Image restoration method based on Schatten-p norm decomposition robustness

Abstract

The invention belongs to the technical field of image processing, and particularly provides an image restoration method based on Schatten-p norm decomposition robustness, and the method comprises the steps: obtaining a to-be-restored image, and converting the to-be-restored image into an original image matrix; preprocessing the converted original image matrix; based on the preprocessed image matrix, decomposing the image matrix into low-rank factors by adopting a matrix decomposition technology; based on the low-rank factor and the original image matrix, constructing a loss function containing a data fitting item and a regularization item, estimating the difference between the image matrix reconstructed through the low-rank factor and the original image matrix on known pixels through the data fitting item, and punishing the complexity of the low-rank factor through the regularization item by adopting a Schatten-p norm; and minimizing the loss function, and determining the optimal image matrix estimation so as to repair the missing or damaged part of the original image matrix. An image is expressed as a low-rank matrix, and then the matrix is constrained by using a Schatten-p norm, so that robust restoration of the image is realized.

Description

Technical Field

[0001] The present invention belongs to the technical field of image processing, and particularly relates to a robust image inpainting method based on Schatten-p norm decomposition. Background Art

[0002] In the fields of image processing and computer vision, image inpainting is an important research direction, which involves restoring damaged or missing image data. Traditional image inpainting methods, such as Poisson image editing and dictionary learning, etc., although can repair images to a certain extent, often show low robustness when facing noise and outliers. In recent years, image inpainting methods based on matrix decomposition have received extensive attention due to their powerful recovery ability and computational efficiency.

[0003] Among them, as an effective regularization tool, the Schatten-p norm has been successfully applied in many machine learning and optimization problems. The Schatten-p norm can constrain the singular values of the matrix, so that the model can better handle noise and outliers during the training process.

[0004] Based on this idea, we propose a robust image inpainting method based on Schatten-p norm decomposition. Summary of the Invention

[0005] In order to overcome the deficiencies of the prior art, the present invention provides a robust image inpainting method based on Schatten-p norm decomposition to solve the problems in the prior art.

[0006] One embodiment of the present invention provides a robust image inpainting method based on Schatten-p norm decomposition, including the following steps:

[0007] a) Obtain the image to be inpainted, and convert the image to be inpainted into an original image matrix;

[0008] b) Preprocess the converted original image matrix to obtain a preprocessed image matrix;

[0009] c) Based on the preprocessed image matrix, use matrix decomposition technology to decompose the image matrix into low-rank factors;

[0010] d) Based on the low-rank factors and the original image matrix, construct a loss function including a data fitting term and a regularization term. Estimate the difference between the image matrix reconstructed by the low-rank factors and the original image matrix on the known pixels through the data fitting term, and use the Schatten-p norm in the regularization term to penalize the complexity of the low-rank factors;

[0011] e) Minimize the loss function to determine the optimal estimate of the image matrix for repairing the missing or damaged parts of the original image matrix.

[0012] In one embodiment, the formula in step d) is:

[0013] (1)

[0014] where M Ω represents the position indicator of the known pixels in the original image matrix, X Ω represents the part of the known pixels in the image matrix reconstructed by the low-rank factors, f p (U, V) is the regularization function of the low-rank factors U and V, λ is the regularization parameter, s.t. X = UV T is the constraint condition, which enforces that the final image matrix X must be reconstructed from the low-rank factors U and V.

[0015] In one embodiment, in the formula of step d), p = 1, 1 / 2, 2 / 3, so the formula of f p (U, V) is:

[0016] (1.1)

[0017] In one embodiment, the RMCS p NF with p = 1, 1 / 2, 2 / 3 is defined as RMCS 1 NF, RMCS 1 / 2 NF and RMCS 2 / 3 NF, and the corresponding forms are as follows:

[0018]

[0019] In one embodiment, in formula (1), the residual matrix E Ω is added, such that E Ω = M Ω - X Ω , and the penalty function is used to transform formula (1) into an unconstrained formula:

[0020] (2)

[0021] where, when μ → +∞, the solution of formula (2) converges to formula (1).

