A four-direction overlapping combined sparse total variation method for restoring cauchy noise image

Abstract

The application discloses a four-direction overlapping combined sparse total variation Cauchy noise image restoration method, relates to the image restoration technical field, and solves the technical problem that the existing image restoration method has a poor Cauchy noise restoration effect and ladder effect after restoration, obtains a degraded image polluted by Cauchy noise; an image degradation model is established based on an overlapping combined sparse non-convex mixed regular term; an improved alternating direction multiplier fast algorithm is proposed to solve the model; and stability analysis is performed on model parameters to obtain a final restoration image; the application can effectively remove Cauchy noise in the image, relieve the ladder effect of the image edge area, and well protect information in a complex texture area of the image, and has great advantages in visual effect and numerical analysis.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of image restoration, in particular to a four-direction overlapping combination sparse total variation Cauchy noise image restoration method. BACKGROUND

[0002] Due to the influence of environment and imaging equipment, etc., image is often accompanied by noise in the process of acquisition, transmission, storage, etc., resulting in image distortion and degradation, therefore, image restoration technology is a very important processing technology in the field of image processing, in recent years, the technology has been widely studied in remote sensing satellite imaging, medical imaging, wireless communication system, etc.

[0003] In the prior art, most image restoration technologies only restore images for Gaussian white noise, while in actual engineering production and application, Cauchy noise is most common, in addition, the existing image restoration method is well applied in retaining image edges and texture details, but at the same time, it also produces a certain step effect, which is also a problem to be solved, therefore, the present application provides a four-direction overlapping combination sparse total variation Cauchy noise image restoration method. SUMMARY

[0004] In order to solve the problems mentioned in the background, the purpose of the present application is to provide a four-direction overlapping combination sparse total variation Cauchy noise image restoration method, which can remove Cauchy noise while retaining image detail information and suppressing the step effect.

[0005] The purpose of the present application can be achieved by the following technical scheme: a four-direction overlapping combination sparse total variation Cauchy noise image restoration method, the method comprising the following steps:

[0006] Step one: obtaining a degraded image contaminated by Cauchy noise;

[0007] Step two: establishing a degradation model based on the overlapping combination sparse mixed regularization term;

[0008] Step three: solving the degradation model by using an improved alternating direction multiplier method;

[0009] Step four: obtaining a final restoration image by using parameter analysis to obtain a stable solution of the degradation model.

[0010] Further, the degradation model is:

[0011] g=Hf+v

[0012] wherein, is an original image, g is a to-be-restored image, H is a convolution operator, and v is Cauchy noise.

[0013] Further, the mixed regular term based on the overlapping combination sparsity is used to establish a formula of the degradation model.

[0014]

[0015] Wherein, <log(γ 2 +(Hf-g) 2 ),1> is a fidelity term, is a quadratic penalty term; λ>0 is a regular parameter; μ is a positive penalty parameter; is an OGSTV regular term; is a non-convex term; ω>0 is a regular parameter of the regular term.

[0016] Further, the degradation image satisfies a periodic boundary condition.

[0017] Further, the process of solving the degradation model by using the improved alternating direction multiplier method comprises:

[0018] A plurality of auxiliary variables z,v x ,v y ,q,s,

[0019]

[0020] s.t.z=Hf,v x =D x f,v y =D y f,q=D 2 f,s=f

[0021] Then, the Lagrange function is obtained by using Lagrange multipliers u1, u2, u3 and u4.

[0022]

[0023] Wherein, β i (i=1, 2, 3, 4, 5) are all positive penalty parameters.

[0024] The present application has the following beneficial effects:

[0025] In the use process of the present application, firstly, a degradation image polluted by Cauchy noise is acquired; a degradation model is established based on a mixed regular term of overlapping combination sparsity; the degradation model is solved by using an improved alternating direction multiplier method; and a stable solution of the degradation model is obtained by using parameter analysis, so that the final restored image is obtained. The design can better maintain the smoothness and detail texture characteristics of the image, remove the Cauchy noise in the maximum extent, solve the staircase effect, and better restore the image clarity. BRIEF DESCRIPTION OF DRAWINGS

[0026] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, for those skilled in the field, other drawings can also be obtained based on these drawings without any creative effort.

