An image compression sensing reconstruction method based on an AMP-PCSBL algorithm

Abstract

The present application is directed to the high-dimensional signal compression sensing reconstruction problem, on the basis of the existing pattern coupling sparse Bayesian learning PCSBL algorithm, the prior information and posterior probability in the signal are regarded as the message in the factor graph, the calculation efficiency of reconstruction is improved by using the approximate message passing AMP framework, an image compression sensing reconstruction method based on message passing pattern coupling sparse Bayesian learning AMP-PCSBL algorithm is proposed, the problem that the existing PCSBL algorithm is low in efficiency and is not suitable for compression reconstruction of high-dimensional signal is solved, and the calculation complexity is reduced to image compression reconstruction experiment proves that the AMP-PCSBL algorithm can greatly improve the calculation efficiency while keeping the reconstruction performance unchanged, and realizes the fast and efficient high-dimensional block sparse image signal reconstruction.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of compressed sensing, and particularly relates to an image compressed sensing reconstruction technology. BACKGROUND

[0002] As an efficient signal processing method, the core idea of compressed sensing technology is that if a signal is sparse in a certain transform domain (such as Fourier transform or wavelet transform), the original signal can be reconstructed from a small amount of non-uniform sampling through an optimization algorithm. This not only reduces the cost and time of data acquisition, but also has a significant advantage in processing high-dimensional data. At present, compressed sensing technology has shown great application potential in many fields: in the field of medical imaging, especially in magnetic resonance imaging (MRI), compressed sensing technology can shorten the scanning time while maintaining image quality, improve image resolution and signal-to-noise ratio, and has important significance for clinical diagnosis and treatment planning; in wireless communication, compressed sensing technology can reduce data transmission, efficiently complete spectrum sensing and channel estimation, and effectively prolong the life cycle of wireless networks; compressed sensing also has important applications in big data processing, which can significantly reduce the amount of data to be processed, making it faster and more accurate.

[0003] As the core content of compressed sensing technology, the reconstruction algorithm determines the application of compressed sensing to a certain extent. The existing compressed sensing reconstruction algorithm and related research mainly target general sparse signals, but in actual applications, sparse signals often have the characteristic of block distribution. Most of the reconstruction algorithms for block sparse signals require prior knowledge of block distribution, so the practicability of the algorithm is poor. The PCSBL algorithm uses the pattern coupling idea to reconstruct sparse signals with unknown block division structure, but the calculation complexity of the algorithm is high, and the algorithm efficiency is insufficient when facing high-dimensional reconstruction problems. SUMMARY

[0004] The technical problem to be solved by the application is to provide an efficient sparse reconstruction method suitable for unknown block division structure.

[0005] The technical solution adopted by the application to solve the above technical problem is an image compressed sensing reconstruction method based on an AMP-PCSBL algorithm, comprising the following steps:

[0006] Image compression step: the data sender unfolds the pixel matrix of the image data, takes the obtained gray value vector as the original signal x, and uses a Gaussian random measurement matrix A to compress and sample the original signal x to obtain an observation vector y;

[0007] Image transmission step: the data sender transmits the observation vector y and the random measurement matrix A to the data receiver;

[0008] Image reconstruction step: The data receiver obtains the estimated value of the gray value vector x from the observation vector y and the Gaussian random measurement matrix A, and then obtains the reconstructed image by pixel recombination;

[0009] The method by which the data receiver obtains the estimated value of the grayscale value vector x from the observation vector y and the Gaussian random measurement matrix A is:

[0010] 1) Initialization: Set the error threshold and the maximum number of iterations, set the current number of iterations t = 0, and set the initial values ​​of each element of the reconstructed estimate Set the variable z m Initial value of Set the variance σ of the observation signal 2 The initial value (σ 2 ) (0) and the variance C of each factor node n Initial value of Set the first element γ0 and the N+2th element γ of the hyperparameter γ N+1 unchanged, γ0=γ N+1 =0, the nth element γ n Initial value of Sequence number n=1,…,N, N is the dimension of the original signal, y m is the mth element of the observation vector y, with serial number m=1,…,M, where M is the dimension of the observation signal y and the superscript (·) indicates the number of iterative updates; set the parameters a, b, c, and d, and set the coupling coefficient β in the range [0,1];

[0011] 2) Update the reconstruction signal:

[0012]

[0013] Among them, A mn is the element in the mth row and nth column of the Gaussian random measurement matrix A;

[0014] 3) Update parameters:

[0015]

[0016]

[0017] Among them, ||·||2 is the L2 norm;

[0018] 4) Update t = t + 1, repeat steps 2) - 3) above until the error limit is met or the maximum number of iterations is reached, and output as an estimate of the grayscale value vector x.

[0019] In practical applications, the higher the calculation efficiency of the algorithm is, the greater the application value is. The beneficial effect of the present application is that the statistical characteristics can be effectively utilized to realize efficient sparse reconstruction of the block sparse signal without relying on a specific block division structure, and the potential of the compressive sensing technology in various application scenarios is improved. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1 is a factor graph of the AMP-PCSBL algorithm;

[0021] Figure 2 is a reconstruction result of a data set image; the first row is an original image; the second row is a reconstruction image of the SBL algorithm and the PCSBL algorithm; and the third row is a reconstruction image of the AMP-SBL algorithm and the AMP-PCSBL algorithm;

[0022] Figure 3 is a comparison of normalized mean square errors of different algorithms in a reconstruction image process;

[0023] Figure 4 is a comparison of time consumed by different algorithms in a reconstruction image process. DETAILED DESCRIPTION

[0024] The present application is based on the idea of message passing theory, and proposes an efficient block sparse signal reconstruction algorithm, which is named as the AMP-PCSBL algorithm. The algorithm uses a message passing process to calculate a posterior probability, simplifies an iteration step through a reasonable approximation under a large-scale assumption, and simultaneously adopts a mode coupling idea to constrain an estimation variance, so that efficient reconstruction of the block sparse signal can be realized without prior knowledge of block division.

[0025] The specific implementation method of the present application is further deduced and described as follows:

[0026] The AMP-PCSBL algorithm assumes that an observation signal y conforms to a Gaussian independent and identical distribution i.i.d model, that is, a posterior probability p of a known original signal x and y is:

[0027]

[0028] x is an original signal, N is a dimension of the original signal x, x is composed of N elements x n , n takes a value of 1 to N, A is a randomly generated Gaussian random measurement matrix, ||·||2 is an L2 norm; σ 2 is a variance of an observation signal, M is a dimension of the observation signal y, y is composed of M elements y m , m is 1 to M; exp is a natural exponential function;

[0029] In order to improve the recovery performance by utilizing the potential block sparse structure in some signals, considering that the sparse patterns of adjacent coefficients are statistically correlated, the idea of ​​pattern coupling is adopted when assuming signal priors. That is, the Gaussian sparse prior of each coefficient is affected not only by its own hyperparameters, but also by the adjacent hyperparameters, as shown in the following formula:

[0030]

[0031] γ is a hyperparameter, which consists of N+2 elements, including γ0, γ n , γ N+1 , γ0 is the first element, γ N+1 is the N+2th element, n=1,…,N, represents a Gaussian distribution, represents the Gaussian distribution function of x, E is the mean of the Gaussian distribution, F is the variance of the Gaussian distribution, and β is a parameter that measures the pattern correlation between the hyperparameters γ, β∈[0,1].

[0032] Assume γ0 = γ N+1 = 0. When β = 0, there is no correlation between the hyperparameters, which is consistent with the traditional Gaussian hierarchical prior model. When β > 0, adjacent hyperparameters jointly affect the sparsity of the coefficients, making the sparse solution tend to structured convergence. In addition, the gamma distribution Gamma is used as the hyperprior for the hyperparameter γ, that is:

[0033]

[0034] Gamma is commonly used for gamma distribution, a is the shape parameter of gamma distribution, b is the scale parameter of gamma distribution, and Γ is the gamma function.

[0035] Since both the posterior of y and the prior of x are Gaussian, the calculation is more efficient and convenient under the AMP framework. The posterior probability of the original signal x is:

[0036] p(x|y)∝p(y|x)p(x;γ)(4)

[0037] ∝ indicates correlation.

