An image denoising method based on Markov robust sparse Bayesian learning

Abstract

The present invention proposes an image denoising method based on Markov robust sparse Bayesian learning. Without increasing computational complexity, the state information of the previous sample is used to correct the state probability of the current sample, thereby improving the robustness of the RSBL algorithm and effectively removing salt and pepper noise from the image. The process of the image denoising method is as follows: the image is divided into pixel blocks of uniform size, a regression model is established between the pixel position and the pixel intensity for each pixel block, and the regression problem is converted into a sparse reconstruction problem. After initializing the parameters involved in the present invention, the perception matrix and observation vector are input, the posterior distribution of the parameters and the sparse vector are iteratively updated, and finally the sparse vector approximate posterior expectation is used as the weight vector of the regression model. The regression model is then used to predict the intensity of the center position of the pixel block. After performing the prediction operation on all pixel blocks, the denoised image is obtained.

Description

Technical Field

[0001] The present 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 that simultaneously performs sampling and data compression. The sparse reconstruction algorithm, as an important component of compressed sensing, can restore the original signal through a Bayesian learning framework.

[0003] In practical applications, a regression model can be established between image pixel positions and pixel intensities. The kernel function matrix corresponds to the perception matrix in the compressed sensing model, the kernel function weight vector corresponds to the sparse vector in the compressed sensing model, and the pixel intensities vector corresponds to the observation vector. After reconstructing the weight vector using a sparse reconstruction algorithm, denoising can be achieved by predicting the pixel intensity of each point using a regression model. While sparse reconstruction algorithms based on traditional Bayesian learning frameworks generally achieve good reconstruction results, they can be affected by impulsive noise in certain environments, leading to performance degradation. Therefore, a solution for removing impulsive noise is needed. 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 images based on robust sparse reconstruction, which is suitable for the impact noise environment and has the characteristics of short duration and high energy of salt and pepper noise in real environments.

[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, which includes the following steps:

[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 , that 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 To distinguish whether it is affected by impact noise, i =1 means the i-th 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 represents z i-1 =0 time z i =0, let parameter π 1,i represents z i-1 =1 when z i =1 probability, π 0,i and π 1,i associated via the correlation parameter x;

[0008] For each pixel block, a regression model is established between the pixel position and the pixel intensity, and the regression problem of the regression model is converted into a sparse reconstruction problem in the compressed sensing model. 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. The algorithm iteratively updates the parameters and the approximate posterior distribution of the sparse signal, 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 salt and pepper noise denoising is achieved by using a regression model to predict image pixel intensity. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 is 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 salt and pepper noise removal using (a) Markov-BG-RSBL algorithm and (b) median filter. 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, the value range of variable i is 1 to M. The observation vector y corresponds to the flag vector z. The impact noise and non-impact noise are expressed using two Gaussian distribution models with different parameters, using the flag 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, and each element s in s is i 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 s i The accuracy of Gamma distribution follows 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 represents z i = 0, the non-impact noise at the i-th pixel obeys Gaussian η0 represents n 0,i The accuracy of follows the gamma distribution Gamma(c0,d0), c0 and d0 are hyperparameters that control the shape of the gamma distribution; n 1,i represents z i= 1, the impulse noise at the i-th pixel obeys the 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 obeys the Beta-Bernoulli distribution.

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

[0022]

[0023] In addition, the algorithm's model 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 represents 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 the parameter π 1,i represents 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 iteratively update each parameter. 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 identity matrix, <> represents the posterior estimate, <> A Indicates the posterior expectation of the probability of subscript A, and defines 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 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] Show that q(η0) is distribution, where:

[0040]

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

[0042]

[0043] Show 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. Therefore 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 of the model belonging to 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 Including 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 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 analyzed and verified the performance of Markov-BG-RSBL under different noise conditions. The experiment set the correlation parameter x = 0.1, the observation vector length M = 100, the sparse vector length N = 200, and the sparsity K = 5. Each round of Monte Carlo experiments was repeated 500 times, and the final results were averaged. The experimental noise was sudden impulse noise and Gaussian noise. It can be seen that under sudden impulse noise, the reconstruction performance of Markov-BG-RSBL was significantly improved compared to BG-RSBL. In the Gaussian environment without impulse noise, the performance was average, with the performance being on par with BG-RSBL in the range of -5dB to 10dB. The reconstruction error was greater below -5dB because the small-probability large-value noise in Gaussian noise affected the Markov-BG-RSBL's judgment of 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 used a kernel function combining a Gaussian kernel and a polynomial kernel, published in Mitra K, Veeraraghavan A, and Chellappa R. Robust RVM regression using a sparse outlier model [C]. 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010, 1887-1894. This kernel function was applied to the Lena image for denoising. The image size was 512×512, and the image depth was 8 bits. To simulate a polluted image, 20% salt and pepper noise was added to the original image. Before denoising, the image was segmented into multiple 6×6 pixel regions, and regression fitting was performed on each region. The inferred parameters were then used to predict the intensity of the central pixel in the region. A 6×6 median filter was used for comparison. The results show that the present invention can effectively preserve the edge details of the Lena image while removing salt and pepper noise.< / z> < / z>

Claims

1. An image denoising method based on Markov robust sparse Bayesian learning, characterized in that: Including steps: 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 , 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 To distinguish whether it is affected by impact noise, i =1 means the i-th 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 represents z i-1 =0 time z i =0, let parameter π 1,i represents z i-1 =1 when z i =1 probability, π 0,i and π 1,i Related by correlation parameter x; π 0,i =π 1,i +x, where x∈[0,1]; For each pixel block, a regression model is established between the pixel position and the pixel intensity, and the regression problem of the regression model is converted into a sparse reconstruction problem in the compressed sensing model. 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 , specifically including calculating 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>), diag means converting the vector into a diagonal matrix, <> means the posterior expectation estimate, T Indicates transposition, 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, where <η0> and <η1> are the precision estimates; and the posterior estimate μ of the sparse vector is 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.< / z> < / z> 2. The method according to claim 1, wherein: 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 methods for distinguishing whether the equipment is affected by impact noise are as follows: Among them, a i is the i-th row of the perception matrix A; s is a sparse vector, and each element s in s is i 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 s i The accuracy of Gamma distribution follows 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 represents z i = 0, the non-impact noise at the i-th 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 represents z i = 1, the impulse noise at the i-th 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, wherein: The state variable z at the i-th pixel i Obeys the Beta-Bernoulli distribution.

4. The method according to claim 3, wherein: 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>), diag means converting the vector into a diagonal matrix, <> means the posterior expectation 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 accuracy 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, and d1 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, and h1 Each element in Complete the control parameters of the Beta distribution shape Updates, Calculate the estimated value of the signature vector z for each element Complete the flag vector estimation <z>Update of 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, perform another round of iteration.

5. 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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