[0022] In one embodiment, after step e), the following steps are further included:

[0023] f) Alternating optimization:

[0024] f1) Initialize the low-rank factor U, the low-rank factor V, and the image matrix estimate X;

[0025] f2) Enter the alternating optimization loop:

[0026] Fix the image matrix estimate X and only update the low-rank factors U and V. By minimizing the parts of the loss function with respect to U and V, obtain the new U and V.

[0027] Fix the low-rank factors U and V and only update the image matrix estimate X. By minimizing the part of the loss function with respect to X, obtain the new X.

[0028] Repeat the alternating update of the low-rank factors U, V and the update of the image matrix estimate X until the stopping condition is satisfied.

[0029] In one embodiment, in the step f2), by minimizing the parts of the loss function with respect to U and V, obtain the new U and V, and by minimizing the part of the loss function with respect to X, obtain the new X. The specific formulas are as follows:

[0030]

[0031] where k represents the current iteration number, and k + 1 represents the next iteration; U k+1 , V k+1 respectively represent the updates of the low-rank factors at the (k + 1)-th iteration, and X k+1 represents the updated low-rank approximation result after the (k + 1)-th iteration, represents the new residual matrix after the (k + 1)-th iteration, and μ is the weight parameter;

[0032] where, for different values of f p (U, V), a closed-form solution in the optimization process can be obtained.

[0033] In one embodiment, after the step f), the following steps are further included:

[0034] g) Algorithm analysis: including computational complexity analysis and convergence analysis;

[0035] Among them, the computational complexity:

[0036] For each iteration, matrix inversion and matrix multiplication operations are adopted;

[0037] For three p values, the computational complexity of updating X k+1 is o(2mnd), and the computational complexities of updating U k+1 and V k+1 are as follows:

[0038] When p = 1, the computational complexities of updating U k+1 and V k+1 are respectively o(d3 +nd 2 +mnd) and o(d 3 +md 2 +mnd);

[0039] When p = 2 / 3, update U k+1 and V k+1 The computational complexities of 3 +nd 2 +mnd) and o(mnd+(m+n)d 2 +2d 3 );

[0040] When p = 1 / 2, update U k+1 and V k+1 The computational complexities of both are o(mnd+(m+n)d 2 +2d 3 );

[0041] Finally, the total computational complexity is the sum of the computational complexities of each step;

[0042] where m represents the number of rows of the matrix, n represents the number of columns of the matrix, and d represents the number of dimensions of the vector;

[0043] where, through convergence analysis, it is determined that is the stable point of the minimization problem.

[0044] The robust image inpainting method based on Schatten-p norm decomposition provided by the above embodiments has the following beneficial effects:

[0045] 1. By representing the image as a low-rank matrix and then using the Schatten-p norm to constrain the matrix, robust inpainting of the image is achieved. Through experiments, it is verified that the method of this application shows good performance in dealing with noise and outliers, and at the same time has high computational efficiency.

[0046] 2. In one of the embodiments, by adopting a robust matrix completion method based on Schatten-p norm decomposition with three p values, effective and efficient inpainting of the image can be achieved. At the same time, to solve the established optimization problem, this application considers a penalty technique to achieve an unconstrained minimization problem, and then provides a multivariable iterative process, computational complexity analysis, and convergence analysis. Moreover, since there is no singular value decomposition calculation, the iterative algorithm of this application can converge faster. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on the structures shown in these drawings.

[0048] Figure 1 The flowchart of a robust image inpainting method based on Schatten-p norm decomposition provided by an embodiment of the present invention;

[0049] Figure 2 The alternative optimization flowchart of a robust image inpainting method based on Schatten-p norm decomposition provided by an embodiment of the present invention. Detailed implementation manners

[0050] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0051] It should be noted that if there are directional indications (such as up, down, left, right, front, back...) involved in the embodiments of the present invention, the directional indications are only used to explain the relative positional relationship and movement conditions between components in a specific posture. If the specific posture changes, the directional indications will also change accordingly.

[0052] In addition, if there are descriptions such as "first" and "second" involved in the embodiments of the present invention, the descriptions of "first" and "second" are only for descriptive purposes and cannot be understood as indicating or implying their relative importance or implicitly indicating the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include at least one of such features. In addition, if "and / or" or "and / or" appears throughout the text, its meaning includes three parallel solutions. Taking "A and / or B" as an example, it includes the A solution, or the B solution, or the solution where A and B are satisfied simultaneously. In addition, the technical solutions between various embodiments can be combined with each other, but it must be based on the fact that those of ordinary skill in the art can implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection required by the present invention.

