Image compression sensing reconstruction method based on MOD-FMLM-SBL algorithm

Abstract

The present application belongs to the compressed sensing technology, provide a kind of based on MOD-FMLM-SBL algorithm's image compressed sensing reconstruction method, including image acquisition step, image compression step, image transmission step and image recovery step;In the sensing process of image transmission step, the influence of the noise precision parameter value on the sensing process is eliminated, only the estimation of the hyperparameter vector is concerned, while the source vector obtains a more sparse inductive prior distribution, and the sensing process is realized by a lower complexity calculation method, so as to realize the fast and efficient sparse signal reconstruction, and successfully realize the compressed sensing reconstruction of the image with better performance.

Description

TECHNICAL FIELD

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

[0002] In image processing, an image can exhibit good sparse characteristics in Fourier transform or wavelet transform, etc. Compressed sensing technology utilizes this characteristic to compress, transmit and restore the image, thereby greatly reducing the hardware requirements and improving the accuracy of the reconstructed image. Single-pixel camera is a good application. Rice University greatly reduces the order of magnitude of the measurement matrix by using compressed sensing technology, reduces the computational complexity, reduces the storage space and ensures the image quality. At the same time, the camera combined with the Bayer filter can also shoot color images. Compressed sensing technology also has important applications in medicine. In real life, due to physiological and physical limitations, the data acquisition cost of nuclear magnetic resonance is extremely expensive. The application of compressed sensing technology can effectively reduce the amount of data acquisition, obtain less and effective data, and thereby reduce the medical expenses of patients. In the process of hyperspectral image processing, compressed sensing technology has also been applied. When the signal collector receives the signal reflected from the ground, different substances reflect different frequencies of waves differently. Combined with the reflected waves of all substances to form a corresponding measurement matrix, the classification of hyperspectral images is realized.

[0003] Sparse Bayesian Learning (SBL) algorithm is an important part of the compressed sensing reconstruction algorithm. In the SBL algorithm, the optimization of the marginal likelihood function is usually realized by the Expectation Maximization (EM) principle. However, the Sparse Bayesian Learning (SBL) algorithm based on the EM process is accompanied by the inverse process of the matrix at each iteration. Generally, the computational complexity of the inverse operation of an N-dimensional square matrix is O(N 3 ). Therefore, when processing data, the traditional SBL algorithm will have an exponential explosion of computational complexity with the increase of data dimension, which greatly hinders the operation speed of the SBL algorithm when processing high-dimensional data. Therefore, many fast SBL algorithms have been proposed.

[0004] In the fast SBL algorithm based on the Fast Marginal Likelihood Maximization (FMLM) idea, the estimation result of the noise precision hyperparameter β introduced to the source vector will affect the marginal likelihood function and the determination of the support set. However, the noise precision hyperparameter β is difficult to handle, and inappropriate values of it will greatly affect the performance of sparse signal reconstruction. SUMMARY

[0005] The technical problem to be solved by the present invention is to provide an image compressed sensing method that can eliminate the influence of noise accuracy parameter values ​​on the source vector and can quickly and efficiently complete sparse signal reconstruction.

[0006] The technical solution adopted by this invention to solve the above-mentioned technical problems is an image compressed sensing reconstruction method based on the MOD-FMLM-SBL algorithm, comprising the following steps:

[0007] Image acquisition steps: The data acquisition party acquires the image to be transmitted and sends it to the data sender;

[0008] Image compression steps: The data sender performs wavelet transform on the image to be transmitted to obtain wavelet coefficients, and uses the measurement matrix to compress and sample the wavelet coefficients to obtain the observation vector y;

[0009] Image transmission steps: The data sender uses the observation vector y and the corresponding measurement matrix Φ as input to the sensing process, obtains the sparse reconstructed wavelet coefficients through sensing, and sends the sparse reconstructed wavelet coefficients to the data receiver to complete the image transmission.

[0010] Image restoration steps: The data receiver recovers the image by using inverse wavelet transform on the sparsely reconstructed wavelet coefficients;

[0011] The perception process specifically includes:

[0012] 1) Obtain the hyperparameter vector α used to determine the source vector w in the SBL algorithm, where the i-th element of α is α_i. i Represented as:

[0013]

[0014] α i =∞, otherwise

[0015] Where M is the length of the observation vector y, c represents the location parameter of the Gamma distribution, and the median value s i q i g i They are respectively:

[0016]

[0017] C -i This is the matrix obtained by removing the i-th column from the intermediate quantity C. Let Φ be the i-th column of the observation matrix Φ, k be the inverse scaling parameter of the noise precision Gamma distribution, and C = I + ΦA be the intermediate quantity. -1 Φ T Let I be the identity matrix, and A be an N*N diagonal matrix whose i-th diagonal element is α.i ;

[0018] 2) α i =∞ The i-th element of the corresponding source vector w is updated to 0, thereby deleting the i-th column in the corresponding observation matrix Φ. Then, steps 1) and 2) are performed sequentially for i = 1, 2, ... N, to obtain an efficient SBL algorithm;

[0019] 3) Obtain the estimated value of the source vector by sparsely reconstructing the source vector w based on the hyperparameter vector α. The estimated value of the source vector As wavelet coefficients for sparse reconstruction.

