CN112950500B - Hyperspectral denoising method based on edge detection low-rank total variation model - Google Patents

Hyperspectral denoising method based on edge detection low-rank total variation model Download PDF

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CN112950500B
CN112950500B CN202110213967.8A CN202110213967A CN112950500B CN 112950500 B CN112950500 B CN 112950500B CN 202110213967 A CN202110213967 A CN 202110213967A CN 112950500 B CN112950500 B CN 112950500B
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蒋俊正
蔡万源
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Abstract

The invention discloses a hyperspectral denoising method based on an edge detection low-rank total variation model, which comprises the steps of constructing an input signal model, and constructing and optimizing the edge detection low-rank total variation model based on the input signal model; solving a first sub-problem divided in the edge detection low-rank fully-variable model by using a singular value shrinkage method; dividing the obtained second subproblems based on the number of wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm; solving the obtained third subproblem by using a soft threshold shrinkage operator, and iterating the obtained results of all the subproblems; and comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value, and improving the denoising effect of the spectral image through experimental verification analysis.

Description

Hyperspectral denoising method based on edge detection low-rank total variation model
Technical Field
The invention relates to the technical field of hyperspectral image data processing, in particular to a hyperspectral denoising method based on an edge detection low-rank total variation model.
Background
The hyperspectral imaging (HSI) technology can acquire two-dimensional images in a wide electromagnetic spectrum range and a higher spectral resolution, and thus is widely applied to the fields of archaeology and art protection, vegetation and water resource control, food quality and safety control, forensic medicine, surgery and diagnosis, crime scene detection, biomedicine, military and the like.
However, due to its unique physical design, HSI is inevitably contaminated by various noises during the acquisition process, and the common types of the contaminated noises include gaussian noise, impulse noise, dead pixel, streak noise, and the like. In recent years, many scientists have proposed various HSI denoising algorithms. Among them, the method for restoring hyperspectral images based on low-rank matrix restoration (LRMR), the method for restoring hyperspectral images based on total variation regularization low-rank matrix decomposition (LRTV), and the like are good in effect. The LRMR method utilizes the low-rank characteristic of the hyperspectral image to represent the main information of the hyperspectral image by using a low-rank matrix, so that a large amount of redundant information in the hyperspectral image is removed. Most of noise is contained in redundant information, so that the denoising effect can be achieved in the process of recovering the hyperspectral image by using the low-rank matrix. However, the denoising method based on low rank only studies the correlation between spectral bands, ignores the spatial correlation of local neighborhood pixels, and thus cannot achieve the optimal denoising effect. The LRTV method describes the spatial correlation and spatial smoothness of local neighborhood pixels on the basis of low-rank characteristics, the denoising effect is better than that of LRMR, but the method has less utilization of the spatial correlation information of the neighborhood pixels, and neglects the protection of the hyperspectral image edge in the smoothing process of the local neighborhood pixels. Therefore, a design method for improving the denoising effect of the spectral image is to be further introduced.
Disclosure of Invention
The invention aims to provide a hyperspectral denoising method based on an edge detection low-rank total variation model, which improves the denoising effect of a spectral image.
In order to achieve the purpose, the invention provides a hyperspectral denoising method based on an edge detection low-rank total variation model, which comprises the following steps:
constructing an input signal model, and constructing and optimizing an edge detection low-rank total variation model based on the input signal model;
solving a first sub-problem divided in the edge detection low-rank fully-variable model by using a singular value shrinkage method;
dividing the obtained second subproblems based on the number of the wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm;
solving the obtained third subproblem by using a soft threshold shrinkage operator, and iterating the obtained results of all the subproblems;
and comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, and calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value.
The method comprises the following steps of constructing an input signal model, constructing and optimizing an edge detection low-rank total variation model based on the input signal model, and comprises the following steps:
sequentially adding the acquired original image with randomly generated sparse noise and Gaussian noise to obtain an input signal model;
and constructing an edge detection low-rank total variation model based on the input signal model, optimizing the edge detection low-rank total variation model by using an augmented Lagrange function method, and dividing main sub-problems, wherein the divided sub-problems comprise a first sub-problem, a second sub-problem and a third sub-problem.
Dividing the obtained second subproblems based on the number of the wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm, wherein the method comprises the following steps of:
dividing the obtained second sub-problem into a plurality of band sub-problems according to the number of the bands of the hyperspectral image;
rewriting and iterating each wave band subproblem, and assigning values to the detected pixel points by using an edge detection operator in the iteration process;
and carrying out iterative calculation on the difference absolute values of the four adjacent domains of all the pixel points which meet the conditions after assignment until the set iterative conditions are met, and obtaining the solution of the corresponding second subproblem.
