CN111598798B - Image restoration method based on low-rank tensor chain decomposition - Google Patents

Image restoration method based on low-rank tensor chain decomposition Download PDF

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CN111598798B
CN111598798B CN202010345730.0A CN202010345730A CN111598798B CN 111598798 B CN111598798 B CN 111598798B CN 202010345730 A CN202010345730 A CN 202010345730A CN 111598798 B CN111598798 B CN 111598798B
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CN111598798A (en
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郑建炜
周力强
陈婉君
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Zhejiang University of Technology ZJUT
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T5/00—Image enhancement or restoration
    • G06T5/77—Retouching; Inpainting; Scratch removal
    • 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
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/10—Image acquisition modality
    • G06T2207/10004—Still image; Photographic image
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Abstract

An image restoration method based on low-rank tensor chain decomposition comprises the steps of obtaining a damaged observation image, and determining a set omega of all unknown pixel points of an area to be restored in the damaged image; constructing an image restoration model TNN-TT based on low-rank tensor kernel norm chain decomposition, introducing total variation, constructing a weighted tensor Schatten P norm as non-convex relaxation of tensor ranks, constructing a weighted tensor Schatten P norm and total variation image restoration model WTS-TT based on low-rank tensor chain decomposition, introducing auxiliary variables, optimizing the image restoration model, solving by adopting an ADMM algorithm, and obtaining a restored image. The application considers the low-rank tensor characteristic of the image, the correlation and the local smoothness, and has good image restoration effect.