[0038] According to the message passing theory, we can get the factor graph as Figure 1 As shown. Among them, x n The elements of the sparse vector representing the original information are called variable nodes; f n is the prior information corresponding to each element; g mis the posterior information of each element of the observation vector y, which is called a factor node. Since all functions on the factor graph are Gaussian, and it can be proved that the messages calculated by the sum-product algorithm are also Gaussian, in subsequent iterations, we only need to care about the mean and variance of the message. The sum-product algorithm can be used to obtain the message expression transmitted between the variable node and the factor node:

[0039]

[0040] Among them, V represents the message, For slave node f n To node x n News; x -n Indicates that except x n All elements in the x vector except ; q, n, l, m are ordinal variables, if Then formula (6) can be simplified as:

[0041]

[0042] μ qm 、v qm Message The mean and variance of the Gaussian distribution of x q is the independent variable of Gaussian distribution; z mn 、C mn Message The mean and variance of the Gaussian distribution, A mn x n is the independent variable of Gaussian distribution; A mn Represents the mth row and nth column of A;

[0043] In message passing, after computing After that, you need to calculate in:

[0044]

[0045] When N is large, considering the limit approximation of large systems, the following approximation is available:

[0046]

[0047] The variance C of the message from the lth factor node to the nth variable node ln The approximation is to replace the message passing representation with the iteration of the n-th variable node itself.

[0048] Then formula (10) can be simplified as:

[0049]

[0050] Then we get the message passed from the variable node to the factor node The mean μ nm and variance v nm for:

[0051]

[0052] Based on the idea of ​​the AMP framework, the message passing process between the above nodes is converted into an iterative process of separate variable nodes. The approximate process is as follows:

[0053] Let the message passing variable μ nm =μ n +δμ nm , message passing variable z mn =z m +δz mn , then according to formula (13):

[0054]

[0055] μ n is a separate variable of the variable node, z m is a separate variable of the factor node, and δ is a preset minimum value;

[0056] According to formula (8):

[0057]

[0058] Substitute (16) into μ n We can get:

[0059]

[0060] Substitute equation (15) into z m We can get:

[0061]

[0062] Where the hyperparameters γ and variance σ 2 Using the expectation maximization EM algorithm for learning, the derivation process is as follows:

[0063] For the posterior probability expectation function Q, we have:

[0064]

[0065] is the expectation, [P] represents the distribution, Indicates that y is the prior and subject to γ,σ 2 The impact of , the expectation of the distribution of P with respect to x.

[0066] For the hyperparameter γ, substituting (2) and (3) into the above formula, we can obtain:

[0067]

[0068] Then in the iteration, the new hyperparameters are obtained as follows:

[0069]

[0070] Due to the mutual coupling between hyperparameters, it is difficult to obtain an analytical solution, so a suboptimal solution is used instead. Assuming γ * Is the optimal solution of formula (21), let the Q function be in γ * The first-order derivative of is 0, so γ * The upper and lower bounds of :

[0071]

[0072]

[0073] Therefore, the suboptimal solution for the hyperparameter γ can be chosen as:

[0074]

[0075] The iterative process of noise variance is similar, assuming that the noise variance follows a gamma distribution:

[0076]

[0077] The update formula is as follows:

[0078]

[0079] In summary, the process of the AMP-PCSBL algorithm can be summarized as follows:

[0080]

[0081]

[0082] Wherein, the superscript (0) indicates the initial value, 1e-10=1*10 -10 , 1e-4=1*10 -4 .

[0083] The effect of the present invention is shown by simulation Figure 2 , Figure 3 and Figure 4 To explain:

[0084] The algorithm's performance was verified through image reconstruction in simulation experiments. The experimental images used were from the MNIST handwritten digit dataset, a widely used dataset in image processing. The dataset images are grayscale images. Because digits occupy only a small portion of the image, the distribution of pixel grayscale values ​​in each image is sparse. When the image's pixel matrix is ​​split into columns, the grayscale values ​​of the columns containing digits are all zero at the edges, with non-zero values ​​only in the center. Combining all these columns into a single pixel column vector reveals that this column vector is not only sparse but also exhibits a blocky distribution. Four handwritten digit images were concatenated to form the experimental image, with an image size of 28×112. After expansion, the resulting signal length was N=3136. The perception matrix was a normalized Gaussian random matrix of size 1882×3136 (the observation vector length was M=1882). To simulate the transmission process, Gaussian random noise was added to the observation vector, with a signal-to-noise ratio range of [-10, 50] dB.

