Method for poisson image restoration based on adaptive euler elastic regularization

Abstract

The application provides a Poisson image restoration working method based on adaptive Euler elasticity regularization, and comprises the following steps: S1, a new Poisson image restoration model is formed by adding an adaptive weighting matrix to Euler elasticity regularization; S2, the new Poisson image restoration model is solved by an alternating direction multiplier method (ADMM), and Poisson image restoration is performed according to a set condition; and S3, the numerical result of Poisson image restoration is evaluated by an evaluation index, and the quality of the restored image is evaluated from the visual effect.

Description

Technical Field

[0001] This invention relates to the field of image processing, and more particularly to a working method for Poisson image restoration (referred to as AEEPR) based on adaptive Euler elastic regularization. Background Technology

[0002] Noise and blur are unavoidable during image imaging, transmission, and display. Removing them through image restoration techniques is essential. Image restoration has received widespread attention in many application areas, including astronomical and medical imaging, computer science, electronics, and engineering. Its goal is to obtain a sharp image u from a degraded image f. This is considered an inverse problem. Due to the lack of some prior information, this inverse problem is ill-posed. Therefore, many regularization techniques have been extensively studied to overcome this shortcoming. The variational regularization problem of image restoration can be constructed as follows:

[0003]

[0004] in Let f represent the degraded image, u represent the image restored from f, and λ > 0 be the term affecting the fit. and regularization terms The regularization parameters that balance between them. Typical examples include total variational (TV) regularization, higher-order TV (HOTV) regularization, generalized total variational (TGV) regularization, and curvature-based regularization (including Euler elasticity (EE), Gaussian curvature, and mean curvature). Among these, the TV regularization term proposed by Rudin, Osher, and Fatemi is one of the most well-known methods and has been widely applied in many other image processing applications, such as decomposition, segmentation, and reconstruction.

[0005] In many practical applications, the acquired images are more susceptible to Poisson noise. Unlike Gaussian noise, Poisson noise is related to image intensity, making Poisson image restoration extremely challenging. Restoration models proposed for Gaussian noise are ineffective at removing Poisson noise.

[0006] To effectively remove Poisson noise, the existing techniques propose TV-based regularization models. Since TV regularization has the ideal edge-preserving ability, it has been widely applied in image processing and computer vision. However, the main drawback of TV regularization is that it often produces the staircasing effect in the flat regions of the recovered image, especially for piecewise smooth images. To overcome the staircasing effect, many improved Poisson image restoration models are proposed. In particular, in the existing techniques, some HOTV-based regularizations are introduced. Compared with TV regularization, these methods have better ability to preserve the features of smooth regions. However, they usually cause the blurring of edges in the recovered image. To preserve the edges while suppressing the staircasing effect, another existing technique proposes a Poisson restoration model based on TGV regularization. Since TGV regularization has the ability to construct piecewise polynomial functions, the proposed model can more accurately describe the intensity variation in smooth regions. In terms of preserving edges and overcoming the staircasing effect, researchers propose a non-convex EE model for Gaussian noise removal. This model can more quickly suppress the high-frequency components of the image. Therefore, it can effectively overcome the staircasing effect, making the denoised image more natural. Due to its non-convexity, non-smoothness and non-linearity, the solution of the EE model is a highly challenging task. To overcome these problems and reduce the computational cost, a series of fast algorithms are studied. Inspired by the advantages of EE regularization, it is considered to use it for Poisson image restoration.

[0007] Generally, an image contains many different features. For traditional TV-based models, the two sub-variables in the gradient operator use the same weight. In this case, these models cannot effectively couple with the local features of the image. To overcome this drawback, Pang et al. propose an anisotropic TV (ATV) based on local information, which adds different weights for each component of the gradient operator. Compared with other TV-based models, ATV can better describe the local features of the image. Another existing technique is a weighted anisotropic total variation (WATV) model for Gaussian image restoration, in which a non-linear monotonically increasing function is used as the weight. In some special images, the texture details in the image have obvious directionality. Therefore, the designed model needs to be able to describe this geometric feature. The directional total variation (DTV) method can effectively couple with the local structure of the image by adding a rotation matrix to rotate the gradient operator. However, when dealing with some special images with multiple dominant directions, using the previous DTV-based model will not achieve the ideal restoration effect. The reason is that a fixed angle parameter in the rotation matrix cannot describe multiple dominant directions at the same time. In addition, by combining the rotation matrix, the adaptive weighting matrix and the TV p norm, an adaptive weighted TV p (AWTV p ) regularization model is proposed. pThe norm is used to promote sparsity by setting p e (0, 1). The adaptive weighting matrix can enhance the diffusion along the tangential direction of the edge. Combined with the rotation matrix, the proposed model can handle images with complex structures that have multiple dominant directions. Unlike traditional TV-based models, the adaptive TV-based models can achieve better restoration results due to the use of some adaptive weighting matrices. In particular, they can effectively preserve image details. However, there are still some problems in restoring smooth images. Considering the superior performance of the EE model in overcoming the staircase effect and preserving structural information, the prior art proposes an adaptive Euler-elasticity regularization model for Gaussian denoising.

