CN111626943A - Total variation image denoising method based on first-order forward and backward algorithm - Google Patents

Total variation image denoising method based on first-order forward and backward algorithm Download PDF

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CN111626943A
CN111626943A CN202010234130.7A CN202010234130A CN111626943A CN 111626943 A CN111626943 A CN 111626943A CN 202010234130 A CN202010234130 A CN 202010234130A CN 111626943 A CN111626943 A CN 111626943A
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杨敏
谈晶圩
吴骁伦
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a first-order forward and backward algorithm-based total variation image denoising method, which is improved on the basis of a traditional total variation model aiming at the defect of non-ideal denoising effect of the traditional algorithm. The experimental results show that: in the aspects of signal to noise ratio and image consistency, the method improves the method, can generate clearer edges and image structures, and improves the denoising performance.

Description

Total variation image denoising method based on first-order forward and backward algorithm
Technical Field
The invention relates to the technical field of image processing, in particular to a fully-variable partial image denoising method based on a first-order forward and backward algorithm.
Background
Image restoration (image restoration) is to restore the original view of the degraded image by using the prior knowledge of the degradation process. The image restoration technology is mainly proposed for "degradation" in the imaging process, which is called "degradation" because of various factors, and the degradation of the image is mainly caused by two factors, namely, the relevant characteristics of the system and noise. The degradation of the image quality can seriously affect the subsequent processing of the image, aiming at the degraded image, firstly, a mathematical degradation model is established in the degradation process according to the original prior knowledge, then, the model is adopted to carry out inverse processing to recover the image, and meanwhile, certain specific algorithms are applied to judge the image recovery effect. The image restoration becomes a hot problem in the field of image processing, and the problem of how to reduce noise points in the image and recover the original image is of great significance to the image processing process, wherein the research on the denoising algorithm optimization and the denoising model is the key of the problem. Nowadays, image restoration is widely applied to various applications, such as the beauty function of each mobile phone camera, the automatic removal of spots such as acne scars, the addition of various filters, the new functions of rain removal, fog removal and the like, and has important application value and research prospect.
The total variation, also called total variation, is the most commonly used one in image restoration, and is most directly and effectively applied to image restoration. The full-variation model is a model for smoothing an image by gradient descent and protecting edges, consists of a regular term and a fidelity term, and can remove noise and be similar to an original image as far as possible. Although the full-variation model has the effect of restoring the image, the model only considers gradient information in the vertical and horizontal directions and does not fully consider neighborhood information of pixels, so that some structural information of the image is ignored, and the image restoring effect is not ideal. Therefore, a method is needed to improve the image restoration effect, solve the disadvantages of the conventional total variation model, and improve the visual effect of the restored image.
Disclosure of Invention
The invention aims to provide a full-variational image denoising method based on a first-order forward and backward algorithm aiming at the defects of the traditional full-variational restoration model, which can generate clearer edges and structures and improve the visual effect of the restored image.
In order to achieve the purpose, the invention adopts the following technical scheme: the method for denoising the fully-variational image based on the first-order forward and backward algorithm comprises the following steps:
a first-order forward and backward algorithm-based total variation image denoising method comprises the following steps:
step 1: constructing a total variation denoising model and converting the model into a dual form;
step 2: converting a fuzzy image to be restored into a two-dimensional matrix, and inputting the two-dimensional matrix into the total variation denoising model;
and step 3: and solving the total variation denoising model through a first-order forward and backward algorithm to obtain an original image of the blurred image.
Further, the total variation denoising model in step 1 is:
Figure BDA0002430395330000021
the dual type is:
Figure BDA0002430395330000022
wherein u is a dual variable of x, g (x) is a regular term of a fully-variant denoising model, and noise is suppressed in the optimization process, f (x) is a fidelity term of the fully-variant denoising model, and is used for keeping the similarity between a denoised image and an observed image,
Figure BDA0002430395330000023
is the dual of f, L is the fuzzy matrix, L*Is the conjugate of L, | · |. non-woven phosphorvRepresenting the v norm, g is the observed image.
Further, the specific steps of step 3 include:
step 3.1: solving the approximate mapping of the function g (x) using an approximation operator;
step 3.2: solving the optimization problem through a forward and backward splitting algorithm, and solving a function g (x) minimum solution as a result of image recovery;
step 3.3: and recovering the original image.
Further, in step 3.1, the approximation operator is:
Figure BDA0002430395330000024
wherein z is the starting point for solving the optimal function g (x), and tau is the step length; the approximation operator can find an approximation of the minimum value of the function g without deviating from the starting point z;
the approximate mapping of function g (x) is:
Figure BDA0002430395330000025
Figure BDA0002430395330000029
where k is the number of iterations starting from 1, when the function g (x) is differentiable,
Figure BDA0002430395330000026
g is a sub-gradient of G, and any x needs to satisfy τ G + (x)k-z) 0; at this time xk＝proxg(z, τ) ═ z- τ G, indicating that the optimal value x of the function is obtained as long as z is opposite to G;
τG+(xk-z) 0 is equivalent to
Figure BDA0002430395330000027
The approximation operator can now be expressed as
Figure BDA0002430395330000028
JτGz is represented as the pre-solver of τ G.
Further, the specific steps of step 3.2 include:
the method comprises the steps that a forward and backward splitting algorithm is applied to solve the minimization problem of a total variation model, a regular term in the total variation model needs to be reconstructed into a differentiable function, an original model needs to be converted into a dual form, and therefore a dual variable u of x is introduced;
the convex original problem form of the total variation denoising model is as follows:
Figure BDA0002430395330000031
applying dual theory knowledge, the convex-dual problem form is:
Figure RE-GDA0002574806340000032
d represents a first-order difference operator, lambda represents a regularization parameter, a first term of the model is called a regularization term, and noise is suppressed in the optimization process; the second term is called a fidelity term and is used for keeping the similarity between the denoised image and the observed image; function iotacIs an indicator function of set c; the solution to the minimization problem of the above equation is trivial;
after the total variation g (x) term is introduced into an approximation operator, the dual form is as follows:
Figure BDA0002430395330000033
then, a forward and backward algorithm is used for iteratively solving a total variation denoising model to find an optimal solution x which is a result of image recovery; the forward backward splitting algorithm is as follows:
the algorithm FBS:
1.For k＝1，2，...,do
2.
Figure BDA0002430395330000034
3.
Figure BDA0002430395330000035
scalar τkDenotes the step size of the kth iteration, prox being the approximate operator introduced.
Through the implementation of the technical scheme, the invention has the beneficial effects that: the image restoration method has the advantages that the advantages of a traditional full-variation restoration model algorithm are kept in the image restoration process, the edge protection performance is further enhanced, the image is more effectively restored, and the image quality index value is improved to some extent.
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FIG. 1 is a block diagram of a process of a first-order forward-backward algorithm-based fully-variational image denoising algorithm according to the present invention.
Fig. 2 is a diagram illustrating the image contrast effect between the image to be restored and the restored image.
Detailed Description
The invention is further described with reference to the following figures and specific examples.
It will be understood by those within the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs, and it is to be understood that such terms as those defined in commonly used dictionaries are to be interpreted as having a meaning that is consistent with their meaning in the context of the prior art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.
As shown in fig. 1 and fig. 2, the fully-variant image denoising algorithm based on the first-order forward-backward algorithm includes the following steps:
step (1): converting the minimization problem of the total variation model into a convex dual problem, and establishing an original dual form;