[0020] This invention proposes a new perception process that uses integration to extract the noise precision hyperparameter β in the original SBL algorithm, replacing the step of finding a point estimate for β in the original algorithm. This allows the improved algorithm to focus only on the estimation of the hyperparameter vector α.

[0021] The beneficial effects of this invention are that it eliminates the influence of noise precision parameter values ​​on the sensing process, obtains a more sparse and induced prior distribution for the source vector, and implements the sensing process through a computationally less complex method, thereby achieving fast and efficient sparse signal reconstruction and successfully realizing compressed sensing reconstruction of images with better performance. Attached Figure Description

[0022] Figure 1 A schematic diagram illustrating the monotonicity of the log-likelihood function with respect to a single hyperparameter;

[0023] Figure 2 The images are the reconstruction results in the examples. The first row shows the image reconstructed by inverse wavelet transform; the second row shows the image reconstructed by the MOD-FMLM-SBL algorithm; and the third row shows the image reconstructed by the FMLM-SBL algorithm. Detailed Implementation

[0024] This invention proposes a fast SBL algorithm, named MOD-FMLM-SBL, based on the idea of ​​Fast Marginal Likelihood Maximization (FMLM). This algorithm introduces a noise precision parameter into the source vector and integrates this parameter, eliminating the influence of the noise precision parameter's value on the algorithm. Simultaneously, the source vector obtains a more sparsity-inducible prior distribution. Furthermore, the MOD-FMLM-SBL algorithm utilizes the matrix inversion lemma to transform the matrix inversion operation in the reconstruction process into a lower-complexity iterative update formula, thereby achieving fast and efficient sparse signal reconstruction. This invention applies the MOD-FMLM-SBL algorithm to image reconstruction, successfully achieving compressed sensing image reconstruction.

[0025] I. The derivation process is as follows:

[0026] The source vector w has a length of N, and the hyperparameter vector α also has a length of N.

[0027] In the SBL algorithm model, the source vector w is controlled only by the hyperparameter α. The MOD-FMLM-SBL algorithm of this invention incorporates a noise accuracy hyperparameter β.

[0028]

[0029] Where p(w|α,β) represents the probability distribution of w under conditions α,β; N is the length of the source vector w; w i Represents the i-th element in w; α i This represents the i-th element in α; Indicates w i It satisfies the condition that the mean is 0 and the variance is . The hyperparameter β follows a Gaussian distribution; the hyperparameter β follows a Gamma distribution, as shown in the following equation.

[0030] p(β)=Gamma(β∣c,k)

[0031] Where c represents the location parameter of the Gamma distribution; k represents the inverse scaling parameter of the Gamma distribution.

[0032] The only difference between the above framework and the original SBL algorithm framework is that the hyperparameter β controls the prior information of the source vector. This improvement facilitates obtaining an analytical expression for the integrals involved in subsequent processes. Given the hyperparameter α and the observation vector y, the likelihood function of the source vector w can be written as:

[0033]

[0034] Where d denotes the differential operator; e is the natural constant; M is the length of the observation vector y, i.e., the number of rows in the observation matrix Φ; Γ is the Gamma function; exp denotes the exponential function with the natural constant e as its base; the superscript T denotes transpose; μ and ∑ are intermediate quantities, and their expressions are as follows:

[0035] μ=∑Φy

[0036] ∑=(Φ T Φ+A) -1

[0037] A is a diagonal matrix of size N*N, where the i-th diagonal element is α. i .

[0038] By integrating the hyperparameter β, it can be found that the likelihood function of the source vector follows a multidimensional Student-t distribution. This distribution not only facilitates obtaining sparse solutions for the source vector but also exhibits better robustness in handling outliers in the observation vector.

