CN111626943B - 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 PDFInfo
- Publication number
- CN111626943B CN111626943B CN202010234130.7A CN202010234130A CN111626943B CN 111626943 B CN111626943 B CN 111626943B CN 202010234130 A CN202010234130 A CN 202010234130A CN 111626943 B CN111626943 B CN 111626943B
- Authority
- CN
- China
- Prior art keywords
- image
- total variation
- denoising
- algorithm
- function
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 238000004422 calculation algorithm Methods 0.000 title claims abstract description 42
- 238000000034 method Methods 0.000 title claims abstract description 30
- 230000009977 dual effect Effects 0.000 claims description 20
- 238000005457 optimization Methods 0.000 claims description 10
- 238000011084 recovery Methods 0.000 claims description 9
- 238000013507 mapping Methods 0.000 claims description 8
- 239000011159 matrix material Substances 0.000 claims description 6
- 238000004364 calculation method Methods 0.000 claims 1
- 230000000694 effects Effects 0.000 abstract description 10
- 230000007547 defect Effects 0.000 abstract description 3
- 238000006731 degradation reaction Methods 0.000 description 7
- 230000015556 catabolic process Effects 0.000 description 5
- 238000012545 processing Methods 0.000 description 5
- 238000005034 decoration Methods 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 238000011160 research Methods 0.000 description 2
- 230000000007 visual effect Effects 0.000 description 2
- OAICVXFJPJFONN-UHFFFAOYSA-N Phosphorus Chemical compound [P] OAICVXFJPJFONN-UHFFFAOYSA-N 0.000 description 1
- 206010039580 Scar Diseases 0.000 description 1
- 230000003796 beauty Effects 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 238000007635 classification algorithm Methods 0.000 description 1
- 238000013145 classification model Methods 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 238000011478 gradient descent method Methods 0.000 description 1
- 238000003384 imaging method Methods 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T5/00—Image enhancement or restoration
- G06T5/70—Denoising; Smoothing
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T10/00—Road transport of goods or passengers
- Y02T10/10—Internal combustion engine [ICE] based vehicles
- Y02T10/40—Engine management systems
Landscapes
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- Theoretical Computer Science (AREA)
- Image Processing (AREA)
Abstract
The invention discloses a first-order forward and backward algorithm-based total variation image denoising method, which is an improvement on the basis of a traditional total variation model aiming at the defect of unsatisfactory 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
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 images and recover original images is of great significance to the image processing process, wherein the key to the problem is to research denoising algorithm optimization and denoising models. 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.
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 total variation model is a model which depends on gradient descent to smooth an image and protect 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 a recovery effect on 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 recovery effect of the image 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 realize the purpose, the invention adopts the following technical scheme: the full-variation image denoising method 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:
the dual type is:
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,
is the dual of f, L is the fuzzy matrix, L * Is the conjugate of L, | · |. non-woven phosphor v Representing 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:
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:
where k is the number of iterations starting from 1, and when the function g (x) is differentiable,
g is the sub-gradient of G, any x needs to satisfy τ G + (x) k -z) 0; at this time x k ＝prox g (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+(x k -z) 0 is equivalent to
The approximation operator can now be expressed as
J τG z 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:
applying dual theory knowledge, the convex-dual problem form is:
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 iota c Is 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:
then, a forward and backward algorithm is used for iteratively solving a total variation denoising model to find an optimal solution x which is the result of image recovery; the forward backward splitting algorithm is as follows:
the algorithm FBS:
1.For k＝1，2，...,do
2.
3.
scalar τ k Denotes 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.
Drawings
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;
general denoising optimization model primitive form:
the convex-dual mode:
given a convex, lower semicontinuous function f, the conjugate function f of f * Is defined as
If it is not
Conjugate function f of f * Is defined as
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:
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 an approximate mapping operator is used in a general denoising optimization model, the model is expressed into an original dual form by using a forward and backward algorithm:
when the function g can be micro-scaled,
g is this gradient of G, and any x needs to satisfy τ G + (x) k -z) 0. At this time x k ＝prox g (z, τ) ═ z- τ G, indicating that the optimal value x of the function can be obtained as long as z is opposite to G.
τG+(x k -z) is in turn equivalent to 0
In this case, the approximation operator can be expressed as
J τG z is expressed as the pre-solver of τ G.
And (3): and solving a minimum solution of the function in the model through a forward and backward splitting algorithm as a result of image recovery. 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.
3.
scalar τ k The step size of the kth iteration is shown, and the step 3 is the step of the approximation 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 total variation restoration model whose convex primitive problem form is:
applying dual theory knowledge, the convex-dual problem form is:
||·|| v representing 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 iota c Is 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 before and after restoration and before restoration of the fully-variational image denoising algorithm based on the first-order forward and backward algorithm is shown in fig. 2, wherein fig. 2(a) shows an image before restoration, fig. 2(b) shows an image after restoration, and experimental results show that the image restoration effect of the fully-variational image restoration original-couple 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 (4)
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: solving the total variation denoising model through a first-order forward-backward algorithm to obtain an original image of the blurred image;
the total variation denoising model in the step 1 is as follows:
the dual type is:
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,
is the dual of f, L is the fuzzy matrix, L * Is the conjugate of L, | · | non-calculation v Representing the v norm, g is the observed image.
2. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 1, 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.
3. The method for denoising a fully-variational image based on a first-order forward-backward algorithm as claimed in claim 2, wherein said approximation operator in step 3.1 is:
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:
where k is the number of iterations starting from 1, when the function g (x) is differentiable,
g is a sub-gradient of G, and any x needs to satisfy τ G + (x) k -z) 0; at this time x k ＝prox g (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+(x k -z) 0 is equivalent to
The approximation operator can then be expressed as
J τG z is represented as the pre-solver of τ G.
4. The method for denoising fully-variational images based on the first-order forward-backward algorithm as claimed in claim 3, wherein the specific steps of step 3.2 include:
the convex original problem form of the total variation denoising model is as follows:
applying dual theory knowledge, the convex-dual problem form is:
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 de-noised image and the observed image; function iota c Is a collectionC, an indication function of the sum; 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:
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.
3.
scalar τ k Represents the step size of the kth iteration, prox is the introduced approximation operator.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010234130.7A CN111626943B (en) | 2020-03-30 | 2020-03-30 | Total variation image denoising method based on first-order forward and backward algorithm |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202010234130.7A CN111626943B (en) | 2020-03-30 | 2020-03-30 | Total variation image denoising method based on first-order forward and backward algorithm |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN111626943A CN111626943A (en) | 2020-09-04 |
| CN111626943B true CN111626943B (en) | 2022-09-06 |
Family
ID=72272503
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202010234130.7A Active CN111626943B (en) | 2020-03-30 | 2020-03-30 | Total variation image denoising method based on first-order forward and backward algorithm |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN111626943B (en) |
Families Citing this family (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112508807B (en) * | 2020-11-26 | 2023-07-21 | 电子科技大学 | Image denoising method based on multi-direction total variation |
| CN113112425B (en) * | 2021-04-08 | 2024-03-22 | 南京大学 | Four-direction relative total variation image denoising method |
Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110084756A (en) * | 2019-04-15 | 2019-08-02 | 闽南师范大学 | A kind of image de-noising method based on the overlapping sparse full variation of group of high-order |
-
2020
- 2020-03-30 CN CN202010234130.7A patent/CN111626943B/en active Active
Patent Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110084756A (en) * | 2019-04-15 | 2019-08-02 | 闽南师范大学 | A kind of image de-noising method based on the overlapping sparse full variation of group of high-order |
Non-Patent Citations (2)
| Title |
|---|
| 全变分模型图像复原的一阶前向后向优化算法研究;陈少利;《中国优秀硕士学位论文全文数据库》;20171026;全文 * |
| 基于四阶微分全变差的图像去噪模型;王璐;《计算机技术与发展》;20160310(第03期);全文 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN111626943A (en) | 2020-09-04 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN107403415B (en) | Compressed depth map quality enhancement method and device based on full convolution neural network | |
| He et al. | Guided image filtering | |
| CN111161360B (en) | Image defogging method of end-to-end network based on Retinex theory | |
| Fang et al. | Single image dehazing and denoising: a fast variational approach | |
| CN103761710A (en) | Image blind deblurring method based on edge self-adaption | |
| CN107301662B (en) | Compression recovery method, device and equipment for depth image and storage medium | |
| Shen et al. | Convolutional neural pyramid for image processing | |
| CN111626943B (en) | Total variation image denoising method based on first-order forward and backward algorithm | |
| Shu et al. | Alternating minimization algorithm for hybrid regularized variational image dehazing | |
| CN108564544A (en) | Image Blind deblurring based on edge perception combines sparse optimization method | |
| CN110415193A (en) | The restored method of coal mine low-light (level) blurred picture | |
| CN108460723B (en) | Bilateral total variation image super-resolution reconstruction method based on neighborhood similarity | |
| CN111353955A (en) | Image processing method, device, equipment and storage medium | |
| CN114820352A (en) | Hyperspectral image denoising method and device and storage medium | |
| Arulkumar et al. | Super resolution and demosaicing based self learning adaptive dictionary image denoising framework | |
| CN115526779A (en) | Infrared image super-resolution reconstruction method based on dynamic attention mechanism | |
| CN111640077A (en) | Simple and efficient fuzzy text picture sharpening processing method | |
| CN110852947B (en) | Infrared image super-resolution method based on edge sharpening | |
| Guo et al. | Image blind deblurring using an adaptive patch prior | |
| CN106033595B (en) | Image blind deblurring method based on local constraint | |
| CN108492264B (en) | Single-frame image fast super-resolution method based on sigmoid transformation | |
| CN116342443A (en) | Near infrared and visible light image fusion method and system | |
| CN109919857B (en) | Noise image completion method based on weighted Schleiden norm minimization | |
| CN109345478B (en) | Image smoothing processing method based on gradient minimization | |
| Yang et al. | Hyperspectral image denoising with collaborative total variation and low rank regularization |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |