CN112241938A - Image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation - Google Patents

Image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation Download PDF

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CN112241938A
CN112241938A CN202010846940.8A CN202010846940A CN112241938A CN 112241938 A CN112241938 A CN 112241938A CN 202010846940 A CN202010846940 A CN 202010846940A CN 112241938 A CN112241938 A CN 112241938A
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CN112241938B (en
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郑建炜
黄娟娟
陈婉君
秦梦洁
徐宏辉
陶星朋
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Zhejiang University of Technology ZJUT
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    • G06T5/77—Retouching; Inpainting; Scratch removal
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Abstract

The image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation comprises the following steps: 1) inputting image to be restored
Figure DDA0002643359830000011
Determining an area to be repaired of an image; 2) constructing a high-order tensor Hankelization and discrete total variation model; 3) combining the step 2) to construct an image restoration model of smooth Tak decomposition and high-order tensor Hank transformation, restoring the color image, and finally reconstructing and outputting a high-quality visual data image
Figure DDA0002643359830000012
The invention has the advantages that: the efficiency of image processing and the accuracy of image restoration are both considered.

Description

Image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation
Technical Field
The invention relates to the field of image processing, in particular to an image restoration method.
Background
With the rapid development of modern network technology, computer communication and sampling technology, the data to be analyzed mostly has a very complex structure. Generally, the image data acquisition process is influenced by various external factors, which results in poor visual quality, such as damage to hardware equipment, light and interference of electromagnetic waves. In this case, too, it may be impossible to directly retrieve the relevant image data due to device or time limitations. Therefore, repairing various blurred, low-resolution, partial-pixel-loss and other images to obtain high-quality visual data is a research content with practical application value.
Image inpainting is a typical image processing ill-posed problem that can be expressed as a missing value estimation problem. The core problem of missing value estimation is how to establish the relationship between known elements and unknown elements, and adding other prior information can effectively solve the image restoration problem, such as local smooth prior, non-local self-similar prior, sparse prior, low rank prior, sparse gradient prior, and the like. In recent years, many scholars have proposed different image restoration algorithms, which are mainly classified into three categories: 1) image restoration based on a variational differential equation; 2) image inpainting based on texture synthesis; 3) and (3) a mixing method. Bertalmia et al first proposed a differential equation-based image restoration method that restores an image by diffusing information of an undamaged region into the interior of a region to be restored by diffusing the boundaries of the region to be restored in different directions. This method has a good repairing effect only on the damages of a small area in the image. Chan et al propose a Total Variation (TV) algorithm, which has the greatest advantage of effectively overcoming the problem of linear filtering that smoothes image edges while suppressing noise, but the greatest drawback of the TV algorithm is that the "discontinuity" principle in human vision cannot be satisfied. The Curvature-Drive Diffusion (CDD) algorithm is an improved algorithm for TV algorithm, and aims to solve the problem of visual discontinuity in TV algorithm. Criminisi et al propose a sample block-based image restoration algorithm, which calculates the priority of a block to be restored by using boundary information of the region to be restored, and then searches for a sample block with the maximum similarity to the block to be restored in the undamaged region of the image to perform filling and restoration. The algorithm has a good repairing effect on a large-area damaged area, but the efficiency of the algorithm is reduced due to the fact that the repairing time is too long.
With the recent development of deep neural network architectures, deep learning methods have great significance in computer vision tasks such as object detection, image classification and image noise reduction. However, the deep learning based method requires a large number of labeled samples, which are difficult to obtain and consume a large amount of computation, so that research and application of the conventional method are still necessary and there is a great room for improvement.
Disclosure of Invention
The present invention provides an image restoration method based on smooth tack decomposition and high-order tensor hank-based method to overcome the above problems in the prior art. ,
in order to solve the visual processing problem of image data distortion, the invention expands the Hank structuring technology into the high-order tensor visual data, fully considers the essential attribute of the image and introduces the Discrete Total Variation (TV)d) The regularization term factor integrates it into a uniform objective function.
