A multi-dimensional signal reconstruction method and system of a tensor sparse optimization deep unfolding network

Abstract

This invention discloses a method and system for multidimensional signal reconstruction using tensor sparse optimization deep unfolding networks, belonging to the fields of signal processing and image processing technology. The method includes: inputting a multidimensional signal tensor to be reconstructed; initializing the multidimensional signal tensor to be reconstructed with sparse coefficient tensors and learnable parameters of the network; based on the multidimensional signal tensor to be reconstructed and the initialization results, continuously optimizing the sparse coefficient tensor through proximal gradient updates learned iteratively by K layers of the network, obtaining the optimized sparse coefficient tensor; reconstructing the reconstructed signal based on the learnable reconstruction operator learned by the K layers of the network and the optimized sparse coefficient tensor; and outputting the reconstructed multidimensional signal tensor. This invention uses tensor modal product at each layer to implement learnable Kronecker-free reconstruction / residual projection operators for analytical proximal gradient updates and soft thresholding, thereby directly learning sparse coefficients in the tensor domain, avoiding explicit vectorization and Kronecker operations.

Description

Technical Field

[0001] This invention relates to a method for multidimensional signal reconstruction using tensor sparse optimized deep unfolding networks, belonging to the fields of signal processing and image processing technology, and particularly to the application of deep unfolding networks and tensor computation in multidimensional signal recovery. Background Technology

[0002] Multidimensional signal reconstruction is one of the core tasks in fields such as image processing, video analysis, and remote sensing. As an important method for solving this problem, sparse recovery can reconstruct multidimensional data from undersampled or degraded observations by mining the sparse priors of the signal in a specific transform domain. Considering the structural characteristics of multidimensional data, tensor sparse representation provides a more natural and efficient representation form for sparse recovery of multidimensional data by constructing cross-modal coupling relationships in the tensor domain.

[0003] Meanwhile, Deep Unfoldable Networks (DUNs), as an effective paradigm, combine the interpretability of optimization algorithms with the expressive power of deep learning by unfolding the traditional iterative solution process into a trainable hierarchical structure, achieving significant success in various imaging inverse problems. Based on the modeling dimension and whether it is unfolded into a learning network, existing sparse recovery methods can be divided into optimization methods, one-dimensional deep unfoldable networks, and multi-dimensional deep unfoldable networks.

[0004] Early optimization-based sparse recovery methods, such as various greedy tracing, convex relaxation optimization, and dictionary learning methods, typically employed vectorized modeling to directly reuse mature vector domain optimization theory. These methods performed well in interpretability and small-scale problems, but generally required numerous iterations and converged slowly. Furthermore, when dealing with multidimensional tensor data, their flattening operations disrupted the intermodal structure, leading to a sharp increase in parameter size and computational cost. To address this, tensor-based optimization methods utilized techniques such as tensor decomposition to directly learn the sparse characteristics of the data in the tensor domain, better preserving the intrinsic relationships of its multidimensional structure. However, limited by the insufficient learning capacity of iterative optimization, their convergence remained slow.

[0005] To overcome the convergence bottleneck of optimization-based methods, one-dimensional deep unfoldable networks (DUNs) approximate iterative solvers within a fixed depth through network learning, significantly accelerating the inference process and improving convergence speed. However, they still suffer from vectorization limitations, severely restricting their scalability in real-world multidimensional problems. To address this, DUNs based on Convolutional Sparse Coding (CSC) replace the global vectorized matrix with local convolution operators, preserving the interpretability of the unfolded network while adapting to the spatial structure of two-dimensional images. Subsequent work further integrates strong sensing modules such as U-Net and Transformer into the unfolded network framework, significantly improving signal reconstruction in complex scenes through strategies such as cross-stage information transfer and global attention mechanisms. However, these methods generally rely on implicit neural network module modeling, sacrificing the native interpretability of the unfolded network, while significantly increasing parameters and computational overhead. Furthermore, they remain within the vector / matrix domain modeling framework, making it difficult to fully explore the cross-modal correlations of multidimensional signals.