After assignment, iterative computation is performed on the four-neighbor domain difference absolute values of all pixel points meeting the condition until a set iteration condition is met, and a solution of a corresponding second subproblem is obtained, wherein the iterative computation comprises the following steps:
turning over the value of the pixel point after assignment to obtain a detection value;
and adding the difference absolute values of the four adjacent domains of each pixel point, calculating the difference between the result value of the constraint condition of the current calculation result and the result value of the constraint condition of the last calculation result, dividing the obtained absolute value of the difference by the result value of the constraint condition of the current calculation result, terminating iteration if the obtained calculated value is smaller than the set iteration condition, and overlapping all frequency band result values to obtain the solution of the second subproblem.
Comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, and calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value, wherein the method comprises the following steps:
if the current calculation result does not meet the set iteration termination condition, solving the first subproblem, the second subproblem, the third subproblem and all other subproblems again until the set iteration termination condition is met, wherein the other subproblems are the problems except the first subproblem, the second subproblem and the third subproblem;
and after the set iteration termination condition is met, comparing the obtained hyperspectral denoised image with the original image to obtain a corresponding peak signal-to-noise ratio and a corresponding structural similarity value.
The invention discloses a hyperspectral denoising method based on an edge detection low-rank total variation model, which comprises the steps of constructing an input signal model, and constructing and optimizing the edge detection low-rank total variation model based on the input signal model; solving a first sub-problem divided in the edge detection low-rank fully-variable model by using a singular value shrinkage method; dividing the obtained second subproblems based on the number of wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm; solving the obtained third subproblem by using a soft threshold shrinkage operator, and iterating the obtained results of all the subproblems; comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value, and denoising the hyperspectral image by using an edge detection four-neighborhood total variation algorithm, so that the smoothness relation among neighborhoods is enhanced, the edges in the hyperspectral image are protected, and the phenomenon that the edges are smoothed to influence the denoising effect is avoided. Through experimental verification and analysis, the method has a better denoising effect.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a schematic step diagram of a hyperspectral denoising method based on an edge detection low-rank total variation model provided by the invention.
Fig. 2 is a clean artwork provided by the present invention.
Fig. 3 is a noisy image provided by the present invention.
FIG. 4 is a denoised LRMR image provided by the present invention.
FIG. 5 is a denoised LRTV image according to the present invention.
FIG. 6 is an EDTV denoised image according to the present invention.
Fig. 7 is a comparison diagram of PSNR of each band provided by the present invention.
Fig. 8 is a comparison diagram of SSIM of each band provided by the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
In the description of the present invention, "a plurality" means two or more unless specifically defined otherwise.
Referring to fig. 1, the present invention provides a hyperspectral denoising method based on an edge detection low-rank total variation model, including the following steps:
s101, constructing an input signal model, and constructing and optimizing an edge detection low-rank total variation model based on the input signal model.
Specifically, an input signal model Y is constructed, wherein Y represents an input noise signal; x represents a clean original image; s represents sparse noise and is used for depicting pulse noise, dead pixels, stripe noise and the like; n represents gaussian noise.
According to an input signal model, establishing an edge detection low-rank total variation model:
Figure BDA0002952439790000041
Figure BDA0002952439790000042
wherein min represents the value of X and S when the following expression reaches the minimum value; i | · | purple wind * The kernel norm is the sum of matrix singular values, is used for convex approximation rank constraint and is used for depicting the low rank characteristic of the hyperspectral image; | X | non-conducting phosphor EDTV Representing the piecewise smoothness of the hyperspectral image; | S | non-woven phosphor 1 Representing sparse noise; s.t. objectto, with constraints indicated later; i | · | live through F Is Frobenius norm, which is the square sum of the absolute values of the matrix elements and then the square of the square; in constraint term
Figure BDA00029524397900000510
The square of the F norm of Gaussian noise is represented, and the term is made to be as small as possible after denoising, so that the denoising effect is achieved; epsilon is a number as small as possible and is used for restricting an optimization term; rank (·) denotes the rank of the matrix; r is the set matrix rank size and is used for constraining optimization items to meet the low-rank property; both τ and λ are regular term parameters.
And (3) according to the established edge detection low-rank total variation model, carrying out optimization solution on the model:
the above problem is solved by using an augmented lagrangian method (hereinafter, referred to as ALM). The above problem was first equivalently rewritten as follows:
Figure BDA0002952439790000051
Figure BDA0002952439790000052
the equivalence of L to X before the equivalent overwrite is convenient for the subsequent use of ALM method.
By adopting the ALM method, the optimized augmented Lagrangian function is as follows:
Figure BDA0002952439790000053
s.t.rank(L)≤r
wherein,
Figure BDA00029524397900000511
expression relating to L, X, S, Lambda 1 ,Λ 2 A minimum function of the function of (a); lambda 1 ,Λ 2 Optimizing a coefficient matrix;<·,·>represents the inner product; μ is a penalty factor and the initial value is set to 1 e-2.
Dividing the problem into a plurality of sub-problems, solving one of the sub-problems in an iterative manner:
Figure BDA0002952439790000054
Figure BDA0002952439790000055
Figure BDA0002952439790000056
Figure BDA0002952439790000057
Figure BDA0002952439790000058
wherein
Figure BDA0002952439790000059
The value of the expression when the minimum value is reached is expressed; k represents the kth iteration; * (k+1) Represents the result of formula after the (k + 1) th iteration; * (k) The results of formula after the k-th iteration are shown.
Thus solving the problem as solving L, X, S three main first through third sub-problems.
S102, solving a first sub-problem divided from the edge detection low-rank total variation model by using a singular value shrinkage method.
Specifically, the first sub-problem with respect to L is solved using a singular value contraction method.
Figure BDA0002952439790000061
For a given matrix W, it is decomposed using singular value decomposition, resulting in:
W＝UE r V * ,E r ＝diag({σ i } 1≤i≤r )
u and V are unitary matrixes obtained by singular value decomposition of W; v * Is a conjugate transpose of V; e r Determining a diagonal matrix for the semi-positive; diag (·) denotes a diagonal matrix whose diagonal elements are constructed as · s; { sigma. } i } 1≤i≤r Representing a set of the first r diagonal elements.
And then using a singular value contraction operator:
Figure BDA0002952439790000062
wherein D is δ (W) is formula
Figure BDA0002952439790000063
When the minimum value is reached and the rank of L is less than r, the value of L is obtained;
D δ (W)＝UD δ (E r )V * ,D δ (E r )＝diag{max((σ i δ),0), where max (x, 0) represents the maximum value for each comparison of the two compared to 0.
S103, dividing the obtained second subproblems based on the number of the wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm.
Specifically, the second sub-problem for X is solved:
Figure BDA0002952439790000071
the problem is decomposed into p band sub-problems according to the number p of bands of the hyperspectral image, i.e.
Figure BDA0002952439790000072
Wherein j is an integer from 1 to p;
Figure BDA0002952439790000073
represents the value of the jth band X after the (k + 1) th iteration;
Figure BDA0002952439790000074
after assignment, iteration calculation is carried out on the four-neighborhood difference absolute values of all the pixel points which meet the condition that the pixel point is a non-edge pixel point and the four neighborhoods of the pixel point are non-edge pixel points until set iteration conditions are met, and a solution of a corresponding second subproblem is obtained.
The problem is solved by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm.
For each sub-problem of the above band, we can rewrite it as an equivalent problem as follows:
Figure BDA0002952439790000075
the solution of the above problem is
Figure BDA0002952439790000076
Wherein, P C For the orthographic projection operator, L (p, q) is a matrix pair operator, wherein
L(p,q) i,j ＝p i,j +q i,j -p i-1,j -q i,j-1 ，i＝1,...,m,j＝1,...,n
p i,j ＝x i,j -x i+1,j ，i＝1,...,m-1,j＝1,...,n
q i,j ＝x i,j -x i,j+1 ，i＝1,...,m,j＝1,...,n-1
Wherein x is *,· The value of a pixel point with the abscissa as x and the ordinate as · is defined;
in the iteration process, X is continuously assigned and is continuously close to the optimal solution, in each iteration process, edge detection is carried out on X by adopting a Sobel edge operator, if the pixel point is detected to be an edge, the value is 1, and if not, the value is 0; the 0, 1 value is turned over to obtain a detection value delta i,j 。
And then, the smoothness of the hyperspectral image is depicted by adding the difference absolute values of four adjacent domains of each pixel point, when an edge is detected, the pixel point does not participate in the smoothness depiction, or when a certain point around the pixel point is an edge, the adjacent domain in the direction does not participate in the smoothness depiction, so that the smoothness depiction is obtained
Figure BDA0002952439790000081
Wherein,
Figure BDA0002952439790000082
when iterated to satisfy this
Figure BDA0002952439790000083
Absolute value division of the value subtracted from the value of the last iteration of the equationAnd when the value is smaller than the preset iteration condition 1e-4, stopping the iteration, and solving to obtain the calculated value X.
