L1 / L2 norm ratio regular minimization image restoration method based on analytical model

Abstract

The invention provides an L1 / L2 norm ratio regular minimization image restoration method based on an analytical model, and belongs to the field of signal and image processing inverse problems. In order to solve the problems that the reconstruction precision is limited and the algorithm robustness is insufficient under the conditions of undersampling, pixel loss, blurring and the like in the existing image restoration technology, an image restoration optimization model is established and solved by utilizing the inherent scale invariance and parameter freedom of an L1 / L2 norm ratio and combining the stability of signal representation under an analytical model. The geometric structure and texture details of the image can be stably recovered from under-sampled or damaged observation data, and the image recovery problem in various typical degradation scenes is effectively solved.

Description

Technical Field

[0001] This invention relates to the field of inverse problems in signal and image processing, and specifically to an image restoration method based on the L1 / L2 norm ratio regularization minimization of an analytical model. Background Technology

[0002] Image restoration is a task that aims to extract data from observation vectors. Restore the original image , For degenerate operators, For noise terms. This task is typically an underdetermined (e.g., compressed sensing, image inpainting) or ill-conditioned (e.g., image deblurring) inverse problem.

[0003] A common approach to solving this type of problem is to introduce sparse priors for regularization. Currently, the most common sparse regularization is constructed within the framework of synthetic models. This model assumes... , For synthesized dictionaries, For sparse vectors, the following problem can be solved by optimizing:

[0004] .

[0005] However, this optimization problem is NP-hard. Therefore, the L0 convex relaxation is usually replaced with the L1 norm. Although convex relaxation methods can achieve efficient solutions, their recovery performance is often inferior to non-convex models, such as Schatten models. Pseudonorm, L1 L2, etc. It should be noted that most non-convex models lack scale invariance property (SIP). In contrast, the L1 / L2 norm ratio has advantages such as SIP and parameter freedom.

[0006] There exists a class of analytical models corresponding to the synthetic model, which assumes the existence of an analytical operator. It is usually a tight frame, making It is a sparse vector. Compared to synthetic models, all atoms in the analytical model participate in the signal representation, resulting in better stability. However, most existing image restoration methods based on analytical models employ the L1 norm or lack a non-convex regularization term (L1) that does not possess SIP. L2). However, issues remain regarding limited reconstruction accuracy and insufficient algorithm robustness when dealing with undersampling and corrupted observation data.

[0007] Current research on the L1 / L2 norm ratio mainly focuses on the synthetic model framework, while research on the analytical model framework is relatively limited. Furthermore, when applying the L1 / L2 norm ratio to the analytical model, the constructed objective function is non-convex and non-smooth, and the variables are inseparable, making it difficult for existing algorithms to solve directly. Summary of the Invention

[0008] Based on this, this invention proposes an image restoration method based on the L1 / L2 norm ratio regularization minimization of an analytical model. The method includes the following steps: S1: Determine the analytical operator and set the parameters and maximum number of iterations; S2: Construct an optimization model, which includes data fidelity terms and regularization terms; S3: Initialize variables and pre-calculate constants related to subsequent iterations; S4: Execute the iterative optimization loop, updating the image variables, inertia parameters, extrapolation weights, extrapolation terms, and auxiliary variables sequentially in each iteration; S5: Determine whether the preset iteration stopping condition is met; if yes, output the final restored image; if not, return to step S4 to execute the next iteration.

[0009] Further, step S1 includes: determining the analytic operator. Set regularization parameters Penalty parameters and convex parameters .

[0010] Further, step S2 includes constructing the optimization model as follows:

[0011] .

[0012] Further, step S3 includes: initializing auxiliary variables. Extrapolated variables Inertial parameters ,in Estimate the initial image. Pre-compute constants. .

[0013] Further, step S4 includes: S41: Solving the linear equation Get the first Image variables in the next iteration S42: Update inertial parameters Calculate extrapolation weights S43: Calculate the extrapolation term The extrapolation term is introduced to utilize historical iteration information and accelerate the convergence speed of the algorithm; S44: Calculate the subgradient. , where sign S45: A sign function that acts on each element of the vector; Update auxiliary variables ,in It is a soft contraction operator. This is the threshold parameter.

[0014] The beneficial effects of this invention are as follows: This invention utilizes the SIP and parameter freedom inherent in the L1 / L2 norm ratio, combined with the stability of signal representation under analytical models, to construct an image restoration optimization model and provides a solution algorithm. Compared to existing technologies, this invention effectively overcomes the limitations in reconstruction accuracy and robustness under conditions such as undersampling and corrupted observation data. It can stably restore the geometric structure and texture details of an image from limited or damaged observation data, significantly improving the quality of image restoration. Attached Figure Description

[0015] Figure 1 Here is the algorithm flowchart;

[0016] Figure 2 The image shown is the convergence curve of the corresponding image "Fruits" in the scenario of Embodiment 1 of the present invention, which shows how the relative error changes with the number of iterations.

