CN111325697A - Color image restoration method based on tensor eigen transformation - Google Patents

Color image restoration method based on tensor eigen transformation Download PDF

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CN111325697A
CN111325697A CN202010144793.XA CN202010144793A CN111325697A CN 111325697 A CN111325697 A CN 111325697A CN 202010144793 A CN202010144793 A CN 202010144793A CN 111325697 A CN111325697 A CN 111325697A
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tensor
matrix
column
eigen
color image
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CN111325697B (en
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刘静
苏立玉
黄开宇
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Xian Jiaotong University
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    • 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
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    • G06T2207/10024—Color image
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Abstract

The invention discloses a color image restoration method based on tensor eigen transformation, which utilizes designed tensor eigen transformation to directly obtain the structural characteristics of an image to be restored and better reserve the internal association between image pixels. The image restoration method is better in the aspect of keeping the internal relation of the pixels, so that the method can restore the detail characteristics of the damaged pixels and can be used for restoring the damaged natural color image.

Description

Color image restoration method based on tensor eigen transformation
[ technical field ] A method for producing a semiconductor device
The invention belongs to the technical field of image processing, and relates to a color image restoration method based on tensor eigen transformation.
[ background of the invention ]
The modern society is in the era of data explosion, and the collection and processing of data promote the development of computer technology, however, different from the past, in the era of big data, the collected data has a more complex structure, and the data can be well represented by using a high-dimensional multi-linear structure. In the receiving and transmitting process of data, the obtained data is likely to be partially damaged due to environmental interference, and even if the data is received and transmitted under ideal conditions, it is desirable to compress the data to improve the utilization rate of resources. Based on the low rank matrix completion model, we can adopt fewer samples to mine the possible values of other unknown samples, and the technology is applied to various recommendation systems. In real life, if an attempt is made to mine an unknown sample, the influence factors are likely to be more, and therefore, the efficiency of solving the unknown sample by using the matrix is not high. The tensor is a high-dimensional multi-linear structure which can well reflect the internal connection between high-order data. On the basis of a low-rank matrix completion model, the low-rank tensor completion model is widely applied to the fields of computer vision, machine learning, data mining, neuroscience and the like.
Image processing is an important research direction of computer vision, wherein image restoration techniques are aimed at restoring damaged pixel values in an image with a high probability. Under a low-rank tensor completion model, a color image can be represented as a third-order tensor, and how to mine the low-rank property of the tensor becomes an important problem of color image completion. The low rank property of the tensor is generally defined by decomposition methods of various tensors, and currently, common tensor decomposition methods include: 1) tucker Decomposition 2) CANDECOMP-PARAFAC (CP) Decomposition 3) Tensor column Decomposition 4) Tensor Singular Value Decomposition 4 (Tensor Singular Value Decomposition). LIU et al (thesis entitled Tensor completing and Estimating Missing Values in Visual Data) respectively expand tensors into mode matrixes according to different modes by using a TUCKER decomposition method, and the low rank of the original Tensor is defined by using the rank of the mode matrixes, but the relevance damage of the method to the Data in the original Tensor is large. Under the definition of a CP decomposition method, the order of Tensor solving is an NP difficult problem, and the CP order of the Bayesian factor estimation Tensor is introduced by ZHAO et al (a paper titled Bayesian Robust test factor for incorporated Multi Data), so that the low order of the original Tensor is defined, but the complexity of the calculation process of the method is higher. Bengua et al (thesis entitled Efficient Tensor composition for Color Image and video recovery) defines a new Tensor generating matrix method by using traditional Tensor column decomposition, and further excavates the low-rank property of the original Tensor according to the rank of the generating matrix under the new method. ZHANG et al (article titled Novel Methods for multilinear Data Completion and De-noised base on sensor-SVD) defines TUBAL rank of Tensor Based on Tensor singular value decomposition method, and the method does not rely on Tensor development, well preserves Data relevance of Tensor, but TUBAL rank has the possibility of large description error in describing Tensor low rank. LI (thesis entitled Low rank document Assembly with Total Variation for Visual Data Inpainting) et al propose embedding the Total Variation method into the Tensor Completion process without the help of Tensor development to define low rank, but the principle process of the method is more complicated, and the applicability to the image with larger damage is not strong.
[ summary of the invention ]
The invention aims to solve the problems in the prior art and provides a color image restoration method based on tensor eigen transformation, which can better save the relevance among color pixels in color image restoration, thereby obtaining a clearer restoration result and achieving the effect that human eyes cannot easily perceive restoration traces.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
a color image restoration method based on tensor eigen transformation comprises the following steps:
step 1, obtaining damaged image, namely to-be-compensated tensor
Figure BDA0002400360220000031
Determining a set of all missing pixels corresponding to the damaged region
Figure BDA0002400360220000032
Setting preset parameters epsilon, β, re,rcSetting the remodeling parameter ρ1,ρ2Determining a set error requirement and a maximum iteration requirement;
step 2, obtaining the full tensor to be compensated by utilizing the tensor eigen transformation tau
Figure BDA0002400360220000033
The corresponding eigen-matrix E, noted:
Figure BDA0002400360220000034
step 3, using a threshold operator Dε,βUpdating the intrinsic matrix E in the step 2, and recording the updated intrinsic matrix as En：
En＝Dε,β(E)
Step 4, introducing auxiliary tensor
Figure BDA0002400360220000035
Using inverse tensor eigentransformations tau-1Determining an updated eigen matrix EnCorresponding tensor
Figure BDA0002400360220000036
Recording as follows:
Figure BDA0002400360220000037
step 5, determining the update tensor
Figure BDA0002400360220000038
If the update tensor pixel exists in the pixel damaged area
Figure BDA0002400360220000039
Order to
Figure BDA00024003602200000310
If the update tensor pixel is not in the pixel damage area
Figure BDA00024003602200000311
Order to
Figure BDA00024003602200000312
Step 6, judging relative error
Figure BDA00024003602200000313
Whether the set error requirement is met or not, or whether the iteration frequency reaches the maximum iteration requirement or not; if the error requirement or the maximum iteration requirement is met, outputting the latest tensor
Figure BDA00024003602200000314
Namely the repaired image; otherwise make
Figure BDA00024003602200000315
And returning to the step 2 to enter an iterative loop.
The invention further improves the following steps:
in step 1, a set of all missing pixels corresponding to the damaged area is determined
Figure BDA00024003602200000316
The specific method comprises the following steps:
step 1.1, reading all pixel points of an image to be restored, enabling all the points with pixel values not being 0 to be known pixel points, and recording the positions of the known pixel points as a set omega;
step 1.2, all the points with the pixel values of 0 are made to be unknown pixel points, and the position set of the unknown pixel points is recorded to be
Figure BDA0002400360220000041
In the step 2, the tensor to be compensated is obtained by utilizing the tensor eigen transformation tau
Figure BDA0002400360220000042
The specific method of the corresponding eigen matrix E is as follows:
step 2.1, using improved tensor column decomposition to complete tensor to be compensated
Figure BDA0002400360220000043
Decomposed into tensor column kernels
Figure BDA0002400360220000044
The concrete expression is as follows:
Figure BDA0002400360220000045
in the above formula, the first and second carbon atoms are,
Figure BDA0002400360220000046
representing the coordinate in the tensor to be complemented as (i)c,jc,kc) The value of the pixel of (a) is,
Figure BDA0002400360220000047
indicating the amount of tension to be compensated
Figure BDA0002400360220000048
The tensor of the reshaped intermediate medium,
Figure BDA0002400360220000049
a b-th front tangent matrix representing an a-th tensor column kernel;
step 2.2, carrying out tensor singular value decomposition on one tensor column core in the step:
Figure BDA00024003602200000410
in the above equation, a tensor column kernel is decomposed into the form of the product of three tensors, where
Figure BDA00024003602200000411
Called f-diagonal tensor, taking the initial value a as 1;
step 2.3, in the Fourier domain, the tensor column in the previous step is centered
Figure BDA00024003602200000412
Corresponding f-diagonal tensor
Figure BDA00024003602200000413
The values in (1) are mapped into a matrix E according to coordinates, and the corresponding coordinate relationship is expressed as:
Figure BDA00024003602200000414
in the above formula, the first and second carbon atoms are,
Figure BDA00024003602200000415
representing tensor column core
Figure BDA00024003602200000416
F-diagonal tensor in Fourier domain, ζ denotes tensor column core pointer, ξ denotes eigenmatrix pointer, IaRepresenting the tensor of the intermediate medium to be complemented
Figure BDA00024003602200000417
The a-th dimension; the tensor column core pointer ζ has a specific value of ζ being 1 when a is 1 or 3, and ζ being i in other cases; the specific values of the eigen matrix pointers are:
Figure BDA0002400360220000051
in the above formula, reAnd rcRespectively representing the edge rank and the center rank in the tensor eigen transformation, wherein the edge rank is always set as re＝1；
Step 2.4, if a<3, taking the intermediate tensor to be compensated
Figure BDA0002400360220000052
A +1 th tensor column core of
Figure BDA0002400360220000053
I.e. let a be a +1, go back to step 2.3 to enter iteration, otherwise determine tensor
Figure BDA0002400360220000054
The eigenmatrix of (a) is E.
The specific method of the improved tensor column decomposition is as follows:
step 2.1.1, first, the tensor corresponding to the color image
Figure BDA0002400360220000055
Remodel into
Figure BDA0002400360220000056
Wherein the setting parameter p is utilized1,ρ2The remodeling relationship of (A) is that,
Figure BDA0002400360220000057
step 2.1.2, introduce the auxiliary temporary tensor
Figure BDA0002400360220000058
Order to
Figure BDA0002400360220000059
Will tensor
Figure BDA00024003602200000510
Reshaped into a corresponding matrix
Figure BDA00024003602200000511
Step 2.1.3, remodeling the matrix obtained in the step 2.1.2
Figure BDA00024003602200000512
Decomposition according to matrix singular values:
C＝U·S·VT(5)
wherein the content of the first and second substances,
Figure BDA00024003602200000513
respectively called left and right singular matrices of the matrix C,
Figure BDA00024003602200000514
is a diagonal matrix;
step 2.1.4, get matrix
Figure BDA00024003602200000515
Front r ofcColumn forming a new matrix
Figure BDA00024003602200000516
Get matrix
Figure BDA00024003602200000517
Front r ofcColumn forming a new matrix
Figure BDA00024003602200000518
Get matrix
Figure BDA00024003602200000519
Front r ofcRow and rcColumns forming a new matrix
Figure BDA00024003602200000520
Step 2.1.5, matrix
Figure BDA00024003602200000521
Remodelling into first volumetric core
Figure BDA00024003602200000522
Determining updated matrices
Figure BDA00024003602200000523
Step 2.1.6, matrix
Figure BDA00024003602200000524
Remolding into a new matrix
Figure BDA00024003602200000525
And is obtained according to the singular value decomposition of the matrix, i.e. the decomposition in equation (5)
Figure BDA00024003602200000526
Get matrix
Figure BDA00024003602200000527
Front r ofcColumn forming a new matrix
Figure BDA0002400360220000061
Get matrix
Figure BDA0002400360220000062
Front r ofcColumn forming a new matrix
Figure BDA0002400360220000063
Get matrix
Figure BDA0002400360220000064
Front r ofcRow and rcColumns forming a new matrix
Figure BDA0002400360220000065
Step 2.1.7, matrix
Figure BDA0002400360220000066
Remodeled to a second tensor column core
Figure BDA0002400360220000067
Determining updated matrices
Figure BDA0002400360220000068
C is to bedRemodeled to a third tensor column core
Figure BDA0002400360220000069
In step 3, the specific method of tensor singular value decomposition is as follows:
step 2.2.1, get a quantitative core
Figure BDA00024003602200000610
Performing three-dimensional Fourier transform to obtain tensor column core in corresponding Fourier domain
Figure BDA00024003602200000611
Taking an initial value s as 1;
step 2.2.2, get tensor column core
Figure BDA00024003602200000612
And recording the matrix as the s-th front section matrix
Figure BDA00024003602200000613
Namely have
Figure BDA00024003602200000614
To pair
Figure BDA00024003602200000615
Performing singular value decomposition of the matrix to obtain left and right singular matrices thereof
Figure BDA00024003602200000616
And diagonal matrix
Figure BDA00024003602200000617
Namely, it is
Figure BDA00024003602200000618
Step 2.2.3 creating tensors
Figure BDA00024003602200000619
Setting the initial value of the newly-built tensor to be 0, and setting the left and right singular matrixes in the previous step
Figure BDA00024003602200000620
And diagonal matrix
Figure BDA00024003602200000621
The s-th front tangent plane matrix given to the newly created tensor, i.e.
Figure BDA00024003602200000622
Step 2.2.4, if s<I3If s is equal to s +1, the procedure returns to step 2.2.2 to enter a loop, otherwise, the three tensors are determined
Figure BDA00024003602200000623
Step 2.2.5, for
Figure BDA00024003602200000624
Respectively carrying out three-dimensional inverse Fourier transform to obtain tensors of real number domain
Figure BDA00024003602200000625
To sum up, a tensor column core is obtained
Figure BDA00024003602200000626
Singular value decomposition of, i.e. having
Figure BDA00024003602200000627
The threshold operator D in the step 3ε,βThe concrete expression is as follows:
Figure BDA00024003602200000628
wherein sign (x) is a sign function, parameter c0,c1,c2Comprises the following steps:
Figure BDA0002400360220000071
wherein ε, β is the preset parameter input in advance, defining the symbol Dε,β(X) represents the thresholding operation on all elements in matrix X.
In said step 4, the inverse tensor eigen-transform τ-1The specific method comprises the following steps:
step 4.1, in the Fourier domain, a tensor eigen transformation method is adopted, and a plurality of corresponding tensor column cores are obtained through calculation of the eigen matrix E
Figure BDA0002400360220000072
F-diagonal tensor of
Figure BDA0002400360220000073
Taking an initial value a as 1;
step 4.2, mixing
Figure BDA0002400360220000074
Is obtained by inverse Fourier transform
Figure BDA0002400360220000075
From the f-diagonal tensor in the real domain
Figure BDA0002400360220000076
And corresponding left and right singular tensors
Figure BDA0002400360220000077
Calculating to obtain tensor column core
Figure BDA0002400360220000078
Figure BDA0002400360220000079
In the above formula, the symbol "+" represents the tensor product;
step 4.3, if a<3, the a +1 f-diagonal tensor is taken
Figure BDA00024003602200000710
If a is equal to a +1, returning to the step 4.2 to enter iteration, otherwise, entering the next step;
step 4.4, according to the obtained plurality of tensor column cores
Figure BDA00024003602200000711
Calculating to obtain the updating tensor corresponding to the eigen matrix E
Figure BDA00024003602200000712
Figure BDA00024003602200000713
In the above formula, subscript i, j, k represents the tensor of the pixel point
Figure BDA00024003602200000714
The position of (1); will update the tensor
Figure BDA00024003602200000715
Reshaped into an original image
Figure BDA00024003602200000716
Tensor of equal size
Figure BDA00024003602200000717
Compared with the prior art, the invention has the following beneficial effects:
the invention provides a novel tensor eigen-transformation, which is not subjected to any tensor development matrix to define the low rank of the tensor, but directly excavates the low rank of the tensor by decomposing the structural characteristics of the tensor. Compared with the prior art, the method has the advantages that based on tensor eigen transformation, the low-rank property of the tensor to be complemented can be solved in an iterative manner, and meanwhile, the relevance among tensor elements is greatly saved; in addition, under a low-rank tensor completion model, the proposed tensor eigen transformation recovers damaged pixels with higher probability by utilizing the relevance between known pixels and the damaged pixels, and the method is better at exploring the tensor internal relation, so that a more accurate repaired image can be obtained, and the effect that the repaired trace is not easily distinguished in the aspect of image detail repair can be achieved.
[ description of the drawings ]
Fig. 1 is a flowchart of a color image restoration method based on tensor eigen transformation according to the present invention;
FIG. 2 is a diagram of a color image defect area according to the present invention
Figure BDA0002400360220000081
And a diagram of a known region Ω, wherein the black area is used to indicate the damaged missing region;
FIG. 3 is a schematic diagram of the tensor eigen-transforms of the present invention;
fig. 4 is a comparison graph of the application effect of the present invention and the prior art, which shows the original image of a color image, a damaged image and an image repaired by each method, and the repaired image has a relative error of recovery attached.
[ detailed description ] embodiments
In order to make the technical solutions of the present invention better understood, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, not all of the embodiments, and are not intended to limit the scope of the present disclosure. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present disclosure. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Various structural schematics according to the disclosed embodiments of the invention are shown in the drawings. The figures are not drawn to scale, wherein certain details are exaggerated and possibly omitted for clarity of presentation. The shapes of various regions, layers and their relative sizes and positional relationships shown in the drawings are merely exemplary, and deviations may occur in practice due to manufacturing tolerances or technical limitations, and a person skilled in the art may additionally design regions/layers having different shapes, sizes, relative positions, according to actual needs.
In the context of the present disclosure, when a layer/element is referred to as being "on" another layer/element, it can be directly on the other layer/element or intervening layers/elements may be present. In addition, if a layer/element is "on" another layer/element in one orientation, then that layer/element may be "under" the other layer/element when the orientation is reversed.
It should be noted that the terms "first," "second," and the like in the description and claims of the present invention and in the drawings described above are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used is interchangeable under appropriate circumstances such that the embodiments of the invention described herein are capable of operation in sequences other than those illustrated or described herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
The invention is described in further detail below with reference to the accompanying drawings:
the prior art cannot utilize the relation between known and unknown pixels to the maximum extent, various generating modes are generally used when defining low rank, the relevance among the pixels is greatly damaged, and the recovered image has certain distortion in visual effect. Therefore, we propose tensor eigen transformation to overcome the defect, as shown in fig. 1, in an embodiment of the present invention, a color image restoration method based on tensor eigen transformation does not need to define the low rank property of the original tensor by means of a tensor matrix-generating approach, directly obtains the low rank property of the original tensor by using the inherent structural features of the tensor eigen transformation, greatly utilizes and stores the inherent relevance between the color image pixels, overcomes the defect of insufficient storage of the pixel relevance in the prior art, and makes the restored color image more accurate and has greater generalization.
Referring to fig. 1, the color image restoration method based on tensor eigen transformation of the present invention includes the following steps:
step 1, obtaining damaged image (i.e. to-be-compensated tensor)
Figure BDA0002400360220000101
) With a pixel of 256, i.e. corresponding to the dimension of the tensor to be compensated
Figure BDA0002400360220000102
Determining a set of all missing pixels corresponding to the damaged region
Figure BDA0002400360220000103
Set suitablyPresetting parameters epsilon, β, rcWherein, the edge rank of the tensor eigen-transform is always set to re1 is ═ 1; setting a remodeling parameter ρ1,ρ2In the embodiment, the parameters are set to be epsilon 0.3, β is 100, and r isc20; remodeling parameter ρ1＝4,ρ24, the maximum iteration number is 500, and the error requirement is less than 1 × 10-3The missing pixel ratio is set to 70%, i.e. 70% of the color image pixels are missing or damaged;
as shown in FIG. 2, to identify the pixel positions to be repaired, a set of all missing pixels corresponding to the damaged area is first determined
Figure BDA0002400360220000104
The concrete setting is as follows:
step 1.1, reading all pixel points of an image to be restored, enabling all the points with pixel values not being 0 to be known pixel points, and recording the positions of the known pixel points as a set omega;
step 1.2, all the points with the pixel values of 0 are made to be unknown pixel points, and the position set of the unknown pixel points is recorded to be
Figure BDA0002400360220000105
Step 2, obtaining the full tensor to be compensated by utilizing the tensor eigen transformation tau
Figure BDA0002400360220000106
The corresponding eigen-matrix E, denoted
Figure BDA0002400360220000107
As shown in FIG. 3, the method uses tensor eigen transformation tau to obtain the tensor to be compensated
Figure BDA0002400360220000108
The corresponding intrinsic matrix E specifically comprises the following steps:
step 2.1, using improved tensor column decomposition to complete tensor to be compensated
Figure BDA0002400360220000109
Decomposed into tensor column kernels
Figure BDA00024003602200001010
The concrete expression is as follows:
Figure BDA00024003602200001011
in the above formula, the first and second carbon atoms are,
Figure BDA0002400360220000111
representing the coordinate in the tensor to be complemented as (i)c,jc,kc) The value of the pixel of (a) is,
Figure BDA0002400360220000112
indicating the amount of tension to be compensated
Figure BDA0002400360220000113
The tensor of the reshaped intermediate medium,
Figure BDA0002400360220000114
a b-th Frontal Slice matrix (Frontal Slice) representing an a-th Tensor column Core (Tensor Train Core);
specifically, the improved tensor column decomposition described in the tensor eigentransform has the following steps:
step 2.1.1, in order to ensure the reconstruction precision, firstly, the tensor corresponding to the color image
Figure BDA0002400360220000115
Remodel into
Figure BDA0002400360220000116
Wherein the setting parameter p is utilized1,ρ2The remodeling relationship of (A) is that,
Figure BDA0002400360220000117
step 2.1.2, introduce the auxiliary temporary tensor
Figure BDA0002400360220000118
Order to
Figure BDA0002400360220000119
Will tensor
Figure BDA00024003602200001110
Reshaped into a corresponding matrix
Figure BDA00024003602200001111
Step 2.1.3, remodeling the matrix obtained in the step 2.1.2
Figure BDA00024003602200001112
The matrix singular value decomposition can be specifically expressed as:
C＝U·S·VT(2)
wherein
Figure BDA00024003602200001113
Respectively called left and right singular matrices of the matrix C,
Figure BDA00024003602200001114
is a diagonal matrix;
step 2.1.4, get matrix
Figure BDA00024003602200001115
Front r ofcColumn forming a new matrix
Figure BDA00024003602200001116
Get matrix
Figure BDA00024003602200001117
Front r ofcColumn forming a new matrix
Figure BDA00024003602200001118
Get matrix
Figure BDA00024003602200001119
Front r ofcRow and rcColumns forming a new matrix
Figure BDA00024003602200001120
Step 2.1.5, matrix
Figure BDA00024003602200001121
Remodelling into first volumetric core
Figure BDA00024003602200001122
Determining updated matrices
Figure BDA00024003602200001123
Step 2.1.6, matrix
Figure BDA00024003602200001124
Remolding into a new matrix
Figure BDA00024003602200001125
And is obtained according to the singular value decomposition of the matrix, i.e. the decomposition in equation (2)
Figure BDA00024003602200001126
Get matrix
Figure BDA00024003602200001127
Front r ofcColumn forming a new matrix
Figure BDA00024003602200001128
Get matrix
Figure BDA00024003602200001129
Front r ofcColumn forming a new matrix
Figure BDA00024003602200001130
Get matrix
Figure BDA0002400360220000121
Front r ofcRow and rcColumns forming a new matrix
Figure BDA0002400360220000122
Step 2.1.7, matrix
Figure BDA0002400360220000123
Remodeled to a second tensor column core
Figure BDA0002400360220000124
Determining updated matrices
Figure BDA0002400360220000125
C is to bedRemodeled to a third tensor column core
Figure BDA0002400360220000126
Step 2.2, carrying out tensor singular value decomposition on one tensor column core in the step, wherein the tensor singular value decomposition is specifically represented as follows:
Figure BDA0002400360220000127
in the above equation, a tensor column kernel is decomposed into the form of the product of three tensors, where
Figure BDA0002400360220000128
Called f-diagonal Tensor (f-diagonal Tensor), takes initial value a as 1;
the tensor singular value decomposition in the tensor eigen transformation specifically comprises the following steps:
step 2.2.1, get a quantitative core
Figure BDA0002400360220000129
Performing three-dimensional Fourier transform to obtain tensor column core in corresponding Fourier domain
Figure BDA00024003602200001210
Taking an initial value s as 1;
step 2.2.2, sheet takingCore of the quantum array
Figure BDA00024003602200001211
And recording the matrix as the s-th front section matrix
Figure BDA00024003602200001212
Namely have
Figure BDA00024003602200001213
To pair
Figure BDA00024003602200001214
Performing singular value decomposition of the matrix to obtain left and right singular matrices thereof
Figure BDA00024003602200001215
And diagonal matrix
Figure BDA00024003602200001216
Namely, it is
Figure BDA00024003602200001217
Step 2.2.3 creating tensors
Figure BDA00024003602200001218
Setting the initial value of the newly-built tensor to be 0, and setting the left and right singular matrixes in the previous step
Figure BDA00024003602200001219
And diagonal matrix
Figure BDA00024003602200001220
The s-th front tangent plane matrix given to the newly created tensor, i.e.
Figure BDA00024003602200001221
Step 2.2.4, if s<I3If s is equal to s +1, the procedure returns to step 2.2.2 to enter a loop, otherwise, the three tensors are determined
Figure BDA00024003602200001222
Step 2.2.5, for
Figure BDA00024003602200001223
Respectively carrying out three-dimensional inverse Fourier transform to obtain tensors of real number domain
Figure BDA00024003602200001224
To sum up, a tensor column core is obtained
Figure BDA00024003602200001225
Singular value decomposition of, i.e. having
Figure BDA00024003602200001226
Step 2.3, in the Fourier domain, the tensor column in the previous step is centered
Figure BDA00024003602200001227
Corresponding f-diagonal tensor
Figure BDA00024003602200001228
The values in (1) are mapped into a matrix E according to coordinates, and the corresponding coordinate relationship can be expressed as:
Figure BDA0002400360220000131
in the above formula, the first and second carbon atoms are,
Figure BDA0002400360220000132
representing tensor column core
Figure BDA0002400360220000133
F-diagonal tensor in Fourier domain, ζ denotes tensor column core pointer, ξ denotes eigenmatrix pointer, IaRepresenting the tensor of the intermediate medium to be complemented
Figure BDA0002400360220000134
The a-th dimension; the specific value of the tensor column core pointer zeta is that when a is equal toζ ═ 1 when 1 or a ═ 3, and ζ ═ i in other cases; the specific value of the eigen matrix pointer is
Figure BDA0002400360220000135
In the above formula, reAnd rcRespectively representing the edge rank and the center rank in the tensor eigen transformation, wherein the edge rank of the method is always set as re＝1；
Step 2.4, if a<3, taking the intermediate tensor to be compensated
Figure BDA0002400360220000136
A +1 th tensor column core of
Figure BDA0002400360220000137
I.e. let a be a +1, go back to step 2.3 to enter iteration, otherwise determine tensor
Figure BDA0002400360220000138
The eigenmatrix of (a) is E.
Step 3, using a threshold operator Dε,βUpdating the intrinsic matrix E in the step 2, and recording the updated intrinsic matrix as EnI.e. En＝Dε,β(E)；
The threshold operator D described in this stepε,βThe method specifically comprises the following steps:
Figure BDA0002400360220000139
wherein sign (x) is a sign function, parameter c0,c1,c2Needs to be determined as
Figure BDA00024003602200001310
ε, β is the preset parameter input in advance, defining the symbol Dε,β(X) represents the thresholding operation on all elements in matrix X.
In the step 4, the step of,introducing an auxiliary tensor
Figure BDA0002400360220000141
Using inverse tensor eigentransformations tau-1Determining an updated eigen matrix EnCorresponding tensor
Figure BDA0002400360220000142
Record as
Figure BDA0002400360220000143
Inverse tensor eigentransform τ in this step-1Is the inverse process of the tensor eigen-transform tau, it is emphasized that the inverse tensor eigen-transform tau-1Subject to τ, otherwise τ-1Will become meaningless; inverse tensor eigentransform τ-1The method comprises the following specific steps:
step 4.1, in the Fourier domain, referring to the tensor eigen transformation method in the formulas (4) and (5), calculating the corresponding tensor column cores by the eigen matrix E
Figure BDA0002400360220000144
F-diagonal tensor of
Figure BDA0002400360220000145
Taking an initial value a as 1;
step 4.2, mixing
Figure BDA0002400360220000146
Is obtained by inverse Fourier transform
Figure BDA0002400360220000147
From the f-diagonal tensor in the real domain
Figure BDA0002400360220000148
And corresponding left and right singular tensors
Figure BDA0002400360220000149
Calculating to obtain tensor column core
Figure BDA00024003602200001410
The expression is as follows:
Figure BDA00024003602200001411
the symbol "+" in the above formula represents the tensor product;
step 4.3, if a<3, the a +1 f-diagonal tensor is taken
Figure BDA00024003602200001412
I.e. let a be a +1, go back to step 4.2 to enter iteration, otherwise go to the next step.
Step 4.4, according to the obtained plurality of tensor column cores
Figure BDA00024003602200001413
Calculating to obtain the updating tensor corresponding to the eigen matrix E
Figure BDA00024003602200001414
The concrete expression is as follows:
Figure BDA00024003602200001415
subscript i, j, k in the above formula represents the tensor of the pixel point
Figure BDA00024003602200001416
The position of (1); in addition, the tensor will be updated
Figure BDA00024003602200001417
Reshaped into an original image
Figure BDA00024003602200001418
Tensor of equal size
Figure BDA00024003602200001419
Step 5, determining the update tensor
Figure BDA00024003602200001420
If the update tensor pixel exists in the pixel damaged area
Figure BDA00024003602200001421
Order to
Figure BDA00024003602200001422
If the update tensor pixel is not in the pixel damage area
Figure BDA00024003602200001423
Order to
Figure BDA00024003602200001424
Step 6, judging relative error
Figure BDA00024003602200001425
Whether the set error requirement is met or not, or whether the iteration frequency reaches the maximum iteration requirement or not; if the error requirement or the maximum iteration requirement is met, outputting the latest tensor
Figure BDA0002400360220000151
(i.e., the repaired image), otherwise let
Figure BDA0002400360220000152
And returning to the step 2 to enter an iterative loop.
As shown in fig. 4, the present embodiment recovers the color image by a tensor eigen transformation-based color image recovery method, and the recovery effect of the recovered image is still better even under an extremely high damage rate, i.e., 70% of the pixels are damaged, and compared with the original image, an accurate recovery result is obtained. For comparison, the low-rank tensor total variation method and the tensor nuclear norm method are respectively set according to default algorithms; the relative error of the recovery of the method of the invention is 10.2%, which is better than 12.6% and 12.8% of the prior art, and the recovery result of the prior art is visually seen to have errors in brightness and detail. The experimental results show that the method provided by the invention has better performance no matter in objective index evaluation or subjective visual effect.
In the embodiment, the low rank property is defined without any method for expanding the tensor into the matrix, but the proposed tensor eigen transformation is directly utilized to form the framework, so that the low rank property of the tensor can be iteratively obtained under the framework, and meanwhile, the relevance among pixels is ensured and an effective recovery image is obtained.
With reference to this embodiment, an algorithm of a color image restoration method based on tensor eigen transformation is given as follows:
inputting: spoiled colour images
Figure BDA0002400360220000153
Suitable preset parameters ε, β, rcWherein r ise1 is ═ 1; setting a remodeling parameter ρ1,ρ2Determining a set error requirement tol and a maximum iteration requirement K;
step 1, initializing a damaged color image
Figure BDA0002400360220000154
Initializing iteration parameter i as 1, initializing relative error
Figure BDA0002400360220000155
Step 2, judging whether the maximum iteration requirement or the specified error requirement is met, if i is less than or equal to K or errori>tol proceeds to the next step, otherwise let
Figure BDA0002400360220000156
And terminate the algorithm;
step 3, obtaining tensor based on the eigen transformation of the tensor
Figure BDA0002400360220000157
E;
step 4, updating the intrinsic matrix E through formulas (6) and (7) to obtain En；
Step 5, obtaining an updated eigen matrix E based on the inverse eigen transformation of the tensornCorresponding tensor of
Figure BDA0002400360220000158
And step 6, updating the damaged color image,
Figure BDA0002400360220000161
obtaining the color image restored in this step
Figure BDA0002400360220000162
Calculating the relative error of the iteration
Figure BDA0002400360220000163
Step 7, enabling the iteration parameter i to be i + 1; update the damaged color image
Figure BDA0002400360220000164
And returning to the step 2;
and (3) outputting: repairing color images
Figure BDA0002400360220000165
The parameters and related mathematical symbols involved in the algorithm are the same as those in the present invention, so that the repeated definition of the parameters and the mathematical symbols is avoided.
The color image restoration method based on tensor eigen transformation described in the embodiment is characterized in that firstly, a damaged color image is read, a set of known pixels and an unknown pixel is identified and defined, and a premise is created for restoring the image by utilizing internal relevance between the known pixels and the unknown pixels; in addition, based on the proposed tensor eigen transformation, the corresponding tensor low-rank property is mined directly according to the structure of the color image, the method greatly preserves the pixel relevance, and under the framework of the tensor eigen transformation, the damaged pixels can be iteratively repaired through simple transformation, so that the missing pixels can be more accurately recovered.
