Parallel chebyshev iterative ADMM compressed sensing fast reconstruction method

Abstract

The present application aims to provide a parallel Chebyshev iterative ADMM compressed sensing fast reconstruction method, which can support reconstruction of signals with arbitrary sparsity. The present application first simplifies the matrix inversion operation with high computational complexity in the ADMM algorithm by using the Chebyshev iteration method. Secondly, in consideration of real-time computing requirements, the data-independent terms in the Chebyshev iterative ADMM calculation steps are separated, the parallelism is improved, and the speed of ADMM reconstruction is improved.

Description

Technical Field

[0001] This invention belongs to the technical fields of signal processing and compressed sensing, and specifically relates to a fast reconstruction method for compressed sensing using parallel Chebyshev iterative ADMM. Background Technology

[0002] Compressed sensing (CS) theory refers to the ability to sample sparse or compressible signals at frequencies far below the Nyquist frequency and accurately reconstruct the signal by designing reconstruction algorithms. It states that as long as the signal is compressible or sparse in a certain transform domain, the transformed high-dimensional signal can be projected onto a low-dimensional space using an observation matrix independent of the transform basis. Then, by solving an optimization problem, the original signal can be reconstructed with high probability from these few projections. It incorporates two properties: uncorrelation and underdeterminacy; the compression and reconstruction of compressed sensing are achieved precisely through these two properties.

[0003] Existing technologies available for reference include:

[0004] 1. Patent No.: CN114125447A, Patent Title: A Fast Reconstruction Method for Compressed Sensing Based on Block Segmentation and Transposition Algorithms

[0005] This invention relates to a compressed sensing fast reconstruction method based on block-based and transposition algorithms, belonging to the field of image processing. The invention first employs a block-based method to divide the original image into blocks. A low-order random measurement matrix is ​​used to globally sample the original signal of each sub-image after block division. Subsequently, transposition operations are performed and reconstruction is completed. This overcomes the poor real-time performance of existing methods when reconstructing large images, effectively improving reconstruction speed while maintaining reconstruction quality. However, this invention involves highly complex matrix inversion calculations, resulting in relatively low reconstruction efficiency compared to the present method, which is not conducive to hardware implementation.

[0006] 2. Patent No.: CN115414055A, Patent Title: A Non-Iterative EEG Signal Compressed Sensing Fast Reconstruction Method Based on Deep Learning

[0007] This invention is a non-iterative, deep learning-based method for rapid reconstruction of EEG signals using compressed sensing, belonging to the field of compressed sensing technology. It includes the following steps: S1, random projection of the EEG signal; S2, preprocessing of the projected measurement signal; S3, signal reconstruction. An improved residual network model is used, which directly passes the data stream to the next layer through identity shortcut connections, thereby mitigating gradient decay caused by multiple stacked nonlinear transformations and improving the network's learning ability and training efficiency. Furthermore, one-dimensional dilated convolution is used to extract feature information from the EEG signal, directly learning the nonlinear mapping relationship between the measured value and the original signal, thus achieving rapid and accurate reconstruction of the EEG signal. This method improves the accuracy and speed of EEG signal reconstruction and enables rapid EEG signal reconstruction. However, this method still involves computationally complex matrix pseudo-inverse operations, therefore, the reconstruction efficiency is lower compared to this method. Summary of the Invention

[0008] Considering that the computational complexity of the Alternating Direction Method of Multipliers (ADMM) convex optimization reconstruction algorithm is largely unaffected by changes in signal sparsity, the ADMM compression reconstruction algorithm has become the mainstream algorithm for CS compression reconstruction of high-sparse signals. The ADMM algorithm can decompose the complex reconstruction problem into multiple simple subproblems, improving the speed of compressed sensing reconstruction through parallel computation. However, the computation process of the subproblems in the ADMM algorithm involves complex matrix multiplication and inversion calculations, and the computational speed needs to be improved.

[0009] Table 1. Computational complexity analysis of the ADMM reconstruction algorithm

[0010]

[0011] Table 1 shows the computational complexity of the ADMM algorithm. The ADMM algorithm has a high single-run computational complexity. Given the relatively limited hardware computing resources of existing IoT information acquisition terminal devices, the high single-run computational complexity prevents the ADMM algorithm from being applied to IoT information acquisition systems.

[0012] To address the aforementioned situation and the shortcomings and deficiencies of existing technologies, this invention proposes a fast compressed sensing reconstruction method using Parallel Chebyshev Iteration ADMM (PCIADMM). This method supports reconstruction of signals with arbitrary sparsity. First, this invention simplifies the computationally complex matrix inversion operation in the ADMM algorithm using the Chebyshev iteration method. Second, considering real-time computing requirements, it separates data-independent terms in the Chebyshev Iteration ADMM calculation steps, improving parallelism and thus increasing the speed of ADMM reconstruction.