[0006] Multidimensional deep expandable networks preserve the complete structure of data by modeling the unfolding algorithm in the tensor domain. Existing methods mainly design networks using tensor low-rank priors, element-level masks (Hadamard), and separable Kronecker factorization dictionaries, balancing structure preservation, learnability, and accelerated convergence, but with trade-offs between expressiveness, generalization, and parameter / computational efficiency. Tensor priors and Hadamard products can improve reconstruction quality, but their explicit modeling capabilities are limited and they are highly dependent on specific tasks. While separability / factorization methods can reduce dimensionality costs, they often require flattening / Kronecker operations in implementation or theoretical analysis, thus retaining some of the limitations of vectorization methods. Summary of the Invention

[0007] The purpose of this invention is to provide a multidimensional signal reconstruction method and system for tensor sparse optimization deep unfolded networks. This method uses tensor modal product at each layer to implement learnable Kronecker-free reconstruction / residual projection operators for analytical proximal gradient updates and soft thresholding, thereby directly learning sparse coefficients in the tensor domain and avoiding explicit vectorization and Kronecker operations. This invention significantly reduces the number of parameters and computational overhead while preserving the interpretability of the optimization model. It achieves performance comparable to state-of-the-art methods in tasks such as synthetic sparse coding and multispectral image denoising, and also features faster convergence speed and better fine-grained texture recovery capabilities.

[0008] To address the aforementioned technical problems, this invention provides a method for multidimensional signal reconstruction using tensor sparse optimized deep unfolded networks, comprising the following steps: Input the multidimensional signal tensor to be reconstructed; The sparse coefficient tensor and network learnable parameters are initialized to obtain the initialization results for the multidimensional signal tensor to be reconstructed. Based on the multidimensional signal tensor to be reconstructed and the initialization results, the sparse coefficient tensor is continuously optimized by updating the proximal gradient through K layers of network iterative learning, and the optimized sparse coefficient tensor is obtained. The reconstructed signal is obtained by reconstructing the multidimensional signal tensor based on the learnable reconstruction operator learned through K-layer iterative learning of the network and the optimized sparse coefficient tensor. Output the reconstructed multidimensional signal tensor.

[0009] Preferably, the initialization of the sparse coefficient tensor and the network learnable parameters includes the following steps: Learnable threshold parameter in network learnable parameters Using theoretical reference step size Initialization; initial weights of the modal product learnable operator Depend on Initialization; Initial coefficient tensor Approximate inverse mapping of the observation tensor through lexicographical transpose The result is obtained by back projection; that is To measure the quality of network reconstruction and constrain the sparsity of coefficients over the observation domain, a method based on MSE and... The total loss function composed of sparse terms weighted together is as follows: (7) in, For sample index, This represents the number of samples within a batch. This represents the total number of elements in a single observation sample. For the network to learn the multi-mode reconstruction operator through the last layer From coefficient tensor The reconstructed signal is recovered. The weighting hyperparameters are used to balance the reconstruction error and the sparsity term.

[0010] Preferably, based on the multidimensional signal tensor to be reconstructed and the initialization results, the sparse coefficient tensor is continuously optimized through K-layer iterative proximal gradient updates to obtain the optimized sparse coefficient tensor. Specifically, this includes the following steps: Using TenSR as the basic signal model; given any multidimensional signal Assume that it satisfies tensor sparsity constraints under a given multidimensional dictionary; The tensor gradient of tensor sparsity constraint is obtained by using the n-mode chain rule, and the iterative update formula of TISTA is obtained based on the idea of ​​proximal gradient descent and soft threshold function. The TISTA iterative process is mapped to a deep unfoldable network structure composed of learnable multimodal operators; The dictionary-related operations in the tensor iteration formula are replaced with learnable modal integration operators, and the fixed threshold is replaced with a learnable threshold to construct the TSR-LISTA network layer update rules; By stacking the update rules K times, we obtain the TSR-LISTA network, which maps the TISTA algorithm in the tensor domain to a learnable multidimensional deep unfoldable network for tensor sparse recovery.

[0011] Preferably, the tensor sparsity recovery problem is expressed as: (1) in, and The dimension representing the observation space and coefficient space ( ), , For the first Model dictionary For sparse representation coefficients in a multidimensional dictionary, yes .

[0012] Preferably, the iterative update formula for TISTA is: (2) in .