[0008] Regarding the optimization problem, some efficient numerical algorithms have been studied, such as the primal-dual algorithm, the split Bregman method, the nonlinear multigrid method, and the alternating direction method of multipliers (ADMM). The present invention adopts the ADMM which has been used in many applications. The main contributions of the present invention are summarized as follows: Models related to Euler-elasticity energy are rarely used for Poisson image restoration. By integrating a new adaptive weighting matrix into Euler-elasticity regularization, the present invention proposes an adaptive Euler-elasticity model with better adaptability and stronger restoration ability. This combination can effectively overcome the staircase effect and preserve image details, especially for some smooth images. SUMMARY

[0009] The present invention aims to at least solve the technical problems existing in the prior art, and particularly innovatively proposes a Poisson image restoration method based on adaptive Euler-elasticity regularization.

[0010] In order to achieve the above-mentioned purpose of the present invention, the present invention provides a Poisson image restoration method based on adaptive Euler-elasticity regularization, comprising the following steps:

[0011] S1, by adding an adaptive weighting matrix to the Euler-elasticity regularization, a new Poisson image restoration model is formed;

[0012] S2, according to the new Poisson image restoration model, the solution is carried out by the alternating direction method of multipliers (ADMM), and the Poisson image restoration is carried out according to the set conditions;

[0013] S3, the numerical results of the Poisson image restoration are evaluated by the evaluation index, and the quality of the restored image is evaluated from the visual effect.

[0014] According to the above technical solution, the S1 comprises:

[0015]

[0016] g(κ(u i,j)) is a specific function of curvature, defined as follows:

[0017] g(κ(u i,j ))=1+α|κ(u i,j )|

[0018] where alpha is a positive constant, and for a two-dimensional curve, the curvature kappa is expressed as a function of u, i.e.

[0019]

[0020] T is an adaptive weighting matrix, which is defined as:

[0021]

[0022] According to the preferred technical scheme, the S2 comprises:

[0023] Three auxiliary variables v, w, and q are introduced to convert the original unconstrained optimization problem into the following constrained optimization problem:

[0024]

[0025] s.t. w=Tv,q=Ku.

[0026] In order to solve the constrained problem, three Lagrange multipliers η1=(η 11 ,η 12 ) T ,η2=(η 21 ,η 22 ) T and η3 are introduced, and then it is converted into a saddle point problem, and the corresponding augmented Lagrangian functional is:

[0027]

[0028] where gamma1, gamma2, and gamma3>0 are three penalty parameters, which take different values.

[0029] According to the preferred technical scheme, the S2 comprises:

[0030] S2-1, input a fuzzy noisy image f;

[0031] S2-2, set parameters: lambda, iota, delta, beta, alpha, nMax, and

[0032] S2-3, initialization: u 0 =f, q 0 , v 0 , w 0 , and n=0;

[0033] S2-4, when n

[0034]

[0035] S2-5, replace n with n+1;

[0036] S2-6, if ||u n -u n-1 ||2 / ||u n ||2≤∈, stop iteration;

[0037] S2-7, end loop;

[0038] S2-8, return u * = u n as the final restored image.

[0039] According to the above technical scheme, preferably, the S2-4 comprises:

[0040] wherein, The subproblem is reformulated as:

[0041]

[0042] According to the optimality condition thereof, the Euler-Lagrange equation is obtained:

[0043]

[0044] Under the periodic boundary condition, the above formula is efficiently solved by the fast Fourier transform (FFT) method:

[0045]

[0046] According to the above technical scheme, preferably, the S2-4 comprises:

[0047] wherein, The subproblem is expressed as

[0048]

[0049] The subproblem has a closed-form solution, and by solving the corresponding quadratic equation, the solution is

[0050]

[0051] According to the above technical scheme, preferably, the S2-4 comprises:

[0052] wherein, The subproblem is reformulated as

[0053]

[0054] Considering its optimality condition, we have

[0055]