The technical scheme adopted by the invention for solving the technical problems comprises the following steps:
the image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation comprises the following steps:
step 1) inputting an image to be restored
Figure RE-GDA0002822316910000021
Determining an image to-be-repaired area and performing blocking operation on the image to-be-repaired area, wherein pixels in the image are divided into known points and unknown points, the known points are points with pixels not being 0 in the image, the unknown points are points with pixels being 0 in the image, and all the unknown points in the image form a set omega;
step 2), constructing a high-order tensor Hankelization and discrete total variation model;
step 3) image restoration constructed by combining the step 2)The model is used for repairing the color image and finally reconstructing and outputting a high-quality visual data image
Figure RE-GDA0002822316910000022
The invention has the following beneficial effects: the Hank structuring technology is expanded to be applied to the tensor field. Considering that the low rank and the smoothness of the matrix also exist in the tensor, firstly, data are embedded into a high-dimensional tensor, and tensor hank is structured through multi-dimensional linear replication and multi-dimensional folding linear operation; and secondly, considering data smoothness, introducing a discrete total variation factor to optimize a model, and finally, better finding the optimal rank through a low-rank incremental algorithm, wherein the algorithm has good convergence and more accurately restores a natural image.
The invention has the advantages that: the efficiency of image processing and the accuracy of image restoration are both considered.
Drawings
FIG. 1 is a schematic view of an area to be repaired;
FIG. 2 is a natural image with a pixel loss rate of 90%;
FIG. 3 is a natural image restored using the present invention;
fig. 4 is a flow chart of a method of the present invention.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
The image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation comprises the following steps:
step 1) inputting an image to be restored
Figure RE-GDA0002822316910000031
Determining an image to-be-repaired area and performing blocking operation on the image to-be-repaired area, wherein pixels in the image are divided into known points and unknown points, the known points are points with pixels not being 0 in the image, the unknown points are points with pixels being 0 in the image, and all the unknown points in the image form a set omega;
step 2), constructing a high-order tensor Hankelization and discrete total variation model;
step 3) repairing the color image by combining the image repairing model constructed in the step 2), and finally reconstructing and outputting a high-quality visual data image
Figure RE-GDA0002822316910000032
The treatment process of the step 2) is as follows:
(2-1) the high-order hank structured image inpainting model is defined as follows:
Figure RE-GDA0002822316910000033
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0002822316910000034
which represents the input of the image to be repaired,
Figure RE-GDA0002822316910000035
a post-repair image is represented as,
Figure RE-GDA0002822316910000036
represents the Frobenius norm;
Figure RE-GDA0002822316910000037
where 1 represents an observable pixel and 0 represents a missing pixel;
Figure RE-GDA0002822316910000038
is defined as
Figure RE-GDA0002822316910000039
Wherein fold(I,τ):
Figure RE-GDA00028223169100000310
Through the fold(I,τ)The input N-order tensor can be constructed into a 2N-order tensor, and the operation can be regarded as multi-dimensional linear replication and multi-dimensional foldingOperating; wherein
Figure RE-GDA00028223169100000311
The matrix is a copy matrix, namely a matrix comprising a plurality of identity matrixes, and the specific form is as follows:
Figure RE-GDA00028223169100000312
(2-2) the discrete total variation model is defined as follows:
Figure RE-GDA00028223169100000313
x represents a two-dimensional image and v represents a gradient;
Figure RE-GDA00028223169100000314
l. representing bilinear interpolation operations, particularly in a grid
Figure RE-GDA0002822316910000041
(n1，n2) The interpolation of (1); a*Representing an accompanying operation; the definition of the discrete operator D is (Dx)1[n1,n2]＝x[n1+1,n2]-x[n1,n2]，(Dx)2[n1,n2]＝x[n1,n2+1]-x[n1,n2]；
Let l of v be1,1,2Norm represents three vector components
Figure RE-GDA0002822316910000042
L of v. a1,1,2Norm sum, i.e.