the general denoising optimization model primitive form:
Figure BDA0002430395330000041
the convex-dual mode:
Figure BDA0002430395330000042
given a convex, lower semicontinuous function f, the conjugate function f of f*Is defined as
Figure BDA0002430395330000043
If it is not
Figure BDA0002430395330000044
Conjugate function f of f*Is defined as
Figure BDA0002430395330000045
Step (2): solving the approximate mapping of the function by using an approximate operator;
g (x) in the total variation model is not microminiature, is not easy to solve and cannot be solved simply by a gradient descent method. Solving the approximate mapping of the objective function with an approximation operator:
Figure BDA0002430395330000046
the approximation operator can find an approximation of the minimum of the function g without deviating from the starting point Z, this operator being called backward gradient descent with a step length τ. The full-variation classification model can be conveniently and efficiently solved by using a forward and backward classification algorithm, the problem of the non-microminiature target convex function is solved, and the simplicity and the effectiveness of the algorithm are kept.
After the general denoising optimization model uses an approximate mapping operator, the model is expressed into an original dual form by using a forward and backward algorithm:
Figure BDA0002430395330000047
Figure BDA00024303953300000411
when the function g can be micro-scaled,
Figure BDA0002430395330000048
g is this gradient of G, and any x needs to satisfy τ G + (x)k-z) 0. At this time xk＝proxg(z, τ) ═ z- τ G, indicating that the optimal value x of the function is obtained as long as z is opposite to G.
τG+(xk-z) is in turn equivalent to 0
Figure BDA0002430395330000049
In this case, the approximation operator can be expressed as
Figure BDA00024303953300000410
JτGz is represented as the pre-solver of τ G.
And (3): and solving a minimization solution of the function in the model through a forward and backward splitting algorithm to obtain an image recovery result. The minimization of the target function is the key of the image recovery effect, and the forward and backward splitting algorithm is as follows:
the algorithm FBS:
1.For k＝1，2，...,do
2.
Figure BDA0002430395330000051
3.
Figure BDA0002430395330000052
scalar τkThe step size of the kth iteration is shown, and the step 3 is the step of approximating the mapping operator.
The forward and backward splitting algorithm is applied to solve the minimization problem of the total variation model, and the regular term in the total variation model needs to be reconstructed into a differentiable function, so that a dual variable u is introduced.
We consider a full-variational restoration model whose convex primitive problem form is:
Figure BDA0002430395330000053
applying dual theory knowledge, the convex-dual problem form is:
Figure RE-GDA0002574806340000054
||·||vrepresenting the v norm, D the first order difference operator, λ the regularization parameter, and g the observed image. The first term of the model, called the regularization term, suppresses noise during the optimization process. The second term is called a fidelity term and is used for keeping the similarity of the denoised image and the observed image. Function iotacIs an indicator function of the set c. The solution to the minimization problem of the above equation is trivial.
And (4): restoring a complete image;
the image comparison effect graph of the image before and after restoration and the image before restoration based on the first-order forward and backward algorithm fully-variational image denoising algorithm is shown in fig. 2, wherein fig. 2(a) shows the image before restoration, fig. 2(b) shows the image after restoration, and experimental results show that the image restoration effect of the fully-variational image restoration orthodox mixed gradient new algorithm is good.
The invention has the advantages that: the method is improved on the basis of the traditional total variation component model aiming at the defect that the denoising effect of the traditional algorithm is not ideal, firstly, the regular term and the fidelity term of the total variation component model are converted into a convex dual problem, and then, the problem is solved by applying a forward and backward splitting algorithm in an iterative mode. The experimental results show that: in the aspects of signal to noise ratio and image consistency, the method improves the method, can generate clearer edges and image structures, and improves the denoising performance. The image restoration method has the advantages that the advantages of a traditional full-variation restoration model algorithm are kept in the image restoration process, the edge protection performance is further enhanced, the image is more effectively restored, and the image quality index value is improved to some extent.
The foregoing is only a partial embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (5)

