Image denoising method based on Markov robust sparse Bayesian learning

Abstract

According to the image denoising method based on Markov robust sparse Bayesian learning provided by the invention, the state probability of the current sample is corrected by using the state information of the previous sample under the condition that the calculation complexity is not increased, the robustness of the RSBL algorithm is improved, and the salt and pepper noise of the image can be effectively removed. The process of the image denoising method is as follows: dividing an image into pixel blocks with the same size, establishing a regression model between the position of a pixel and the intensity of the pixel for each pixel block, converting the regression problem into a sparse reconstruction problem, initializing parameters related to the method, inputting a sensing matrix and an observation vector, and obtaining a sparse reconstruction model; and carrying out iterative updating on the posteriori distribution of the parameters and the sparse vectors, finally taking the sparse vector approximate posteriori expectation as a weight vector of a regression model, predicting the intensity of the central position of the pixel block by using the regression model, and carrying out prediction operation on all the pixel blocks to obtain a denoised image.

Description

Technical Field

[0001] The invention relates to digital image processing technology, in particular to an image denoising technology based on Markov robust sparse Bayesian learning. Technical Background

[0002] Compressed sensing is a theoretical framework for simultaneous sampling and data compression. As an important component of compressed sensing, the sparse reconstruction algorithm can restore the original signal through a Bayesian learning framework.

[0003] In practical applications, a regression model can be established between the image pixel position and pixel intensity, where the matrix composed of the kernel function corresponds to the perception matrix in the compressed sensing model, the weight vector of the kernel function corresponds to the sparse vector in the compressed sensing model, and the vector composed of pixel intensities corresponds to the observation vector. After reconstructing the weight vector through the sparse reconstruction algorithm, denoising can be achieved by predicting the pixel intensity of each point through the regression model. Generally, the sparse reconstruction algorithm based on the traditional Bayesian learning framework can achieve good reconstruction results, but in some special environments, it will be affected by impact noise, resulting in performance degradation, so it is necessary to design a solution to remove impact noise. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a method for removing salt and pepper noise of an image based on robust sparse reconstruction suitable for an impulsive noise environment, in view of the fact that in a real environment, the salt and pepper noise on the image meets the characteristics of short-duration and high-energy impulsive noise.

[0005] The technical solution adopted by the present invention to solve the above technical problems is an image denoising method based on Markov robust sparse Bayesian learning, called Markov-BG-RSBL algorithm, comprising the steps of:

[0006] The image is divided into pixel blocks of uniform size. The total number of pixels in each pixel block is M. The vector composed of the pixel intensities of each pixel in the image block is used as the observation vector y; the i-th element in the observation vector y is the observation value y of the i-th pixel i , which is the pixel intensity of the i-th pixel in the image;

[0007] The impact noise and non-impact noise are expressed using two Gaussian distribution models with different parameters, using the flag variable z at the i-th pixel in the flag vector z i For the observed value y i Whether it is affected by impact noise or not, i =1 means that the ith pixel is subject to impact noise, z i = 0 means that the i-th pixel is subject to non-impact noise; let the current flag variable z i The previous flag variable z i-1The influence of 0,i Indicates z i-1 =0 time z i =0, let parameter π 1,i Indicates z i-1 =1 when z i =1 probability, π 0,i and π 1,i associated via correlation parameter x;

[0008] A regression model is established between the pixel position and the pixel intensity for each pixel block, and the regression problem of the regression model is converted into a sparse reconstruction problem in the compressed sensing model, wherein the matrix formed by the kernel function of the regression model corresponds to the perception matrix in the compressed sensing model, and the weight vector of the kernel function of the regression model corresponds to the sparse vector in the compressed sensing model;

[0009] Solve the posterior estimate μ of the sparse vector based on the perception matrix A, the observation vector y, and the correlation parameter x s , and the posterior estimate μ of the sparse vector s The weight vector of the kernel function of the regression model is used to predict the pixel intensity of each point in the pixel block. After the prediction operation is performed on all pixel blocks, the denoised image is obtained.

[0010] The present invention models noise as a Markov-Gaussian distribution, describes the impact noise and normal noise by two variances, uses the state information of the previous sample to correct the state probability of the current sample, and designs a new robust sparse Bayesian learning algorithm, which iteratively updates parameters and the approximate posterior distribution of sparse signals, and finally uses the approximate posterior expectation of the sparse signal as the estimated value of the kernel function weight.