[0017] Figure 3 The image shown is the convergence curve of the corresponding image "Flinstones" in the scenario of Embodiment 2 of the present invention, which shows how the relative error changes with the number of iterations.

[0018] Figure 4 The image shown is the convergence curve of the "Female" image in the scenario of Embodiment 3 of the present invention, illustrating how the relative error changes with the number of iterations. Detailed Implementation

[0019] Before introducing specific embodiments, the solution principle of the L1 / L2 norm ratio regularization minimization image restoration method based on analytical models proposed in this invention will be explained in detail.

[0020] This invention aims to solve the model

[0021] ,

[0022] because The indivisibility of the components makes directly solving the above problem quite difficult. This invention addresses this by introducing auxiliary variables. Furthermore, by utilizing the quadratic penalty function technique, the above optimization problem is transformed into the following optimization model (i.e., the optimization model described in the claims):

[0023]

[0024] For this model, the present invention designs the following iterative solution scheme: using an alternating minimization strategy, the complex problem is decomposed into two subproblems that are solved alternately.

[0025] First, by fixing auxiliary variables Update image variables Due to the image variables The objective function is a quadratic programming problem, which can be obtained by solving the linear equation derived from the first-order optimal conditions (corresponding to step S41). Since the computational complexity of inversion is too high, this invention employs the conjugate gradient method to solve this linear equation in all embodiments. Secondly, by fixing image variables... Update auxiliary variables Due to the fact that... The objective function is neither convex nor smooth, and this invention employs the Difference of Convex Algorithm (DCA) to solve it. Specifically, it will discuss... The objective function can be written in the following form:

[0026] ,

[0027] Using the DCA solution framework, this problem can be written as:

[0028] ,

[0029] in This is a subdifferential operator. To utilize historical iteration information to accelerate algorithm convergence, this invention introduces an extrapolation mechanism. The extrapolation term calculation method corresponds to steps S42 and S43 in the specification. Specifically, S42 is executed first to update the inertial parameters. And calculate the extrapolation weights Next, step S43 is executed to calculate the extrapolation term. Based on this, the subgradient will be calculated. In Replace with This yields the specific calculation formula for step S44 in the instruction manual. Specifically, the sign function (sign() mentioned in S44) Specifically defined as:

[0030] .

[0031] By refining the formula, we can obtain: ,

[0032] This problem has a closed-form solution, namely the specific calculation formula in step S45. The soft contraction operator mentioned in S45 is specifically defined as follows: , It represents the Hadamardi (or Hadama) stack.

[0033] To verify the effectiveness of the method proposed in this invention, the method (denoted as Algorithm 1) is compared with the Smooth Monotone Fast Iterative Shrinkage-Thresholding Algorithm (SMFISTA) and the Projected Fast Iterative Shrinkage-Thresholding Algorithm (Projected Fast Iterative Shrinkage-Thresholding Algorithm) based on analytical models. The results were compared with the sorted L1 / L2 (denoted as Sorted-L1 / L2) based on the synthetic model.

[0034] In all the following embodiments, the parsing operator described in step S1 All were set to contourlet transform. To ensure the fairness of the experimental comparison, the comparison algorithm based on the analytical model and the method proposed in this invention used the same analytical operators. For comparison algorithms based on synthetic models, a synthetic dictionary is set. At the same time, select a size of The standard test image is scaled to [0, 1] and then vectorized to obtain the original image signal. To simulate a real sampling process, a uniform mean of 0 and a variance of 0 were introduced into the observed data. Additive white Gaussian noise And then according to the formula Generate observation data (of which) (Specific degeneracy operators in each embodiment). To evaluate the convergence performance of the algorithm, the first... The relative error of the next iteration is ,in For the original image, For the first The restored image obtained in the next iteration. After setting the parameters in step S1, image restoration is performed according to the following unified steps:

[0035] S2: Construct an optimization model;

[0036] S3: Initialize auxiliary variables Extrapolated variables Inertial parameters ,in Image estimation is performed using an all-zero initialization. Pre-computed constants are used. ;

[0037] S4: Update the image variables, inertial parameters, extrapolation weights, extrapolation terms, and auxiliary variables sequentially according to the calculation formulas in steps S41 to S45;

[0038] S5: Determine whether the preset iteration stop condition is met; if yes, output the final restored image; if no, return to step S4 to execute the next iteration.