[0039] Integrating the hyperparameter β in the marginal likelihood function p(y|α,β) yields a new marginal likelihood function, which is:

[0040]

[0041] Where C is a constant; intermediate quantity C = I + ΦA -1 Φ T I is the identity matrix. Using the FMLM concept, C can be decomposed into:

[0042]

[0043] Among them, C -i This is the matrix obtained by removing the i-th column from C; Let be the i-th column of Φ. Using the properties of matrices, the determinant |C| of matrix C and its inverse operation Ci... -1 It can be represented in the following form:

[0044]

[0045] Therefore, the marginal likelihood function is written as

[0046]

[0047] Where, α -i To remove the i-th element α from α i The subsequent hyperparameter vector; intermediate quantity s i q i g i Defined as follows

[0048]

[0049] Similarly, considering the partial derivative of the log-likelihood function with respect to the hyperparameter α, and setting it to 0, we have

[0050]

[0051] The solution to the above equation is

[0052]

[0053] α i =∞, otherwise

[0054] This is the likelihood function. The stationary points of the likelihood function. According to mathematical knowledge, stationary points are not necessarily extreme points of the function. In order to explore the properties of these stationary points, it is necessary to consider the monotonicity of the likelihood function.

[0055] For ease of calculation, it will be compared with α iReplace irrelevant items.

[0056]

[0057] Using the above substitutions, the log-likelihood function The second derivative with respect to the hyperparameter α can be written as

[0058]

[0059] Consider the two stationary points obtained in the previous text for the above equation.

[0060] when When noting that the second term in the numerator on the right side of the second derivative equation is 0, we have...

[0061]

[0062] As can be seen, the above equation is always negative, therefore the likelihood function l(α) i It has a unique maximum value. This maximum value is the first stationary point.

[0063] When α i When α = ∞, the second and all other higher-order derivatives of the likelihood function are 0 at that point. However, according to the first derivative of the likelihood function, when α... i As we approach infinity, the sign of the first derivative of the log-likelihood function changes from... Decision. The classification and discussion of U are as follows:

[0064] (1) When U>0, At this point, the first derivative of the log-likelihood function is at α i When α = ∞, it is a negative value, which indicates that when α i As the log-likelihood function decreases, it will gradually increase to its unique maximum value, which is still the first stationary point.

[0065] (2) When U < 0, At this point, the log-likelihood function is monotonically increasing on its domain, α i =∞ is the maximum point of the likelihood function.

[0066] (3) When U = 0, At this point, the maximum point of the log-likelihood function is still the first stationary point.

[0067] In summary, the log-likelihood function with respect to a single hyperparameter α i Monotonicity such as Figure 1 As shown.

[0068] Reviewing the hierarchical prior model of the SBL algorithm, we know that let α i =∞ is equivalent to letting w i =0, thus Φ of column i can be... (basis functions) The stationary point formula is removed from the observation matrix Φ. Therefore, the stationary point formula controls the basis functions. The increments and decrements are then performed. By sequentially performing the above operations on i = 1, 2, ..., N, an efficient SBL algorithm can be obtained.

[0069] II. Based on the above derivation, this invention proposes an image compressed sensing reconstruction method based on the MOD-FMLM-SBL algorithm, comprising the following steps:

[0070] Image acquisition steps: The data acquisition party acquires the image to be transmitted and sends it to the data sender;

[0071] Image compression steps: The data sender performs wavelet transform on the image to be transmitted to obtain wavelet coefficients, and uses the measurement matrix to compress and sample the wavelet coefficients to obtain the observation vector y;

[0072] Image transmission steps: The data sender uses the observation vector y and the corresponding measurement matrix Φ as input to the sensing process, obtains the sparse reconstructed wavelet coefficients through sensing, and sends the sparse reconstructed wavelet coefficients to the data receiver to complete the image transmission.

[0073] The specific details of the perception process are as follows:

[0074] 1) Obtain the hyperparameter vector α used to determine the source vector w in the SBL algorithm, where the i-th element of α is α_i. i Represented as:

[0075]

[0076] α i =∞, otherwise

[0077] Where M is the length of the observation vector y, and the intermediate value s i q i g i They are respectively:

[0078]

[0079] α -i To remove the i-th element α from α i The subsequent hyperparameter vector, C -i This is the matrix obtained by removing the i-th column from the intermediate quantity C. Let Φ be the i-th column of the observation matrix Φ, k be the scale parameter of the Gamma distribution, and the intermediate quantity C = I + ΦA. -1 Φ T Let I be the identity matrix, and A be an N*N diagonal matrix whose i-th diagonal element is α. i ;

[0080] 2) α i =∞ The i-th element of the corresponding source vector w is updated to 0, thereby deleting the i-th column in the corresponding observation matrix Φ. Steps 1) and 2) are completed sequentially for i = 1, 2, ... N.