[0011] The beneficial effect of the present invention is that a sparse reconstruction result with higher accuracy is obtained without increasing the computational complexity, and finally the salt and pepper noise denoising is realized by predicting the image pixel intensity using a regression model. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A hierarchical Bayesian model of the Markov-BG-RSBL algorithm of the present invention;

[0013] Figure 2 Comparison of reconstruction errors between the Markov-BG-RSBL algorithm and the BG-RSBL algorithm in a burst noise environment;

[0014] Figure 3 Comparison of reconstruction errors between the Markov-BG-RSBL algorithm and the BG-RSBL algorithm in a Gaussian noise environment;

[0015] Figure 4The image denoising results include (a) the image contaminated by salt and pepper noise, (b) the denoised image using the Markov-BG-RSBL algorithm, and (c) the denoised image using the median filter.

[0016] Figure 5 Detailed comparison of (a) Markov-BG-RSBL algorithm and (b) median filter for removing salt and pepper noise. DETAILED DESCRIPTION

[0017] The following is the derivation and description of the Markov-BG-RSBL algorithm proposed in the present invention. The Markov-BG-RSBL algorithm model is as follows: Figure 1 As shown:

[0018] Divide the image into pixel blocks of the same size, determine the perception matrix A and the total number of pixel blocks M of the image, and use the image to be denoised as the observation vector: y = [y1, y2, ..., y M ] T , T Indicates transposition. The i-th element in the observation vector y is the observation value y i , that is, the pixel value of the i-th pixel block in the image, and the value range of variable i is 1 to M. The observation vector y corresponds to the sign vector z. The impact noise and non-impact noise are expressed using two Gaussian distribution models with different parameters, using the sign variable z i For the observed value y i Distinguish whether it is affected by impact noise.

[0019]

[0020] Among them, a i is the i-th row of the perception matrix A; s is a sparse vector, let each element s in s i They obey a Gaussian distribution with zero mean and unknown variance and are independent of each other. The elements s i The probability density of is expected to be 0, and the variance is Gaussian distribution δ i Indicates i The accuracy of follows the gamma distribution Gamma(a,b), where a and b are hyperparameters that control the shape of the gamma distribution; each δ i The precision vector v that makes up the sparse vector s; n 0,i Indicates z i = 0, the non-impact noise at the i-th pixel follows the Gaussian η0 represents n 0,i The accuracy of follows the gamma distribution Gamma(c0,d0), where c0 and d0 are hyperparameters that control the shape of the gamma distribution; n 1,i Indicates z i= 1, the impulse noise at the i-th pixel follows a Gaussian distribution η1 represents n 1,i The accuracy of z follows the gamma distribution Gamma(c1,d1), where c1 and d1 are hyperparameters that control the shape of the gamma distribution; i represents the state variable at the i-th pixel, which follows the Beta-Bernoulli distribution.

[0021] The conditional likelihood function of the observation vector y is established as:

[0022]

[0023] In addition, the model of this algorithm assumes that the current state probability z i Subject to the previous state probability z i-1 The influence of. Let the parameter π 0,i Indicates z i-1 =0 time z i =0 follows the Beta distribution Beta(g0,h0), where g0 and h0 are hyperparameters that control the shape of the Beta distribution; let parameter π 1,i Indicates z i-1 =1 when z i =1 follows the Beta distribution Beta(g1,h1), where g1 and h1 are hyperparameters that control the shape of the Beta distribution. 0,i and π 1,i By correlation parameter x, we have:

[0024] π 0,i =π 1,i +x (3)

[0025] where x∈[0,1].

[0026] The variational inference framework is used to obtain the approximate posterior distribution q of each parameter, and each parameter is iteratively updated. The approximate posterior probability can be obtained by the following formula:

[0027]

[0028] where q(θ i ) represents the variable θ subvariable θ i The distribution of When the KL divergence is equal to 0, q(θ) is equal to the posterior p(θ,y), so updating q(θ) can obtain the approximate posterior distribution of the unknown parameters, and we can get:

[0029] (1) Approximate posterior distribution of s q(s)

[0030]

[0031] Among them, ∝ is the proportional symbol, D z=diag( <z>)、D 1-z =I-diag( <z>) and D δ = diag(<δ>). diag means converting the vector into a diagonal matrix, I is the unit matrix, <> means the posterior estimate, <> A It means to find the posterior expectation of the probability of subscript A, and define D=<η0>D 1-z +<η1>D z It can be seen that the approximate posterior distribution of s conforms to the characteristics of Gaussian distribution, and its expected vector μ s and the covariance matrix Φ s It is expressed as:

[0032]

[0033] (2) Approximate posterior distribution of δ q(δ)

[0034]

[0035] in It can be seen that q(δ) is distribution, where the gamma distribution shape control parameter

[0036]

[0037] (3) Approximate posterior distribution of η0 q(η0)

[0038]

[0039] It shows that q(η0) is Distribution, where:

[0040]

[0041] (4) Approximate posterior distribution of η1 q(η1)

[0042]

[0043] It shows that q(η1) is Distribution, where:

[0044]

[0045] (5) Approximate posterior distribution of z q(z)

[0046]

[0047] It can be seen that:

[0048]

[0049] C is a constant coefficient. So z i The posterior expectation of is:

[0050] (6)π 0,i The approximate posterior distribution q(π 0,i )

[0051]

[0052] It can be seen that q(π 0,i ) is Beta Distribution, where:

[0053]

[0054] (7)π 1,i The approximate posterior distribution q(π 1,i )

[0055]

[0056] It can be seen that q(π 1,i )for Distribution, where:

[0057]

[0058] From the properties of Beta distribution, we know that:

[0059]

[0060] Where Ψ(·) represents the Digamma function. y is the observation vector Observation of the model of the Markov-BG-RSBL algorithm. s i 、n i 、z i and z i-1 Hidden variable belonging to the Markov-BG-RSBL algorithm, n i Include 0,i ,n 1,i δ i ,η0,η1,π 0,i , π 1,i Parameters of the model belonging to the Markov-BG-RSBL algorithm. a, b, c0, d0, c1, d1, g0, h0, g1, h1 are Hyperparameters of the model belonging to the Markov-BG-RSBL algorithm.

[0061] In summary, the process of the Markov-BG-RSBL algorithm is summarized as follows:

[0062]

[0063]

[0064] The posterior estimate μ of the sparse vector s As the weight vector of the kernel function of the regression model, the regression model is then used to predict the pixel intensity of each point in the pixel block, and the denoised image is obtained after the prediction operation is performed on all pixel blocks. Figure 2 , Figure 3 and Figure 4 illustrate:

[0065] The simulation experiment first analyzes and verifies the performance of Markov-BG-RSBL under different noises. The experiment sets the correlation parameter x = 0.1, the observation vector length M = 100, the sparse vector length N = 200, and the sparsity K = 5. The number of Monte Carlo experiments in each round is 500, and the final result is averaged. The experimental noise is burst impact noise and Gaussian noise. It can be seen that in the burst impact noise environment, the reconstruction performance of Markov-BG-RSBL is significantly improved compared with BG-RSBL; in the Gaussian environment without impact noise, the performance is average, and the performance is the same as BG-RSBL in the range of -5dB to 10dB. When it is lower than -5dB, the reconstruction error is larger. The reason is that the large numerical noise with a small probability in Gaussian noise affects Markov-BG-RSBL's judgment on subsequent samples. Among them, BG-RSBL is published in Rong J, Zhang J, Duan H. Robust sparse Bayesian learning based on the Bernoulli-Gaussian model of impulsive noise [J]. Digital Signal Processing, 2023, 136: 104013.

[0066] The image denoising experiment uses a kernel function composed of a Gaussian kernel and a polynomial kernel, which is published in Mitra K, Veeraraghavan A, Chellappa R. Robust RVM regression using sparse outlier model [C]. 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010, 1887-1894. The kernel function is used to denoise the Lena image. The image size is 512×512, and the image depth is 8 bits. In order to simulate the polluted image, 20% salt and pepper noise is added to the original image. Before denoising, the image is divided into many 6×6 pixel areas, and regression fitting is performed on each area, and then the inferred parameters are used to predict the intensity of the central pixel of the area. The experiment uses a 6×6 median filter as a comparison. From the results, it can be seen that the present invention can better retain the edge details of the Lena image under the premise of removing the salt and pepper noise.< / z> < / z>

