A tensor restoration method fusing non-convex low-rank minimization and deep prior

Abstract

The application discloses a tensor restoration method fusing non-convex low-rank minimization and deep prior, and comprises the following steps: acquiring an observed tensor, an index set corresponding to the observed tensor, a prior set and a reversible transformation; inputting the acquired data into a constructed tensor restoration model to output an initial restoration tensor; the construction of the tensor restoration model comprises the following steps: performing differential processing on a multi-order tensor to obtain a gradient tensor; constructing a TCTV regular term according to a tensor non-convex norm corresponding to the gradient tensor; building a tensor restoration model based on the TCTV regular term; and inputting the initial restoration tensor into a convolutional neural network model to output a final restoration tensor. The method enhances the restoration capability of images, and the restoration performance is more significant in a limit environment, plays a key role in dealing with high-dimensional data loss or damage, and helps to ensure the accuracy of downstream tasks based on image processing, such as face recognition and target tracking.

Description

Technical Field

[0001] This invention relates to the field of image processing technology, and more specifically to a tensor recovery method that integrates non-convex low-rank minimization and depth prior. Background Technology

[0002] Tensor recovery is a widely used data processing method across many fields, encompassing tensor data completion and denoising. A tensor is a mathematical representation of a multidimensional array, encompassing generalizations from scalars (0-dimensional), vectors (1-dimensional), and matrices (2-dimensional) to higher dimensions. Compared to matrices, tensors retain more structural information and offer better performance when handling multidimensional data. In actual data acquisition, due to complex situations such as occlusion, noise interference, and packet loss during transmission, data is often incomplete or contaminated by noise. Therefore, it is necessary to complete missing data or recover the original signal through denoising methods. Both of these tasks involve reconstructing high-dimensional tensors from partial observations or damaged data.

[0003] Tensor completion is a technique for inferring and recovering missing data from partial observation data. The mainstream method is based on low-rank decomposition models of tensors. The low-rank property of tensors is similar to that of matrices; they can be approximated by decomposing them into multiple simple factors. Techniques based on this idea include CP (CANDECOMP / PARAFAC) decomposition and Tucker decomposition. CP decomposition approximates the tensor by decomposing it into the outer product of a set of vectors, while Tucker decomposition represents the tensor by the core tensor and its product with a set of factors. These decomposition methods are widely used in theory and practice, effectively recovering the original tensor from incomplete observation data. However, tensor decomposition is not unique and has high computational complexity, posing significant challenges for low-rank tensor recovery methods when dealing with large-scale datasets.

[0004] Another commonly used tensor completion method is recovery techniques based on compressed sensing. Compressed sensing theory was initially applied to signal processing, utilizing the sparsity of data to recover signals from a small amount of sampled data. In tensor recovery, compressed sensing methods leverage the sparse representation or low-rank property of tensors to reconstruct the original tensor from partial observations. These methods typically require the tensor to be sparse in some transform domain, such as Fourier transform or discrete cosine transform, to achieve good recovery results with limited observation data. However, compressed sensing methods rely on the prior assumption of tensor sparsity, which is difficult to satisfy in certain application scenarios.

[0005] Furthermore, the nuclear norm minimization method, a generalization of matrix completion, is also widely applied in the field of tensor completion. Nuclear norm minimization utilizes the multilinear low-rank structure of tensors to achieve recovery by minimizing the tensor's nuclear norm. The advantage of this method is that it can utilize the multidimensional structural information of tensors, thus offering better recovery capabilities than simple matrix completion techniques. However, nuclear norm minimization has high computational complexity, and the tensor nuclear norm is not the most compact convex relaxation of the tensor rank, resulting in poor performance in tensor recovery.