[0085] The experiment uses SBL algorithm, PCSBL algorithm and AMP-SBL algorithm as comparison. The image reconstruction results when the signal-to-noise ratio is 50dB are as follows Figure 2 The results of calculating the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) for the reconstructed images are shown in Table 1. It can be seen that the SBL algorithm achieves the best reconstruction results, while the AMP-PCSBL algorithm performs almost identically to the SBL algorithm. In contrast, the PCSBL algorithm loses some detail in the reconstructed image, while the AMP-SBL algorithm exhibits significant additional noise. The PSNR and SSIM of both algorithms are also poor.

[0086] The comparison results of the normalized mean square error (NMSE) of image reconstruction are as follows: Figure 3 As shown in Figure 2, as the signal-to-noise ratio increases, the NMSE of the SBL and AMP-PCSBL algorithms is significantly lower, outperforming the other algorithms by an order of magnitude. The performance gap between the AMP-PCSBL algorithm and the SBL algorithm is relatively small.

[0087] The running time comparison results of image reconstruction are as follows Figure 4 As shown in Figure 2, it can be seen that the computational complexity of the PCSBL algorithm and the SBL algorithm is high and the running time is too long, while AMP-PCSBL is the most efficient among the four algorithms, reducing the computational complexity from Reduced to

[0088] In summary, it can be seen that the AMP-PCSBL algorithm proposed in the present invention greatly improves the calculation speed of the algorithm at the cost of a slight loss of reconstruction accuracy, and realizes efficient reconstruction of block-sparse signals.

[0089] Table 1

[0090] Method name PSNR / dB SSIM SBL 40.7434 0.8163 PCSBL 30.5285 0.5999 AMP-SBL 28.1854 0.6283 AMP-PCSBL 37.4058 0.7083

Claims

1. An image compression sensing reconstruction method based on the AMP-PCSBL algorithm, comprising the steps of: Image compression step: The data sender expands the pixel matrix of the image data and uses the obtained grayscale value vector as the original signal x. The original signal x is compressed and sampled using the Gaussian random measurement matrix A to obtain the observation vector y. Image transmission step: The data sender transmits the observation vector y and the random measurement matrix A to the data receiver; Image reconstruction step: The data receiver obtains the estimated value of the gray value vector x from the observation vector y and the Gaussian random measurement matrix A, and then obtains the reconstructed image by pixel recombination; in, The method by which the data receiver obtains the estimated value of the grayscale value vector x from the observation vector y and the Gaussian random measurement matrix A is: 1) Initialization: Set the error threshold and the maximum number of iterations, set the current number of iterations t = 0, and set the initial values ​​of each element of the reconstructed estimate Set the variable z m Initial value of Set the variance σ of the observation signal 2 The initial value (σ 2 ) (0) and the variance C of each factor node n Initial value of Set the first element γ0 and the N+2th element γ of the hyperparameter γ N+1 unchanged, γ0=γ N+1 =0, the nth element γ n Initial value of Sequence number n=1,…,N, N is the dimension of the original signal, y m is the mth element of the observation vector y, with serial number m=1,…,M, where M is the dimension of the observation signal y and the superscript (·) indicates the number of iterative updates; set the parameters a, b, c, and d, and set the coupling coefficient β in the range [0,1]; 2) Update the reconstruction signal: Among them, A mn is the element in the mth row and nth column of the Gaussian random measurement matrix A; 3) Update parameters: Among them, ||·||2 is the L2 norm; 4) Update t = t + 1, repeat steps 2) - 3) above until the error limit is met or the maximum number of iterations is reached, and output as an estimate of the grayscale value vector x.

2. The method according to claim 1, wherein: set up 3. The method according to claim 1, wherein: Set parameters a=0.5, b=1*10 -10 , c=d=1*10 -4 .

4. A computer program product comprising a computer program / instructions, characterized in that When the computer program or instruction is executed by a processor, the steps of the method according to claim 1 are implemented.

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