Claims

1. An image denoising method based on Markov robust sparse Bayesian learning, characterized in that: Includes steps: The image is divided into pixel blocks of the same size. The total number of pixels in each pixel block is M. The vector composed of the pixel intensities of each pixel in the image block is used as the observation vector y. The i-th element in the observation vector y is the observation value yi of the i-th pixel, that is, the pixel intensity of the i-th pixel in the image. The impact noise and non-impact noise are expressed using two Gaussian distribution models with different parameters, using the flag variable z at the i-th pixel in the flag vector z i For the observed value y i Whether it is affected by impact noise or not, i =1 means that the ith pixel is subject to impact noise, z i = 0 means that the i-th pixel is subject to non-impact noise; let the current flag variable z i The previous flag variable z i-1 The influence of 0，i Indicates z i-1 =0 time z i =0, let parameter π 1，i Indicates z i-1 =1 when z i =1 probability, π 0，i and π 1，i associated via correlation parameter x; A regression model is established between the pixel position and the pixel intensity for each pixel block, and the regression problem of the regression model is converted into a sparse reconstruction problem in the compressed sensing model, wherein the matrix formed by the kernel function of the regression model corresponds to the perception matrix in the compressed sensing model, and the weight vector of the kernel function of the regression model corresponds to the sparse vector in the compressed sensing model; Solve the posterior estimate μ of the sparse vector based on the perception matrix A, the observation vector y, and the correlation parameter x s , and the posterior estimate μ of the sparse vector s The weight vector of the kernel function of the regression model is used to predict the pixel intensity of each point in the pixel block. After the prediction operation is performed on all pixel blocks, the denoised image is obtained.

2. The method according to claim 1, characterized in that: The impact noise and non-impact noise are expressed using two Gaussian distribution models with different parameters, using the corresponding flag variable z in the flag vector z i For the observed value y i The specific method to distinguish whether it is affected by impact noise is: Among them, a i is the i-th row of the perception matrix A; s is a sparse vector, let each element s in s i They obey a Gaussian distribution with zero mean and unknown variance and are independent of each other. The elements s i The probability density of is expected to be 0, and the variance is Gaussian distribution δ i Indicates i The accuracy of follows the gamma distribution Gamma(a, b), where a and b are hyperparameters that control the shape of the gamma distribution; each δ i The precision vector δ that makes up the sparse vector s; n 0，i Indicates z i = 0, the non-impact noise at the ith pixel has an expectation of 0 and a variance of Gaussian distribution η0 represents n 0,i The accuracy of follows the gamma distribution Gamma(c0,d0), where c0 and d0 are hyperparameters that control the shape of the gamma distribution; n 1,i Indicates z i = 1, the impulse noise at the ith pixel has an expectation of 0 and a variance of Gaussian distribution η1 represents n 1,i The accuracy of follows the gamma distribution Gamma(c1,d1), where c1 and d1 are hyperparameters that control the shape of the gamma distribution.

3. The method according to claim 2, characterized in that: The state variable z at the i-th pixel i It follows the Beta-Bernoulli distribution.

4. The method according to claim 3, characterized in that: Solve for the posterior estimate μ of the sparse vector s The specific method is: Receive the input perception matrix A, observation vector y and correlation parameter x; initialize hyperparameters a, b, c0, d0, c1, d1, g0, h0, g1, h1; Iteration steps: Compute the posterior estimate μ of the sparse vector s and the covariance matrix Φ of the sparse vector s , μ s =Φ s A T Dy,Φ s =(A T DA+D δ ) -1 ; Among them, the intermediate quantity D δ =diag(<δ>), where diag means converting the vector into a diagonal matrix, <> means the posterior expected estimate, and the intermediate quantity D = <η0>D 1-z +<η1>D z , D z =diag( <z>)、D 1-z =I-diag( <z> ), I is the identity matrix;< / z> < / z> Calculate the gamma distribution shape control parameters based on hyperparameters a and b and Calculate the precision vector estimate <δ> for each <δ i >, Complete the update of the precision vector estimate <δ>; Calculate the gamma distribution shape control parameters using hyperparameters c0, d0, c1, d1 T represents transpose; Using the Gamma Distribution Shape Control Parameters Update the accuracy estimates <η0> and <η1>, where Calculate the Beta distribution shape control parameters using hyperparameters g0, h0, g1, h1 Each element in Complete the control parameters of the Beta distribution shape Update, Calculate the elements of the estimated value of the signature vector z Complete the flag vector estimation <z>Update of; where q(z i =1) and q(z i =0) indicates z i =1 and z i = 0 posterior distribution probability;< / z> Determine whether the iteration stop condition is met. If so, output the posterior estimate μ of the current sparse vector s , otherwise, another round of iteration is performed.

5. The method according to claim 1, characterized in that: Parameter π 0，i and parameter π 1，i By correlation parameter x, we have: π 0，i =π 1，i +x, where x∈[0,1].

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

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