[0006] Tensor Principal Component Analysis (TPCA) is a widely used denoising technique for high-dimensional data processing, aiming to extract the original low-dimensional structural information from noisy observation data. It achieves denoising by assuming that the original tensor data has a low-rank structure, while the noise component often exhibits high-rank components. TPCA's goal is to separate the low-rank principal components from the observation data, suppress the influence of high-rank noise, and thus recover a tensor close to the original signal. In practice, Multilinear Singular Value Decomposition (HOSVD) is a commonly used method. HOSVD decomposes the tensor along different dimensions, extracting the principal components in each dimension, a process similar to PCA's singular value decomposition (SVD) of a matrix. HOSVD effectively utilizes the multidimensional structure of tensors, helping to separate low-dimensional signals from highly noisy data. Furthermore, the nuclear norm minimization method for tensors can also constrain their low-rank nature by minimizing the nuclear norm, further removing noise. Nuclear norm minimization techniques help maintain the low-rank structure of tensors, suppress the complex high-rank components introduced by noise, and improve denoising performance.

[0007] In recent years, learning-based tensor recovery techniques have gradually become a research hotspot in the field of tensor recovery, demonstrating significant performance advantages when dealing with complex, high-dimensional data. Unlike traditional model-driven methods such as low-rank decomposition and compressed sensing, learning-based tensor recovery methods primarily rely on data-driven strategies. Through machine learning techniques such as deep learning, they automatically extract high-order structural information from the data, thereby more effectively recovering tensors. Convolutional Neural Networks (CNNs), as a common deep learning model, can effectively capture local features in tensor data. By designing a neural network architecture suitable for tensor inputs, CNNs can learn potential high-dimensional structures from partial observation data, thus recovering the complete tensor. Learning-based tensor recovery methods have high flexibility and adaptability, do not rely on explicit low-rank or sparsity assumptions, and can automatically learn complex data distributions and structures through large amounts of training data. However, these methods typically require a large amount of labeled data for training, and the model training and inference processes are relatively complex, placing high demands on computational resources.

[0008] Tensor recovery technology has been widely applied in various fields. In image and video processing, it is used for image inpainting, super-resolution reconstruction, and video completion; in medical imaging, it helps repair missing MRI data or CT images; in recommender systems, it is used to recover latent preferences from sparse user-item interaction matrices; and in wireless communication and signal processing, it is used for compressed sensing, signal reconstruction, and multidimensional channel estimation. Furthermore, tensor recovery technology is also applied to social network analysis, environmental monitoring, and multimodal data fusion, helping to extract valuable information from partial observation data.

[0009] The significance of tensor recovery technology lies in its ability to efficiently handle the prevalent problems of incomplete and noisy high-dimensional data in real-world applications. For many applications, the direct use of missing or noisy data can lead to erroneous or inefficient decisions. Through tensor completion and denoising techniques, key information can be recovered by maximizing the use of partially observed data, thereby improving model performance. This has significant practical value in applications such as recommender systems, medical diagnosis, and video restoration. The successful application of tensor recovery technology also demonstrates that low-rank and sparse structures in data are common phenomena, reflecting an underlying order or regularity. Fully utilizing these structures can not only reduce computational complexity but also significantly improve the accuracy of data processing.

[0010] The existing technique, "Guaranteed Tensor Recovery Fused Low-rankness and Smoothness," discloses a precise tensor recovery method that integrates low-rank and smoothness. However, this technique suffers from the following drawbacks: 1. The tensor kernel norm is not the most compact convex relaxation of the tensor rank, thus limiting its application in tensor recovery models. 2. The model-based method in this existing technique struggles to capture abstract prior information from tensor data. 3. Model-based methods are not only complex to tune, but also inferior to learning-based methods in terms of recovery performance and speed.

[0011] Therefore, how to enhance the ability to restore images, reduce the complexity of parameter tuning, and uncover the deeper data value of tensor data is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0012] In view of the above problems, the present invention provides a tensor recovery method that integrates non-convex low-rank minimization and depth prior, so as to at least solve some of the technical problems mentioned in the background art.

[0013] To achieve the above objectives, the present invention adopts the following technical solution:

[0014] This invention provides a tensor recovery method that integrates non-convex low-rank minimization and depth prior, comprising the following steps:

[0015] Obtain the observation tensor and the corresponding index set, prior set, and invertible transformation of the observation tensor; input the obtained data into the constructed tensor recovery model, and output the initial recovery tensor;

[0016] The construction steps of the tensor recovery model include: performing difference processing on multi-order tensors to obtain gradient tensors; constructing TCTV regularization terms based on the tensor non-convex norm corresponding to the gradient tensors; and building a tensor recovery model based on the TCTV regularization terms.