1. A first-order forward and backward algorithm-based total variation image denoising method is characterized by comprising the following steps:
step 1: constructing a total variation denoising model and converting the model into a dual form;
step 2: converting a fuzzy image to be restored into a two-dimensional matrix, and inputting the two-dimensional matrix into the total variation denoising model;
and step 3: and solving the total variation denoising model through a first-order forward and backward algorithm to obtain an original image of the blurred image.
2. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 1, wherein the fully-variational denoising model in step 1 is:
Figure FDA0002430395320000011
the dual type is:
Figure FDA0002430395320000012
wherein u is a dual variable of x, g (x) is a regular term of a fully-variant denoising model, and noise is suppressed in the optimization process, f (x) is a fidelity term of the fully-variant denoising model, and is used for keeping the similarity between a denoised image and an observed image,
Figure FDA0002430395320000013
is the dual of f, L is the fuzzy matrix, L*Is the conjugate of L, | · |. non-woven phosphorvRepresenting the v norm, g is the observed image.
3. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 2, wherein the specific steps of step 3 comprise:
step 3.1: solving the approximate mapping of the function g (x) using an approximation operator;
step 3.2: solving the optimization problem through a forward and backward splitting algorithm, and solving a function g (x) minimum solution as a result of image recovery;
step 3.3: and recovering the original image.
4. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 3, wherein the approximation operator in step 3.1 is:
Figure FDA0002430395320000014
wherein z is the starting point for solving the optimal function g (x), and tau is the step length;
the approximate mapping of function g (x) is:
x(k)＝▽f*(-L*u(k))
Figure FDA0002430395320000015
where k is the number of iterations from 1, when the function G (x) is differentiable, G is ▽ G (x), G is the sub-gradient of G, and any x needs to satisfy τ G + (x)k-z) 0; at this time xk＝proxg(z, τ) ═ z- τ G, indicating that the optimal value x of the function is obtained as long as z is opposite to G;
τG+(xk-z) ═ 0 is equivalent to τ ▽ g (x)k)+(xk-z) is 0, in which case the approximation operator can be expressed as xk＝(τ▽g+I)-1z＝Jτ▽gz，JτGz is represented as the pre-solver of τ G.
5. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 4, wherein the specific steps of step 3.2 comprise:
the convex original problem form of the total variation denoising model is as follows:
Figure RE-FDA0002574806330000023
applying dual theory knowledge, the convex-dual problem form is:
Figure RE-FDA0002574806330000024
d represents a first-order difference operator, lambda represents a regularization parameter, a first term of the model is called a regularization term, and noise is suppressed in the optimization process; the second term is called a fidelity term and is used for keeping the similarity between the denoised image and the observed image; function iotacIs an indicator function of set c; the solution to the minimization problem of the above equation is trivial;
after the total variation g (x) term is introduced into an approximation operator, the dual form is as follows:
Figure RE-FDA0002574806330000025
then, a forward and backward algorithm is used for iteratively solving a total variation denoising model to find an optimal solution x which is a result of image recovery; the forward backward splitting algorithm is as follows:
the algorithm FBS:
1.For k＝1，2，...,do
2.
Figure RE-FDA0002574806330000026
3.
Figure RE-FDA0002574806330000027
scalar τkDenotes the step size of the kth iteration, prox being the approximate operator introduced.
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CN112508807A (en) * 2020-11-26 2021-03-16 电子科技大学 Image denoising method based on multi-direction total variation
CN113112425A (en) * 2021-04-08 2021-07-13 南京大学 Four-direction relative total variation image denoising method

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112508807A (en) * 2020-11-26 2021-03-16 电子科技大学 Image denoising method based on multi-direction total variation
CN112508807B (en) * 2020-11-26 2023-07-21 电子科技大学 Image denoising method based on multi-direction total variation
CN113112425A (en) * 2021-04-08 2021-07-13 南京大学 Four-direction relative total variation image denoising method
CN113112425B (en) * 2021-04-08 2024-03-22 南京大学 Four-direction relative total variation image denoising method

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