[0081] 3) Obtain the estimated value of the source vector by sparsely reconstructing the source vector w based on the hyperparameter vector α. The estimated value of the source vector The wavelet coefficients are used for sparse reconstruction. Image restoration steps: The data receiver uses the inverse wavelet transform to recover the image from the sparsely reconstructed wavelet coefficients.

[0082] III. Simulation Analysis

[0083] This experiment verifies the algorithm's performance through image reconstruction. A hybrid compressed sensing framework is used, applied to the wavelet transform of the image. The wavelet transform employs the "symmlet8" wavelet basis, with a coarsest scale parameter of 3 and a finest scale parameter of 7. The wavelet coefficients, i.e., the source vector length, are N = 16320, and the observation vector length is M = 10672. The image observation matrix Φ is taken from a uniform spherical distribution with a size of 10672 × 16320.

[0084] This experiment performed compressed sensing reconstruction on images of Indor, Mondrian, and Lena, with image sizes of 256×256 pixels, 512×512 pixels, and 512×512 pixels, respectively, all with a bit depth of 8. These images have been widely used in [3,4] and other literature to evaluate the performance of the SBL algorithm, and can be obtained online at websites such as http: / / sparselab.stanford.edu / and http: / / descai.ugr.es / cvg / adimagenes / .

[0085] The experiment used the FMLM-SBL algorithm for comparison. The image reconstruction results are as follows: Figure 2 As shown in the figure, the first row displays the reconstruction results of the three images using inverse wavelet transform, with the observation vector length set to M = N = 16384, i.e., linear reconstruction, which can be considered the theoretically best performance achievable by the compressed sensing reconstruction algorithm in this invention. The second row shows the images reconstructed using the MOD-FMLM-SBL algorithm, demonstrating that the MOD-FMLM-SBL algorithm reconstructed the three images with high quality. The third row shows the reconstruction results of the FMLM-SBL algorithm.

[0086] This experiment evaluates the algorithm's reconstruction performance from three aspects: normalized root-mean-square error (nRMSE), runtime required to reach convergence, and sparsity of the reconstruction results. The results are shown in Table 1. Except for linear reconstruction, the optimal values ​​for each image reconstruction metric are highlighted in bold. It can be seen that the MOD-FMLM-SBL algorithm proposed in this invention significantly accelerates the convergence speed while substantially reducing the sparsity of the reconstruction results, at the cost of a small loss in accuracy.

[0087] Table 1. Performance comparison of FMLM-SBL and MOD-FMLM-SBL algorithms for image reconstruction.

[0088]

Claims

1. An image compressed sensing reconstruction method based on the MOD-FMLM-SBL algorithm, characterized in that, The MOD-FMLM-SBL algorithm is an FMLM-SBL algorithm that introduces a noise precision parameter into the source vector and integrates this noise precision parameter using an integral, including the following steps: Image acquisition steps: The data acquisition party acquires the image to be transmitted and sends it to the data sender; Image compression steps: The data sender performs wavelet transform on the image to be transmitted to obtain wavelet coefficients, and uses the measurement matrix to compress and sample the wavelet coefficients to obtain the observation vector y; Image transmission steps: The data sender uses the observation vector y and the corresponding measurement matrix Φ as input to the sensing process, obtains the sparse reconstructed wavelet coefficients through sensing, and sends the sparse reconstructed wavelet coefficients to the data receiver to complete the image transmission. Image restoration steps: The data receiver recovers the image by using inverse wavelet transform on the sparsely reconstructed wavelet coefficients; The perception process specifically includes: 1) Obtain the hyperparameter vector α used to determine the source vector w in the MOD-FMLM-SBL algorithm, where the i-th element α is... i Represented as: Where M is the length of the observation vector y, c represents the location parameter of the Gamma distribution, and the median value s i q i g i They are respectively: C -i This is the matrix obtained by removing the i-th column from the intermediate quantity C. Let Φ be the i-th column of the observation matrix Φ, k be the inverse scaling parameter of the Gamma distribution of the noise vector, and C = I + ΦA be the intermediate quantity. -1 Φ T Let I be the identity matrix, and A be an N*N diagonal matrix whose i-th diagonal element is α. i ; 2) α i =∞ The i-th element of the corresponding source vector w is updated to 0, thereby deleting the i-th column in the corresponding observation matrix Φ, and then steps 1) and 2) are completed sequentially for i = 1, 2, ... N; 3) Obtain the estimated value of the source vector by sparsely reconstructing the source vector w based on the hyperparameter vector α. The estimated value of the source vector As wavelet coefficients for sparse reconstruction.

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