Figure RE-GDA0002822316910000043
The discrete total variation model can therefore be redefined as:
Figure RE-GDA0002822316910000044
and fully utilizing low-rank complementary information and potential smoothing characteristics.
The treatment process of the step 3) is as follows:
(3-1) constructing an image restoration model based on smooth Tak decomposition and high-order tensor Hank transformation, which is defined as follows
Figure RE-GDA0002822316910000045
Figure RE-GDA0002822316910000046
In the formula, λ represents a balance parameter,
Figure RE-GDA0002822316910000047
represents a decomposition factor U(n)The (J, R) th entry of (A), and
Figure RE-GDA0002822316910000048
(3-2) solving equation (6) is dependent on variables
Figure RE-GDA0002822316910000049
Least Squares (ALS) optimization can be used, algorithm 1 describes the main process of ALS,
algorithm 1:
inputting: data to be repaired
Figure RE-GDA00028223169100000410
Nuclear tensor dimension (R)1,…,R2N)
And (3) outputting: decomposition factor
Figure RE-GDA00028223169100000411
Nuclear tensor
Figure RE-GDA00028223169100000412
3.2.1 initializing factorization factor U(2N)And nuclear tensor
Figure RE-GDA00028223169100000413
3.2.2 when N is 1, …,2N
3.2.3
Figure RE-GDA00028223169100000414
3.2.4U(n)And algorithm 2.
3.2.5
Figure RE-GDA00028223169100000415
(3-3) Algorithm 1 describes a traditional ALS-based Take factorization whose computational and storage bottleneck is the update factor matrix U(n)To solve the sub-problem U(n)We update it as follows
Figure RE-GDA0002822316910000051
Due to the complexity of equation (7), an Alternating Proximal Gradient Method (APG) algorithm is used to solve the above equation,
Figure RE-GDA0002822316910000052
let G (v) | | v | | non-phosphor1,1,2，C＝-L*Thus can obtain
Figure RE-GDA0002822316910000053
Thus, it can translate into a dual problem:
Figure RE-GDA0002822316910000054
final subproblem U(n)As described in algorithm 2;
and 2, algorithm:
3.3.1τ,μ＞0；θ∈[0,1]；k＝0
3.3.2 initialization of U(0),v(0),
Figure RE-GDA0002822316910000055
3.3.3
Figure RE-GDA0002822316910000056
3.3.4
Figure RE-GDA0002822316910000057
3.3.5
Figure RE-GDA0002822316910000058
3.3.6k＝k+1
Wherein proxσAnd proxτIs mapped to
Figure RE-GDA0002822316910000059
(3-4) in addition, the tach-based method can obtain a satisfactory effect by minimizing the rank of the tach, but it is difficult to set an appropriate rank (R)1,…,R2N). In our method, we minimize the following objective function, and this process is to obtain an approximate minimum of a sufficiently low rank
Figure RE-GDA0002822316910000061
Wherein ε represents an error threshold; order to
Figure RE-GDA0002822316910000062
Rn′To representThe best rank of the constraint is,
E(1)≥E(2)≥…≥E(Rn′-1)≥ε≥E(Rn′) (13)
finally, an image restoration model based on smooth Tak decomposition and high-order tensor Hank transformation is optimized by using low-rank increment, and the method is described in algorithm 3
Algorithm 3:
inputting: data to be repaired
Figure RE-GDA0002822316910000063
Ω，ε
And (3) outputting: decomposition factor
Figure RE-GDA0002822316910000064
Nuclear tensor
Figure RE-GDA0002822316910000065
3.4.1 initialization
Figure RE-GDA0002822316910000066
3.4.2
Figure RE-GDA0002822316910000067
3.4.3n′←n，Rn′←Rn
3.4.4 until convergence condition:
Figure RE-GDA0002822316910000068
(3-5) Tak decomposition factor reconstructed output
Figure RE-GDA0002822316910000069
(3-6) similarly, the Hank structured tensor
Figure RE-GDA00028223169100000610
The inverse hank structuring of (a) can be expressed as:
Figure RE-GDA00028223169100000611
wherein
Figure RE-GDA00028223169100000612
(3-7) finally, outputting high-quality image visual data
Figure RE-GDA00028223169100000613
And finishing image restoration.