[0013] Preferably, mapping the TISTA iterative process to a deep unfoldable network structure composed of learnable modal product operators specifically includes the following steps: Substituting the tensor gradient into its update formula (2), expanding the gradient term and simplifying it, we obtain the first... Modal product expansion of tensor coefficient updates in the next iteration: (3) Both terms in the gradient term contain dictionary transpose operations. These are combined to extract the common residual projection term, which is then used to rewrite the gradient term. (4) Among them, step size The soft threshold operator determines the magnitude of gradient descent. This ensures the sparsity constraint.

[0014] Preferably, the dictionary-related operations in the tensor iteration formula are replaced with learnable modal integration operators, and the fixed threshold is replaced with a learnable threshold to construct the TSR-LISTA network layer update rule, specifically including the following steps: Replace the dictionary-related operations in equation (4) with learnable parameters, and set a fixed threshold. Replace with learnable threshold We obtain the update rules for the k-th layer of TSR-LISTA; (5) in, and denoted as the learnable Kronecker product-free reconstruction operator and the residual projection operator, respectively.

[0015] Preferably, the and Both are achieved through modal product, and their specific definitions are as follows: (6) in, These are the multidimensional learnable weight parameters within the operator; Characterizing the self-constraint relationship of the coefficient tensor of the k-th layer along each dimension enables adaptive learning of cross-modal correlation; Observations The residuals between the reconstruction results and the coefficient domain are mapped back to the coefficient domain to adaptively adjust the cross-modal error feedback. For the step size parameter L, it is set as a global constant as a theoretical reference step size for initializing the network parameters. .

[0016] Preferably, the set of learnable parameters is denoted as... Optimization is achieved through end-to-end training.

[0017] This invention also provides a multidimensional signal reconstruction system using tensor sparse optimized deep unfolded networks, comprising: The acquisition module is used to input the multidimensional signal tensor to be reconstructed; The initialization module is used to initialize the sparse coefficient tensor and the learnable parameters of the network for the multidimensional signal tensor to be reconstructed, and obtain the initialization results. The optimization module is used to continuously optimize the sparse coefficient tensor based on the multidimensional signal tensor to be reconstructed and the initialization results, through the proximal gradient update learned by K layers of the network, and obtain the optimized sparse coefficient tensor. The reconstruction module is used to reconstruct the reconstructed signal based on the learnable reconstruction operator and the optimized sparse coefficient tensor after K-layer iterative learning of the network, and obtain the reconstructed multidimensional signal tensor. The output module is used to output the reconstructed multidimensional signal tensor.

[0018] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention proposes a novel multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks. The proposed TSR-LISTA network constructs Kronecker-free reconstruction and residual projection operators through tensor n-modulus product, iteratively unfolding TISTA into a learnable multi-layer network structure, thus achieving proximal gradient updates in the native tensor domain. The key technical points and protection points of this invention are as follows: Kronecker-free operator design based on n-modulus product: Learnable reconstruction and residual projection operators are constructed by tensor n-modulus product, which completely avoids vectorization and Kronecker product operation, preserves the complete structure of multidimensional signal, and significantly reduces parameter coupling and computational redundancy.

[0019] Interpretable Deep Unfolding Architecture: Each step of the TISTA iterative algorithm is precisely mapped to a layer of the network, and the operation of each layer has a clear correspondence with the classic proximal gradient update, thus preserving the interpretability of the optimization model.

[0020] Modal-level parameter learning mechanism: Network parameters are learned modally, and the number of parameters grows linearly with the tensor order, rather than exponentially like the vectorization method, which greatly improves the scalability of the model in high-order tensor scenarios. Attached Figure Description

[0021] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0022] Figure 1 This is a network structure diagram of a multidimensional signal reconstruction method based on tensor sparse optimization depth expansion network. Figure 2 This is a flowchart of a multidimensional signal reconstruction method based on tensor sparse optimization depth unfolding network; Figure 3 This is a comparison of the sparse coding curves of each model. The vertical axis represents the NMSE of the sparse code, and the horizontal axis represents the number of iterations or network layers. (a) The figure shows a size of... Two-dimensional patch ( (a) In case (b), the figure represents a size of 3D cube ( )Condition; Figure 4 This is a comparison of the reconstruction quality (mean PSNR, in dB) of various depth-expandable algorithms on the Columbia multispectral image dataset with FLOPs; Figure 5The images are (a) an image of the 'cloth' band at 430 nm (top) and an image of the 'chart and stuffed toy' band at 670 nm (bottom); (b)-(l) the recovery results of the methods in Table 1 on images contaminated with Gaussian noise (σ = 30). The calibrated areas in each image were magnified 5 times and had the same contrast for easy observation of details. Detailed Implementation