[0056] where is the identity operator, and let v = [v1, v2] T , w = [w1, w2] T , η i = [η i1 , η i2 ] T (i = 1, 2), then v n +1 satisfy the following linear equations:

[0057]

[0058] Therefore, the explicit solution of v n+1 is obtained by simple calculation:

[0059]

[0060] According to the preferred technical scheme, the S2-4 comprises:

[0061] wherein, The w sub-problem can be expressed as

[0062]

[0063] Then its closed-form solution is obtained by a soft thresholding operator

[0064]

[0065] According to the preferred technical scheme, the S3 comprises:

[0066] The quality of the restored image is evaluated using two evaluation indexes, namely the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM), which are calculated as follows:

[0067]

[0068]

[0069] where I i,j represents the pixel value of the initial clean image, u i,j represents the pixel value of the restored image, Max represents the maximum pixel value of the clean image I, μ I and μ u represent the local average values of the images I and u, σ I and σ uσ represents the standard deviation of each. Iu is the covariance between the clean image I and the restored image u. c1 and c2 are two constants used to avoid a denominator of 0. Generally, a larger PSNR value indicates less noise after denoising, while a larger SSIM value reflects a higher similarity between the restored image u and the clean image I. n The relative error is defined as follows:

[0070]

[0071] According to the preferred embodiment of the above technical solution, it also includes:

[0072] when At this point, the AEEPR model is simplified to a Poisson denoising model, called AEEPD, where K is a known fuzzy matrix. The unit operator is represented as:

[0073]

[0074] Here, f is a noisy image contaminated with Poisson noise, g(κ(u i,j )) is a specific function of curvature, and T is an adaptive weighting matrix;

[0075] To solve this minimization problem, two auxiliary variables, v and w, are used, and it is rewritten as a constrained minimization problem as follows:

[0076]

[0077] st w = Tv

[0078] Then, by introducing two Lagrange multipliers η1 and η2, the above equation is reconstructed into a saddle-point problem, whose augmented Lagrange functional is:

[0079]

[0080] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0081] An adaptive Euler elastic Poisson image restoration model is proposed by adding an adaptive weighting matrix to the Euler elastic regularization. The proposed model is solved using an efficient ADMM, where all subproblems have explicit solutions. Experimental results on natural and synthetic images, compared to other state-of-the-art methods, demonstrate the effectiveness and superiority of the proposed method. Subsequent research extends the proposed regularization method to other problems in image processing and computer vision, such as image segmentation, hyperspectral unmixing, and hyperspectral image fusion.

[0082] Additional aspects and advantages of the application will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following and the attendant drawings or can be learned by practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0083] The above and / or additional aspects and advantages of the present application will become apparent and be readily appreciated from the following description, including the accompanying drawings, wherein:

[0084] Figure 1 is a general flowchart of the present application;

[0085] Figure 2 is a clear diagram of the data experiment of the present application;

[0086] Figure 3 is a restored image and a local enlarged view of a specific experiment Peppers image of the present application;

[0087] Figure 4 is a restored image and a local enlarged view of a specific experiment Lena image of the present application;

[0088] Figure 5 is a restored image and a residual image of a specific experiment Dot image of the present application;

[0089] Figure 6 is a restored image and a residual image of a specific experiment Shape image of the present application;

[0090] Figure 7 is a graph of the experiment of the present application under a Gaussian blur and a noise level of σ = 1;

[0091] Figure 8 is a graph of the experiment of the present application under a motion blur and a noise level of σ = 5;

[0092] Figure 9 is a restored image and a graph of the experiment of the present application under a Barbara image in a noise environment;

[0093] Figure 10 is a restored image and a graph of the experiment of the present application under a Triangle image in a noise environment. DETAILED DESCRIPTION

[0094] Embodiments of the present application are described in detail below with reference to the attached drawings, which are incorporated by reference in their entirety. The embodiments described below are examples of one specific embodiment of the present application and are not intended to limit the scope of the application. Rather, these embodiments are included to explain the present application and to provide additional examples for the practice thereof. It should be appreciated by those skilled in the art that the present application can be embodied in other forms without departing from the spirit or scope of the application.

[0095] As Figure 1As shown, the present application discloses a kind of Poisson image restoration working method based on adaptive Euler elasticity regularization, comprising the following steps:

[0096] S1, by adding adaptive weighting matrix to Euler elasticity regularization, form new Poisson image restoration model;

[0097] S2, according to new Poisson image restoration model, by alternating direction multiplier method (ADMM) for solution, according to the condition set is carried out Poisson image restoration;

[0098] S3, by evaluating index to evaluate the numerical results of Poisson image restoration, and the quality of the recovered image is evaluated from visual effect.