[0017] The initial recovery tensor is input into the convolutional neural network model, and the final recovery tensor is output.

[0018] Furthermore, the TCTV regular expression is represented as follows:

[0019]

[0020] Among them, ||X|| TCTV Represents the TCTV regularization term; the symbol ":=" indicates the definition; m represents the basis of the prior set; Γ represents the prior set; k represents the specific direction for calculating the gradient; ||G k || p G represents the gradient tensor k The corresponding tensor is a non-convex norm; p denotes an invertible transformation.

[0021] Furthermore, the tensor recovery model includes a tensor completion model and a tensor denoising model;

[0022] The tensor completion model is represented as follows:

[0023]

[0024] The tensor denoising model is expressed as follows:

[0025]

[0026] Where X represents a multi-order tensor; Ω represents the index set corresponding to the true tensor X0 to be recovered; Ω ⊥ K represents the orthogonal complement of the index set Ω; K represents the restriction to Ω. ⊥ To fill in the missing index value; δ Ω (K) represents the indicator function; ▽ k (·) denotes the difference operator in the multi-order tensor X along each mode; P(·) denotes the projection operator; β denotes the trade-off parameter; M denotes the observation tensor with sparse noise E; ||E||1 denotes the use of the 1 norm to characterize sparsity.

[0027] Furthermore, the output initial recovery tensor specifically includes:

[0028] The tensor recovery model is solved using the alternating direction multiplier method. If the relative error between the intermediate recovery tensor of the current round and the observed tensor is less than a preset value, then the intermediate recovery tensor of the current round is used as the initial recovery tensor.

[0029] Furthermore, the initial recovery tensor is input into the convolutional neural network model, specifically including:

[0030] For the completion task, the initial recovery tensor is randomly overlaid with 20% of the pixel values ​​at the missing positions, and fixed random noise of the same size as the initial recovery tensor is added before being input into the convolutional neural network model.

[0031] For the noise reduction task, fixed random noise of the same size as the initial recovery tensor is added to the initial recovery tensor and then input into the convolutional neural network model.

[0032] Furthermore, the convolutional neural network model incorporates a plug-and-play attention module.

[0033] Furthermore, it also includes:

[0034] Based on the objective function of the completion task or the noise reduction task, the mean square error loss is calculated according to the final recovered tensor and the observed tensor in the current round;

[0035] The mean squared error loss is used for backpropagation to update the parameters of the convolutional neural network, thereby optimizing the convolutional neural network model.

[0036] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a tensor recovery method that integrates non-convex low-rank minimization and depth prior, which has the following beneficial effects:

[0037] This invention integrates two major categories of tensor restoration techniques: tensor restoration models and neural network learning. It aims to improve the framework's restoration performance by characterizing explicit prior information through non-convex regularization terms and capturing implicit prior information through neural networks. This method enhances image restoration capabilities, exhibiting significantly improved performance in extreme environments and playing a crucial role in addressing the loss or corruption of high-dimensional data. It also helps ensure the accuracy of downstream image processing tasks such as face recognition and object tracking.

[0038] The tensor recovery model constructed in the first stage of this invention encompasses tensor completion and tensor denoising techniques, providing an effective solution to data loss or noise contamination issues. By preserving the high-dimensional structural features of tensor data, this type of method can extract potential global information from incomplete or noisy data, thereby achieving more accurate recovery. Its advantages not only lie in filling in data gaps and improving data integrity, but also in enhancing the accuracy and reliability of data during the denoising process.

[0039] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0040] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0041] Figure 1 This is a schematic diagram of the tensor recovery method that integrates non-convex low-rank minimization and depth prior, provided in an embodiment of the present invention.

[0042] Figure 2 A schematic diagram of the two-stage tensor recovery model GTTR structure provided in an embodiment of the present invention.

[0043] Figure 3 This is a schematic diagram illustrating the differences in the first-stage recovery results provided in an embodiment of the present invention.