Description

Image restoration method based on low-rank tensor chain decomposition
Technical Field
The application relates to the field of image processing, in particular to an image restoration method based on low-rank tensor chain decomposition.
Background
With the advancement of human society and the development of technology, people acquire hundreds of millions of pieces of information including images, voices, texts and the like through digital media every day. It is counted that about 75% of the information acquired by humans is obtained by the visual perception system. The image is one of the main carriers of visual transmission and plays a vital role in communication and visual perception of people.
Some information is inevitably lost in the process of acquiring, storing and transmitting the image, for example, when the remote sensing image is acquired, stripe noise is generated in the acquired remote sensing image or useful information is blocked by cloud layers due to factors such as damage of a sensor or weather influence and the like. Image restoration refers to restoration reconstruction of a damaged image or removal of redundant objects in the image, and the image restoration has become an indispensable preprocessing step before subsequent application of the image.
Around the problem of image restoration, researchers at home and abroad have proposed many restoration methods. The first is a low rank based approach. The high-dimensional image can be regarded as a natural tensor, effectively recovering the image with low rank performance in all dimensions. The second category is methods based on local smoothness. In addition to low rank priors, actual image data often exhibits piecewise smooth properties. Total Variation (TV) is capable of characterizing segment smoothness properties well and proved to be excellent in image restoration. The third class is correlation-based methods. Tensors, while capable of recording information of each dimension of a high-dimensional image in more detail than a matrix, still remain an open problem how to define the exact characterization of each dimension's correlation, classical tensor decomposition methods are CP decomposition, tucker decomposition and tensor chain decomposition (Tensor Train decomposition, TT).
Disclosure of Invention
In order to further improve the effect of image restoration, the application provides an image restoration method based on low-rank tensor chain decomposition.
The technical scheme adopted for solving the technical problems is as follows:
an image restoration method based on low-rank tensor chain decomposition, the method comprising the steps of:
step 1) obtaining a damaged observation image and determining a set omega of all unknown pixel points of an area to be repaired in the damaged image;
step 2) taking low rank property and correlation of all dimensionalities of the image tensor into consideration, and constructing an image restoration model based on low rank tensor kernel norm chain decomposition by combining tensor kernel norm and tensor chain decomposition, and marking the model as a TNN-TT model;
step 3) introducing a total variation which can well describe the segment smoothness and is proved to be excellent in image restoration, taking a TNN-TT model as a framework, constructing a weighted tensor Schatten P norm as a non-convex relaxation of tensor rank, constructing a weighted tensor Schatten P norm and total variation image restoration model based on low-rank tensor chain decomposition, and marking the model as a WTS TV-TT model;
step 4) introducing auxiliary variables for eliminating the correlation between matrix dataOptimizing the WTS-TT of the image restoration model;
step 5) solving the image restoration optimization model by adopting an ADMM algorithm to obtain a restored image
Further, the processing procedure of the step 2) is as follows:
first, considering the low rank of all dimensions of an image, the tensor kernel norms employed are defined as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,representing an image to be restored, wherein I 1 ,I 2 ,...,I N Is the dimension of each order of tensors, +.>Representing real numbers, alpha i Is a non-negative weight and satisfies +.> Is tensor->An expansion matrix of an ith dimension;
in consideration of the correlation of the images, the definition of the adopted tensor chain decomposition is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,is a tensor kernel representing a tensor chain, x 1 R represents the reduction and multiplication operation of tensor chain kernel n Represents the nth rank, and is specifically defined as R 0 ＝R N ＝1；
Further constructing an image restoration TNN-TT model based on low-rank tensor kernel norm chain decomposition:
where gamma is the coefficient of the low rank norm,an image representing a breakage;
further, the processing procedure of the step 3) is as follows:
the introduction of a total variation that is capable of characterizing segment smoothness well and that has proved to be excellent in image restoration is defined as follows:
TV(x)＝∑ i＝1 ∑ j＝1 |x(i,j)| (4)
where || represents the L2 norm and x (i, j) represents the pixel value in the image;
the weighted Schatten P norm of the i-th layer matrix is defined as follows:
wherein r is (i) Is a matrixRank of->Is a matrix->The j th singular value,/, for>Is a spatial weight matrix, w i Is the i-th element of the weight w;
thereby obtaining tensorsSchatten P-norm:
wherein p (p is more than or equal to 0 and less than or equal to 1) is a parameter of Schatten-p norm;
further constructing a WTS-TT model of weight tensor Schatten P norm and total variation image restoration based on low-rank tensor chain decomposition:
where τ is a non-negative canonical parameter, λ i Is non-negative weight and satisfiesD (i) Representation (I) i -1)-by-I i Matrix of [ D ] (i) ] j,j ＝1，[D (i) ] j,j+1 = -1, and D (i) Other elements in (2) are equal to 0;
further, the processing procedure of the step 4) is as follows:
introducing auxiliary variablesThe WTS-TT optimization of the image restoration model is as follows:
further, the processing procedure of the step 5) is as follows:
(5-1)establishment ofSub-problems under ADMM algorithm:
wherein mu 1 Is a penalty coefficient, Λ is a lagrangian multiplier,is the Frobenius norm; (5-2) establishment ofSub-problems under ADMM algorithm:
wherein mu 2 Is a penalty coefficient, Φ is a Lagrangian multiplier;
(5-3) establishment ofSub-problems under ADMM algorithm:
wherein mu 3 Is a penalty coefficient, Γ is the Lagrangian multiplier;
(5-4) establish the sub-problem of χ under the ADMM algorithm:
wherein mu 4 Is a penalty coefficient, ψ is the Lagrangian multiplier;
(5-5) updating the lagrangian multiplier:
(5-6) updating penalty coefficients:
μ 1 ＝κμ 1 ,μ 2 ＝κμ 2 ,μ 3 ＝κμ 3 ,μ 4 ＝κμ 4 (14)
where k.epsilon.1, 1.5 is a constant.
The beneficial effects of the application are as follows: the image restoration method comprises the steps of (1) taking low-rank tensor characteristics and correlation of images into consideration (2) taking local smoothness of the images into consideration (3) and achieving good image restoration effect.
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FIG. 1 is a schematic illustration of an image tensor chain decomposition of the present application;
fig. 2 is a flow chart of the method of the present application.
Detailed Description
The following description of the embodiments of the present application will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present application, but not all embodiments. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used herein in the description of the application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.
An image restoration method based on low-rank tensor chain decomposition, the method comprising the steps of:
step 1) obtaining a damaged observation image and determining a set omega of all unknown pixel points of an area to be repaired in the damaged image;
step 2) taking low rank property and correlation of all dimensionalities of the image tensor into consideration, and constructing an image restoration model based on low rank tensor kernel norm chain decomposition by combining tensor kernel norm and tensor chain decomposition, and marking the model as a TNN-TT model;
step 3) introducing a total variation which can well describe the segment smoothness and is proved to be excellent in image restoration, taking a TNN-TT model as a framework, constructing a weighted tensor Schatten P norm as a non-convex relaxation of tensor rank, constructing a weighted tensor Schatten P norm and total variation image restoration model based on low-rank tensor chain decomposition, and marking the model as a WTS TV-TT model;
step 4) introducing auxiliary variables for eliminating the correlation between matrix dataOptimizing the WTS-TT of the image restoration model;
step 5) solving the image restoration optimization model by adopting an ADMM algorithm to obtain a restored image
Further, the processing procedure of the step 2) is as follows:
first, considering the low rank of all dimensions of an image, the tensor kernel norms employed are defined as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,representing an image to be restored, wherein I 1 ,I 2 ,...,I N Is the dimension of each order of tensors, +.>Representing real numbers, alpha i Is a non-negative weight and satisfies +.> Is tensor->An expansion matrix of an ith dimension;
the definition of the tensor chain decomposition used, considering the correlation of the images, is as follows:
wherein, the liquid crystal display device comprises a liquid crystal display device,is a tensor kernel representing a tensor chain, x 1 R represents the reduction and multiplication operation of tensor chain kernel n Represents the nth rank, and is specifically defined as R 0 ＝R N ＝1。
Further constructing an image restoration TNN-TT model based on low-rank tensor kernel norm chain decomposition:
where gamma is the coefficient of the low rank norm,an image representing a breakage;
further, the processing procedure of the step 3) is as follows:
the introduction of a total variation that is capable of characterizing segment smoothness well and that has proved to be excellent in image restoration is defined as follows:
TV(x)＝∑ i＝1 ∑ j＝1 |x(i,j)| (4)
where || represents the L2 norm and x (i, j) represents the pixel value in the image;
the weighted Schatten P norm of the i-th layer matrix is defined as follows:
wherein r is (i) Is a matrixRank of->Is a matrix->The j th singular value,/, for>Is a spatial weight matrix, w i Is the i-th element of the weight w;
thereby obtaining tensorsSchatten P-norm:
wherein p (p is more than or equal to 0 and less than or equal to 1) is a parameter of Schatten-p norm;
further constructing a WTS-TT model of weight tensor Schatten P norm and total variation image restoration based on low-rank tensor chain decomposition:
where τ is a non-negative canonical parameter, λ i Is non-negative weight and satisfiesD (i) Representation (I) i -1)-by-I i Matrix of [ D ] (i) ] j,j ＝1，[D (i) ] j,j+1 = -1, and D (i) Other elements in (2) are equal to 0;
further, the processing procedure of the step 4) is as follows:
introducing auxiliary variablesThe WTS-TT optimization of the image restoration model is as follows:
further, the processing procedure of the step 5) is as follows:
(5-1) establishment ofSub-problems under ADMM algorithm:
wherein mu 1 Is a penalty coefficient, Λ is a lagrangian multiplier,is the Frobenius norm;
(5-2) establishment ofSub-problems under ADMM algorithm:
wherein mu 2 Is a penalty coefficient, Φ is a Lagrangian multiplier;
(5-3) establishment ofSub-problems under ADMM algorithm:
wherein mu 3 Is a penalty coefficient, Γ is the Lagrangian multiplier;
(5-4) establish the sub-problem of χ under the ADMM algorithm:
wherein mu 4 Is a penalty coefficient, ψ is the Lagrangian multiplier;
(5-5) updating the lagrangian multiplier:
(5-6) updating penalty coefficients:
μ 1 ＝κμ 1 ,μ 2 ＝κμ 2 ,μ 3 ＝κμ 3 ,μ 4 ＝κμ 4 (14)
where k.epsilon.1, 1.5 is a constant.
The application considers the low-rank tensor characteristic of the image, the correlation and the local smoothness, and has good image restoration effect.
The embodiments described in the present specification are merely examples of implementation forms of the inventive concept, and the scope of protection of the present application should not be construed as being limited to the specific forms set forth in the embodiments, and the scope of protection of the present application and equivalent technical means that can be conceived by those skilled in the art based on the inventive concept.

