Satellite image inpainting method based on two-fold hankel matrix of gaussian filter

Abstract

The application discloses a Gauss-filtering-based two-fold Hankel matrix method for satellite image inpainting, comprising the following steps: 1. after an image is acquired, performing a sliding block Gauss filtering on the image, and constructing a two-fold Hankel matrix of the image by using a reduction filter convolution relationship; 2. establishing a two-fold Hankel matrix low-rank constraint by using a Geman-based kernel norm weighting scheme to construct an image inpainting model; and 3. on the basis of the two-fold Hankel matrix low-rank constraint image inpainting model, using an ADMM method-based inpainting algorithm to solve. The method directly performs a Gauss filtering on the image to replace time-consuming image block segmentation, uses the Geman-based kernel norm weighting scheme to perform a two-fold Hankel matrix low-rank constraint, so that the image inpainting time of the method is obviously superior to that of a two-fold Hankel matrix low-rank inpainting method based on image blocks, and the method has the characteristics of good visual effect and high inpainting precision.

Description

Technical Field

[0001] This invention relates to the field of image restoration technology, specifically to a method for restoring satellite images based on a two-fold Hankel matrix using Gaussian filtering. Background Technology

[0002] In modern national defense informationization, satellite imagery is a crucial medium for transmitting various types of information. However, satellite images are susceptible to damage during the imaging process due to hardware malfunctions, laser irradiation, and abnormal electromagnetic radiation from space. Typical damaged satellite images exhibit features such as random missing lines and striped damage. Electromagnetic pulse noise interference is unavoidable during image transmission, storage, and at the receiving end. In civilian applications, satellite remote sensing images may be obscured by artificial advertising characters. To effectively improve the information utilization rate of contaminated / damaged satellite remote sensing images, restoration is necessary. Satellite image restoration is an ill-posed problem due to the lack of a proper reference standard. Traditional restoration methods, such as filtering, interpolation, and optimization methods using prior information regularization constraints, have potential for further improvement in restoration effectiveness. In recent years, popular deep learning neural network frameworks have also been applied to image restoration, but their generalization ability is poor due to limitations in training datasets.

[0003] Visible light satellite images generally exhibit local smoothness, so their representation matrices typically possess significant low-rank properties. Low-rank prior information demonstrates excellent performance in signal optimization processing fields such as image denoising, restoration, reconstruction, and deblurring. Existing image restoration methods based on low-rank matrices primarily utilize low-rank prior information in the following ways: inherent low-rank properties of the signal itself, such as the inherent low-rank properties of the matrix, local patch information similarity, and frame-to-frame similarity in video; and low-rank properties of the structural matrix based on the signal transform domain, such as using the reduction filter relationship. If the signal is smoothed in the time domain and corresponds to sparseness in the Fourier domain, a Hankel-type structural low-rank matrix can be constructed in the Fourier domain. In addition to low-rank properties, early impulse interference image restoration assumed that the image has sparse representations in different domains, such as the difference domain and wavelet domain, and then constrained the sparsity prior information to restore the image from the noisy data. Since low-rank and sparsity each play irreplaceable roles in constrained optimization problems, image restoration schemes that combine constrained sparsity and low-rank prior information are also being proposed.

[0004] The paper “Kyong H., Ye J.. Annihilating filter-based low-rank Hankel matrix approach for image in painting[J]. IEEE Transactions on Image Processing, 2015, 24(11): 3498-511” proposes a patch-based low-rank Hankel matrix inpainting method. The method divides the image into multiple patches and uses a uniform regularization parameter to constrain the low-rank regularization term. However, the inpainting effect of different patches does not reach their respective optimal solutions, and there is also a time consumption problem.

[0005] Therefore, there is an urgent need to design a satellite image restoration method based on Gaussian filtering and a two-fold Hankel matrix to solve the problems existing in the above-mentioned technologies. Summary of the Invention

[0006] To address the aforementioned problems, this invention aims to provide a satellite image restoration method based on Gaussian filtering and a two-fold Hankel matrix. This method directly applies Gaussian filtering to the image instead of time-consuming image segmentation, and uses a Geman-based nuclear norm weighting scheme to constrain the low-rank property of the two-fold Hankel matrix. This makes the image restoration time significantly faster than image segmentation-based two-fold Hankel matrix low-rank restoration methods, and it features better visual restoration effects and higher restoration accuracy.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] A method for satellite image inpainting based on Gaussian filtering and the Hankel matrix, including

[0009] Step 1. After acquiring the image, perform a sliding Gaussian filter on the image to construct the image's two-fold Hankel matrix;

[0010] Step 2. Based on the bifold Hankel matrix, use the Geman-based nuclear norm weighting scheme to establish low-rank constraints on the bifold Hankel matrix and construct an image restoration model;

[0011] Step 3. Based on the low-rank constraint of the two-fold Hankel matrix, the image restoration model is solved using an restoration algorithm based on the ADMM method.

