3.1. Low-rank Visual Art Image Restoration Based on Progressive Regularization

LRM restoration has been extensively applied in many fields such as image restoration and deblurring. In an image data matrix, each row or column can be expressed by other rows or columns, meaning that this matrix contains a lot of redundant information ^[16]. In a sense, image restoration is a typical LRM restoration problem. The image restoration is displayed in Fig. 1.

Fig. 1. Image recovery process.

In Fig. 1, the image restoration is to repair and eliminate blurring of the image. Although image restoration techniques have been able to effectively repair or eliminate blurring in images, there is still a significant gap in clarity in the restored and the original image due to various reasons. Therefore, the study introduces a weight update strategy. The weighted kernel norm minimization method is used for matrix filling, as shown in Eq. (1).

In Eq. (1), $\Omega$ represents a binary indicator matrix of the same size as $Y$. $p_{\Omega}(Y)$ represents the Hadamard product of the indicator matrix and the observation matrix $Y$. This limitation indicates that the estimation matrix $X$ matches the lossless information of the observation matrix $Y$. Adjacency matrix is a matrix used to describe the nearest neighbor relationships between nodes. This algorithm takes a one-dimensional array to store all points in the graph, while a two-dimensional array stores the relationships between points ^[17]. The adjacency matrix is specifically shown in Fig. 2.

Fig. 2. Adjacency matrix diagram.

In Fig. 2, $+1$ indicates mutual perception, while blank indicates inability to perceive each other. For an undirected graph, its adjacent matrices must be symmetric, its main diagonal must be 0, and its sub diagonal may not necessarily be 0. In an undirected graph, the degree of any vertex refers to the $i$-th non-zero element quantity. In a directed graph, adjacent matrices may not necessarily be symmetric. Among them, the number of outgoing lines of node $i$ in a directed graph is the non-zero element quantity in row $i$. In a directed graph, the outgoing lines of node $i$ are non-zero elements ^[18]. LRM recovery takes the rank of the matrix as the sparse measure. The linear measurement is shown in Eq. (2).

In Eq. (2), $y$ represents the observed data. $A$ represents the linear operator. $X \in R^{m \times n}$, $y \in R^p$. The definition of the linear operators is specifically shown in Eq. (3).

In Eq. (3), $\langle A_i, X \rangle$ represents the inner product of the matrix. From a mathematical perspective, the low-rank restoration is reduced to a rank minimization, as displayed in Eq. (4).

In Eq. (4), $rank(X)$ signifies the rank function of matrix $X$. The update process of the weight vector is set to the following form, as shown in Eq. (5).

In Eq. (5), $\varepsilon_l$ represents a sequence that gradually decreases in the range of 0 to 1 and approaches 0. Traditional regularization algorithms require $\varepsilon$ to take a small optimal solution and maintain a constant during iteration. This algorithm takes $\varepsilon_l = 1$ as the initial value, increasing from high to low. The maximum value includes some suboptimal solutions first. During the decreasing process, it continuously shrinks to gradually approach the optimal solution. This regularization method can improve the sparsity of rank and achieve better results. A block-based image restoration algorithm is introduced to address the non-local self-similarity characteristics of images. This algorithm mainly includes similar block matching, matrix low-rank approximation, image reconstruction, and iterative diffusion. In the block matching stage, the missing image $Y$ is first divided into $N$ overlapping blocks. For each block, the similarity measure is used to find other blocks that are similar to it. Next, each similar block is transformed into a column vector form, and a block matrix is formed by connecting all column vectors and stacking them. The process of building a block matrix is shown in Fig. 3.

Fig. 3. The process of building a block matrix.

In Fig. 3, in the process of building block matrix, the art image is first divided into several small blocks, which are then searched for other similar blocks in the image through similarity search algorithm. Each similar block is converted into column vector form, and these column vectors are connected and stacked to form block matrix. In this process, the adjacency matrix plays an important role, which describes the adjacency relationship between nodes. For image restoration, nodes here can be image blocks, and the adjacency matrix reflects the similarity or correlation between these image blocks. In this way, the adjacency matrix helps the algorithm identify and exploit similar areas in the image, thereby preserving the structure and texture information of the image during the repair process. In image restoration, adjacency matrix not only helps to search and match similar blocks, but also helps to optimize the combination of image blocks in the iterative process, so that the repaired image can better retain the features and styles of the original image.

3.2. Image Restoration Based on Singular Value Entropy Function

Singular value arrangement in image restoration uses singular value decomposition to decompose an image, obtain a set of singular values, and arrange these singular values in descending order. In image restoration, singular value permutation is used to restore or reconstruct damaged or missing image parts. Specifically, singular value decomposition can decompose a complex image matrix into several simple matrices, of which the singular value matrix is an important component. These singular values can reflect important features and structural information. Therefore, the main features can be preserved by retaining larger singular values. For smaller singular values, they can be ignored or set to zero to remove noise, fill missing parts, or perform other image restoration operations ^[19,20]. Singular value decomposition and arrangement are shown in Fig. 4.

Fig. 4. Singular value decomposition and arrangement.