The embodiments described in this specification are merely illustrative of implementations of the inventive concept and the scope of the present invention should not be considered limited to the specific forms set forth in the examples but rather by equivalents thereof as may occur to those skilled in the art upon consideration of the inventive concept.

Claims (3)

1. An image restoration method based on smooth Tak decomposition and high-order tensor Hank transformation comprises the following steps:
step 1), inputting an image to be repaired
Figure RE-FDA0002822316900000011
Determining an image to-be-repaired area and carrying out blocking operation on the image to-be-repaired area, wherein pixels in the image are divided into known points and unknown points, the known points are points with pixels not being 0 in the image, and the unknown points are points with pixels being 0 in the image; all unknown points in the image form a set omega;
step 2), constructing a high-order tensor Hankelization and discrete total variation model;
step 3), the color image is repaired by combining the image repairing model constructed in the step 2), and finally the high-quality visual data image is reconstructed and output
Figure RE-FDA0002822316900000012
2. The image restoration method according to claim 1, wherein the image restoration method based on smooth Tak decomposition and high-order tensor Hank decomposition comprises: the step 2) specifically comprises the following steps:
(2-1) the high-order hank structured image inpainting model is defined as follows:
Figure RE-FDA0002822316900000013
in the formula (I), the compound is shown in the specification,
Figure RE-FDA0002822316900000014
which represents the input of the image to be repaired,
Figure RE-FDA0002822316900000015
a post-repair image is represented as,
Figure RE-FDA0002822316900000016
represents the Frobenius norm;
Figure RE-FDA0002822316900000017
where 1 represents an observable pixel and 0 represents a missing pixel;
Figure RE-FDA0002822316900000018
is defined as
Figure RE-FDA0002822316900000019
Wherein fold(I,τ):
Figure RE-FDA00028223169000000110
Through the fold(I,τ)The input N-order tensor can be constructed into a 2N-order tensor, and the operation can be regarded as a multi-dimensional linear copying operation and a multi-dimensional folding operation; wherein
Figure RE-FDA00028223169000000111
Is a copyA matrix, that is, a matrix including a plurality of identity matrices, is specifically formed by:
Figure RE-FDA00028223169000000112
(2-2) the discrete total variation model is defined as follows:
Figure RE-FDA0002822316900000021
x represents a two-dimensional image and v represents a gradient;
Figure RE-FDA00028223169000000218
representing bilinear interpolation operations, particularly in grids
Figure RE-FDA0002822316900000023
(n1,n2) The interpolation of (1); a*Representing an accompanying operation; the definition of the discrete operator D is (Dx)1[n1,n2]＝x[n1+1,n2]-x[n1,n2]，(Dx)2[n1,n2]＝x[n1,n2+1]-x[n1,n2]；
Let l of v be1,1,2Norm represents three vector components
Figure RE-FDA00028223169000000217
L of1,1,2Norm sum, i.e.
Figure RE-FDA0002822316900000025
The discrete total variation model can therefore be redefined as:
Figure RE-FDA0002822316900000026
and fully utilizing low-rank complementary information and potential smoothing characteristics.