Calculating value X obtained from each wave band j And (5) overlapping and reducing to obtain complete X.
And S104, solving the obtained third sub-problem by using a soft threshold shrinkage operator, and iterating the obtained results of all the sub-problems.
Specifically, a soft threshold shrinkage operator is used to solve the third sub-problem with respect to S.
Figure BDA0002952439790000084
Shrinking operators by soft threshold
Figure BDA0002952439790000085
Wherein x ∈ R, Δ > 0; then the solution to this step sub-problem is:
Figure BDA0002952439790000086
iterate to obtain Λ 1
Figure BDA0002952439790000091
Iterate to obtain Λ 2
Figure BDA0002952439790000092
And S105, comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, and calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value.
Specifically, the penalty factor is set such that ρ is 1.5 and μ is obtained for each amplification step max 1e6, iterate μ.
μ( k+1 )＝min(ρμ,μ max ) Wherein min (·) represents comparing values of · and taking the smaller value of the two.
Judging an iteration termination condition, if the condition is met, stopping iteration, wherein the iteration termination condition is as follows:
Figure BDA0002952439790000093
wherein | · | purple ∞ The matrix is an infinite norm and represents the maximum value of the sum of absolute values of row vectors of the matrix; epsilon 1 ，ε 2 Is a number as small as possible, and is a constraint parameter.
And if the iteration termination condition is not met, continuously returning to solve the first subproblem again, and performing a new iteration until the termination condition is met to obtain the solution of the materialized model.
And performing peak signal-to-noise ratio (PSNR) and Structural Similarity (SSIM) parameter calculation on the resolved hyperspectral denoised image X and a clean picture without noise, and evaluating the superiority of the contrast denoising effect.
Example analysis:
the hyperspectral image input in this example is a clean hyperspectral image of indiana, X, which has 224 wavebands and has a dimension of 145X 224. Adding noise artificially, wherein the added noise comprises sparse noise (including salt and pepper noise) and Gaussian noise, and outputting Y after adding the noise, wherein tau is 0.015; λ is
Figure BDA0002952439790000094
Where M, N is the hyperspectral image dimension, this example M145, N145; r is 10; epsilon 1 ＝ε 2 1 e-6; the initial value of μ is 1 e-2.
Sending the three-dimensional hyperspectral image after noise addition into the model of the technical scheme, and obtaining a contrast image before and after noise removal as shown in figures 2-6, wherein figures 2-6 are gray level images which represent black depth, and the gray level value of the gray level image is smaller when the image is darker; wherein, fig. 2 shows a clean original image, fig. 3 shows a denoised image, fig. 4 shows an image denoised by LRMR method, fig. 5 shows an image denoised by LRTV method, and fig. 6 shows an image denoised by the technical scheme; according to the comparison between fig. 2 and fig. 3, it can be observed that the experiment adds more serious noise, and the effectiveness of the technical scheme can be further embodied; according to fig. 4, it can be observed that there are still fuzzy points in the image, and the denoising effect is not good enough compared with fig. 5 and 6; observing fig. 5 and fig. 6, it is observed that the denoising effects of the two are similar globally in fig. 6, but when the detail aspect is carefully observed, the denoising effect of fig. 6 is better than that of fig. 5, that is, it can be observed that the denoising effect of the model in the technical scheme is good. Compared with a method (LRMR) for restoring a hyperspectral image based on low-rank matrix restoration proposed by Hongyan Zhang and a method (LRTV) for restoring a hyperspectral image based on holomorphic regular low-rank matrix decomposition proposed by Wei He, the PSNR and SSIM parameters of each wave band after denoising in the technical scheme are compared with the prior art scheme to obtain a graph 7-8, the graph 7 is observed, a curve (solid line) positioned at the top of the image is a PSNR parameter curve of the technical scheme, a curve (dotted line) positioned in the middle is a PSNR parameter curve of the technical scheme, a curve (dotted line) positioned at the bottom is a PSNR parameter curve of the technical scheme, and a linear type corresponding to the technical scheme is arranged at the upper right of the image; the higher the PSNR value is, the better the denoising effect is; according to the observation of FIG. 7, the denoising effect of each wave band of the PSNR is better than that of the LRMR in the technical scheme; the PSNR basically meets the condition that the denoising effect of all bands is superior to that of an LRTV technical scheme (the existing individual band is slightly lower than that of the LRTV technical scheme, but the slightly poor band is almost the same as that of the LRTV technical scheme), and compared with the LRTV technical scheme, the PSNR avoids the deterioration phenomenon of the denoising effect of the individual band; observing fig. 8, the curve (dotted line) at the top of the image is the SSIM parameter curve of the technical scheme, the curve (dotted line) at the middle is the SSIM parameter curve of the LRTV technical scheme, the curve (solid line) at the bottom is the SSIM parameter curve of the LRMR technical scheme, and there is a linear type corresponding to the technical scheme on the upper right of the image; the closer the SSIM parameter is to 1, the closer the similarity of the SSIM parameter and the original image structure is, namely the better the denoising effect is, and the conclusion consistent with the observation of the graph 7 can be observed; in conclusion, the technical scheme is superior to the prior technical scheme.
Averaging the PSNR and SSIM of each wave band in the technical scheme and the prior scheme, and recording the average as a primary result; repeating the experiment for 30 times, and obtaining results for 30 times; the PSNR and SSIM parameters obtained by averaging the results of 30 times are shown in table 1, and it can be observed that the present technical solution is superior to the above existing solutions.
Table 130 times comparison table of denoising effect of each scheme
Figure BDA0002952439790000101
The invention discloses a hyperspectral denoising method based on an edge detection low-rank total variation model, which comprises the steps of constructing an input signal model, and constructing and optimizing the edge detection low-rank total variation model based on the input signal model; solving a first sub-problem divided in the edge detection low-rank fully-variable model by using a singular value shrinkage method; dividing the obtained second subproblems based on the number of wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm; solving the obtained third subproblem by using a soft threshold shrinkage operator, and iterating the obtained results of all the subproblems; comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value, and denoising the hyperspectral image by utilizing an edge detection four-neighborhood total variation algorithm, thereby not only enhancing the smoothness relation between neighborhoods, but also protecting the edges in the hyperspectral image and avoiding the edges from being smoothed to influence the denoising effect. Through experimental verification and analysis, the method has a better denoising effect.
While the invention has been described with reference to a preferred embodiment, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (2)

1. A hyperspectral denoising method based on an edge detection low-rank total variation model is characterized by comprising the following steps:
constructing an input signal model, and constructing and optimizing an edge detection low-rank total variation model based on the input signal model;
solving a first sub-problem divided in the edge detection low-rank fully-variable model by using a singular value shrinkage method;
dividing the obtained second subproblems based on the number of wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm;
iterating the obtained results of all the sub-problems by using a soft threshold shrinkage operator;
comparing the current iteration result with a set iteration termination condition until the iteration termination condition is met, and calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value;
an input signal model is built, and an edge detection low-rank total variation model is built and optimized based on the input signal model, and the method comprises the following steps:
constructing an input signal model Y which is X + S + N, wherein Y represents an input noise signal; x represents a clean original image; s represents sparse noise; n represents Gaussian noise;
according to an input signal model, establishing an edge detection low-rank total variation model:
Figure FDA0003756278960000011
Figure FDA0003756278960000012
wherein min represents the value of X and S when the formula reaches the minimum value; i | · | purple wind * Is a kernel norm, refers to the sum of matrix singular values, is used for convex approximate rank constraint,the low-rank characteristic is used for describing a hyperspectral image; | X | non-conducting phosphor EDTV Representing the piecewise smoothness of the hyperspectral image; | S | non-woven phosphor 1 Representing sparse noise; s.t. objectto, with constraints indicated later; i | · | purple wind F Is Frobenius norm, which is the square sum of the absolute values of the matrix elements and then the square of the square; in constraint term
Figure FDA0003756278960000013
Represents the square of the F-norm of gaussian noise; ε is used to constrain the optimization term; rank (·) denotes the rank of the matrix; r is the set matrix rank size and is used for constraining optimization items to meet the low-rank property; both tau and lambda are regular term parameters;
according to the established edge detection low-rank total variation model, optimizing and solving the model:
solving the problems by adopting an augmented Lagrange function method; the above problem is first equivalently rewritten as follows:
Figure FDA0003756278960000021
Figure FDA0003756278960000022
l is equivalent to X before equivalent rewriting;
by adopting the ALM method, the optimized augmented Lagrangian function is as follows:
Figure FDA0003756278960000023
s.t.rank(L)≤r
wherein, mini (L, X, S, Lambda) 1 ,Λ 2 ) Expression relating to L, X, S, Lambda 1 ,Λ 2 A minimum function of the function of (a); lambda 1 ,Λ 2 Optimizing a coefficient matrix;<·,·>represents the inner product; mu is a penalty factor, and the initial value is set to be 1 e-2;
dividing the problem into a plurality of sub-problems, and solving one of the sub-problems in an iterative manner:
Figure FDA0003756278960000024
Figure FDA0003756278960000025
Figure FDA0003756278960000026
Λ 1 (k+1) ＝Λ 1 (k) +μ(Y-L (k+1) -S (k+1) )
Λ 2 (k+1) ＝Λ 2 (k) +μ(X (k+1) -L (k+1) )
wherein
Figure FDA0003756278960000029
The value of the expression when the minimum value is reached is expressed; k represents the kth iteration; * (k+1) Represents the result of formula after the (k + 1) th iteration; * (k) Represents the result of formula after the k-th iteration;
thus solving L, X, S the three main first to third sub-problems;
dividing the obtained second subproblems based on the number of the wave bands of the hyperspectral image, and solving all the second subproblems by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm, wherein the method comprises the following steps:
solving a second sub-problem with respect to X:
Figure FDA0003756278960000027
the problem is decomposed into p band sub-problems according to the number p of bands of the hyperspectral image, i.e.
Figure FDA0003756278960000028
Wherein j is an integer from 1 to p; x j (k+1) Represents the value of the jth band X after the (k + 1) th iteration;
Figure FDA0003756278960000031
after assignment, carrying out iterative computation on the four-neighbor domain difference absolute values of all the pixels which meet the condition that the pixel is a non-edge pixel and the four neighborhoods of the pixel are non-edge pixels until a set iteration condition is met, and obtaining a solution of a corresponding second subproblem;
solving the problem by using an iterative gradient-based fast edge detection four-neighborhood total variation algorithm;
for the sub-problem of each band described above, we can rewrite it as an equivalent problem, as follows:
Figure FDA0003756278960000032
the solution of the above problem is
Figure FDA0003756278960000033
Wherein, P C For the orthographic projection operator, L (p, q) is a matrix pair operator, wherein
L(p,q) i,j ＝p i,j +q i,j -p i-1,j -q i,j-1 ，i＝1,...,m,j＝1,...,n
p i,j ＝x i,j -x i+1,j ，i＝1,...,m-1,j＝1,...,n
q i,j ＝x i,j -x i,j+1 ，i＝1,...,m,j＝1,...,n-1
Wherein x is *,· The value of a pixel point with the abscissa as x and the ordinate as · is defined;
continuously evaluating the value of X in the iteration process, continuously approaching to the optimal solution, carrying out edge detection on X by adopting a Sobel edge operator in each iteration process, if the pixel point is detected to be an edge, setting the value to be 1, and otherwise, setting the value to be 0; the 0, 1 value is turned over to obtain a detection value delta i,j ；
And then, the smoothness of the hyperspectral image is depicted by adding the difference absolute values of four adjacent domains of each pixel point, when an edge is detected, the pixel point does not participate in the smoothness depiction, or when a certain point around the pixel point is an edge, the adjacent domain in the direction does not participate in the smoothness depiction, so that the smoothness depiction is obtained
Figure FDA0003756278960000034
Wherein,
Figure FDA0003756278960000035
when iterated to satisfy this
Figure FDA0003756278960000036
When the absolute value of the subtraction of the value and the value of the formula of the previous iteration is divided by the value of the current iteration to be less than the preset iteration condition 1e-4, stopping the iteration, and solving to obtain a calculated value X;
calculating value X obtained from each wave band j And (5) overlapping and reducing to obtain complete X.
2. The hyperspectral denoising method based on the edge detection low-rank total variation model according to claim 1, wherein comparing the current iteration result with a set iteration termination condition until the iteration termination condition is satisfied, and calculating a corresponding peak signal-to-noise ratio and a corresponding structural similarity value comprises:
if the current calculation result does not meet the set iteration termination condition, solving the first subproblem, the second subproblem, the third subproblem and all other subproblems again until the set iteration termination condition is met, wherein the other subproblems are the problems except the first subproblem, the second subproblem and the third subproblem;
and after the set iteration termination condition is met, comparing the obtained hyperspectral denoised image with the original image to obtain a corresponding peak signal-to-noise ratio and a corresponding structural similarity value.
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