[0012] Preferably, the process of constructing the bifold Hankel matrix of the image in step 1 includes:

[0013] Step 1.1 After acquiring the satellite image, select a Gaussian slider of size w×w (set w=9 based on experience or select an appropriate w value based on the actual situation, generally 6<w<20) to preprocess the image items to be repaired;

[0014] Step 1.2 Constructs a two-fold Hankel matrix for the preprocessed, smoother image.

[0015] (1) According to the characteristics of satellite images: Generally, an image X has smoothness in the time domain and sparsity in the frequency domain. Therefore, there exists a reduction filter T such that F(X)·F(T)=0, where F(·) represents performing a Fourier transform on it.

[0016] (2) According to the frequency domain multiplication corresponding to the time domain convolution, there exists X*T = 0, where * indicates convolution calculation. The process of calculating convolution X*T is equivalent to multiplying the two-dimensional convolution matrix composed of the pixel values ​​of image X with the one-dimensional vector composed of T. Mathematically, truncating the beginning and end of this convolution matrix yields a low-rank bifold Hankel matrix HX, where H represents the code bifold Hankel matrix operation operator;

[0017] Preferably, the process of constructing the image inpainting model by constraining the low rank of the two-fold Hankel matrix in step 2 includes:

[0018] Step 2.1 After obtaining the bifold Hankel matrix, construct a satellite image inpainting model based on the low-rank constraint term of the bifold Hankel matrix using Gaussian filtering.

[0019]

[0020] Where X represents the image to be solved, X * Let Y represent the optimal solution image of the model, Y represent the image to be repaired, and Ω represent the set of undamaged pixel locations. H represents the Gaussian filtering operator, and H represents the code-fold Hankel matrix operator.

[0021] Step 2.2 Use the Genman weighted nuclear norm to convexly approximate the number of non-zero singular values; introduce it into model (2) to obtain:

[0022]

[0023] in, Representation matrix The i-th singular value, let the matrix be... The size is m×n, l=min(m,n), i=1,2,...,l, g(·) represents the expression for singular values ​​δ i A function that performs Geman weighting. γ is the weighted balance coefficient, γ > 0;

[0024] Preferably, step 3, the process of solving using the ADMM-based repair algorithm, includes...

[0025] Step 3.1 First, introduce auxiliary variables. Model (3) becomes,

[0026]

[0027] Among them, Z * as auxiliary variables Z The optimal solution, μ is the parameter of the introduced regularization constraint term and μ > 0, and L is the introduced residual variable.

[0028] Step 3.2 Solve the unknown variables X, Z, L in model (4) alternately and iteratively, solving them sequentially in the t-th iteration:

[0029]

[0030]

[0031]

[0032] Among them, X t-1 Z t-1 ,L t-1 This represents the value at the (t-1)th iteration, initialized with... Z 0 and L 0 It is an all-zero matrix.

[0033] Preferably, the Z t The solution process includes

[0034] (1) When t = 1: Solve for Z 1 In the middle, order

[0035] (2) When t≠1: First, for Performing singular value decomposition on the matrix yields... Where S represents a diagonal matrix with singular values ​​on its diagonal;

[0036] Then let S i express Let S be the i-th singular value of the matrix. newi =g(S i ),in

[0037] Finally, S newi By inserting the elements into the diagonal positions of the diagonal matrix, we obtain a new diagonal matrix S. new Zt =U*S*V H .

[0038] Preferably, the X t The solution process includes

[0039] X t =H -1 (Z t -L t-1 ),

[0040]

[0041] Among them, H -1 This represents the inverse operator of the code bifold Hankel matrix.

[0042] The beneficial effects of this invention are: This invention discloses a satellite image restoration method based on Gaussian filtering and a two-fold Hankel matrix. Compared with the prior art, the improvement of this invention lies in:

[0043] This invention proposes a satellite image inpainting method based on a Gaussian filter and a bifold Hankel matrix. The method first applies a sliding Gaussian filter to the image, then constructs the bifold Hankel matrix. A Geman-based nuclear norm weighting scheme is used to constrain the low-rank property of the bifold Hankel matrix, constructing an image inpainting model. Finally, an algorithm based on the ADMM method is used to solve the problem. In the solution process, this method directly applies Gaussian filtering to the image instead of time-consuming patch segmentation, resulting in a significantly faster image inpainting time compared to patch-based bifold Hankel matrix low-rank inpainting methods. Furthermore, the use of a Geman-based nuclear norm weighting scheme for low-rank constraint of the bifold Hankel matrix leads to good visual restoration effects and high restoration accuracy. Experimental results show that the proposed method can improve the visual effect and numerical indicators of satellite image inpainting under conditions such as random missing rows (stripes), electromagnetic pulse interference, and advertising character coverage. It can provide high-precision images for applications such as satellite image multi-source fusion and satellite image target detection and recognition, demonstrating the advantages of good restoration effect and high restoration accuracy. Attached Figure Description

[0044] Figure 1 This is a flowchart of the satellite image restoration method based on Gaussian filtering using a two-fold Hankel matrix, as described in this invention.

[0045] Figure 2 This is a diagram showing the smoothness and frequency domain sparsity of satellite images in this invention.

[0046] Figure 3 This is a graph of the function g(·) of this invention.

[0047] Figure 4 The images shown are the original diagram of Embodiment 2 of the present invention and the corresponding diagrams to be repaired under three different damage conditions.

[0048] Figure 5 This is a comparison of the repair effects when there are random missing rows (stripes) of damage in Embodiment 2 of the present invention.

[0049] Figure 6 This is a comparison chart of the repair effect under pulse interference in Embodiment 2 of the present invention.

[0050] Figure 7 This is a graph showing the convergence of algorithm error in Embodiment 2 of the present invention.

[0051] Among them: Figure 2 In the image, Figure (a) shows an original RGB satellite image used to demonstrate smoothness features, and Figure (b) shows... Figure 2 Figure (a) shows an R-channel plot, and Figure (c) shows the R-channel plot. Figure 2 (b) After performing a two-dimensional FFT transformation, a frequency domain map is obtained, which is used to show the sparsity characteristics in the frequency domain;

[0052] exist Figure 4 In the middle, Figure (a) shows the original Figure 1 Figure (b) shows the original Figure 2 Figure (c) shows the original Figure 3 Figure (d) shows the original Figure 1 The image to be repaired, Figure (e) shows the original image. Figure 2 The image to be repaired, Figure (f) shows the original image. Figure 3 The image to be repaired;

[0053] exist Figure 5 In the image, Figure (a) shows the image repaired using Gaussian filtering when there are random missing lines (stripes); Figure (b) shows the image repaired using TV method when there are random missing lines (stripes); Figure (c) shows the image repaired using Hankel patch bifolding method when there are random missing lines (stripes); and Figure (d) shows the image repaired using extraction method when there are random missing lines (stripes).

[0054] exist Figure 6 In the figure, Figure (a) shows the image restored by Gaussian filtering under impulse interference, Figure (b) shows the image restored by TV method under impulse interference, Figure (c) shows the image restored by Hankel patch bifold under impulse interference, and Figure (d) shows the image restored by extraction method under impulse interference.

[0055] exist Figure 7 In the diagram, the vertical axis represents the RLNE value at the current iteration, and the horizontal axis represents the number of iterations. Detailed Implementation

[0056] To enable those skilled in the art to better understand the technical solutions of the present invention, the technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0057] Example 1: Refer to Appendix Figure 1-7 The method shown is a two-fold Hankel matrix satellite image inpainting method based on Gaussian filtering, including...

[0058] Step 1. After acquiring the image, first apply a sliding Gaussian filter to the image to construct the image's two-fold Hankel matrix.

[0059] Various defects or omissions encountered during satellite imaging or transmission can be viewed as high-frequency components in the frequency domain, such as electromagnetic pulse interference and random missing lines. Gaussian sliding block filtering can smooth the image and suppress local abrupt pixel changes caused by defects.

[0060] Step 1.1 After acquiring the satellite image, select a Gaussian slider with a size of 9*9 to preprocess the image item to be repaired;

[0061] Step 1.2 Then, a two-fold Hankel matrix is ​​constructed on the preprocessed, smoother image. Due to the subtraction filter relationship, this two-fold Hankel matrix has low-rank prior features. The construction process of the two-fold Hankel matrix includes:

[0062] like Figure 2 As shown: (1) For RGB satellites Figure 2 One of the R channels in (a) is as follows Figure 2 (b) The frequency domain is obtained after performing a two-dimensional FFT transformation. Figure 2 (c) According to the principle of the reduction filter, if the two-dimensional image X is sparse in the frequency domain, then there exists a reduction filter T such that F(X)·F(T)=0, where F(·) represents performing a Fourier transform on it; According to the convolution theorem: frequency domain multiplication corresponds to time domain convolution, then there exists X*T=0, where * represents convolution calculation. If X is a two-dimensional matrix composed of images, then a two-fold Hankel matrix HX can be obtained according to the convolution calculation process, where H represents the code two-fold Hankel matrix operation operator;

[0063] Step 2. Construct an image inpainting model by using a Geman-based nuclear norm weighting scheme to establish low-rank constraints on the bifold Hankel matrix.

[0064] Step 2.1 After obtaining the bifold Hankel matrix, construct a satellite image inpainting model based on the low-rank constraint term of the bifold Hankel matrix using Gaussian filtering.

[0065]

[0066] Where x represents the image to be solved, X * Let Y represent the optimal solution image of the model, Y represent the image to be repaired, and Ω represent the set of undamaged pixel locations. H represents the Gaussian filtering operator, and H represents the code-fold Hankel matrix operator.

[0067] Step 2.2 The low-rank property of a matrix is ​​generally constrained by minimizing the number of non-singular values ​​of the matrix, i.e., the zero norm of the singular values. To facilitate the solution, the Genman weighted nuclear norm is used to convexly approximate the number of non-zero singular values; it is introduced into model (2) to obtain:

[0068]

[0069] in, Representation matrix The i-th singular value, assuming the matrix If the size is m×n, then l=min(m,n), i=1,2,...,l, and g(·) represents the expression for singular values ​​δ. i A function that performs Geman weighting. γ>0; when g(·) is a singular value zero-norm function, g(δ i )=#δ i , where # indicates that the number δ following it is calculated. i Is it zero? If δ i If it is non-zero, then denote it as #δ i =1; when g(·) is a singular value nuclear norm function, g(δ) i )=|δ i |, that is, directly take δ i The value; where # indicates that the number δ following it is calculated. i Is it zero, if δ i Non-zero is represented by #δ i =1;

[0070] Suppose there are 100 singular values ​​ranging from 0 to 2, with equal intervals of 0.01. The curves of the g(·) function under the 0 norm, nuclear norm, and Geman-weighted nuclear norm for these singular values ​​are respectively as follows: Figure 3 As shown, the Geman weighted nuclear norm approximates the number of non-zero singular values ​​better than the nuclear norm.

[0071] Step 3. Solve using the ADMM-based repair algorithm.

[0072] Step 3.1 For the proposed repair model (3), the Alternating Direction Multiplier Method (ADMM) is used for algorithm design. First, auxiliary variables are introduced. Model (3) becomes,

[0073]

[0074] Among them, Z * For the optimal solution of auxiliary variable Z, μ is the parameter of the introduced regularization constraint term, and L is the introduced residual variable;

[0075] Step 3.2 Alternating Iterative Solution of Model (4) Unknown variables X, Z, L, solve the following problems sequentially in the t-th iteration:

[0076]

[0077]

[0078]

[0079] Among them, X t-1 Z t-1 ,L t-1 This represents the value at the (t-1)th iteration, initialized with... Z 0 and L 0 Given a matrix consisting entirely of zeros, solve for Z. t The specific process includes:

[0080] (1) When t = 1, that is, when solving Z 1 season

[0081] (2) When t≠1, firstly, for Performing singular value decomposition on the matrix yields... Where S represents a diagonal matrix with singular values ​​on its diagonal; then, let S i express Let S be the i-th singular value of the matrix. newi =g(Si), where Finally, S newi By inserting the elements into the diagonal positions of the diagonal matrix, we obtain a new diagonal matrix S. new Then Z t =U*S*V H ;

[0082] Solve for X t :

[0083] X t =H -1 (Z t -L t-1 ),

[0084] and

[0085] Among them, H- 1 This represents the inverse operator of the code bifold Hankel matrix.

[0086] Example 2: To verify the effectiveness of the method proposed in Example 1, experimental results and analysis are presented below to validate the above method.

[0087] In this embodiment, the satellite images are sourced from the publicly available dataset DOTA-v.2.0, and are provided by the China Resources Satellite Data and Application Center, satellite GF-1, satellite GF-2, etc.; the images are obtained using... Figure 4 The experimental results are presented using three original satellite images shown in (a), (b), and (c); the three images are damaged under conditions of random missing lines, electromagnetic pulse interference, and advertising text coverage, respectively. Figure 4 As shown in (d), (e), and (f);

[0088] The experimental repair effects of various methods were compared, namely: 1. Untreated; 2. Repair method based on ADMM solution under two-dimensional finite difference regularization constraint (TV method); 3. Patch bifold Hankel method; 4. Gaussian filtering method; 5. Interpolation method (mean interpolation); 6. Repair method based on ADMM solution under direct matrix low-rank constraint; 7. Bifold Hankel method based on Geman weighted nuclear norm under Gaussian filtering; where 7 is the method proposed in Embodiment 1 of this invention (hereinafter referred to as the proposed method);

[0089] In the above comparison methods, the parameters are set to their optimal values. In the method proposed in this embodiment, the Gaussian filter sliding window is set to a size of 9×9, the two-fold Hankel matrix is ​​set to a size of 9×9 for the reduction filter, the Geman weighting parameter γ is set to 6, the regularization parameter μ is set to 0.001, the maximum number of algorithm iterations is set to 50, and the algorithm convergence tolerance tol is set to 10. -6 ;

[0090] All simulation experiments in this embodiment are based on a Windows 10 system and MATLAB R2016a environment, with a computer featuring an Intel Core i7 CPU 3.6GHz and 32GB of RAM. The image restoration quality is evaluated using relative least normalized error (PSNR), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM) for numerical comparison. RLNE is defined as follows.

[0091]

[0092] Where χ * The image is represented by χ, which indicates the undamaged image.

[0093] Since its introduction in 2004, the SSIM image quality assessment metric has been widely used for image enhancement evaluation due to its excellent integration of human visual evaluation elements such as contrast and image structure; its definition is:

[0094]

[0095] Where μ 1，j and μ 2，j It is image X and The mean of the j-th local window, while σ 1，j and σ 2，j This is the corresponding standard deviation; σ 12，j Represents covariance; constants ∈1 and ∈2 are used to avoid instability. Generally, the larger the SSIM value, the better the repair quality, with a maximum value of 1.

[0096] The numerical comparisons of RLNE, PSNR, and SSIM evaluation indices after satellite image restoration using seven methods under conditions of random missing lines (stripes), electromagnetic pulse interference, and deliberate advertising character contamination are shown in Tables 1, 2, and 3, respectively. It can be seen that the method proposed in this embodiment has advantages over other methods. Compared with the Hankel method of tile bifolding, the average RLNE is improved by 0.94%, the average PSNR by 1.32 dB, and the average SSIM by 1.41%.

[0097] The computation time of the proposed method is compared with that of the tile bifold Hankel method, as shown in Table 4. It can be seen that the method proposed in Embodiment 1 of the present invention can greatly reduce the computation time based on the tile bifold Hankel method.

[0098] Figure 5 , 6 The paper presents a comparison of visual restoration methods using Gaussian filtering, TV method, Hankel method with tile bifolding, and the proposed method. It is evident that the proposed method in Embodiment 1 of this invention can improve the visual restoration effect compared with Gaussian filtering and TV method.

[0099] The bifold Hankel method based on Geman weighted nuclear norm under Gaussian filtering proposed in Embodiment 1 of this invention exhibits the following convergence properties: Figure 7 As shown, this demonstrates that the proposed algorithm can converge effectively;

[0100] Table 1: Comparison of RLNE, PSNR, and SSIM evaluation indices for seven remediation methods in cases of random, missing (stripe-like) contamination.

[0101]

[0102] The " / " indicates that there is no repair effect;

[0103] Table 2: Comparison of RLNE, PSNR, and SSIM evaluation indices under seven repair methods during impulse interference.

[0104]

[0105]

[0106] The " / " indicates that there is no repair effect;

[0107] Table 3: Comparison of RLNE, PSNR, and SSIM evaluation indicators under seven repair methods when there is human-caused advertising character pollution.

[0108]

[0109] The " / " indicates that there is no repair effect;

[0110] Table 4: Comparison of Calculation Time for Repair Methods

[0111]

[0112] In summary, the method proposed in Embodiment 1 of this invention can significantly reduce the time of the tile-based bifold Hankel method. Furthermore, due to the use of Gaussian filtering and weighting schemes, the visual repair effect and accuracy are also improved.

[0113] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for satellite image inpainting based on a Gaussian filter and a two-fold Hankel matrix, characterized in that: include Step 1. After acquiring the image, perform a sliding Gaussian filter on the image to construct the image's two-fold Hankel matrix; Step 2. Based on the bifold Hankel matrix, use the Geman-based nuclear norm weighting scheme to establish low-rank constraints on the bifold Hankel matrix and construct an image restoration model; Step 3. Based on the low-rank constraint of the two-fold Hankel matrix, the image restoration model is solved using an restoration algorithm based on the ADMM method; The process of constructing the bifold Hankel matrix of the image described in step 1 includes: Step 1.1 After acquiring the satellite image, select a size of... A Gaussian slider is used to preprocess the image to be repaired; Step 1.2 Constructs a two-fold Hankel matrix for the preprocessed, smoother image. (1) Based on the characteristics of satellite images: images If something exhibits smoothness in the time domain and sparsity in the frequency domain, then a reduction filter exists. , making ,in This indicates that a Fourier transform is performed on it; (2) According to the frequency domain multiplication corresponding to the time domain convolution, there exists , Represents convolution calculation; convolution The calculation process is equivalent to converting the image... Multiplying a two-dimensional convolution matrix composed of pixel values ​​by a one-dimensional vector T; pruning the beginning and end of this convolution matrix yields a low-rank, two-fold Hankel matrix. ,in Indicates the bifold Hankel matrix operation operator; Step 2 describes the process of constructing an image inpainting model using the low-rank constraint of the bifold Hankel matrix. Step 2.1 After obtaining the bifold Hankel matrix, construct a satellite image inpainting model based on the low-rank constraint term of the bifold Hankel matrix using Gaussian filtering. （2） in, The graph to be solved is represented. The image represents the optimal solution of the model. This indicates the image to be repaired. This represents the set of locations of undamaged pixels. This represents the Gaussian filtering operator. Indicates the bifold Hankel matrix operation operator; Step 2.2 Use the Genman weighted nuclear norm to convexly approximate the number of non-zero singular values; introduce it into model (2) to obtain: （3） in, Representation matrix The Given a matrix with singular values, let the matrix... Size is , , , Represents singular values A function that performs Geman weighting. , For weighted balance coefficients, ; Step 3 describes the process of solving the problem using the ADMM-based repair algorithm. Step 3.1 First, introduce auxiliary variables. Model (3) becomes, (4) in, as auxiliary variables The optimal solution. The parameters of the introduced regularization constraint term and , The introduced residual variable is... ; Step 3.2 Alternating iterative solution of unknown variables in model (4) Solve sequentially in the t-th iteration: （5） （6） ； in, This represents the value at the (t-1)th iteration, initialized with... , and It is a matrix consisting entirely of zeros.

2. The method for satellite image restoration based on Gaussian filtering and a two-fold Hankel matrix as described in claim 1, characterized in that: The aforementioned The solution process includes (1) Time: Solving In the middle, order ; (2) At the time: First of all Performing singular value decomposition on the matrix yields... ,in Represents a diagonal matrix whose diagonal values ​​are singular; Then let express The first of the matrix Let a singular value, ,in ; Finally Insert it into the diagonal position of the diagonal matrix to obtain a new diagonal matrix. ,get .

3. The method for satellite image restoration based on Gaussian filtering and a two-fold Hankel matrix as described in claim 1, characterized in that: The aforementioned The solution process includes ， ； in, This represents the inverse operator of the code bifold Hankel matrix.

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