[0023] Numerous specific details are set forth in the following description to provide a full understanding of the invention. However, the invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0024] The terminology used in one or more embodiments of this specification is for the purpose of describing particular embodiments only and is not intended to be limiting of the one or more embodiments of this specification. The singular forms “a,” “described,” and “the” as used in one or more embodiments of this specification and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used in one or more embodiments of this specification refers to and includes any or all possible combinations of one or more associated listed items.

[0025] It should be understood that although the terms first, second, etc., may be used to describe various information in one or more embodiments of this specification, such information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, first may also be referred to as second without departing from the scope of one or more embodiments of this specification, and similarly, second may also be referred to as first. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to a determination."

[0026] The present invention will now be described in further detail with reference to the accompanying drawings: Definitions of abbreviations and key terms in this invention: Multidimensional signal reconstruction: The process of recovering the original multidimensional signal from undersampled, noise-contaminated, or otherwise degraded observations. Tensor sparse representation: Represents a multidimensional signal as a modal product of a multidimensional dictionary and a sparse core tensor, directly mining the sparsity characteristics of the signal in the tensor domain. Deep expandable networks: Map each step of a traditional iterative optimization algorithm to a layer of a neural network, forming a trainable end-to-end network structure that combines the interpretability of optimization algorithms with the expressive power of deep learning. n Modular product: The multiplication operation between a tensor and a matrix in a specific mode. It is one of the basic operations in the tensor field and can completely preserve the multidimensional structure of the tensor. Proximal gradient descent: an optimization algorithm that combines gradient descent and proximal operators, suitable for solving convex optimization problems with regularization terms. Soft threshold operator: An operator used in proximal gradient descent to implement L1 regularization, which sets elements smaller than a threshold to zero, thus achieving sparsity constraints. Kronecker-free: Avoids Kronecker product operations, thus preventing parameter coupling and computational redundancy.

[0027] This invention provides a method for multidimensional signal reconstruction using tensor sparse optimized deep unfolded networks, the network structure of which is as follows: Figure 1 As shown, this network expands the Tensor Iterative Soft Thresholding Algorithm (TISTA) into a K-layer learnable deep network structure, and uses n-modulus to construct Kronecker-free reconstruction operators and residual projection operators, thereby achieving efficient reconstruction of multidimensional signals.

[0028] The overall process of this method is as follows: Figure 2 As shown, the first input is the multidimensional signal tensor to be reconstructed. Initialize the sparse coefficient tensor And network parameters; then optimize the sparse coefficient tensor layer by layer through K-layer iterative proximal gradient updates. Specifically, within each layer, the input tensor Learnable Kronecker-free reconstruction and residual projection operators implemented with modal product The system interacts to model the reconstruction error and uses backpropagation via residual connections to update the coefficients, thereby generating a sparse coefficient tensor sequence layer by layer. Finally, the output sparsity coefficients Through the learned reconstruction operator Reconstruction yields the recovered signal .

[0029] Main steps: S1 Input the multidimensional signal tensor to be reconstructed S2 Initializes the sparse coefficient tensor and network learnable parameters S3 optimizes the sparse coefficient tensor through K-layer iterative proximal gradient updates. S4 recovers the reconstructed signal from the optimized sparse coefficients using the reconstruction operator of the final layer. S5 outputs the reconstructed multidimensional signal tensor I. In-depth exploration of TISTA (TSR-LISTA core module) Step S3 is the core step of the reconstruction method proposed in this invention, and its main contents are described as follows: This invention uses TenSR as the basic signal model. Given any multidimensional signal... Assuming that it satisfies tensor sparsity constraints under a given multidimensional dictionary, the tensor sparsity recovery problem can be expressed as: (1) in and The dimension representing the observation space and coefficient space ( ), , For the first Model dictionary For sparse representation coefficients in a multidimensional dictionary, yes .

[0030] The tensor gradient of equation (1) is obtained using the n-mode chain rule, and the iterative update formula of TISTA is obtained based on the idea of ​​proximal gradient descent and soft threshold function: (2) in .