[0099] Because the model related to Euler elasticity energy is rarely used for Poisson image restoration.The present application integrates new adaptive weighting matrix into Euler elasticity regularization, the present application proposes an adaptive Euler elasticity model with better adaptability and stronger restoration ability.This combination effectively overcomes the step effect and preserves image details, especially for some smooth images.

[0100] It is known that TV regularization is proposed for removing Gaussian noise, which has been used in many fields such as image restoration, image inpainting, medical image reconstruction, etc.Especially, the TV-based Poisson restoration model is described as:

[0101]

[0102] where K represents a known blur matrix, λ>0 is a regularization parameter, 1 Ω indicates the indicator function on the discrete grid of image region Ω.As one of the most popular regularizations, TV performs very well in handling piecewise constant images.That is, it can better preserve image edges in the restoration process.However, it can produce step effect, especially for piecewise smooth images.Although this, due to its advantages, it has been widely applied to various image processing problems.

[0103] AWDTV p model

[0104] Prior art AWDTV p model uses an adaptive weighted directional TV p regularization for Poisson restoration, and the AWDTV p model is constructed as follows:

[0105]

[0106] where K represents a known blur matrix; p subscript indicates l pnorm, the superscript p denotes p-th power, T β (f) and R -θ respectively represent an adaptive weighting matrix and a rotation matrix, which are defined as follows:

[0107] and Here, denotes a Gaussian kernel, ι is a parameter, β(f) is used to enhance diffusion, and θ is an affine angle. In particular, when β(f) = β is set, where β is a constant. Otherwise, β(f) = 1 is set.

[0108] For processing images with obvious directionality, it is very useful to add a rotation matrix to rotate the gradient operator. When an image has multiple dominant directions, it is necessary to further combine the adaptive weighting matrix with the rotation matrix to obtain better restoration results. In addition, the l p norm used in the model is to promote gradient sparsity.

[0109] 1 Proposed model

[0110] Although the TV-based model can effectively preserve image edges, it cannot restore smooth images well. It usually produces a staircase effect in flat areas. To overcome this problem, some high-order regularization models have been proposed, including the popular curvature regularization. Inspired by the successful application of curvature regularization, a new Poisson image restoration model is constructed by adding an adaptive weighting matrix to the Euler-elasticity regularization.

[0111] To obtain high-quality restored images, the present application proposes the following Poisson restoration model based on adaptive Euler-elasticity regularization:

[0112]

[0113] g(κ(u i,j )) is a specific function of curvature, defined as follows:

[0114] g(κ(u i,j )) = 1 + α|κ(u i,j )|,

[0115] where α is a positive constant. For a two-dimensional curve, the curvature κ is expressed as a function of u, i.e.

[0116]

[0117] T is an adaptive weighting matrix, which is defined as:

[0118]

[0119] To effectively characterize the diffusion, β(f) is used to represent the smooth and edge regions. If i = 0 and β(f) = 1, T will be the identity matrix. In this case, the proposed model is related to the Euler-elastic model. In addition, if a = 0 and β(f) = 1, the proposed model is also related to the ATV model.

[0120] 2 Proposed algorithm

[0121] Since the minimization problem is non-convex and non-smooth, it will face some numerical difficulties in solving. To solve this problem, the present invention designs an efficient alternating direction method of multipliers (ADMM), which can decompose a large global problem into a series of smaller local sub-problems, and then use the local solution to calculate the solution of the original problem.

[0122] The present invention first introduces three auxiliary variables v, w, q, and converts the original unconstrained optimization problem into the following constrained optimization problem:

[0123]

[0124] s.t. w = Tv, q = Ku.

[0125] To solve this constraint, three Lagrange multipliers η1 = (η 11 ,η 12 ) T , η2 = (η 21 ,η 22 ) T and η3 are introduced, and then converted into a saddle point problem. The corresponding augmented Lagrangian functional is:

[0126]

[0127] where γ1, γ2, γ3 > 0 are three penalty parameters, which take different values. The above formula includes four sub-variables, so when using ADMM, each sub-variable needs to be solved alternately until the termination condition is reached; see the following algorithm.