[0044] Figure 4 This is a schematic diagram of the convolutional neural network model structure provided in an embodiment of the present invention. Detailed Implementation

[0045] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0046] This invention discloses a tensor recovery method that integrates non-convex low-rank minimization and depth prior. See [link to relevant documentation]. Figure 1 As shown, it includes:

[0047] Phase 1: Obtain the observation tensor and its corresponding index set, prior set, and invertible transformation; input the acquired data into the constructed tensor recovery model and output the initial recovered tensor; the construction steps of the tensor recovery model include: performing difference processing on the multi-order tensor to obtain the gradient tensor; constructing the TCTV regularization term based on the tensor non-convex norm corresponding to the gradient tensor; and building the tensor recovery model based on the TCTV regularization term.

[0048] The second stage involves inputting the initial recovery tensor into the convolutional neural network model and outputting the final recovery tensor.

[0049] In the field of image processing technology, the aforementioned observation tensor is the observation image, which refers to an image that is known, partially missing, or affected by noise in the true tensor to be recovered. It usually contains some known pixel values ​​and some unknown or damaged pixel values. The aforementioned index set refers to the set of indexes of known pixel values ​​in the observation image, which specifies which pixel positions have known values. The aforementioned prior set is a set of predefined information reflecting the structural properties of the image. This information usually reflects the smoothness or low rank of the image in different directions. The aforementioned reversible transformation is used to transform the image to another domain for better processing and recovery.

[0050] This invention constructs a generalized two-stage tensor recovery model (GTTR) (e.g., Figure 2 As shown, this method integrates two major categories of tensor restoration techniques: tensor restoration models and neural network learning. It aims to improve the framework's restoration performance by characterizing explicit prior information through non-convex regularization terms and capturing implicit prior information through neural networks. This method enhances image restoration capabilities, exhibiting significantly improved performance in extreme environments and playing a crucial role in addressing the loss or corruption of high-dimensional data. It also helps ensure the accuracy of downstream image processing tasks such as face recognition and object tracking.

[0051] In this embodiment of the invention, the method further includes updating the convolutional neural network model: based on the objective function of the completion or denoising task, the mean squared error loss is calculated according to the final recovered tensor and the observed tensor in the current round; the mean squared error loss is used for backpropagation to update the parameters of the convolutional neural network, thereby optimizing the convolutional neural network model. The process stops after a preset number of iterations, and based on the final network parameters, the most realistic image possible is recovered, further improving the recovery performance of the two-stage tensor restoration model GTTR. See also... Figure 2 As shown, the intermediate observations θ0 represents the initial recovery tensor; θ0 represents the initial parameters of the network. This means that backpropagation is performed based on the gradient calculated from the loss function to update the network parameters; This represents a new image generated by receiving a fixed noise input Z through parameters on the current network f.

[0052] Next, the above content will be explained in detail.

[0053] In the first stage described above, the construction steps of the tensor recovery model specifically include the following:

[0054] (1) Perform a difference operation on the tensor to obtain the gradient tensor:

[0055] For multi-order tensors By performing a difference operation on each mode, the gradient tensor G is obtained. k (k = 1, 2, ..., d); where d represents the d-th dimension of the tensor; n d The length of the d-th dimension is represented by k; k represents the specific direction in which the gradient is calculated. For example, when k = d, it means that the gradient is calculated along the d-th dimension.

[0056] (2) Design a unified tensor nonconvex norm:

[0057] Since the tensor kernel norm is not the tightest convex relaxation of the tensor rank, a series of non-convex norms (including L) are used in the embodiments of the present invention. p Further approximations are made using the following norms: SCAD norm, Logarithm norm, MCP norm, Capped L1 norm, ETP norm, Geman norm, and Laplace norm. For this purpose, for a d-order tensor... Under a specific invertible transformation P (such as the discrete Fourier transform or the discrete cosine transform), this invention defines a unified tensor nonconvex norm, expressed as:

[0058]

[0059] Among them, ||·|| non Represents a specific nonconvex norm of the matrix; l refers to the scaling factor corresponding to a specific P transformation; i d It is a variable, ranging from 1 to n. d ;X p This represents a front slice of a tensor under a specific P-transformation.