In this embodiment, a computer device is provided, which may be a terminal. The computing device generally includes a processor, a memory, a network interface, and an input-output device. The processor of the computer device provides computational support for the method of the invention; the memorizer provides a built-in operating system, a running environment of the program and a stored computer program for the method; the network interface provides network connection and communication exchange functions for the method; the input device is used for inputting an image to be restored, and can be a keyboard, a mouse or the like specifically; the output device is used for presenting the repaired image, and may specifically be a display screen or the like. More specifically, the software platform of the present embodiment uses MATLAB R2015 a; the hardware platform uses INTEL CORE CPU and memory 4 GB; the experimental method is the method, the low-rank tensor total variation method and the tensor nuclear norm method.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (7)

1. A color image restoration method based on tensor eigen transformation is characterized by comprising the following steps:
step 1, obtaining damaged image, namely to-be-compensated tensor
Figure FDA0002400360210000011
Determining a set omega of all missing pixels corresponding to the damaged region⊥Setting preset parameters epsilon, β, re,rcSetting the remodeling parameter ρ1,ρ2Determining a set error requirement and a maximum iteration requirement;
step 2, obtaining the full tensor to be compensated by utilizing the tensor eigen transformation tau
Figure FDA0002400360210000012
The corresponding eigen-matrix E, noted:
Figure FDA0002400360210000013
step 3, using a threshold operator Dε,βUpdating the intrinsic matrix E in the step 2, and recording the updated intrinsic matrix as En：
En＝Dε,β(E)
Step 4, introducing auxiliary tensor
Figure FDA0002400360210000014
Using inverse tensor eigentransformations tau-1Determining an updated eigen matrix EnCorresponding tensor
Figure FDA0002400360210000015
Recording as follows:
Figure FDA0002400360210000016
step 5, determining the update tensor
Figure FDA0002400360210000017
If the update tensor pixel exists in the pixel damaged area omega⊥Let us order
Figure FDA0002400360210000018
If the update tensor pixel is not in the pixel damaged area omega⊥Let us order
Figure FDA0002400360210000019
Step 6, judging relative error
Figure FDA00024003602100000110
Whether the set error requirement is met or not, or whether the iteration frequency reaches the maximum iteration requirement or not; if the error requirement or the maximum iteration requirement is met, outputting the latest tensor
Figure FDA00024003602100000111
Namely the repaired image; otherwise make
Figure FDA00024003602100000112
And returning to the step 2 to enter an iterative loop.
2. The tensor eigen transformation-based color image restoration method as recited in claim 1, wherein in the step 1, a set Ω of all missing pixels corresponding to a damaged area is determined⊥The specific method comprises the following steps:
step 1.1, reading all pixel points of an image to be restored, enabling all the points with pixel values not being 0 to be known pixel points, and recording the positions of the known pixel points as a set omega;
step 1.2, all the points with the pixel values of 0 are made to be unknown pixel points, and the position set of the unknown pixel points is recorded to be omega⊥。
3. The method for restoring a color image based on tensor eigen transformation as claimed in claim 1, wherein in the step 2, the tensor to be compensated is obtained by using tensor eigentransformation τ
Figure FDA0002400360210000021
The specific method of the corresponding eigen matrix E is as follows:
step 2.1, using improved tensor column decomposition to complete tensor to be compensated
Figure FDA0002400360210000022
Decomposed into tensor column kernels
Figure FDA0002400360210000023
The concrete expression is as follows:
Figure FDA0002400360210000024
in the above formula, the first and second carbon atoms are,
Figure FDA0002400360210000025
representing the full sheet to be supplementedThe coordinate in the quantity is (i)c,jc,kc) The value of the pixel of (a) is,
Figure FDA0002400360210000026
indicating the amount of tension to be compensated
Figure FDA0002400360210000027
The tensor of the reshaped intermediate medium,
Figure FDA0002400360210000028
a b-th front tangent matrix representing an a-th tensor column kernel;
step 2.2, carrying out tensor singular value decomposition on one tensor column core in the step:
Figure FDA0002400360210000029
in the above equation, a tensor column kernel is decomposed into the form of the product of three tensors, where
Figure FDA00024003602100000210
Called f-diagonal tensor, taking the initial value a as 1;
step 2.3, in the Fourier domain, the tensor column in the previous step is centered
Figure FDA00024003602100000211
Corresponding f-diagonal tensor
Figure FDA00024003602100000212
The values in (1) are mapped into a matrix E according to coordinates, and the corresponding coordinate relationship is expressed as:
Figure FDA00024003602100000213
in the above formula, the first and second carbon atoms are,
Figure FDA00024003602100000214
representing tensor column core
Figure FDA00024003602100000215
F-diagonal tensor in Fourier domain, ζ denotes tensor column core pointer, ξ denotes eigenmatrix pointer, IaRepresenting the tensor of the intermediate medium to be complemented
Figure FDA00024003602100000216
The a-th dimension; the tensor column core pointer ζ has a specific value of ζ being 1 when a is 1 or 3, and ζ being i in other cases; the specific values of the eigen matrix pointers are:
Figure FDA0002400360210000031
in the above formula, reAnd rcRespectively representing the edge rank and the center rank in the tensor eigen transformation, wherein the edge rank is always set as re＝1；
Step 2.4, if a<3, taking the intermediate tensor to be compensated
Figure FDA0002400360210000032
A +1 th tensor column core of
Figure FDA0002400360210000033
I.e. let a be a +1, go back to step 2.3 to enter iteration, otherwise determine tensor
Figure FDA0002400360210000034
The eigenmatrix of (a) is E.