[0039] Peak signal-to-noise ratio (PSNR, dB) and structural similarity (SSIM) were selected to evaluate the image restoration quality. All experimental data shown in the tables below are the average results of 50 independent runs of each algorithm under the same conditions. Bold data in the tables represents the optimal value for each evaluation metric.

[0040] Example 1: This example aims to illustrate the invention at a lower sampling rate ( Under these conditions, the original image is recovered from the observation data. The specific steps are as follows:

[0041] S1: Settings ,in: Represents the dimension (total number of pixels) of the original image signal; Indicates the dimension of the observation; This represents the random sign inversion operator, used to randomly flip the sign of pixel values ​​in an input image signal. Represents the discrete cosine transform; Represents a random selection operator, used to select from... Randomly selected from the transformation coefficients Each coefficient. Setting. , , as well as .

[0042] Using the above parameter settings, the image is iteratively solved according to the aforementioned general image restoration process (i.e., general steps S2 to S5), and the restored image is finally output.

[0043] For the compressed sensing recovery task under the aforementioned low sampling rate conditions, the experimental results of the method of this invention and various comparative algorithms are shown in Table 1.

[0044] Table 1 Comparison of Compressed Sensing Recovery Experiment Results

[0045]

[0046] Example 2: This example aims to illustrate how the present invention recovers the original image from observation data in scenarios with high levels of blur. The specific steps are as follows:

[0047] S1: Set the size to Furthermore, a Gaussian blur kernel with a standard deviation of 6 is used as the degradation operator. .set up , , as well as .

[0048] Using the above parameter settings, the image is iteratively solved according to the aforementioned general image restoration process (i.e., general steps S2 to S5), and the restored image is finally output.

[0049] For the aforementioned degradation scenario of Gaussian blur, the experimental results of image deblurring using the method of this invention and various comparison algorithms are shown in Table 2.

[0050] Table 2 Comparison of Image Deblurring Experiment Results

[0051]

[0052] Example 3: This example aims to illustrate the present invention in the case of random loss of image pixels. In this scenario, the specific steps to recover the original image from the observation data are as follows:

[0053] S1: Set random loss Pixel mask as a degradation operator .set up , , as well as .

[0054] Using the above parameter settings, the image is iteratively solved according to the aforementioned general image restoration process (i.e., general steps S2 to S5), and the restored image is finally output.

[0055] Regarding the aforementioned random pixel loss The experimental results of image restoration using the method of this invention and various comparison algorithms are shown in Table 3 for the damaged scenarios.

[0056] Table 3 Comparison of Image Restoration Experiment Results

[0057]

[0058] Experimental results show that, under the above parameter settings, the method of the present invention has higher PSNR and SSIM than the comparison algorithm in image restoration tasks such as compressed sensing restoration, image deblurring, and image inpainting.

[0059] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. An image restoration method based on analytical model-based L1 / L2 norm ratio regularization minimization, characterized in that, include: S1: Determine the analytic operator And set the parameters and the maximum number of iterations. The algorithm parameters include regularization parameters. Penalty parameters and convex parameters ; S2: Construct an optimization model that includes data fidelity terms and regularization terms; S3: Initialize the variables and pre-calculate the constants related to subsequent iterations; S4: Execute the iterative optimization loop, updating the image variables sequentially in each iteration. Inertial parameters Extrapolation weights Extrapolation terms and auxiliary variables ; S5: Determine whether the preset iteration stop condition is met; if yes, output the final restored image; if no, return to step S4 to execute the next iteration.

2. The image restoration method based on the L1 / L2 norm ratio regularization minimization according to claim 1, characterized in that, The recovery optimization model described in step S2 is specifically represented as follows: , in and Let L1 and L2 represent the norm and L2 norm, respectively.

3. The image restoration method based on the L1 / L2 norm ratio regularization minimization according to claim 2, characterized in that, Step S3 specifically includes initializing auxiliary variables. Extrapolated variables Inertial parameters ,in For initial image estimation; pre-compute constants .

4. The image restoration method based on the L1 / L2 norm ratio regularization minimization according to claim 3, characterized in that, The iterative optimization loop in step S4 specifically includes the following sub-steps: S41: Solving linear equations , obtained the The result of the iteration ; S42: Update the inertial parameters according to the following formula extrapolation weights : ， ； S43: Calculate extrapolation terms ; S44: Calculate the subgradient ,in This is the sign function that acts on each element of the vector; S45: Update auxiliary variables ,in It is a soft contraction operator. This is the threshold parameter.

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