[0128] Alternating direction method of multipliers (ADMM) for solving

[0129] 1: Input: a blurred noisy image f

[0130] 2: Parameters: λ, i, δ, β, a, nMax, and

[0131] 3: Initialization: u 0 = f, q 0 , v 0 , w 0 ,​ and n = 0

[0132] 4: replace n by n + 1

[0133]

[0134] 5: replace n by n + 1

[0135] 6: if ||u n - u n-1 ||2 / ||u n ||2≤∈, stop iteration

[0136] 7: end loop

[0137] 8: return u * = u n as the final restored image.

[0138] In this algorithm, the efficiency of solving the optimization problem depends on how to solve the sub-problems efficiently. The above sub-problems all have explicit solutions.

[0139] The sub-problem is reformulated as:

[0140]

[0141] According to its optimality condition, the Euler-Lagrange equation is obtained:

[0142]

[0143] Under the periodic boundary condition, the above formula is solved efficiently by the fast Fourier transform (FFT) method:

[0144]

[0145] The sub-problem is reformulated as

[0146]

[0147] The sub-problem has a closed-form solution. By solving the corresponding quadratic equation, its solution is

[0148]

[0149] The sub-problem is reformulated as

[0150]

[0151] Consider its optimality condition, which is

[0152]

[0153] where is the identity operator, and let v = [v1, v2] T , w = [w1, w2] T , η i = [η i1 , η i2 ] T (i = 1, 2), then v n +1 satisfy the following linear system of equations:

[0154]

[0155] Thus, the explicit solution of v n+1 is obtained by simple calculation:

[0156]

[0157] The w subproblem can be represented as

[0158]

[0159] Then its closed-form solution is obtained by the soft-thresholding operator

[0160]

[0161] The proposed Poisson image restoration model is compared with the TV model and the AWTV p model. In particular, two types of blur and two noise levels are tested to demonstrate the superiority of the proposed model. In addition, the proposed method is applied to Poisson image denoising and compared with the state-of-the-art curvature regularization model (TAC). Numerical experiments are carried out by MATLAB on a Windows 10 (64-bit) desktop computer with a 2.90 GHz Intel(R) Core(TM) i7-10700 CPU and 32 GB RAM.

[0162] To evaluate the quality of the restored image, two evaluation indicators, i.e., the peak signal-to-noise ratio (PSNR) and the structural similarity (SSIM), are used. Their definitions are as follows:

[0163]

[0164]

[0165] where I i,j denotes the pixel value of the initial clean image, u i,j denotes the pixel value of the restored image, and Max represents the maximum pixel value of the clean image I. μI and μ u σ represents the local average values ​​of images I and u. I and σ u σ represents the standard deviation of each. Iu is the covariance between the clean image I and the restored image u. c1 and c2 are two constants used to avoid a denominator of zero. Generally, a larger PSNR value indicates less noise after denoising, while a larger SSIM value reflects a higher similarity between the restored image u and the clean image I. n The relative error is defined as follows:

[0166]

[0167] In all experiments, when the relative error R(u) n ≤10 -4 The iteration will terminate when the number of iterations reaches 500, where the superscript n represents the restored image after n iterations.

[0168] In the experiment, the selected test images included natural images (“Peppers” (256×256), “Lena” (256×256), “Barbara” (512×512)) and synthetic images (“Dot” (256×256), “Shape” (128×128), “Triangle” (254×214)), such as Figure 2 The images shown are clear pictures used in the data experiments of this invention. a is a picture of Peppers, b is a picture of Lena, c is a picture of a Dot shape, d is a picture of a Shape shape, e is a picture of Barbara, and f is a picture of a Triangle shape; these are used for image restoration and noise reduction experiments.

[0169] Figure 3 Images show the restored "Peppers" images and magnified local images obtained using different restoration models when Gaussian blur and noise level σ = 1. In the images, a is the degraded image, b is the restoration result using the TV method, and c is the AWDTV image. p The image shows the result of the method restoration; d represents the result of the AEEPR method restoration.

[0170] Figure 4 Images show the restored "Lena" image and magnified local images obtained using different restoration models when motion blur and noise level σ = 5. In the images, a is the degraded image, b is the restoration result using the TV method, and c is the AWDTV image. p Image d shows the result of the method restoration; d is the image of the result of the AEEPR method restoration.

[0171] Figure 5Figures of "Dot" restored images and corresponding residual images by different restoration models under Gaussian blur and noise level σ = 5. Where a is the TV method restoration effect image; b is the AWD TV p method restoration effect image; c is the AEEPR method restoration effect image.

[0172] Figure 6 Figures of "Shape" restored images and corresponding residual images by different restoration models under motion blur and noise level σ = 5. Where a is the TV method restoration effect image; b is the AWD TV p method restoration effect image; c is the AEEPR method restoration effect image.

