance is equal difference series of-1/512, the wavelet transformation layer number of the wavelet transformation base is set to be 5 layers (the highest layer number with the data length of 512), wherein the measurement matrix is a Gaussian random matrix, and the reconstruction algorithm adopts an Orthogonal Matching Pursuit (OMP) algorithm.

The relationship curve between the peak signal-to-noise ratio (PSNR) of the image reconstruction before and after the suppression matrix and the sampling rate is obtained as shown in fig. 8 and 9. As can be seen from fig. 8 and 9, under the condition of the same sampling rate, the peak signal-to-noise ratio of the image reconstruction is improved to a certain extent compared with that before the improvement, wherein when the sampling rate is between 0.15 and 0.45, the method has a better reconstruction effect, and the sampling rate interval is also the most common sampling rate interval for compressed sensing image reconstruction engineering application, so that the method can be put into engineering practice more quickly. When the sampling rate is 0.25, the global and local images of Lena and Fingerprint are respectively used for simulation experiments, and the reconstruction effect is as shown in fig. 10, fig. 11, fig. 12 and fig. 13.

As can be seen from fig. 10 and 11, the signal-to-noise ratio and the reconstruction accuracy of the improved image are significantly improved compared with those of the image before the improvement, and particularly for the texture images of the fingerprint type, the texture detail information of the fingerprint can hardly be acquired before the improvement, and the fingerprint cannot be identified. After the global image is subjected to simulation reconstruction, the local region of the image is extracted and reconstructed, and the reconstruction effect is as shown in fig. 12 and 13.

As can be seen from fig. 12 and 13, the improved method greatly improves the reconstruction effect of the local image, the image is smoother, the noise is less, and compared with the global image, the accuracy and the signal-to-noise ratio of the local image reconstruction of the small data are higher, so that the amplified detail information can be basically and clearly presented.

The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.

It should be understood by those skilled in the art that the above embodiments are only for illustrating the present invention and are not to be used as a limitation of the present invention, and that the changes and modifications of the above embodiments are within the scope of the claims of the present invention as long as they are within the spirit and scope of the present invention.

Claims

1. A wavelet sparse basis optimization method in image reconstruction based on compressed sensing is characterized by comprising the following steps: constructing a suppression matrix which is easy to realize and aims at wavelet domain coefficients, and making sparse coefficients of non-sparse original signals after wavelet transformation become sparse, so that measurement values obtained by a measurement matrix accurately reconstruct sparse coefficients in the wavelet domain through a reconstruction algorithm, and finally reconstructing original signals through the sparse coefficients through wavelet inverse transformation;

the principle basis of the wavelet sparse basis optimization method in image reconstruction based on compressed sensing is as follows: the method comprises the steps of thinning an original signal through a wavelet sparse basis, observing the distribution condition of coefficients in a wavelet domain, wherein the wavelet coefficients generally show a gradually decreasing trend, so that the sparsity of the signal is increased by inhibiting small coefficients, and then constructing a diagonal inhibition matrix W which has the same dimension as that of the sparse signal, wherein diagonal elements are an arithmetic series with a first term of 1 and a tolerance of-1/n, and off-diagonal elements are all 0;

such a suppression matrix W is used to suppress small coefficients in the wavelet domain to improve the sparsity of the wavelet coefficients, i.e., s '═ Ws, where s and s' are the wavelet coefficients before and after optimization, respectively, and the wavelet after optimization is changed to Ψ^-1＝WΨ₀ ^-1W is reversible, Ψ₀The original wavelet transform basis is such that the original wavelet coefficients are multiplied by a suppression matrix, such that the further back the wavelet coefficients are arranged, the smaller the wavelet coefficients are, and the suppression matrixThe suppression degree of the wavelet coefficient is higher, and the sparsity of the wavelet coefficient is improved to a great extent by suppressing the small coefficient, so that the wavelet coefficient becomes sparse after passing through a suppression matrix, and the sparse wavelet coefficient is favorably reconstructed by a signal reconstruction algorithm;

the method designs an easily realized wavelet coefficient inhibiting matrix aiming at the characteristic of the coefficient distribution arrangement of the original signals in a wavelet domain, and can directly make the product of the matrix and the wavelet transformation matrix become a part of a sparse change base, so that the original wavelet domain coefficient becomes sparse, thereby being beneficial to the reconstruction of the original signals and being very easy to carry out theoretical analysis;

the wavelet sparse basis optimization method based on compressed sensing in image reconstruction effectively improves the peak signal-to-noise ratio and the structural similarity of a reconstructed image, improves the reconstruction precision of a high-texture image such as a fingerprint, enhances the detail reconstruction capability of the image, and has a sampling rate interval which is 0.15 to 0.45 and is the most common sampling rate interval in compressed sensing image reconstruction engineering application, so that the improved technology can be conveniently put into engineering practice application and has practical significance.