[0060] (3) Construct the TCTV regularization term based on the tensor nonconvex norm corresponding to the gradient tensor:

[0061] Unlike existing technologies, this invention does not characterize prior information on the original tensor X, but rather on the gradient tensor. Based on this, this invention constructs a TCTV regularization term using a unified tensor nonconvex norm corresponding to the gradient tensor, which simultaneously characterizes the low-rank and smoothness of the gradient tensor. This TCTV regularization term is expressed as:

[0062]

[0063] Among them, ||X|| TCTV Represents the TCTV regularization term; the symbol ":=" indicates the definition; m represents the basis of the prior set; Γ represents the prior set; k represents the specific direction for calculating the gradient; ||G k || p G represents the gradient tensor k The corresponding tensor is a non-convex norm; p denotes an invertible transformation.

[0064] The prior set Γ is specifically set according to different types of data. Taking images as an example, it is generally assumed that images have local smoothness in the spatial direction. Here, the spatial direction is represented by the numbers 1 and 2, so for images, Γ = {1,2}. Hyperspectral images, in addition to spatial smoothness, also have spectral smoothness, so Γ is set to {1,2,3}, where the number 3 represents the spectral direction. Color videos, in addition to spatial smoothness, also have temporal smoothness, so Γ can be set to {1,2,4}, where the number 4 represents the temporal direction.

[0065] (4) Based on the TCTV regularization term, a tensor recovery model is constructed; the tensor recovery model in this embodiment of the invention includes a tensor completion model (TC) and a tensor denoising model (TRPCA); wherein:

[0066] The tensor completion model is represented as:

[0067]

[0068] The tensor denoising model is represented as:

[0069]

[0070] Where X represents a multi-order tensor; Ω represents the index set corresponding to the true tensor X0 to be recovered; Ω ⊥ K represents the orthogonal complement of the index set Ω; K represents the restriction to Ω. ⊥ The value is used to fill in the missing index position; δ Ω (K) represents the indicator function; ▽ k (·) denotes the difference operator in each mode of the multi-order tensor X; P(·) denotes the projection operator; β denotes the trade-off parameter; M denotes the observed tensor with sparse noise E; ||E||1 denotes the use of the 1 norm to characterize sparsity;

[0071] Where the indicator function δ Ω (K) is defined as:

[0072]

[0073] In the first stage described above, after constructing the tensor recovery model, the gradient tensor, augmented Lagrange multipliers, and other variables used in solving the tensor recovery model are initialized; the observation tensor and its corresponding index set, prior set, and invertible transformation are obtained; and the obtained data are input into the constructed tensor recovery model.

[0074] The Alternating Directional Multiplier Method (ADMM) is used to alternately control other variables and solve for the required variables. If the relative error between the intermediate recovery tensor and the observed tensor in the current round is less than the preset value, the algorithm is considered to have converged, and the intermediate recovery tensor in the current round is used as the initial recovery tensor.

[0075] In the data transformation from the first stage to the second stage, for the completion task, experiments revealed that approximately 80% of the pixel values ​​had very small errors compared to the true values, while 20% of the pixel values ​​still had significant errors (e.g., Figure 3 (As shown). Therefore, in this embodiment of the invention, the initial restored tensor output from the first stage is again randomly masked by 20% of the pixel values ​​at the missing positions, and fed together with fixed random noise (of the same size as the initial restored tensor) into the convolutional neural network model in the second stage as the observation tensor for further completion. For the denoising task, the initial restored tensor and the fixed random noise (of the same size as the initial restored tensor) are fed together into the convolutional neural network model in the second stage for further denoising. The impact of the two stages on the GTTR restoration performance is shown in Table 1 below:

[0076] Table 1 Ablation Experiment Results of Tensor Recovery Model and Convolutional Neural Network Model

[0077] Data types index Phase 1 Phase Two Overall effect Color images Peak signal-to-noise ratio 24.71 24.97 27.09 Magnetic resonance imaging Peak signal-to-noise ratio 27.21 28.22 29.19 Grayscale video Peak signal-to-noise ratio 22.65 22.74 23.47