[0173] Figure 7 Figures of PSNR, relative error and energy versus CPU time of AEEPR model under Gaussian blur and noise level σ = 1. Where a is the PSNR versus CPU time of AEEPR model; b is the relative error versus CPU time of AEEPR model; c is the energy versus CPU time of AEEPR model.

[0174] Figure 8 Figures of PSNR, relative error and energy versus CPU time of AEEPR model under motion blur and noise level σ = 5. Where a is the PSNR versus CPU time of AEEPR model; b is the relative error versus CPU time of AEEPR model; c is the energy versus CPU time of AEEPR model.

[0175] Figure 9 Figures of "Barbara" restored images by TAC and AEEPD in (a) (c); their corresponding residual images in (b) (d); figures of PSNR, relative error and energy versus CPU time in (e) (f) (g) (noise level σ = 1).

[0176] Figure 10 Figures of "Triangle" restored images by TAC and AEEPD in (a) (c); their corresponding residual images in (b) (d); figures of PSNR, relative error and energy versus CPU time in (e) (f) (g) (noise level σ = 5).

[0177] There are some parameters in the proposed model and algorithm for Poisson image restoration. For all parameters, the trial-and-error method is used to adjust step by step to determine the optimal parameters that produce the highest PSNR value. In the experimental process, first, set several values of λ to a larger range, such as [a, b]. Then, find the corresponding λ value by the maximum PSNR value obtained to determine the appropriate subset By repeating this strategy several times, the subset [a i , b imore suitable parameter values are found. When the difference between consecutive PSNR values is less than 0.01, the corresponding parameter value is set to the selected value of λ. In turn, the same strategy is used to select the other parameters until the PSNR values are almost constant during the loop. The same strategy is adopted to adjust the parameters of the other two methods. During the parameter adjustment, the algorithm terminates when the termination criterion is satisfied. It is worth noting that during the parameter adjustment process, it is found that only a few parameters have a significant impact on the results. Generally, the regularization parameter λ, which is used to control the balance between the fitting term and the regularization term, plays a major role in the restoration results. In addition, the main function of ι is to control the local adaptivity, δ is the standard deviation, and the weighting parameter β can enhance the diffusion. The positive parameter α can balance the influence of the curvature term on the length.

[0178] The proposed AEEPR is compared with TV and AWDTV p for Poisson image restoration. In the experiments, the Matlab functions and fspecial('motion', 10, 45) are used to generate Gaussian blur and motion blur, respectively. In addition, to test the effectiveness of the proposed method under different noise levels, Poisson noise is added using the Matlab program poissrnd(B / σ)*σ. Here, B represents the blurred image degraded from the original clear image I. In the experiments, two noise levels σ = 1 and 5 are tested.

[0179] Table 1 lists the optimal parameter values of different restoration models under different blur kernels and noise levels. The number of parameters used in the AEEPR model is less than that of the AWDTV p model. In these two models, the optimal values of the same parameters are taken in the same range. Table 2 records the PSNR and SSIM values of the three methods for two natural images and two synthetic images, and the corresponding CPU time and iteration number are listed in Table 3.

[0180] From Table 2, it is observed that AEEPR and AWDTV p obtain almost the same PSNR and SSIM values, which are much higher than TV. For example, AEEPR improves the PSNR of TV by 2.46 when processing the Peppers image with Gaussian blur and noise level σ = 1. In addition, Table 3 shows that AEEPR spends less CPU time than AWDTV p to obtain almost the same PSNR values and SSIM, especially in the case of higher noise levels with the same blur.

[0181] To evaluate the visual quality of the restored images obtained by different methods, the restored images and local magnified images of two natural images are given in Figure 2 and 3 , while the restored images and local magnified images of two synthetic images are given in Figure 4 and5 The image shows the restored and residual images (fu) of two synthesized images. These images show that all methods remove a significant amount of noise and achieve good restoration results. However, the TV method produces a noticeable staircase effect in flat areas, especially for the "Peppers" and "Lena" images. Clearly, AEEPR and AWDTV... p The method effectively overcomes the staircase effect while preserving a great deal of image detail, such as neat edges and smoother areas. This is particularly evident in close-ups of the nose and head areas in the "Lena" image. Figure 4 and Figure 5 This indicates that AEEPR and AWDTV p It removes more blur and noise than TV. Overall, AEEPR performs better in restoring degraded images and requires less CPU time.

