Hyperspectral image denoising method based on global variation and local gradient product

Abstract

The invention relates to a hyperspectral image denoising method based on global variation and a local gradient product, and belongs to the technical field of image processing. The method comprises the following steps: S1, obtaining a hyperspectral image data set and cutting the hyperspectral image data set; s2, preprocessing the hyperspectral image serving as the data set; s3, establishing a degradation model for the processed hyperspectral image data, introducing a low-rank and local gradient product for modeling, and solving an optimization problem through an alternating direction multiplier method (ADMM) after parameter initialization; and S4, obtaining a denoised hyperspectral image after iterative optimization, and completing denoising. The performance of the method is superior to that of other hyperspectral image denoising methods. According to the method provided by the invention, the details of the image can be finely recovered, and various different mixed noises can be removed.

Description

Technical Field

[0001] The present invention belongs to the technical field of image processing, and relates to a hyperspectral image denoising method based on global variational and local gradient product. Background Art

[0002] A hyperspectral image is a three-dimensional cube with hundreds of spectral bands, as well as rich image information and spectral information, and has a wide range of applications. For example, classification, clustering, anomaly or target detection, and change detection, etc. However, hyperspectral images are affected by sensor vibration, signal transmission, atmospheric interference, and other factors during the acquisition process, and will inevitably be contaminated by various noises, including Gaussian noise, salt and pepper noise, stripe noise, dead line noise, and mixed noise. These noises not only reduce the quality of the image, but also limit the effectiveness of its subsequent tasks. As an indispensable preprocessing step, hyperspectral image denoising is becoming increasingly important for various subsequent tasks and has received extensive attention.

[0003] In recent decades, many scholars have proposed a large number of hyperspectral image denoising methods. Currently, the existing hyperspectral image denoising methods can be roughly divided into three categories: model-driven, data-driven, and model-data collaborative-driven methods.

[0004] In recent years, with the booming development of deep learning, many methods have been proposed and applied in the field of hyperspectral image denoising. These methods adaptively adjust the trainable parameters of external data and labels through a deep learning framework. These methods do not require careful adjustment of key data, but due to their inability to well combine the physical characteristics of hyperspectral images, the interpretability of these networks is not strong. Model-driven denoising algorithms use predefined prior knowledge to characterize hyperspectral images, construct a hyperspectral image denoising model, and then obtain the denoised image by solving the optimization problem for each parameter in the model. So far, non-local similarity, total variation (TV), sparse representation, and low-rank (LR) models have been widely applied to hyperspectral image denoising. Using a single model for denoising cannot obtain good denoising effects. For example, constructing a model through non-local similarity may cause the image to be over-smoothed when removing noise. It has been found in practice that using TV and LR can well characterize the physical characteristics of hyperspectral images, but these methods mainly study the global characteristics of images and lack attention to the local structural information of hyperspectral images. In fact, local structural information can also reflect the texture characteristics of different regions of the image and is of great significance for maintaining spatial details and processing mixed noise. To address the above problems, the present invention proposes a hyperspectral image denoising method based on global variational and local gradient product. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to provide a hyperspectral image denoising method based on global variation and local gradient product, and establish a new hyperspectral image denoising framework, which takes into account both the LR information of global variation and the local structure information; adopt a new LR model, approximate the LR property of global variation through the γ-norm, and combine the l 1，1 norm of the sparse term. Among them, the γ-norm is introduced to approximate the rank of the hyperspectral image, which overcomes the problem of unbalanced penalty for different singular values in the nuclear norm. At the same time, the l 1，1 norm of the sparse term is introduced to further characterize the sparsity of the noise in the hyperspectral image; a new local structure of the hyperspectral image is defined, and it is combined with the LR framework as a regularization term for the local structure information of the hyperspectral image. This method can better characterize the physical properties of the hyperspectral image and can better retain the edge information and texture features of the image during the noise removal process.

[0006] To achieve the above object, the present invention provides the following technical solutions:

[0007] A hyperspectral image denoising method based on global variation and local gradient product, the method comprising the following steps: S1: Obtain a hyperspectral image dataset and crop it; S2: Preprocess the hyperspectral image as the dataset; S3: Establish a degradation model for the processed hyperspectral image data, introduce low rank and local gradient product for modeling, and solve the optimization problem by the alternating direction method of multipliers (ADMM) after parameter initialization; S4: After iterative optimization, obtain the denoised hyperspectral image to complete the denoising.

[0008] Further, in step S1, a drone platform equipped with a hyperspectral camera is used to collect hyperspectral remote sensing images, and then the collected hyperspectral images are cropped and divided.

[0009] Further, in step S2, preprocess the hyperspectral image as the dataset. First, perform a normalization operation on the obtained hyperspectral image to eliminate the influence of data dimensions and make the data indicators comparable; then convert the three-dimensional data model of the hyperspectral image into a two-dimensional matrix, so that each column of the two-dimensional matrix corresponds to the spectral information of a band of the hyperspectral image.

[0010] Further, in step S3, model the preprocessed hyperspectral data, and obtain the model by modeling its two-dimensional matrix. The formula is as follows:

[0011] Y = X + S + N

[0012] Among them, Y represents the hyperspectral image with noise, X is the finally obtained clean hyperspectral image, and S and N represent sparse noise and Gaussian noise respectively. The sizes of Y, X, S, and N are all mn×p, where m and n are the sizes of the spatial dimensions of the hyperspectral image, and p is the number of bands of the hyperspectral image, corresponding to the operation of preprocessing into a two-dimensional matrix in step S2.

[0013] Furthermore, the low-rank and local gradient product are introduced as regularization terms to construct the model. This model consists of two parts, namely the low-rank model and the local gradient product regularization term. The low-rank model usually consists of two parts. The first part is the low-rank representation of the noise-free image X, and the second part is the related representation of the sparse term S. The common low-rank model applied to hyperspectral image denoising is Robust Principal Component Analysis (RPCA), which can accurately separate the LR variable X and the sparse variable S by solving a very convenient convex optimization. The most common form is to minimize the combination of the nuclear norm and the l1 norm, and the formula is as follows:

[0014]

[0015] Among them, represents the nuclear norm of matrix X, that is, the sum of the singular values of X, ||S||1 := ∑ i,j |S i,j | represents the l1 norm of S.

[0016] The low-rank model of the present invention has made improvements to both of these variables. In the LR variable X, the γ norm is used instead of the nuclear norm for rank approximation. This norm overcomes the unbalanced penalty of different singular values in the nuclear norm and can better approximate the rank of the hyperspectral image. The formula is as follows:

[0017]

[0018] Among them, σ i (X) represents the singular value of matrix X, and γ is a variable.

[0019] In the process of characterizing the hyperspectral image noise, in order to better describe the row sparsity of the hyperspectral image noise, the l 1，1 norm is introduced. Therefore, combined with the above improvement of the LR variable X, a further low-rank model is obtained, and the formula is as follows:

[0020]

[0021] Among them, x i represents the i-th column of X.

[0022] On this basis, the difference operator ▽X is introduced. This is a linear operator with an approximate full rank. Such operators are obtained by performing difference operations on the original data and will also inherit the low-rank characteristics of the original data. The formula is as follows:

[0023] rank(G i ) = rank(X), (i = 1, 2, 3).

[0024] Among them, rank represents the rank of the matrix.

[0025] After combining the introduced difference operator with the above low-rank model, a complete low-rank model is formed. At the same time, it is combined with the previously established degradation model, and Gaussian noise N is added to obtain the final low-rank model. The formula is as follows:

[0026]

[0027] Among them, δ is a constant related to Gaussian noise M, G i represents the gradient of each dimension, and S represents sparse noise.

[0028] Furthermore, after introducing the low-rank model, in order to describe the local characteristics of hyperspectral images, the non-local gradient product (LGP) is introduced as a regularization term. LGP can show that a clear image has a clearer continuous boundary than a noisy image. The continuous boundary of a clear image has a larger image gradient, and at the same time, the direction of the image gradient is more consistent. In addition, the inner product between these gradients is also a large value. The above characteristics are conducive to helping distinguish clear images from images contaminated by noise. Therefore, it is represented as a regularization term in hyperspectral images. The formula is as follows:

[0029]

[0030] Among them, and are the gradients of the three dimensions of the hyperspectral image respectively, X is the image vector of the corresponding hyperspectral image. P is a corresponding matrix of size m×n×p, and · represents the dot product.

[0031] At the same time, it is worth noting that most of the LGP terms of images are mainly positive, and a small number of negative values are all close to 0. Therefore, approximating the LGP term by the l1 norm can make mathematical operations easier. The formula is as follows:

[0032]

[0033] Among them, and They are the gradients of the three dimensions of the hyperspectral image respectively. X is the image vector of the corresponding hyperspectral image. P is a corresponding matrix of size m×n×p, and · represents the dot product, and tr represents the trace of the matrix.

[0034] Then, the regularization term LGP is combined with the previous low-rank model to obtain a complete hyperspectral image denoising model, and the formula is as follows:

[0035]

[0036] Among them, G i represents the gradients of each dimension, S represents the sparse noise, and X represents the clear image.

[0037] Furthermore, based on this optimization model, the alternating direction multiplier method is used to solve the model, and the formula is as follows:

[0038]

[0039] Among them, G i represents the gradients of each dimension, S represents the sparse noise, X represents the clear image, λ1 and λ2 are parameters for balancing each regularization term, Γ i (i = 1, 2, 3, 4) are Lagrange multipliers, μ is the penalty parameter, and <,> is the inner product of two matrices.

[0040] Then, the above optimization problem is solved by fixing other parameters and transforming the characteristic parameters into the following sub-problems, and the formula is as follows:

[0041]

[0042] Among them, G i represents the gradients of each dimension, S represents the sparse noise, X represents the clear image, LGP is the regularization term of local gradient product, Γ i (i = 1, 2, 3, 4) are Lagrange multipliers, and μ is the penalty parameter.

[0043] Furthermore, the sub-problems are analyzed for each variable to solve the sub-problems. In the calculation of G, the objective function is a combination of a concave function and a convex function, and this internal structure can be solved by convex difference (DC) programming. The DC algorithm decomposes a non-convex function into the difference of the sum of two convex functions. First, we perform a singular value decomposition (SVD) on to obtain Udiag{σ A}V T , and then through the DC algorithm, we obtain a closed-form solution, and the formula is as follows:

[0044]

[0045] Among them, is the gradient of f(·) at σ k . Through multiple iterations, it will converge to a local optimal point σ * , and finally obtain the solution for updating G, with the formula as follows:

[0046] G t+1 = U diag{σ *} V T

[0047] where diag is the diagonal of the matrix.

[0048] In the calculation of S, it can be obtained through the soft thresholding operator, with the formula as follows:

[0049]

[0050] By solving S through the above method, the solution of S in the above formula can be obtained, with the formula as follows:

[0051]

[0052] In the calculation of X, fix G and S and combine the above formulas to obtain the simplest formula for solving X, with the formula as follows:

[0053]

[0054] First, take the derivative of the regularization term LGP, and its derivative can be expressed as:

[0055]

[0056] For the convenience of subsequent description, represent this derivative formula as:

[0057]

[0058] Then, solve for the variable X, which can be regarded as solving the following linear system, with the formula as follows:

[0059]

[0060] where represents the transpose operator of

[0061] Finally, diagonalize the above formula using the three-dimensional fast Fourier transform (FFT) matrix. Specifically, perform Fourier transforms on both sides of the equation and utilize the convolution theorem to obtain the closed-form solution of X t+1 , with the formula as follows:

[0062]

[0063] After all variables are updated, the Lagrange multipliers are also updated, and the formula is as follows:

[0064]

[0065] where ρ is a constant.

[0066] Furthermore, the iteration stop conditions are set as follows:

[0067]

[0068] where ε1 and ε2 are very small constants. After the iteration conditions are met, the updated variable X is reconverted from a two-dimensional matrix to three-dimensional data and output. At this time, the obtained X is the image after denoising the hyperspectral image.

[0069] The beneficial effects of the present invention are as follows:

[0070] The hyperspectral image denoising algorithm based on global variation and local gradient product proposed by the present invention combines the low-rank structure with the local gradient product, which helps to extract edge information and texture features in the hyperspectral image, enhances the ability to approximate the rank of the clean image, and significantly improves the denoising ability. The present invention proposes a new type of LR model, which approximates the LR property of global variation through the γ-norm and combines it with the l 1，1 norm of the sparse term, significantly improving the ability of the low-rank model to approximate the rank. The present invention designs a new local structure information of the hyperspectral image, which can capture local structure information. Experimental results on three hyperspectral image datasets show that the proposed model is highly effective and capable of removing different types of mixed noise, and is superior to current advanced hyperspectral denoising methods.

[0071] Other advantages, objectives, and features of the present invention will be described to some extent in the subsequent specification, and to some extent, will be obvious to those skilled in the art based on the study of the following text, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be achieved and obtained through the following specification. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] In order to make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be described in detail preferably with reference to the accompanying drawings, where:

[0073] Figure 1 is the flowchart of the method of the present invention;

[0074] Figure 2 is the algorithm diagram of the present invention;

[0075] Figure 3 is the schematic diagram of the principle of the method of the present invention;

[0076] Figure 4 Visualization results of different methods on the Pavia City hyperspectral image dataset, where (a)Original, (b)Noisy, (c)SMDS-Net, (d)MAC-Net, (e)LRMR, (f)LRTV, (g)LLRSSTV, (h)E3DTV, (i)NGMeet, (j)NLSSR, (k)RCTV, (l)ours. Detailed implementation manners

[0077] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings.

[0078] Figure 1 This is the flowchart of the method of the present invention. The present invention provides a hyperspectral image denoising method based on global variational and local gradient product. As shown in the figure, a three-dimensional hyperspectral image is obtained as a dataset in the image acquisition stage. The model framework for hyperspectral image denoising is as Figure 3 shown. It combines the low-rank model and the local structure regularization term of the hyperspectral image, can extract the global information and local information of the hyperspectral image, and has a good effect on retaining edge information and texture features. The model framework consists of two parts: the low-rank model and the local gradient product. The first part of the low-rank model is the low-rank representation of the noise-free image X, and the second part is the related representation of the sparse term S. First, the γ-norm is used to replace the nuclear norm for rank approximation. This norm overcomes the unbalanced penalty of different singular values in the nuclear norm and can better approximate the rank of the hyperspectral image. Then, in order to better describe the row sparsity of the hyperspectral image noise, the l 1，1 norm is introduced. The local gradient product designed by the present invention is used to characterize the local features of the hyperspectral image. The LGP can show that the clear image has a clearer continuous boundary than the noisy image, which is beneficial to help distinguish the clear image and the image contaminated by noise. Specifically, the technical solution of the present invention includes the following contents:

[0079] 1. Data acquisition: A drone platform equipped with a hyperspectral camera is used to collect hyperspectral remote sensing images, and then the collected hyperspectral images are cropped and divided.

[0080] 2. Data preprocessing: First, the obtained hyperspectral images are normalized to eliminate the influence of data dimensions and make the data indicators comparable; then the three-dimensional data model of the hyperspectral image is converted into a two-dimensional matrix, so that each column of the two-dimensional matrix corresponds to the spectral information of a band of the hyperspectral image.

[0081] 3. Initialize the required parameters, such as Figure 2As shown below. First, a hyperspectral image needs to be input. Then, two penalty parameters are included in this algorithm, which are negative values respectively; and parameters ε1 and ε2 for setting the iteration stop are set. At the same time, to prevent the algorithm from iterating indefinitely, the maximum iteration value μ is set. max And its iteration count is set to 0. Finally, parameter X is set to a random matrix of size (mn, p), and parameters S and the Lagrange multipliers are set to matrices of all zeros.

[0082] 4. Input the processed image into the algorithm. This algorithm includes a low-rank model and a local gradient product. After combining these two, the overall model of the present invention is obtained, as Figure 3 shown below. The formula is as follows:

[0083]

[0084] where λ1 and λ2 are parameters for balancing each regularization term, Γ i (i = 1, 2, 3, 4) are Lagrange multipliers, μ is a penalty parameter, and <,> is the inner product of two matrices.

[0085] 5. Perform variable updates in the model. This model needs to update variables G, X, S, and the Lagrange multiplier Γ i . The optimization problem is solved by fixing other parameters and transforming the characteristic parameters into the following sub-problems. The formula is as follows:

[0086]

[0087]

[0088] where G i represents the gradient in each dimension, S represents the sparse noise, X represents the clear image, LGP is the regularization term of the local gradient product, Γ i (i = 1, 2, 3, 4) are Lagrange multipliers, and μ is a penalty parameter.

[0089] In calculating G, the objective function is a combination of a concave function and a convex function. This internal structure can be solved by convex difference (DC) programming. The DC algorithm decomposes a non-convex function into the difference of the sum of two convex functions. First, we perform a singular value decomposition (SVD) on to obtain Udiag{σ A}V T , and then through the DC algorithm, we obtain a closed-form solution. The formula is as follows:

[0090]

[0091] where is f(·) at σ kThe gradient at converges to a local optimum σ through multiple iterations. * , and finally the solution for updating G is obtained, with the formula as follows:

[0092] G t+1 = U diag{σ *}V T

[0093] where diag is the diagonal of the matrix.

[0094] In the calculation of S, it can be obtained through the soft-thresholding operator, with the formula as follows:

[0095]

[0096] By solving S through the above method, the solution of S in the above formula can be obtained, with the formula as follows:

[0097]

[0098] In the calculation of X, fix G and S and combine the above formulas to obtain the simplest formula for solving X, with the formula as follows:

[0099]

[0100] First, take the derivative of the regularization term LGP, and its derivative can be expressed as:

[0101]

[0102] For the convenience of subsequent description, represent this derivative formula as:

[0103]

[0104] Then, solve for the variable X, which can be regarded as solving the following linear system, with the formula as follows:

[0105]

[0106] where, represents the transpose operator of.

[0107] Finally, diagonalize the above formula using the three-dimensional fast Fourier transform (FFT) matrix. Specifically, perform Fourier transforms on both sides of the equation and use the convolution theorem to obtain the closed-form solution of X t+1 , with the formula as follows:

[0108]

[0109] After all variables are updated, the Lagrange multiplier is also updated, with the formula as follows:

[0110]

[0111] Among them, ρ is a constant.

[0112] 6. Determine whether the iteration can stop. The iteration stop condition is set as follows:

[0113]

[0114] Among them, ε1 and ε2 are very small constants. When the iteration condition is not satisfied, return to the updated variable stage to continue the update. When the iteration condition is satisfied, the updated variable X is reconverted from a two-dimensional matrix to three-dimensional data and output. At this time, the obtained X is the image after denoising the hyperspectral image.

[0115] Such as Figure 4 is the experimental result of the denoising model described in the present invention on an open-source hyperspectral image dataset Pavia City. It can be seen that the noise in this dataset is well removed. The denoising effect of the present invention can be further illustrated by comparative experiments. The method of the present invention and other existing methods MAC-Net, LRMR, LRTV, E3DTV, NGMeet, NLSSR, RCTV are compared on the hyperspectral image dataset, and the peak signal-to-noise ratio (PSNR), structural similarity (SSIM) and spectral angle (SAM) are calculated respectively. Among them, the larger the PSNR and SSIM, the smaller the relative distortion and the higher the quality of the hyperspectral image; the smaller the SAM, the higher the quality of the hyperspectral image. Table 1 shows the denoising results of different methods in the Pavia City dataset:

[0116] Table 1 Comparison of the proposed method with various methods on the Pavia City hyperspectral image dataset

[0117]

[0118] The method of the present invention achieves the best accuracy on this dataset. Figure 4 The visualization results of several hyperspectral image denoising methods on a dataset are given. It can be seen that the performance of the method described in the present invention is better than other hyperspectral image denoising methods. The method proposed by the present invention can finely restore the details of the image and remove various different mixed noises.

[0119] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified without departing from the purpose and scope of the present technical solution, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A hyperspectral image denoising method based on global variation and local gradient product, characterized by: The method comprises the following steps: S1: Get the hyperspectral image dataset and crop it; S2: preprocessing the hyperspectral image as a dataset; S3: A degradation model is established for the processed hyperspectral image data, low rank and local gradient product are introduced for modeling, and the optimization problem is solved by the alternating direction multiplier method (ADMM) after parameter initialization; S4: After iterative optimization, the denoised hyperspectral image is obtained and denoising is completed.

2. The hyperspectral image denoising method based on global variation and local gradient product according to claim 1, characterized in that: In step S1, a UAV platform equipped with a hyperspectral camera is used to collect hyperspectral remote sensing images, and then the collected hyperspectral images are cropped and divided.

3. The hyperspectral image denoising method based on global variation and local gradient product according to claim 2, characterized in that: In step S2, the hyperspectral image as a data set is preprocessed. First, the obtained hyperspectral image is normalized to eliminate the influence of the data dimension and make the data indicators comparable; then the three-dimensional data model of the hyperspectral image is converted into a two-dimensional matrix, so that each column of the two-dimensional matrix corresponds to the spectral information of a band of the hyperspectral image.

4. The hyperspectral image denoising method based on global variation and local gradient product according to claim 3 is characterized by: In step S3, the preprocessed hyperspectral data is modeled and a model is obtained through its two-dimensional matrix modeling. The formula is as follows: Y=X+S+N Among them, Y represents the hyperspectral image with noise, X is the final clean hyperspectral image, S and N represent sparse noise and Gaussian noise respectively, and the sizes of Y, X, S and N are all mn×p, where m and n are the sizes of the spatial dimensions of the hyperspectral image, and p is the number of bands of the hyperspectral image.

5. The hyperspectral image denoising method based on global variation and local gradient product according to claim 4, characterized in that: The low-rank model is introduced. The low-rank model usually consists of two parts, where the first part is the low-rank representation of the noise-free image X, and the second part is the related representation of the sparse term S. The formula is as follows: st||YXS||≤δ G i =▽ i (X),i=1,2,3. Where δ is a constant related to the Gaussian noise M, G i represents the gradient of each dimension, and S represents sparse noise.

6. The method for denoising hyperspectral images based on global variation and local gradient product according to claim 5, characterized in that: After the low-rank model is introduced, in order to describe the local characteristics of the hyperspectral image, the non-local gradient product (LGP) is introduced as a regularization term, and the formula is as follows: Among them, , and are the gradients of the three dimensions of the hyperspectral image, respectively, X is the image vector of the corresponding hyperspectral image, P is a corresponding matrix of size m×n×p, and · is represented as the dot product; At the same time, it is worth noting that the LGP items of most images are mainly positive values, and a small number of negative values ​​are also close to 0. Therefore, approximating the LGP item by the norm can make mathematical operations easier. The formula is as follows: Among them, , and are the gradients of the three dimensions of the hyperspectral image, respectively, and X is the image vector of the corresponding hyperspectral image. P is a corresponding matrix of size m×n×p, and · is represented as the dot product, and tr represents the trace of the matrix. Then, the LGP regularization term is combined with the previous low-rank model to obtain a complete hyperspectral image denoising model, as shown in the following formula: Among them, represents the gradient of each dimension, represents sparse noise, and represents a clear image.

7. The method for denoising a hyperspectral image based on global variation and local gradient product according to claim 6, characterized in that: Based on the optimization model, the alternating direction multiplier method is used to solve the model. The formula is as follows: Among them, represents the gradient of each dimension, represents sparse noise, represents a clear image, and are parameters that balance each regularization term, is the Lagrange multiplier, the penalty parameter of, and is the inner product of the two matrices; Then, the above optimization problem is solved by fixing other parameters and transforming the characteristic parameters into the following sub-problems, as follows: Among them, G i represents the gradient of each dimension, S represents sparse noise, X represents a clear image, LGP is the regularization term of the local gradient product, Γ i (i=1,2,3,4) is the Lagrange multiplier and μ is the penalty parameter.

8. The method for denoising hyperspectral images based on global variation and local gradient product according to claim 7, characterized in that: Analyze the subproblems for each variable to solve the subproblems. In the calculation of G, the objective function is a combination of concave and convex functions. This inherent structure can be solved by convex difference (DC) programming. The DC algorithm decomposes a non-convex function into the difference of the sum of two convex functions. First, we Perform singular value decomposition (SVD) and obtain Udiag{σ A }V T , and then we get a closed-form solution through the DC algorithm, the formula is as follows: in, is f(·) in σ k The gradient at, through multiple iterations, it will converge to a local optimal point σ * , and finally get the solution to update G, the formula is as follows: G t+1 =Udiag{σ * }V T Where diag is the diagonal of the matrix. In the calculation of S, it can be obtained by the soft threshold shrinkage operator, the formula is as follows: By solving S using the above method, we can get the solution of the above formula S, which is as follows: In the calculation of X, fix G and S and combine the above formulas to obtain the simplest formula for solving X, which is as follows: First, the regularization term LGP is derived, and its derivative can be expressed as: For the convenience of subsequent description, the derivative formula is expressed as: Then, solving for the variable X can be considered as solving the following linear system, with the following formula: Among them, T represents the transpose operator of ▽; Finally, the above formula is diagonalized using the three-dimensional fast Fourier transform (FFT) matrix. Specifically, Fourier transform is performed on both sides of the equation, and the convolution theorem is used to obtain X t+1 The closed-form solution of is as follows: After all variables are updated, the Lagrange multiplier is also updated. The formula is as follows: Here, ρ is a constant.

9. The method for denoising a hyperspectral image based on global variation and local gradient product according to claim 8, characterized in that: The iteration stop condition is set as follows: Among them, ε1 and ε2 are constants with very small values. When the iteration condition is met, the updated variable X is converted from a two-dimensional matrix into three-dimensional data and output. At this time, the obtained X is the image after the hyperspectral image is denoised.

Images