[0053] Refer to Figure 1, one embodiment of the present invention provides a robust image inpainting method based on Schatten-p norm decomposition, including the following steps:

[0054] Step a) Obtain the image to be inpainted, and convert the image to be inpainted into the original image matrix X;

[0055] Step b) Preprocess the converted original image matrix X to obtain the preprocessed image matrix X; operations such as denoising, normalization, and filling missing values are used to process the original image matrix X. Each element in the image matrix is a pixel value in the image to be inpainted. This step is the basis of image inpainting because it converts the image problem into a matrix problem, enabling the application of matrix decomposition and matrix completion techniques.

[0056] Step c) Based on the preprocessed image matrix, use matrix decomposition technology to decompose the image matrix into low-rank factors U and V; in this step, first decompose the preprocessed image matrix into a low-rank matrix to capture the global structure information of the image. This step aims to capture the global structure information of the image while removing the influence of noise and outliers, providing a basis for the subsequent inpainting process. Then further decompose the low-rank matrix into two low-rank factors, providing a basis for subsequent alternating optimization.

[0057] Step d) Based on the low-rank factors U and V and the original image matrix X, construct a loss function that includes a data fitting term and a regularization term. Estimate the difference between the image matrix reconstructed by the low-rank factors and the original image matrix at the known pixels through the data fitting term, and use the Schatten-p norm in the regularization term to penalize the complexity of the low-rank factors;

[0058] Among them, the formula in step d) is:

[0059] (1)

[0060] Among them, M Ω represents the position indicator of the known pixels in the original image matrix, X Ω represents the part of the known pixels in the image matrix reconstructed by the low-rank factors, f p (U, V) is the regularization function of the low-rank factors U and V, λ is the regularization parameter, s.t. X = UV T is the constraint condition, which forces the final image matrix X to be reconstructed from the low-rank factors U and V.

[0061] In the formula of step d), p = 1, 1 / 2, 2 / 3, so the formula of f p (U, V) is:

[0062] (1.1)

[0063] For RMCS with p = 1, 1 / 2, 2 / 3 p Define NF as RMCS 1 NF, RMCS 1 / 2 NF and RMCS 2 / 3 NF, correspondingly, has the following form:

[0064]

[0065] In formula (1), it can be seen that due to

[0066]

[0067] the existence of, directly optimizing formula (1) poses a very great challenge; to provide an efficient solution, a residual matrix E is added Ω , such that E Ω = M Ω - X Ω , and the penalty function is used to transform formula (1) into an unconstrained formula:

[0068] (2)

[0069] where, when μ → +∞, the solution of formula (2) will converge to formula (1); specifically, μ can be set to a large value.

[0070] Compared with the existing Unifying matrix completion method, from the perspective of the model, RMCS p NF has two more decomposition forms of the Schatten - p norm than the existing Unifying matrix completion method, namely p = 1 / 2, 2 / 3 (p = 1 for the existing Unifying matrix completion method); in addition, in terms of the error term, RMCS p NF adopts the l p norm with three p values (p = 1, 1 / 2, 2 / 3), while Unifying adopts the l 1 norm constraint. Therefore, the model proposed in this application is more general and robust than the Unifying model. From the optimization perspective, RMCS p NF uses alternating iterative optimization to avoid auxiliary variables, while Unifying uses the ADMM algorithm that includes auxiliary variables.

[0071] Step e) Minimize the loss function to determine the best estimate of the image matrix to repair the missing or damaged part of the original image matrix.

[0072] where, referring to Figure 2 , it also includes the following steps:

[0073] Step f) Alternating optimization:

[0074] Step f1) Initialize the low-rank factor U, the low-rank factor V, and the image matrix estimate X;

[0075] Step f2) Enter the alternating optimization loop:

[0076] Step f2.1) Fix the image matrix estimate X and only update the low-rank factors U and V. By minimizing the parts of the loss function with respect to U and V, obtain the new U and V;

[0077] Step f2.2) Fix the low-rank factors U and V and only update the image matrix estimate X. By minimizing the part of the loss function with respect to X, obtain the new X;

[0078] Step f2.3) Repeat the alternating update of the low-rank factors U, V and the update of the image matrix estimate X until the stopping condition is met.

[0079] Different from the iteration rules of ADMM, before giving the basic optimization steps for directly or alternately updating the original variables, auxiliary variables are not introduced. Specifically, fix the variables of the k-th iteration, that is and ignore the variables that are not relevant to the updated variables. The variables of the (k + 1)-th iteration can be obtained from formula (2); in step f2), by minimizing the parts of the loss function with respect to U and V, obtain the new U and V, and, by minimizing the part of the loss function with respect to X, obtain the new X. The specific formulas are as follows:

[0080]

[0081] where k represents the current iteration number, and k + 1 represents the next iteration; U k+1 , V k+1 respectively represent the updates of the low-rank factors of the (k + 1)-th iteration, and X k+1 represents the updated low-rank approximation result after the (k + 1)-th iteration, represents the new residual matrix after the (k + 1)-th iteration, and μ is the weight parameter;

[0082] where for different values of f p (U, V), a closed-form solution in the optimization process can be obtained.

[0083] (1) Update for U k+1 :

[0084] For different p values, fix the variables of the k-th iteration and ignore the variables that are not related to U in f p (U, V k ), the following formula can be obtained:

[0085]

[0086] The following will give the closed - form solution for the selected p - value:

[0087] (1.1) When p = 1 or 2 / 3, the formula

[0088]

[0089] is actually a least - squares problem. In this problem, calculate the derivative with respect to U and set it to. Thus, we obtain respectively:

[0090] (a)

[0091] (b)

[0092] (1.2) When p = 1 / 2, the formula

[0093]

[0094] can be written as an equivalent optimization problem, expressed as:

[0095]

[0096] where:

[0097]

[0098] Calculate the derivative with respect to U in the formula of the equivalent optimization problem, set it to 0, and we get:

[0099] (c)

[0100] (2) For V k+1 The update of:

[0101] Similar to the update rule of U k+1 , by solving

[0102]

[0103] for different p - values to update V k+1 , and then we obtain the following specific formula:

[0104]

[0105] By incorporating this formula into the above formula, we get the analytical solutions for different p - values:

[0106] (2.1) For p = 1 in fp(V), we get:

[0107] (d)

[0108] 2.1) For \(p = 1 / 2\) and \(p = 2 / 3\) in \(fp(V)\), we respectively obtain:

[0109] (e)

[0110] (f)

[0111] Among them:

[0112]

[0113] (3) Update for \(X\) k+1 :

[0114] Fix the variable Update \(X\) k+1 Actually, it is to solve a least - squares problem. Therefore, take the derivative of \(X\) in the objective function

[0115]

[0116] and set it equal to 0 to obtain the optimal solution, which is:

[0117] (g)

[0118] (4) Update for :

[0119] Fix the variable \(X\) k+1 , and update it by solving the formula:

[0120]

[0121] to update

[0122] (4.1) When \(p = 1\), apply the soft - thresholding shrinkage operator to the above formula, and a closed - form solution can be obtained as follows:

[0123] (h)

[0124] where \(Shrinkage[a,b]=\text{sign}(a)\cdot\max(|a|-b,0)\).

[0125] (4.2) When \(p = 1 / 2\), the closed - form solution of the above formula is obtained through the following formula:

[0126] (i)

[0127] where \(\gamma = 2 / u\), \(H\) γ (A) Solve it through the following formula:

[0128]

[0129] Among them,

[0130] (4.3) When p = 2 / 3, the closed-form solution of the above formula is obtained through the following formula:

[0131] (j)

[0132] Among them, γ = 2 / μ, T γ (A) Solve through the following formula:

[0133]

[0134] Among them,

[0135] Based on the above update steps, the following summarizes the alternating optimization algorithm given in this embodiment to solve the matrix completion problem:

[0136] Solve RMCS p NF model:

[0137] Input: M, r, p ∈ {1, 2 / 3, 1 / 2}, λ and μ;

[0138] Output: X* ← X k+1 .

[0139] Initialize U = rand(m, d), V = rand(n, d), W 1 = I, W 2 = I, k = 0;

[0140] Loop condition: Not reaching the termination condition

[0141] 1: If p = 1

[0142] Update U according to (a) k+1 ;

[0143] Update V according to (d) k+1 ;

[0144] Update X according to (g) k+1 ;

[0145] Update according to (h)

[0146] 2: If p = 2 / 3

[0147] Update U according to (b) k+1 ;

[0148] Update V according to (f) k+1 ;

[0149] Update

[0150]

[0151] Update X according to (g) k+1 ;

[0152] Update according to (j)

[0153] 3: If p = 1 / 2:

[0154] Update U according to (c) k+1 ;

[0155] Update V according to (e) k+1 ;

[0156] Update

[0157]

[0158] Update

[0159]

[0160] Update X according to (g) k+1 ;

[0161] Update according to (i)

[0162] 4: Check the convergence condition:

[0163]

[0164] 5: k ← k + 1.

[0165] End the loop

[0166] During the iteration, the stopping criterion is as follows:

[0167]

[0168] where ε is a relatively small tolerance value, set ε = 10 -3 . The iteration stops when the stopping criterion is met.

[0169] where it also includes the following steps:

[0170] Step g) Algorithm analysis: including computational complexity analysis and convergence analysis;

[0171] where the computational complexity:

[0172] For each iteration, matrix inversion and matrix multiplication operations are used;

[0173] Update X for three p-values k+1 The computational complexity of updating U k+1 and V k+1 is as follows:

[0174] When p = 1, the computational complexities of updating U k+1 and V k+1 are o(d 3 + nd 2 + mnd) and o(d 3 + md 2 + mnd);

[0175] When p = 2 / 3, the computational complexities of updating U k+1 and V k+1 are o(d 3 + nd 2 + mnd) and o(mnd + (m + n)d 2 + 2d 3 );

[0176] When p = 1 / 2, the computational complexities of updating U k+1 and V k+1 are both o(mnd + (m + n)d 2 + 2d 3 )

[0177] Finally, the total computational complexity is the sum of the computational complexities of each step;

[0178] where m represents the number of rows of the matrix, n represents the number of columns of the matrix, and d represents the number of dimensions of the vector;

[0179] where, through convergence analysis, is determined to be the stable point of the minimization problem.

[0180] Regarding convergence analysis:

[0181] Theorem: Assume that formula (2) defines the multivariable objective function H(X, U, V, E), and

[0182] and

[0183]

[0184] hold.

[0185] At three p-values, the variables are respectively produced by the algorithm

[0186] in the k-th iteration, then:

[0187] (i) (Sufficient reduction condition) In each , there exists a positive constant C such that:

[0188]

[0189] (ii) (Convergence of subsequence) There exist subsequences of variables:

[0190] and such that as j → +∞, and:

[0191] (ii1) where, satisfies the KKT conditions.

[0192] Proof: To prove (i), combine the formula

[0193]

[0194] with the update rule in the formula

[0195]

[0196] to conclude from the (k + 1)-th iteration variable generated:

[0197] (i1)

[0198] (i2) naturally holds.

[0199] Since is strongly convex with respect to X in the minimization of in k+1 there exists a positive constant such that

[0200] (i3)

[0201] In addition, from it can be obtained that:

[0202] (i4)

[0203] Then, by summing both sides of formulas (i1) to (i4), it can be inferred that the sequence has a non-increasing property for each in a finite number of iterations.

[0204] To prove (ii), from

[0205]

[0206] It is concluded from: the sequence {X k} is bounded. In addition, according to the Bolzano - Weirstrass theorem, there exists at least one matrix X * and a subsequence such that:

[0207] Based on the formula relationship: X k = U k (V k ) T and it is further deduced that is also bounded. Similarly, it is not difficult to achieve: Meanwhile, holds.

[0208] In the formula

[0209]

[0210] set k = k j , and calculate their derivatives using the corresponding variables respectively, which means:

[0211] (ii2)

[0212] wherein, is the non - differentiable symbol. If is differentiable,

[0213] Therefore, when j → +∞, the formula (ii2) is guaranteed to hold from the specific representation formulas in formula (1) and formula (1.1). From the conclusions, assumptions, and similar processing methods given above, it can be easily achieved that the formula (ii1) holds naturally. Therefore, it is determined that is the stable point of the minimization problem.

[0214] The above - described embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present application, and should all be included within the protection scope of the present application.

Claims

1. A robust image restoration method based on Schatten-p norm decomposition, characterized in that: The steps include: a) Obtaining the image to be repaired and converting the image to be repaired into the original image matrix; b) preprocessing the transformed original image matrix to obtain a preprocessed image matrix; c) based on the preprocessed image matrix, using matrix decomposition technology to decompose the image matrix into low-rank factors; d) Based on the low-rank factor and the original image matrix, a loss function including a data fitting term and a regularization term is constructed. The difference between the image matrix reconstructed by the low-rank factor and the original image matrix at known pixels is estimated through the data fitting term. The Schatten-p norm is used through the regularization term to penalize the complexity of the low-rank factor. e) Minimize the loss function to determine the best image matrix estimate to repair the missing or damaged parts of the original image matrix.

2. The robust image restoration method based on Schatten-p norm decomposition according to claim 1, characterized in that: The formula in step d) is: Among them, M Ω Represents the position indicator of the known pixel in the original image matrix, X Ω represents the part of the image matrix with known pixels reconstructed by the low-rank factor, f p (U, V) is the regularization function of the low-rank factor U, V, λ is the regularization parameter, stX = UV T As a constraint, it is mandatory that the final image matrix X must be reconstructed from the low-rank factors U and V.

3. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 2, characterized in that: In the formula of step d), p = 1, 1 / 2, 2 / 3, so f p The formula for (U, V) is:

4. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 3, characterized in that: The RMCS with p = 1, 1 / 2, 2 / 3 p NF is defined as RMCS1NF, RMCS 1 / 2 NF and RMCS 2 / 3 NF, the corresponding form is as follows:

5. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 2, characterized in that: In formula (1), add the residual matrix E Ω , so that E Ω =M Ω -X Ω , and use the penalty function to transform formula (1) into an unconstrained formula: When μ→+∞, the solution of formula (2) converges to formula (1).

6. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 5, characterized in that: After step e), the following steps are also included: f) Alternating Optimization: f1) Initialize low-rank factor U, low-rank factor V and image matrix estimate X; f2) Enter the alternating optimization loop: The image matrix is ​​fixed and estimated X is updated, and only the low-rank factors U and V are updated. The new U and V are obtained by minimizing the part of the loss function about U and V. Fix the low-rank factors U and V, only update the image matrix to estimate X, and obtain the new X by minimizing the part of the loss function about X; Repeat the alternating updating of the low-rank factors U, V and the updating of the image matrix estimate X until the stopping condition is met.

7. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 6, characterized in that: In the step f2), new U and V are obtained by minimizing the part of the loss function with respect to U and V, and new X is obtained by minimizing the part of the loss function with respect to X. The specific formula is as follows: Among them, k represents the current iteration number, k+1 represents the next iteration; U k+1 , V k+1 Represent the update of the low-rank factor of the k+1th iteration, X k+1 represents the updated low-rank approximation result after the k+1th iteration, represents the new residual matrix after the k+1th iteration, μ is the weight parameter; in, For different values ​​of f p (U, V), a closed-form solution can be obtained in the optimization process.

8. The robust image restoration method based on Schatten-p norm decomposition as claimed in claim 7, characterized in that: After step f), the method further comprises the following steps: g) Algorithm analysis: including computational complexity analysis and convergence analysis; Among them, the computational complexity is: For each iteration, matrix inversion and matrix multiplication operations are performed; For three p values, update X k+1 The computational complexity is o(2mnd), update U k+1 and V k+1 The computational complexity is as follows: When p = 1, update U k+1 and V k+1 The computational complexity is o(d 3 +nd 2 +mnd) and o(d 3 +md 2 +mnd); When p = 2 / 3, update U k+1 and V k+1 The computational complexity is o(d 3 +nd 2 +mnd) and o(mnd+(m+n)d 2 +2d 3 ); When p = 1 / 2, update U k+1 and V k+1 The computational complexity is o(mnd+(m+n)d 2 +2d 3 ) Finally, the total computational complexity is the sum of the computational complexity of each step; Among them, m represents the number of rows of the matrix, n represents the number of columns of the matrix, and d represents the number of dimensions of the vector; Among them, through convergence analysis, it is determined is the stable point of the minimization problem.

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