[0182] Finally, the convergence of the proposed algorithm is demonstrated through experiments. The cases of Gaussian blur and noise level σ = 1, and motion blur and noise level σ = 5 are considered. Furthermore, all four images are processed through the following model. Conduct testing. Figure 6 and Figure 7 In the study, all PSNR values ​​increased rapidly with increasing CPU time, while their energy and relative error decreased accordingly. Therefore, it can be concluded that AEEPR is convergent.

[0183]

[0184] Table 1. Selected parameter sets for different restoration models under different fuzzy kernels and noise levels.

[0185]

[0186]

[0187] Table 2. PSNR and SSIM values ​​obtained from different restoration models.

[0188]

[0189] Table 3 CPU time and number of iterations for different restoration models

[0190] The specific implementation of Poisson noise reduction is as follows:

[0191] First, the adaptive Euler elasticity model is constructed. When the state is both fuzzy and noisy, the model is called the Poisson restoration model; that is, the entire restoration process mentioned above.

[0192] when Only noise, the model is Poisson denoising model, AEEPR model is simplified as Poisson denoising model, called AEEPD, where K is a known blur matrix, denotes the identity operator, written as:

[0193]

[0194] Here, f is the noisy image contaminated by Poisson noise. g(kappa(u i,j )) is some specific function of the curvature, and T is an adaptive weighting matrix.

[0195] To solve this minimization problem, two auxiliary variables v, w are introduced, and it is rewritten as the following constrained minimization problem:

[0196]

[0197] s.t. w = Tv

[0198] Then two Lagrange multipliers η1, η2 are introduced, and the above equation is reformulated as a saddle point problem, whose augmented Lagrangian functional is

[0199]

[0200] When using ADMM, three sub-problems need to be minimized, fixing other variables at each iteration, and updating two Lagrange multipliers. It is worth noting that the fixed point strategy is used to solve the u sub-problem.

[0201] The proposed curvature regularization based variational model can be described as

[0202]

[0203] In particular, two auxiliary variables v and q are used to reformulate the above equation as the following constrained optimization problem:

[0204]

[0205] s.t. q i,j = u i,j ,

[0206] where g(kappa(u i,j )) = 1 + alpha | kappa(u i,j ) | The corresponding augmented Lagrangian functional of the above equation is given by

[0207]

[0208] Here, η1 and η2 are Lagrange multipliers, and μ1 and μ2 are two penalty parameters. There are three subproblems: v, q, and u. Therefore, it is necessary to solve these three subproblems alternately and iteratively, updating the two Lagrange multipliers, until the termination condition is met.

[0209] The v subproblem is a typical nonlinear minimization problem, and its solution is obtained through Newton's method, thus having...

[0210]

[0211] The q subproblem has a closed-form solution, given by the following equation.

[0212]

[0213] Here, we need to solve a quadratic equation to obtain the solution to the q subproblem.

[0214] Based on the optimality condition, the Euler-Lagrange equation for the u subproblem can be easily derived. Assuming periodic boundary conditions, the equation is then solved using the Fast Fourier Transform (FFT), yielding...

[0215]

[0216] In the experiment, select Figure 2 The “Barbara” and “Triangle” images shown are used as test images to demonstrate the competitive performance of AEEPD relative to the TAC method. Test data were generated by adding Poisson noise at noise levels σ = 1 or 5. The relevant numerical results of Poisson denoising are shown in Table 4. Specifically, in… Figure 9 The image shows the restored "Barbara" image and the corresponding residual image when the noise level σ = 1. At this point, the optimal parameter values ​​for the TAC method are λ = 38, α = 0.27, μ1 = 0.39, μ2 = 7, while the parameters for AEEPD are set to λ = 22, α = 0.24, ι = 0.0019, δ = 0.1, β = 0, γ1 = 0.006, γ2 = 2.7. Furthermore, Figure 9 The image shows the relevant visual results for the "Triangle" image at a noise level of σ = 5. Here, for the TAC method, λ = 2.5, α = 0.45, μ1 = 0.36, μ2 = 8 are chosen. For AEEPD, λ = 5.14, α = 0.1, ι = 0.0002, δ = 0.5, β = 0, γ1 = 0.0035, γ2 = 0.1.

[0217]

[0218] Table 4 Numerical results of Poisson denoising

[0219] from Figure 9 andFigure 10 In general, the restored image by TAC is not as natural as AEEPD, and more noise is removed by AEEPD. This is due to the use of adaptive weighting matrix. It is observed from Table 4 that the AEEPD method obtains higher PSNR and SSIM values while spending less CPU time than the TAC method. Moreover, this is easily seen from the PSNR versus CPU time plots in Figure 9 and Figure 10 The relative error and energy versus CPU time plots also indicate the convergence of the algorithm.