[0078] In the second stage described above, untrained and randomly initialized convolutional neural networks (including ResNet, U-Net, Skip, etc.) are used, with fixed random noise of the same size as the initial recovery tensor as the input for each iteration. Plug-and-play attention modules (including CBAM, SENET, SKNET, GCB, scSE) are integrated into the convolutional neural network model to enhance its feature extraction capabilities. Specifically, the insertion positions of the attention modules in the convolutional neural network model are as follows: Figure 4 As shown; the structure of the convolutional neural network model after incorporating the attention module includes a residual block (ResBlock) and a U-net downsampling block (UnetDownBlock); where:

[0079] In the residual block, the input data sequentially passes through the first batch normalization layer, the first 3x3 convolutional layer, the first ReLU activation function, the second batch normalization layer, the second 3x3 convolutional layer, and the first attention layer. The output of the first attention layer is added to the input data of the first batch normalization layer and then output through the second ReLU activation function. In the U-net downsampling block, the input data sequentially passes through the max pooling layer, the third 3x3 convolutional layer, the fourth 3x3 convolutional layer, and the second attention layer.

[0080] The results of the ablation experiments on the attention module are shown in Table 2 below:

[0081] Table 2 Ablation Experiment Results of Attention Module

[0082] Network architecture non-attention mechanism CBAM SENET SKNET GCB scSE U-Net 35.44 35.53 35.52 35.49 35.49 35.55 ResNet18 35.36 35.40 35.49 35.44 35.49 35.42 ResNet34 35.38 35.45 35.44 35.48 35.48 35.48 skip 35.55 35.72 35.66 35.74 35.74 35.74

[0083] Next, specific embodiments will be used to verify the effectiveness of the tensor recovery method that integrates non-convex low-rank minimization and depth prior provided by the present invention.

[0084] The method provided by this invention can be applied to multiple fields such as image processing, knowledge graphs, and signal processing. Taking image restoration as an example, all experiments were completed in MATLAB (R2021a) and PyCharm. The computer was configured with an AMD Ryzen 7 5800H CPU with Radeon Graphics (16 CPUs) and 16GB of RAM.

[0085] In the tensor completion task, the experimental datasets used were two color image datasets (BSD and USC-SIPI), a medical magnetic resonance imaging (MRI) spine dataset, and a grayscale video dataset. Completion was performed under six different missing rate conditions: 70%, 80%, 90%, 95%, 99%, and 99.5%. The recovery was evaluated using three common metrics: PSNR, SSIM, and FSIM. The average experimental results on the color image dataset are shown in Table 3. Comparison with other completion methods shows that the two-stage tensor restoration model GTTR in this invention achieved the best completion effect, improving PSNR by approximately 0.68 dB, 1.27 dB, and 0.65 dB on the color image, MRI, and grayscale video datasets, respectively.

[0086] Table 3. Completion results under different missing rates

[0087]

[0088]

[0089] In the tensor denoising task, salt-and-pepper noise ratios of 0.6 and 0.8 were added, and the specific recovery results are shown in Table 4. By comparing with other denoising methods, it can be seen that the two-stage tensor recovery model GTTR in this invention has the best denoising performance. Under high-noise environments with added noise ratios of 0.6 and 0.8, the PSNR is improved by approximately 4.07 dB and 9.37 dB, respectively.

[0090] Table 4. Noise reduction results under different noise conditions

[0091]

[0092] In summary, this invention provides a tensor restoration method that integrates non-convex low-rank minimization and depth prior. In the first stage, a non-convex regularization term based on the gradient tensor is proposed. Since low-rank and smooth structures are consistent between the original tensor and the gradient tensor, this invention uses a generalized tensor non-convex norm to characterize the low-rank and smooth prior information of the gradient tensor. The use of the gradient tensor helps improve the spatial smoothness of the data, and the non-convex regularization provides a more accurate tensor rank approximation. Then, this invention inputs the restoration results of the first stage into the neural network model of the second stage. Multiple plug-and-play attention modules are inserted into the traditional neural network structure, improving the network's feature extraction capability. Through iterative optimization, the neural network of the second stage further improves the image restoration effect. Actual tensor restoration experiments demonstrate that this technique significantly improves the model's restoration performance, robustness, flexibility, and generalization ability.