[0031] Building upon this foundation, and to further enhance the learnability and expressive power of the model, this invention is inspired by deep expandable networks such as LISTA, and maps the TISTA iterative process to a deep expandable network structure composed of learnable modal product operators.

[0032] First, substitute the tensor gradient into its update formula (2), expand the gradient term, and simplify to obtain the first... Modal product expansion of tensor coefficient updates in the next iteration: (3) Among them, step size The soft threshold operator determines the magnitude of gradient descent. This ensures sparsity constraints. Furthermore, noting that both terms in the gradient term contain dictionary transpose operations, this invention merges them and extracts the common "residual projection" term to rewrite the gradient term: (4) The update process can be clearly broken down into two parts: residual calculation. Back projection with residuals This form is consistent with the vector form of the ISTA structure, but it is executed entirely in the tensor domain. To expand TISTA into a learnable network, this invention replaces the dictionary-related operations in equation (4) with learnable parameters and uses a fixed threshold. Replace with learnable threshold The k-th layer of TSR-LISTA is obtained. The update rules (corresponding to) Figure 1 (a stage in) (5) in, and Let represent the learnable Kronecker-free reconstruction operator and the residual projection operator, respectively, both implemented through modal product, a native and efficient tensor operation. Their specific definitions are as follows: (6) These are the multidimensional learnable weight parameters within the operator. Characterizing the self-constraint relationship of the coefficient tensor of the k-th layer along each dimension enables adaptive learning of cross-modal correlation; Observations The residuals between the reconstruction results and the coefficient domain are mapped back to the coefficient domain, adaptively adjusting the cross-modal error feedback. For the step size parameter L, it is fully absorbed into the learnable residual projection operator. In this invention, it is set as a global constant as a theoretical reference step size for initializing network parameters. That is, Stacking equation (5) K times yields the TSR-LISTA network shown in Figure 1. This network maps the TISTA algorithm in the tensor domain to a learnable, multidimensional, deep, unfoldable network for tensor sparse recovery. The set of learnable parameters is denoted as... The proposed framework is optimized through end-to-end training. While retaining the interpretability of traditional optimization algorithms, it utilizes Kronecker product tensor modeling, achieving better parameter efficiency and computational potential compared to existing methods.

[0033] II. Network Initialization and Loss Function In step S2, some parameters of the network need to be initialized and the loss function required for training needs to be set. The main contents are described below: To ensure that the network both adheres to the parsing semantics of ISTA and has good trainability, a learnable threshold parameter can be used. Using theoretical reference step size Initialization; initial weights of the modal product learnable operator Depend on Initialization, typically using a DCT / ODCT dictionary; initial coefficient tensor Approximate inverse mapping of the observation tensor through lexicographical transpose By performing a back projection, we obtain... To measure the quality of network reconstruction and constrain the sparsity of coefficients in the observation domain, this paper uses a method based on MSE and The total loss function composed of sparse terms weighted together is as follows: (7) in, For sample index, This represents the number of samples within a batch. This represents the total number of elements in a single observation sample. For the network to learn the multi-mode reconstruction operator through the last layer From coefficient tensor The reconstructed signal is recovered. To balance the reconstruction error and the weighting hyperparameters of the sparse term; , , These represent the total loss function, the fidelity term of the loss function, and the sparse constraint term, respectively.

[0034] III. Algorithm Complexity Analysis A major advantage of this invention is that it achieves performance comparable to existing methods while having lower runtime and storage overhead. Therefore, the algorithm complexity of the proposed method will be analyzed from the perspectives of both parameter count and computational cost.

[0035] for The observation tensor of order, its magnitude The observation / coefficient dimension is and The vectorization dimensions are respectively , TSR-LISTA learns modal operators per layer. The number of parameters is counted in "modules," not in a vectorized count of a single large matrix. Each layer's training parameters include... , and threshold Therefore, the parameter scale for each layer is: If a tensor threshold is used, If a scalar threshold is used, Compared to vectorization The parameters of TSR-LISTA vary with the modulus. Linear growth is more suitable for high-level scenarios.