4. The tensor eigenconversion-based color image restoration method as recited in claim 3, wherein the specific method of the improved tensor column decomposition is as follows:
step 2.1.1, first, the tensor corresponding to the color image
Figure FDA0002400360210000035
Remodel into
Figure FDA0002400360210000036
Wherein the setting parameter p is utilized1,ρ2The remodeling relationship of (A) is that,
Figure FDA0002400360210000037
step 2.1.2, introduce the auxiliary temporary tensor
Figure FDA0002400360210000038
Order to
Figure FDA0002400360210000039
Will tensor
Figure FDA00024003602100000310
Reshaped into a corresponding matrix
Figure FDA00024003602100000311
Step 2.1.3, remodeling the matrix obtained in the step 2.1.2
Figure FDA00024003602100000312
Decomposition according to matrix singular values:
C＝U·S·VT(5)
wherein the content of the first and second substances,
Figure FDA00024003602100000313
respectively called left and right singular matrices of the matrix C,
Figure FDA00024003602100000314
is a diagonal matrix;
step 2.1.4, get matrix
Figure FDA00024003602100000315
Front r ofcColumn forming a new matrix
Figure FDA00024003602100000316
Get matrix
Figure FDA00024003602100000317
Front r ofcColumn forming a new matrix
Figure FDA00024003602100000318
Get matrix
Figure FDA00024003602100000319
Front r ofcRow and rcColumns forming a new matrix
Figure FDA00024003602100000320
Step 2.1.5, matrix
Figure FDA00024003602100000321
Remodelling into first volumetric core
Figure FDA00024003602100000322
Determining updated matrices
Figure FDA00024003602100000323
Step 2.1.6, matrix
Figure FDA00024003602100000324
Remolding into a new matrix
Figure FDA00024003602100000325
And is obtained according to the singular value decomposition of the matrix, i.e. the decomposition in equation (5)
Figure FDA0002400360210000041
Get matrix
Figure FDA0002400360210000042
Front r ofcColumn forming a new matrix
Figure FDA0002400360210000043
Get matrix
Figure FDA0002400360210000044
Front r ofcColumn forming a new matrix
Figure FDA0002400360210000045
Get matrix
Figure FDA0002400360210000046
Front r ofcRow and rcColumns forming a new matrix
Figure FDA0002400360210000047
Step 2.1.7, matrix
Figure FDA0002400360210000048
Remodeled to a second tensor column core
Figure FDA0002400360210000049
Determining updated matrices
Figure FDA00024003602100000410
C is to bedRemodeled to a third tensor column core
Figure FDA00024003602100000411
5. The tensor eigen transformation-based color image restoration method according to claim 3 or 4, wherein in the step 3, the specific method of tensor singular value decomposition is as follows:
step 2.2.1, get a quantitative core
Figure FDA00024003602100000412
Performing three-dimensional Fourier transform to obtain tensor column core in corresponding Fourier domain
Figure FDA00024003602100000413
Taking an initial value s as 1;
step 2.2.2, get tensor column core
Figure FDA00024003602100000414
And recording the matrix as the s-th front section matrix
Figure FDA00024003602100000415
Namely have
Figure FDA00024003602100000416
To pair
Figure FDA00024003602100000417
Performing singular value decomposition of the matrix to obtain left and right singular matrices thereof
Figure FDA00024003602100000418
And diagonal matrix
Figure FDA00024003602100000419
Namely, it is
Figure FDA00024003602100000420
Step 2.2.3 creating tensors
Figure FDA00024003602100000421
Setting the initial value of the newly-built tensor to be 0, and setting the left and right singular matrixes in the previous step
Figure FDA00024003602100000422
And diagonal matrix
Figure FDA00024003602100000423
The s-th front tangent plane matrix given to the newly created tensor, i.e.
Figure FDA00024003602100000424
Step 2.2.4, if s<I3If s is equal to s +1, the procedure returns to step 2.2.2 to enter a loop, otherwise, the three tensors are determined
Figure FDA00024003602100000425
Step 2.2.5, for
Figure FDA00024003602100000426
Respectively carrying out three-dimensional inverse Fourier transform to obtain tensors of real number domain
Figure FDA00024003602100000427
To sum up, a tensor column core is obtained
Figure FDA00024003602100000428
Singular value decomposition of, i.e. having
Figure FDA00024003602100000429
6. The tensor eigenconversion-based color image restoration method as recited in claim 1, wherein the threshold operator D in the step 3 is set toε,βThe concrete expression is as follows:
Figure FDA0002400360210000051
wherein sign (x) is a sign function, parameter c0,c1,c2Comprises the following steps:
Figure FDA0002400360210000052
wherein ε, β is the preset parameter input in advance, defining the symbol Dε,β(X) represents the thresholding operation on all elements in matrix X.
7. The tensor eigenconversion-based color image restoration method as recited in claim 1, wherein in the step 4, the inverse tensor eigenconversion τ-1The specific method comprises the following steps:
step 4.1, in the Fourier domain, a tensor eigen transformation method is adopted, and a plurality of corresponding tensor column cores are obtained through calculation of the eigen matrix E
Figure FDA0002400360210000053
F-diagonal tensor of
Figure FDA0002400360210000054
Taking an initial value a as 1;
step 4.2, mixing
Figure FDA0002400360210000055
Is obtained by inverse Fourier transform
Figure FDA0002400360210000056
From the f-diagonal tensor in the real domain
Figure FDA0002400360210000057
And corresponding left and right singular tensors
Figure FDA0002400360210000058
Calculating to obtain tensor column core
Figure FDA0002400360210000059
Figure FDA00024003602100000510
In the above formula, the symbol "+" represents the tensor product;
step 4.3, if a<3, the a +1 f-diagonal tensor is taken
Figure FDA00024003602100000511
If a is equal to a +1, returning to the step 4.2 to enter iteration, otherwise, entering the next step;
step 4.4, according to the obtained plurality of tensor column cores
Figure FDA00024003602100000512
Calculating to obtain the updating tensor corresponding to the eigen matrix E
Figure FDA00024003602100000513
Figure FDA00024003602100000514
In the above formula, subscript i, j, k represents the tensor of the pixel point
Figure FDA00024003602100000515
The position of (1); will update the tensor
Figure FDA00024003602100000516
Reshaped into an original image
Figure FDA0002400360210000061
Tensor of equal size
Figure FDA0002400360210000062
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