[0027] Figure 1 is a flowchart of the present application;

[0028] Figure 2 is an effect diagram before repair of the present application;

[0029] Figure 3 is a denoising effect diagram of the present application with noise level ξ = 0.02 and image size of 256 x 256;

[0030] Figure 4 is a denoising effect diagram of the present application with noise level ξ = 0.02 and image size of 450 x 450;

[0031] Figure 5 is a denoising effect diagram of the present application with noise level ξ = 0.02 and image size of 512 x 512;

[0032] Figure 6 is a denoising effect diagram of the present application with noise level ξ = 0.04 and image size of 256 x 256;

[0033] Figure 7 is a denoising effect diagram of the present application with noise level ξ = 0.04 and image size of 450 x 450. DETAILED DESCRIPTION

[0034] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without any creative effort fall within the protection scope of the present application.

[0035] As shown in Figures 1-7 , a four-direction overlapping combined sparse total variation Cauchy noise image restoration method, the method comprises the following steps:

[0036] Step one: obtaining a degraded image contaminated by Cauchy noise;

[0037] Step two: establishing a degradation model based on an overlapping combined sparse mixed regular term;

[0038] Step three: solving the degradation model by using an improved alternating direction multiplier method;

[0039] Step four: using parameter analysis to obtain the stable solution of the degradation model to obtain the final restored image.

[0040] It should be further explained that in the specific implementation process, the degradation model is:

[0041] g = Hf + v (1)

[0042] wherein, is the original image, g is the image to be recovered, H is a convolution operator, and v is a Cauchy noise.

[0043] It should be further explained that in the specific implementation process, the formula for establishing the degradation model based on the mixed regular term of overlapping combination sparsity is:

[0044]

[0045] wherein, <log(γ 2 +(Hf-g) 2 ),1> is a fidelity term, is a quadratic penalty term; it should be further explained that in the specific implementation process, can solve the non-convexity problem based on the Cauchy distribution fidelity term, λ>0 is a regular parameter, it should be further explained that in the specific implementation process, the parameter is used to balance the fidelity term and the regular term, and μ is a positive penalty parameter; is an OGSTV regular, it should be further explained that in the specific implementation process, can suppress the staircase effect; is a non-convex term, it should be further explained that in the specific implementation process, can smooth the texture area while retaining sharp edge information; ω>0 is a regular parameter of the regular term.

[0046] It should be further explained that in the specific implementation process, the degraded image satisfies the periodic boundary condition.

[0047] It should be further explained that in the specific implementation process, the process of solving the degradation model by using the improved alternating direction multiplier method comprises:

[0048] introducing a plurality of auxiliary variables z, v x ,v y , q, s,

[0049]

[0050] Then the Lagrange multipliers u1, u2, u3, u4 are used to obtain the Lagrange function as:

[0051]

[0052] where, β i (i = 1, 2, 3, 4, 5) are positive penalty parameters.

[0053] The improved alternating direction multiplier method is used in the example to split the constrained optimization problem into several sub-problems for solving, and the specific solution is as follows:

[0054] 1) v x , v y Sub-problem

[0055]

[0056]

[0057] 2) q sub-problem

[0058]

[0059] Since formula (7) is a non-convex second-order problem, an iterative reweighting algorithm is used to minimize this problem, so that the weight l1 problem is obtained as follows:

[0060]

[0061] In order to prevent the denominator from being 0, the present application sets ε as a very small number, and let Therefore, formula (8) can be converted into a one-dimensional problem:

[0062]

[0063] 3) z sub-problem

[0064]

[0065] In order to facilitate understanding, the present application sets

[0066]

[0067] Obtained:

[0068]

[0069] Where K' and K" are the gradient matrix and Hessian matrix of K, respectively.