[0093] The tensor recovery model constructed in the first stage of this invention encompasses tensor completion and tensor denoising techniques, providing an effective solution to data loss or noise contamination issues. By preserving the high-dimensional structural features of tensor data, this type of method can extract potential global information from incomplete or noisy data, thereby achieving more accurate recovery. Its advantages lie not only in filling data gaps and improving data integrity but also in enhancing the accuracy and reliability of data during denoising. The application of tensor recovery technology helps enhance the reliability and accuracy of data analysis results, which is particularly important for fields such as image processing, video restoration, and recommendation systems. It significantly improves the performance of applications in these fields, facilitates the extraction of deeper data value from tensor data, and provides a more solid foundation for decision-making. The tensor recovery model can also effectively address the challenges of high-dimensional data, improve data processing efficiency, and reduce computational resource consumption, thereby optimizing resource allocation while ensuring data quality.

[0094] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0095] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A tensor recovery method that integrates non-convex low-rank minimization and depth prior, characterized in that, Includes the following steps: Obtain the observation tensor and its corresponding index set, prior set, and invertible transformation; input the acquired data into the constructed tensor reconstruction model and output the initial reconstruction tensor; the observation tensor is the observation image; the index set refers to the set of indices of known pixel values ​​in the observation image; the prior set is a predefined set of information reflecting the structural properties of the image; Reversible transformations are used to convert images to another domain for processing and recovery; The construction steps of the tensor recovery model include: performing difference processing on multi-order tensors to obtain gradient tensors; constructing TCTV regularization terms based on the tensor non-convex norm corresponding to the gradient tensors; and building a tensor recovery model based on the TCTV regularization terms. The initial recovery tensor is input into the convolutional neural network model, and the final recovery tensor is output. The TCTV regular expression is represented as follows: in, This represents the TCTV regular expression. m Denotes the basis of the prior set; Represents the prior set; k Indicates the specific direction of the gradient calculation; G represents the gradient tensor k The corresponding tensor is a non-convex norm; p denotes an invertible transformation; The tensor recovery model includes a tensor completion model and a tensor denoising model; The tensor completion model is represented as follows: The tensor denoising model is expressed as follows: Where X represents a multi-order tensor; Indicates the tensor to be recovered The corresponding index set; Represents the index set The orthogonal complement of the set; K represents the restriction on To fill in the missing index values; Indicates an indicator function; This represents the difference operator in a multi-order tensor X along each mode; Represents the projection operator; represents the trade-off parameter; M represents the observation tensor with sparse noise E; This indicates that the 1-norm is used to characterize sparsity.

2. The tensor recovery method according to claim 1, which integrates non-convex low-rank minimization and depth prior, is characterized in that... The output initial recovery tensor specifically includes: The tensor recovery model is solved using the alternating direction multiplier method. If the relative error between the intermediate recovery tensor of the current round and the observed tensor is less than a preset value, then the intermediate recovery tensor of the current round is used as the initial recovery tensor.

3. The tensor recovery method according to claim 1, which integrates non-convex low-rank minimization and depth prior, is characterized in that; The initial recovery tensor is input into the convolutional neural network model, specifically including: For the completion task, the initial recovery tensor is randomly overlaid with 20% of the pixel values ​​at the missing positions, and fixed random noise of the same size as the initial recovery tensor is added before being input into the convolutional neural network model. For the noise reduction task, fixed random noise of the same size as the initial recovery tensor is added to the initial recovery tensor and then input into the convolutional neural network model.

4. The tensor recovery method according to claim 1, which integrates non-convex low-rank minimization and depth prior, is characterized in that... The convolutional neural network model incorporates a plug-and-play attention module.

5. The tensor recovery method according to claim 1, which integrates non-convex low-rank minimization and depth prior, is characterized in that... Also includes: Based on the objective function of the completion task or the noise reduction task, the mean square error loss is calculated according to the final recovered tensor and the observed tensor in the current round; The mean squared error loss is used for backpropagation to update the parameters of the convolutional neural network, thereby optimizing the convolutional neural network model.

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