[0220] While the embodiments of the application have been illustrated and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made therein without departing from the spirit and scope of the application, which is defined by the following claims and their equivalents.

Claims

1. A method for Poisson image restoration based on adaptive Euler elastical regularization, characterized in that, Comprising the following steps: S1, forming a new Poisson image restoration model by adding an adaptive weighted matrix to Euler elasticity regularization; The Poisson image restoration model: g(κ(u i,j )) is a specific function of the curvature, defined as follows: g(κ(u i,j )) = 1 + a | κ(u i,j )| Wherein α is a positive constant; for a two-dimensional curve, the curvature κ is expressed as a function of u, that is T is an adaptive weighted matrix, which is defined as: In order to effectively depict diffusion, β(f) is used to represent the smooth and edge regions; if ι=0 and β(f)=1, T will become a unit matrix; S2, solving by alternating direction multiplier method (ADMM) according to the new Poisson image restoration model, and performing Poisson image restoration according to the set conditions; Three auxiliary variables v, w, q are introduced, and the original unconstrained optimization problem is converted into the following constrained optimization problem: To solve this constraint, three Lagrange multipliers η1 = (η 11 ,η 12 ) T , η2 = (η 21 ,η 22 ) T and η3 are introduced, which are then transformed into a saddle point problem; the corresponding augmented Lagrangian functional is Wherein γ1, γ2, γ3>0 are three penalty parameters, which have different values; the above formula includes four sub variables, so when using ADMM, each sub variable needs to be solved by alternating iteration until the termination condition is reached; for solving alternating direction method of multipliers (ADMM); S2-1, input a fuzzy noisy image f; S2-2, set parameters: λ, i, δ, β, α, nMax, and S2-3, initialization: u 0 = f, q 0 , v 0 , w 0 , and n = 0; S2-4, when n S2-5, replace n with n+1; S2-6, if ||u n - u n-1 ||2 / ||u n ||2≤∈, stop iteration; S2-7, end the loop; S2-8, return u * = u n as the final restored image; The subproblem has an explicit solution, This sub-problem is restated as: According to its optimality condition, the Euler Lagrange equation is obtained: Under the periodic boundary condition, the above formula is efficiently solved by fast Fourier transform (FFT) method: This sub-problem is formulated as The subproblem has a closed form solution; by solving the corresponding quadratic equation, its solution is v the sub-problem is re-expressed as Considering its optimality condition, for where is the identity operator, and let v = [vi, v2] T , w = [wi, w2] T , η i = [η i1 , η i2 ] T (i = 1, 2), then v n+1 satisfies the following linear system of equations: Thus, by simple calculation we obtain v n+1 Explicit solution of the equation: w sub-problem is represented as Then its closed form solution is calculated by a soft threshold operator S3, evaluating the numerical results of Poisson image restoration by evaluation index, and evaluating the quality of the restored image from the visual effect.

2. The method of Poisson image restoration based on adaptive Euler elastical regularization according to claim 1, characterized in that, The S3 includes: The quality of the restored image is evaluated using two evaluation indexes, namely peak signal-to-noise ratio (PSNR) and structural similarity (SSIM), which are calculated as follows: where I i,j represents the pixel value of the initial clean image, u i,j represents the pixel value of the recovered image, Max represents the maximum pixel value of the clean image I, μ I and μ u represent the local mean values of the image I and u, σ I and σ u represent the respective standard deviations, σ Iu is the covariance between the clean image I and the recovered image u, c1 and c2 are two constants used to avoid the denominator being zero, the larger the PSNR value, the smaller the noise after denoising, and the larger the SSIM value, the higher the similarity between the recovered image u and the clean image I, u n The relative error of u is defined as follows:

3. The method of claim 1, wherein the method is based on adaptive Euler-elastic regularization of Poisson image restoration. Also includes: When The Poisson image restoration AEEPR model is reduced to a Poisson denoising model, called AEEPD, when K is a known blur matrix, denotes the identity operator, written as: Here, f is the noisy image contaminated by Poisson noise, g(k(u i,j )) is some specific function of curvature, and T is an adaptive weighting matrix. In order to solve the minimization problem, two auxiliary variables v, w are used, which are rewritten into the following constrained minimization problem: Then two Lagrange multipliers η1, η2 are introduced, and the above formula is reconstructed into a saddle point problem, and its augmented Lagrange functional is

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