[0070] 4) s sub-problem

[0071]

[0072] Through sample mapping, the minimum value can be obtained:

[0073]

[0074] 5) f sub-problem

[0075]

[0076] Since the function f is quadratic, the solution method is as follows:

[0077] Af k+1 =B (16)

[0078] Wherein:

[0079]

[0080] Here, since we use the periodic boundary condition, the optimal solution of formula (16) is obtained by using FFT:

[0081]

[0082] 6) Update each variable

[0083]

[0084] In use, initialize each variable, and set k=0,

[0085] The variable is

[0086] Second step: according to formula (5), (6), (9), (12), (14) and (18), start loop calculation of each variable, k=k+1;

[0087] Third step: loop until the stop condition is met

[0088] Fourth step: end the loop and return f

[0089] In order to verify the effectiveness of the embodiment, the proposed model and other classical models are compared, and the peak signal-to-noise ratio (PSNR), structural similarity (SSIM) and calculation time are introduced as three indicators for evaluating the quality of the restored image, wherein the larger the value of PSNR, the higher the image quality, the value of SSIM ranges from 0 to 1, and the closer the value to 1, the closer the restored image is to the original image, and the shorter the time, the faster the image restoration speed.

[0090] The present application is compared with TV, OGSTV, HTVAM model to verify the effectiveness of the model described in the example, and in the experiment, each parameter value is continuously adjusted to obtain the best visual effect and the maximum PSNR and SSIM value.

[0091] Figure 2These are the original image renderings used in this embodiment, where the first row of three images has a size of 256×256; the second row of three images has a size of 450×450; and the third row of three images has a size of 512×512. Figure 3 , 4 The image sizes used in embodiments 5 are 256×256, 450×450, and 512×512, respectively. The three images show the overall denoising effect and a magnified view at the same noise level (ξ=0.02). Table 1 shows the PSNR, SSIM, and TIME values ​​for the three different image sizes at the same noise level (ξ=0.02). It should be further noted that in the actual implementation, the visual effect of the TV model is worse than the other three, failing to effectively protect image edges and exhibiting a noticeable step effect, such as... Figure 4 The eye structure; although OGSTV and HTVAM can preserve edge information well, there are still a few noise points in the relevant magnified local images; the present invention can achieve a visual effect similar to OGSTV and HTVAM on the basis of cleanly removing noise points; in terms of data analysis, compared with OGSTV and HTVAM, the PSNR and SSIM of this model are slightly higher, but the running time is greatly improved when processing large-size images.

[0092] Table 1. Numerical analysis of denoising using different models for images of different sizes at the same noise level (ξ=0.02)

[0093]

[0094] Figure 6 and Figure 7 The image sizes used in the embodiments are 256×256 and 450×450, respectively. Table 2 shows the PSNR, SSIM and TIME values ​​of images of different sizes at the same noise level (ξ=0.04).

[0095] Table 2. Numerical analysis of denoising using different models for images of different sizes at the same noise level (ξ=0.04)

[0096]

[0097] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A method for recovering a Cauchy noise image by four-directional overlapping combined sparse total variation, characterized in that, The method comprises the following steps: Step one: obtaining a degraded image polluted by Cauchy noise; The degraded image satisfies a periodic boundary condition; Step two: establishing a degradation model based on a mixed regular term of overlapping combination sparsity; The formula for establishing the degradation model based on the mixed regular term of overlapping combination sparsity is: where <log(γ 2 +(Hf-g) 2 ),1> is the fidelity term, is the quadratic penalty term; λ > 0 is the regularization parameter; μ is the positivity parameter; is the OGSTV regularization; is the non-convex term; ω > 0 is the regularization parameter of the regularization term; Step three: solving the degradation model by using an improved alternating direction multiplier method; The process for solving the degradation model by using the improved alternating direction multiplier method comprises: Introducing auxiliary variables z, v x y q, s, the formula of the degenerated model is​ s.t. z = Hf,v x = D x f,v y = D y f,q = D 2 f,s = f Then, a Lagrange function is obtained by using a Lagrange operator u1, u2, u3, u4, u5 as follows: where β i (i = 1, 2, 3, 4) are positive penalty parameters; Step four: obtaining a final restored image by using parameter analysis to obtain a stable solution of the degradation model.

2. The four-directional overlapping combined sparse total variation Cauchy noise image restoration method according to claim 1, characterized in that, The degradation model is: g = Hf + v wherein, is the original image, g is the image to be recovered, H is a convolution operator, and v is a Cauchy noise.

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