In image processing, scratches are equivalent to collision noise at different positions in the spatial domain, making the disorder of pixel values in the image more severe. On the contrary, due to the random loss of pixel values following a uniform distribution, the confusion caused is not significant. For some systems with high chaos, the entropy minimization method can be effectively improved. To address this issue, an image repair method based on the entropy weight is designed to further optimize scratches. The specific definition of entropy function is shown in Eq. (6).

In Eq. (6), $x \in R$. Information theory suggests that the maximum entropy of a random variable exists when it has a uniform distribution. In other words, if the numerical value of a random variable is small, its entropy value is lower. That is, its vector $x$ is sparser. Based on the low-rank approximation, the singularity performance of the repaired image can be improved, resulting in better sparsity and more concentrated main information, thereby improving the quality of image restoration. Therefore, the study introduces another rank approximation method, which is the entropy function of singular values. For an LRM $X$, the rank minimization approximation is equal to solving the minimization problem of the corresponding singular value entropy function. This approach can better capture and utilize the low-rank structure of matrices, thereby optimizing the image restoration effect. The rank minimization approximation is specifically shown in Eq. (7).

Under certain fidelity limitations, the equivalent result of the above problem is displayed in Eq. (8).

The entropy function minimization can be transformed into a weighted kernel norm minimization problem. This entropy-based weight is applied to the image restoration framework to perform block processing on the image. Unlike global low-rank, image restoration is achieved by utilizing the low-rank of similar block matrices and solving the weighted kernel norm minimization problem corresponding to entropy gradient weights. It better capture the local structure and maintain the integrity of these structures during the repair process, thereby improving the repair effect. The consensus weight method can cause the repair result to be overly smooth, so the weight value is specifically shown in Eq. (9).

During the iteration process, an iterative regularization step is introduced to reinforce the improved estimation results by back-projecting the residual image onto the estimated image. Specifically, at the beginning of each iteration, the estimated image output from the previous iteration is updated and then used as the input image for the next iteration. This method can utilize the information from the previous iteration to gradually optimize the image, thereby improving estimation accuracy and stability, as shown in Eq. (10).

In Eq. (10), $\hat{X}$ represents the repair of damaged images. $\delta$ is an adjustment parameter. Peak Signal-to-Noise Ratio (PSNR) is an objective standard used to assess image or video quality. PSNR is frequently used to compare the quality differences between original images and compressed, processed, or reconstructed images, as shown in Eq. (11).

Parameters $M$ and $\delta$ directly affect the effectiveness of the algorithm. The impact parameters $M$ and $\delta$ on PSNR is shown in Fig. 5.

Fig. 5. Effect of δ on PSNR.

Finally, it is decided to choose $M$ as 5 and $\delta$ as 0.4 to complete the image restoration task. Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) are taken as evaluation metrics to measure the error in the repair results. MAE is a commonly used error measure that represents the average absolute deviation in the repaired and the true. A small MAE indicates that the repair result is approached to the true value and the repair effect is better. The MAE is shown in Eq. (12).

RMSE considers the variance and mean of the repair error. Compared with MAE, RMSE is more sensitive to outliers. A small RMSE value indicates better stability of the repair result and better repair effect. The RMSE is shown in Eq. (13).

The steps of an LRM image restoration method ground on asymptotic regularization singular value function roughly include image partitioning, constructing similar block groups, singular value decomposition, singular value contraction, reconstructing LRM, image block aggregation, etc., as shown in Fig. 6.

Fig. 6. The specific process steps of the algorithm.

In Fig. 6, the image to be restored is divided into overlapping or non-overlapping blocks during the image restoration process. For each image block, other similar blocks are searched and combined into a matrix. The matrix composed of each similar block group is subjected to singular value decomposition. Asymptotic regularization singular value function is used to contract singular values. The singular values are adjusted based on their size and preset thresholds or regularization parameters. The adjusted singular values and corresponding left and right singular vectors are used to reconstruct an LRM. All reconstructed low-rank matrices are converted back into image blocks and placed back in their corresponding positions in the image. If the blocks are overlapping, weighted averaging may be necessary to fuse the overlapping regions. The above process is repeated until the image quality no longer significantly improves. The weight updating strategy uses the weighted kernel norm minimization method to fill the matrix, in which the weight vector gradually decreases in the iterative process, the initial value is large to contain part of the suboptimal solution, and then gradually decreases to approximate the optimal solution. This process enhances the sparsity of the rank, so as to obtain better repair effect. The regularization term effectively restores the low-rank structure of the image and removes noise and outliers by introducing low-rank constraints and sparse constraints. At the same time, the robustness and generalization ability of the algorithm are enhanced, so that it can adapt to different types of visual art image restoration tasks. In the process of restoration, these parameters interact with the low-rank components of the image, and the damaged image is decomposed into low-rank components and sparse components through the low-rank approximation model. The low-rank matrix is estimated by the iterative optimization algorithm, and the progressive regularization strategy is used to gradually approximate the original image. Finally, the repaired low-rank components and sparse components are synthesized to obtain the repaired image.