3. The image restoration method according to claim 1, wherein the image restoration method based on smooth Tak decomposition and high-order tensor Hank decomposition comprises: the step 3) specifically comprises the following steps:
(3-1) constructing an image restoration model based on smooth Tak decomposition and high-order tensor Hank transformation, which is defined as follows
Figure RE-FDA0002822316900000027
Figure RE-FDA0002822316900000028
In the formula, λ represents a balance parameter,
Figure RE-FDA0002822316900000029
represents a decomposition factor U(n)The (J, R) th entry of (A), and
Figure RE-FDA00028223169000000210
(3-2) solving equation (6) is dependent on variables
Figure RE-FDA00028223169000000211
Least Squares (ALS) optimization can be used, algorithm 1 describes the main process of ALS,
algorithm 1:
inputting: data to be repaired
Figure RE-FDA00028223169000000212
Nuclear tensor dimension (R)1,…,R2N)
And (3) outputting: decomposition factor
Figure RE-FDA00028223169000000213
Nuclear tensor
Figure RE-FDA00028223169000000214
3.2.1 initializing factorization factor U(2N)And nuclear tensor
Figure RE-FDA00028223169000000215
3.2.2 when N is 1, …,2N
3.2.3
Figure RE-FDA00028223169000000216
3.2.4 U(n)And algorithm 2.
3.2.5
Figure RE-FDA0002822316900000031
(3-3) Algorithm 1 describes a traditional ALS-based Take factorization whose computational and storage bottleneck is the update factor matrix U(n)To solve the sub-problem U(n)We update it as follows
Figure RE-FDA0002822316900000032
Due to the complexity of equation (7), an Alternating Proximal Gradient Method (APG) algorithm is used to solve the above equation,
Figure RE-FDA0002822316900000033
let G (v) | | v | | non-phosphor1,1,2，C＝-L*Thus can obtain
Figure RE-FDA0002822316900000034
Thus, it can translate into a dual problem:
Figure RE-FDA0002822316900000035
final subproblem U(n)As described in algorithm 2;
and 2, algorithm:
3.3.1τ,μ＞0；θ∈[0,1]；k＝0
3.3.2 initialization of U(0),v(0),
Figure RE-FDA0002822316900000036
3.3.3
Figure RE-FDA0002822316900000037
3.3.4
Figure RE-FDA0002822316900000038
3.3.5
Figure RE-FDA0002822316900000039
3.3.6 k＝k+1
Wherein proxσAnd proxτIs mapped to
Figure RE-FDA00028223169000000310
(3-4) in addition, the tach-based method can obtain a satisfactory effect by minimizing the rank of the tach, but it is difficult to set an appropriate rank (R)1,…,R2N). In our method, we minimize the following objective function, and this process is to obtain an approximate minimum of a sufficiently low rank
Figure RE-FDA0002822316900000041
Wherein ε represents an error threshold; order to
Figure RE-FDA0002822316900000042
Rn′The optimal rank of the constraint is represented,
E(1)≥E(2)≥…≥E(Rn′-1)≥ε≥E(Rn′) (13)
finally, an image restoration model based on smooth Tak decomposition and high-order tensor Hank transformation is optimized by using low-rank increment, and the method is described in algorithm 3
Algorithm 3:
inputting: data to be repaired
Figure RE-FDA0002822316900000043
Ω，ε
And (3) outputting: decomposition factor
Figure RE-FDA0002822316900000044
Nuclear tensor
Figure RE-FDA0002822316900000045
3.4.1 initialization
Figure RE-FDA0002822316900000046
3.4.2
Figure RE-FDA0002822316900000047
3.4.3 n′←n，Rn′←Rn
3.4.4 until convergence condition:
Figure RE-FDA0002822316900000048
(3-5) Tak decomposition factor reconstructed output
Figure RE-FDA0002822316900000049
(3-6) similarly, the Hank structured tensor
Figure RE-FDA00028223169000000410
The inverse hank structuring of (a) can be expressed as:
Figure RE-FDA00028223169000000411
wherein
Figure RE-FDA00028223169000000412
(3-7) finally, outputting high-quality image visual data
Figure RE-FDA00028223169000000413
And finishing image restoration.
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