Claims (5)

1. An image restoration method based on low-rank tensor chain decomposition comprises the following steps:
step 1) obtaining a damaged observation image and determining a set omega of all unknown pixel points of an area to be repaired in the damaged image;
step 2) taking low rank property and correlation of all dimensionalities of the image tensor into consideration, and constructing an image restoration model based on low rank tensor kernel norm chain decomposition by combining tensor kernel norm and tensor chain decomposition, and marking the model as a TNN-TT model;
step 3) introducing a total variation which can well describe the segment smoothness and is proved to be excellent in image restoration, taking a TNN-TT model as a framework, constructing a weighted tensor Schatten P norm as a non-convex relaxation of tensor rank, constructing a weighted tensor Schatten P norm and total variation image restoration model based on low-rank tensor chain decomposition, and marking the model as a WTS TV-TT model;
step 4) introducing auxiliary variablesAnd->Optimizing the WTS-TT of the image restoration model;
and 5) solving the image restoration optimization model by adopting an ADMM algorithm to obtain a restored image.
2. The method for image restoration based on low-rank tensor chain decomposition according to claim 1, wherein said method for constructing an image restoration TNN-TT model based on low-rank tensor kernel norm chain decomposition in step 2) comprises:
where gamma is the coefficient of the low rank norm,image representing breakage->Representing an image to be restored, wherein I 1 ,I 2 ,...,I N Is the dimension of each order of tensors, +.>Representing real numbers, alpha i Is a non-negative weight and satisfies +.> Is tensor->An expansion matrix of the i-th dimension, I.I * Kernel norms representing tensors, +.>Is a tensor kernel representing a tensor chain, x 1 R represents the reduction and multiplication operation of tensor chain kernel n Represents the nth rank, and is specifically defined as R 0 ＝R N ＝1。
3. The method for repairing an image based on low-rank tensor chain decomposition according to claim 1, wherein said step 3) is a method for constructing a weighted tensor Schatten P norm and total variation image repairing WTSTV-TT model based on low-rank tensor chain decomposition, comprising:
first, a weighted Schatten P norm of the i-th layer matrix is defined:
wherein r is (i) Is a matrixRank of->Is a matrix->The j th singular value,/, for>Is a matrix of spatial weights that are to be used,w i is the i-th element of the weight w;
thereby obtaining tensorsSchatten P-norm:
wherein p (p is more than or equal to 0 and less than or equal to 1) is a parameter of Schatten-p norm;
further constructing a WTS-TT model of weight tensor Schatten P norm and total variation image restoration based on low-rank tensor chain decomposition:
where τ is a non-negative canonical parameter, λ i Is non-negative weight and satisfiesD (i) Representation (I) i -1)-by-I i Matrix of [ D ] (i) ] j,j ＝1，[D (i) ] j,j+1 = -1, and D (i) Is equal to 0.
4. The image restoration method based on low-rank tensor chain decomposition according to claim 1, wherein said method of optimizing said image restoration model WTSTV-TT in step 4) is as follows:
introducing auxiliary variablesThe WTS-TT optimization of the image restoration model is as follows:
5. the image restoration method based on low-rank tensor chain decomposition according to claim 1, wherein said step 5) adopts an ADMM algorithm to solve said image optimization restoration model comprises the following steps:
(5-1) establishment ofSub-problems under ADMM algorithm:
wherein mu 1 Is a penalty coefficient, Λ is a lagrangian multiplier,is the Frobenius norm;
(5-2) establishment ofSub-problems under ADMM algorithm:
wherein mu 2 Is a penalty coefficient, Φ is a Lagrangian multiplier;
(5-3) establishment ofSub-problems under ADMM algorithm:
wherein mu 3 Is the penalty coefficient, Γ is the LagrangianA day multiplier;
(5-4) establishment ofSub-problems under ADMM algorithm:
wherein mu 4 Is a penalty coefficient, ψ is the Lagrangian multiplier;
(5-5) updating the lagrangian multiplier:
(5-6) updating penalty coefficients:
μ 1 ＝κμ 1 ,μ 2 ＝κμ 2 ,μ 3 ＝κμ 3 ,μ 4 ＝κμ 4 (11)
where k.epsilon.1, 1.5 is a constant.
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