4.1. Performance of LRM Based on Asymptotic Regularization Singular Value Function

The LRM based on asymptotic regularization singular value function (Algorithm 1) is compared with other image restoration algorithms. Comparative indicators include relative error, running time, stability, precision, recall, and F1 value. The comparison algorithms include image restoration algorithm based on sparse representation (Algorithm 2), image restoration algorithm ground on matched sample blocks (Algorithm 3), the single-step compression perception (Algorithm 4), and the convolutional neural network (Algorithm 5). Table 1 displays the results.

In Table 1, the relative error of Algorithm 1 was 0.001, the running time was 28.54s, the stability was 88%, the accuracy was 93.64%, the recall was 92.36%, and the F1 value was 92.51. Overall, various indicator parameters are relatively excellent. The relative error, running time, stability, precision, recall, and F1 value of the other four algorithms are lower and longer than the LRM based on asymptotic regularization singular value function. Therefore, it indicates that the LRM based on asymptotic regularization singular value function has better performance. The Receiver Operating Characteristic (ROC) curve depicts the balance in true positive rate and false positive rate under different classification thresholds. The Area Under the Curve (AUC) quantifies the overall performance of the ROC curve, with values closer to 1 indicating better performance. The ROC, AUC values, and curve generation time of the LRM based on asymptotic regularization singular value function are shown in Fig. 7.

Fig. 7. Comparison of ROC, AUC values and specific running time.

In Fig. 7(a), the ROC curve of the LRM based on asymptotic regularization singular value function was good, with an AUC value close to 1, indicating good performance. The running time in Fig. 7(b) ranged from 20 to 50 minutes, indicating that the algorithm had a high complexity, a large dataset, and more computing resources, but it was within a controllable range. The RMSE of the LRM ground on asymptotic regularization singular value function is further evaluated. The other parameters remain unchanged, only changing the size of the missing rate to evaluate the performance of Algorithm 1 and Algorithm 2 under different missing rates, as shown in Fig. 8.

Fig. 8. RMSE comparison of two algorithms.

In Fig. 8, the minimum RMSE of Algorithm 1 was 0.1082, and the minimum RMSE of Algorithm 2 was 0.1897, indicating that Algorithm 1 had higher accuracy in reconstructing image data. The RMSE of the two algorithms increases as the sample size decreases. Although the sample size is decreasing, Algorithm 1 can still maintain relatively stable performance. This further demonstrates the stability and superiority of the proposed LRM based on asymptotic regularization singular value function.

The LRM algorithm based on asymptotically regularized singular value function is compared with other advanced image restoration algorithms. Comparison algorithms include Wavelet Deep Neural Network (WDNN), Reference Constrained Image Restoration Algorithm, Wavelet deep neural Network (RCIR), Incremental Transformer Algorithm (IT), and Efficient Low-Rank Matrix Factorization (ELRMF). The accuracy and recall rates of the five algorithms are shown in Fig. 9.

Fig. 9. Accuracy and recall of five algorithms.

In Fig. 9(a), as iterations increased, the accuracy of the algorithm stabilized above 0.9 after 50 iterations, which showed the algorithm efficiency and its stability. When predicting positive samples, the algorithm mistakenly identifies the repaired area as undamaged area less often, reduces unnecessary modifications, and maintains the original features of the image. Fig. 9(b) further demonstrates the advantage of the algorithm in terms of recall rate. That is, the algorithm is able to identify most of the actual damaged areas, reducing the risk of missing damaged parts of the original image. In visual art image restoration, any omission can cause the image to lose its artistic value and historical value.

4.2. Image Restoration Effect

To assess the effectiveness of LRM based on asymptotic regularization singular value functions in image restoration, this study applies them to images with 50% missing data and analyzes the PSNR of five image restoration algorithms, as shown in Table 2.

In Table 2, Algorithm 1 had a relatively high PSNR on different images, with an average of 0.93. However, Algorithms 2-5 only had 0.69, 0.74, 0.71, and 0.73, respectively. It indicates that Algorithm 1 has better image restoration quality. The difference from the original image is relatively small. Algorithms 1-3 are used to recover artificial data containing 20%, 30%, 40%, and 50% random deletion rates. 30 independent repeated experiments are conducted. Fig. 10 displays the standard deviation of the PSNR obtained.

Fig. 10. Image repair under different deletion rate.

In Fig. 10, Figs. 10(a)-10(d) show the random deletion rates of 20%, 30%, 40%, and 50%, respectively. Algorithm 1 showed the highest accuracy in restoring image data, with an average restoration accuracy of over 85%. This means that Algorithm 1 can effectively recover lost information and maintain high accuracy when dealing with artificial datasets containing missing data with different proportions. The study further analyzes the impact of different node numbers on the recognition rate of Algorithm 1 on two different datasets, as shown in Fig. 11.

Fig. 11. The influence of different number of nodes on different datasets.

Fig. 11(a) displays the recognition results of the Face Scrub dataset. Fig. 11(b) displays the recognition results of the Tiny Images Dataset dataset. Algorithm 1 performed better on both datasets. As the number of nodes increased, the final recognition rate stabilized at over 80%. The recognition rates of Algorithm 2 and 3 were not high. This once again indicates that the LRM based on asymptotic regularization singular value function is more effective in the image restoration.