[0036] The core computation of TSR-LISTA consists of two n-mode products for each layer. The coefficient tensors for each layer are counted according to the standard of n-mode products. The main FLOPs order of magnitude for a complete reconstruction, residual, and back projection process can be approximated as follows: The final one. This represents minor overhead such as element-wise thresholding, addition / division, etc. In the case of uniform modulus (… ),have ,but: Threshold and other minor terms are The main cost of vectorization In comparison, the above formula shows a significant improvement in exponentiation; at the same time, the n-modulo product operation is more conducive to parallelization and memory saving.

[0037] The beneficial effects of this invention are as follows: To verify the effectiveness of the proposed multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks, this invention evaluates the convergence performance and sparse coding accuracy of the proposed TSR-LISTA algorithm in simulation experiments. Subsequently, its reconstruction accuracy and complexity are evaluated on a three-dimensional multispectral image (MSI) denoising task. This invention uses the sparse code normalized mean square error (NMSE) to evaluate the sparse coding accuracy, which is defined as... This invention uses Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity (SSIM) to evaluate its reconstruction accuracy in the MSI denoising task, and evaluates its complexity by calculating the number of learnable parameters (Params), the average number of floating-point operations (FLOIPs) required to process each patch, and the test time required to test a single image (Test time).

[0038] Figure 3 The comparison results of the sparse coding curves of each model are presented. It can be seen that the sparse coding reconstruction effect of TSR-LISTA is comparable to that of Tensor LISTA, but its convergence speed in the intermediate layers of the iteration is faster than all other methods, demonstrating the best coding convergence characteristics. Tables 1 and 2 show the average PSNR, SSIM, and algorithm complexity results of different methods for different noise levels on the test multispectral image set, respectively. Figure 4This is a comprehensive comparison chart of the reconstruction accuracy and complexity of various models under the same task. It can be seen that TSR-LISTA outperforms most of the compared methods and is comparable to the current best DUN (TDU-DLNet) in reconstruction accuracy. However, thanks to the Kronecker-free n-modulus design, TSR-LISTA's structure is more concise and efficient: its parameter count and FLOPs are only 1 / 20 and 1 / 8 of TDU-DLNet, respectively. This simplicity and efficiency give TSR-LISTA an optimal balance between accuracy and complexity, making it more suitable for efficient reconstruction of large-scale multidimensional signals in real-world scenarios. Table 1 shows the subjective results of each method for MSI denoising. Figure 5 As shown, it can be observed that for fine-grained texture restoration of images, such as dense stripes in a 'chart and stuffed toy', the proposed TSR-LISTA significantly outperforms other methods. This is mainly due to the fact that modal product operations fully preserve the structural characteristics of multidimensional data in the tensor domain, thus more effectively protecting cross-modal local details and texture information.

[0039] Table 1: Average PSNR and SSIM results for different methods at different noise levels on the test multispectral image set: Table 2: Computational complexity and testing time (seconds) of the expandable networks in Table 1 on the MSI dataset. This invention proposes a multidimensional signal reconstruction method using tensor sparse optimization deep unfolded networks. By unfolding the tensor iterative soft thresholding algorithm (TISTA) into a K-layer learnable deep network structure, and using n-modulus products to construct Kronecker-free reconstruction operators and residual projection operators, efficient reconstruction of multidimensional signals can be achieved.

[0040] This invention can also adjust the number of iteration layers K of the network, selecting an appropriate number of layers based on the complexity and accuracy requirements of different tasks; use different initialization dictionaries, such as wavelet dictionaries and learned dictionaries, to replace the DCT / ODCT dictionary; or combine other image priors such as nonlocal self-similarity priors and low-rank priors to further improve reconstruction performance. The method of this invention can also be extended to higher-order tensor data, such as four-dimensional video data and five-dimensional medical image data.

[0041] To better illustrate the technical effects of the present invention, the present invention provides the following specific embodiments to illustrate the above technical process: Example 1: A method for multidimensional signal reconstruction using tensor sparse optimization deep unfolded networks. 1. Dataset In simulation experiments, this invention applies the proposed method to solve the sparse coding problem of two-dimensional matrices and three-dimensional tensors. Specific data settings are as follows; for 2D data, this paper sets... , Sparse coefficient matrix A fixed sparsity of 10% (random distribution of non-zero elements) is used, with non-zero values ​​sampled from a standard normal distribution. dictionary matrix Elements are sampled independently from The observation tensor is obtained through the formula... Generation. For 3D data, this paper sets... , The remaining settings are similar to those for 2D. Based on the above settings, a test set containing 1000 samples is generated.

[0042] In the MSI denoising experiments, this invention uses the Columbia multispectral image database to evaluate the performance of the TSR-LISTA model. This dataset contains 32 scenes, each with... Spatial resolution and 31 bands (400–700 nm). Each image is cropped along its center. and at the noise level Gaussian white noise was added. This paper randomly sampled 15,000 pixels of size [value missing] from the 'beads' scene. The cubes are used for network training, and these samples are subjected to simple flipping data augmentation before training.

[0043] 2. Comparison Algorithm In simulation experiments, this invention compares representative ISTA-based optimization methods TenSR, 1D DUNLISTA, and MD DUN Tensor-LISTA. In MSI denoising experiments, this invention compares optimization methods TenSR, 1D DUNsISTA-Net, CSCNet-DF, MD3, HLISTA-CPSS, EBT-LISTA, TDU-DLNet, and MD DUNs Tensor ADMM-Net, Tensor FISTA-Net, and Tensor LISTA.

[0044] 3. Experimental Setup In the simulation experiments, for the learning-based method, the unified network layer number was 16, the optimizer was Adam, and the initial learning rate was 1e-3. In the MSI denoising experiment, each input sample was normalized to [value missing] during training. The training was performed using the PyTorch framework running on an NVIDIA GeForce RTX 4070. The optimizer used was Adam, and the initial learning rate was set to... The network has 16 layers and uses... A complete dictionary.

[0045] In the several embodiments provided by this invention, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative. For instance, the division of modules, units, or units is merely a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units, modules, or components may be combined or integrated into another device, or some features may be ignored or not executed.

[0046] The units may or may not be physically separate. The components shown as units can be one or more physical units, meaning they can be located in one place or distributed in multiple different locations. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0047] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0048] In particular, according to embodiments disclosed in this invention, the processes described above with reference to the flowcharts can be implemented as computer software programs. For example, embodiments of this disclosure include a computer program product comprising a computer program carried on a computer-readable medium, the computer program containing program code for performing the methods shown in the flowcharts. In such embodiments, the computer program can be downloaded and installed from a network via a communication component, and / or installed from a removable medium. When the computer program is executed by a central processing unit (CPU), it performs the functions defined in the methods of this invention. It should be noted that the computer-readable medium described above in this invention can be a computer-readable signal medium or a computer-readable storage medium, or any combination of the two. The computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof.

[0049] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, can be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0050] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions within the technical scope disclosed in the present invention should be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for multidimensional signal reconstruction using tensor sparse optimization deep unfolded networks, characterized in that, Includes the following steps: Input the multidimensional signal tensor to be reconstructed; The sparse coefficient tensor and network learnable parameters are initialized to obtain the initialization results for the multidimensional signal tensor to be reconstructed. Based on the multidimensional signal tensor to be reconstructed and the initialization results, the sparse coefficient tensor is continuously optimized by updating the proximal gradient through K layers of network iterative learning, and the optimized sparse coefficient tensor is obtained. The reconstructed signal is obtained by reconstructing the multidimensional signal tensor based on the learnable reconstruction operator learned through K-layer iterative learning of the network and the optimized sparse coefficient tensor. Output the reconstructed multidimensional signal tensor.

2. The multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks according to claim 1, characterized in that, Initialize the sparse coefficient tensor and the network's learnable parameters, specifically including the following steps: Learnable threshold parameter in network learnable parameters Using theoretical reference step size Initialization; initial weights of the modal product learnable operator Depend on Initialization; Initial coefficient tensor Approximate inverse mapping of the observation tensor through lexicographical transpose The result is obtained by back projection; that is To measure the quality of network reconstruction and constrain the sparsity of coefficients over the observation domain, a method based on MSE and... The total loss function composed of sparse terms weighted together is as follows: (7) in, For sample index, This represents the number of samples within a batch. This represents the total number of elements in a single observation sample. For the network to learn the multi-mode reconstruction operator through the last layer From coefficient tensor The reconstructed signal is recovered. The weighting hyperparameters are used to balance the reconstruction error and the sparsity term.

3. The multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks according to claim 2, characterized in that, Based on the multidimensional signal tensor to be reconstructed and the initialization results, the sparse coefficient tensor is continuously optimized through K-layer iterative proximal gradient updates to obtain the optimized sparse coefficient tensor. The specific steps include: Using TenSR as the basic signal model; given any multidimensional signal Assume that it satisfies tensor sparsity constraints under a given multidimensional dictionary; The tensor gradient of tensor sparsity constraint is obtained by using the n-mode chain rule, and the iterative update formula of TISTA is obtained based on the idea of ​​proximal gradient descent and soft threshold function. The TISTA iterative process is mapped to a deep unfoldable network structure consisting of learnable modal integration operators; The dictionary-related operations in the tensor iteration formula are replaced with learnable modal integration operators, and the fixed threshold is replaced with a learnable threshold to construct the TSR-LISTA network layer update rules; By stacking the update rules K times, we obtain the TSR-LISTA network, which maps the TISTA algorithm in the tensor domain to a learnable multidimensional deep unfoldable network for tensor sparse recovery.

4. The multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks according to claim 3, characterized in that, The tensor sparsity recovery problem can be expressed as: (1) in, and The dimension representing the observation space and coefficient space ( ), , For the first Model dictionary For sparse representation coefficients in a multidimensional dictionary, yes .

5. The multidimensional signal reconstruction method of tensor sparse optimized deep unfolded network according to claim 4, characterized in that: The iterative update formula for TISTA is as follows: (2) in .

6. The multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks according to claim 5, characterized in that, Mapping the TISTA iterative process to a deep unfoldable network structure composed of learnable modal integration operators specifically includes the following steps: Substituting the tensor gradient into its update formula (2), expanding the gradient term and simplifying it, we obtain the first... Modal product expansion of tensor coefficient updates in the next iteration: (3) Both terms in the gradient term contain dictionary transpose operations. These are combined to extract the common residual projection term, which is then used to rewrite the gradient term. (4) Among them, step size The soft threshold operator determines the magnitude of gradient descent. This ensures the sparsity constraint.

7. The multidimensional signal reconstruction method using tensor sparse optimized deep unfolded networks according to claim 6, characterized in that, The dictionary-related operations in the tensor iteration formula are replaced with learnable modal integration operators, and the fixed threshold is replaced with a learnable threshold to construct the TSR-LISTA network layer update rules. The specific steps include: Replace the dictionary-related operations in equation (4) with learnable parameters, and set a fixed threshold. Replace with learnable threshold We obtain the update rules for the k-th layer of TSR-LISTA; (5) in, and denoted as the learnable Kronecker product-free reconstruction operator and the residual projection operator, respectively.

8. The multidimensional signal reconstruction method of tensor sparse optimized deep unfolded network according to claim 7, characterized in that: The and Both are achieved through modal product, and their specific definitions are as follows: (6) in, These are the multidimensional learnable weight parameters within the operator; Characterizing the self-constraint relationship of the coefficient tensor of the k-th layer along each dimension enables adaptive learning of cross-modal correlation; Observations The residuals between the reconstruction results and the coefficient domain are mapped back to the coefficient domain to adaptively adjust the cross-modal error feedback. For the step size parameter L, it is set as a global constant as a theoretical reference step size for initializing the network parameters. .

9. The multidimensional signal reconstruction method of tensor sparse optimized deep unfolded network according to claim 8, characterized in that: The set of learnable parameters is denoted as Optimization is achieved through end-to-end training.

10. A multidimensional signal reconstruction system for tensor sparse optimization deep unfolded networks, used to implement the multidimensional signal reconstruction method for tensor sparse optimization deep unfolded networks as described in any one of claims 1-9, characterized in that, include: The acquisition module is used to input the multidimensional signal tensor to be reconstructed; The initialization module is used to initialize the sparse coefficient tensor and the learnable parameters of the network for the multidimensional signal tensor to be reconstructed, and obtain the initialization results. The optimization module is used to continuously optimize the sparse coefficient tensor based on the multidimensional signal tensor to be reconstructed and the initialization results, through the proximal gradient update learned by K layers of the network, and obtain the optimized sparse coefficient tensor. The reconstruction module is used to reconstruct the reconstructed signal based on the learnable reconstruction operator and the optimized sparse coefficient tensor after K-layer iterative learning of the network, and obtain the reconstructed multidimensional signal tensor. The output module is used to output the reconstructed multidimensional signal tensor.

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