[0013] This invention presents a fast compressed sensing reconstruction method based on parallel Chebyshev iterative ADMM. Solving the ADMM algorithm involves large-scale matrix multiplication and inversion operations. To address the high computational complexity of the ADMM algorithm, this invention uses the Chebyshev iteration method to reduce the high-complexity matrix multiplication and inversion operations to low-complexity multiplication and addition operations. Furthermore, by decomposing the data dependencies in the iterative calculation, the order of the Chebyshev iterative solution is reduced, resulting in an optimized algorithm with a parallelism of 2: Parallel Chebyshev Iteration ADMM (PCIADMM).

[0014] The specific technical solution adopted by this invention to solve its technical problem is as follows:

[0015] A parallel Chebyshev iterative ADMM compressed sensing fast reconstruction method includes the following steps:

[0016] Step S1: Acquire the raw signal and perform preprocessing;

[0017] Step S2: Based on compressed sensing theory, the preprocessed input signal is mapped into a low-dimensional measurement signal;

[0018] Step S3: The measured signal is rapidly reconstructed using the parallel Chebyshev iterative ADMM fast reconstruction method, and the reconstructed signal is obtained further.

[0019] The parallelism of the ADMM fast reconstruction method based on parallel Chebyshev iteration is 2. It reduces the high-complexity matrix multiplication and inversion operations to low-complexity multiplication and addition operations by decomposing the data dependencies of the iterative calculation, and reduces the order of the Chebyshev iterative solution.

[0020] Further, in step S2:

[0021] Let the signal x of length N and sparsity K be defined as:

[0022] x=Ψα (1)

[0023] In the formula, Ψ is an N×N sparse transformation basis matrix, and α is an N×1 sparse transformation coefficient vector with exactly K non-zero coefficients.

[0024] Using an M×N dimensional observation matrix Φ, the signal x is compressed into an M-dimensional observation vector y, i.e.:

[0025] y=Φx=ΦΨα=Θα (2)

[0026] In the formula, Θ=ΦΨ is an M×N dimensional sensing matrix.

[0027] Furthermore, step S3 specifically includes the following steps:

[0028] Step S31: Initialize all variables and parameters:

[0029]

[0030]

[0031] Where I is a unit vector, η and β are parameters related to the eigenvalues ​​of matrix A, and λ max , λ min These are the maximum and minimum eigenvalues ​​of the matrix, ω1 and ω2, respectively. Here, z represents the Chebyshev parameter, u represents the dual variable, the superscript indicates the number of ADMM iterations, the subscript indicates the number of Chebyshev iterations, and ρ represents the penalty coefficient.

[0032] Step S32: Execute the compressed sensing ADMM reconstruction solution algorithm based on the l1 regularization model:

[0033]

[0034] The signal α for the (t+1)th iteration is obtained by Chebyshev iteration, and then the final estimated value of the signal α is obtained by iteratively solving the problem in the order of α→z→u.

[0035] Step S32 specifically includes the following steps:

[0036] Step S321: Calculate the signal α for the (t+1)th iteration using Chebyshev iteration, including:

[0037] Step S3211: Update the calculation of Chebyshev equation parameters:

[0038] Let c t+1 =Θ T y+ρ(z t -u t );

[0039] Step S3212: Solving for initial values ​​using Chebyshev iteration:

[0040] Update signal Calculation:

[0041] Update residual vector Calculation:

[0042] Update correction vector Calculation:

[0043] Step S3213: Parallel Chebyshev Iterative ADMM Accelerated Computation:

[0044] Step S32131: Calculation of Chebyshev parameters:

[0045]

[0046]

[0047]

[0048]

[0049] Step S32132: Parallel computation of the corrected variables:

[0050]

[0051]

[0052] Step S32133: Calculate the residual vector for the j-th iteration:

[0053]

[0054] Step S32134: Update the iteration count j = j + 2. If the iteration count j ≤ J, return to step S3213; otherwise, proceed to the next step.

[0055] Step S322: Calculate the reconstructed signal

[0056]

[0057] Step S323: Calculate the auxiliary variable: z t+1 =S(α) t+1 +u t ,1 / ρ);

[0058] Step S324: Update dual variable: u t+1 =u t +(α t+1 -z t+1 );

[0059] Step S325: Update the iteration count t = t + 1. If the iteration count t ≤ T, return to step S321; otherwise, end the calculation.

[0060] Compared with the prior art, the present invention and its preferred embodiments have the following advantages:

[0061] 1. As a convex optimization reconstruction algorithm, the computational complexity of this method is basically unaffected by changes in signal sparsity. Therefore, this algorithm can support the reconstruction of signals with arbitrary sparsity.

[0062] 2. By using the Chebyshev iteration method, the high-complexity matrix multiplication and inversion operations are reduced to low-complexity multiplication and addition operations, thereby improving the reconstruction speed.

[0063] 3. By decomposing the data dependencies in the iterative calculations, the order of the Chebyshev iterative solution was reduced, improving parallelism. Therefore, the reconstruction speed was further improved.

[0064] 4. Experiments demonstrate the performance of the parallel Chebyshev iterative optimization PCIADMM algorithm. It can be applied to IoT information acquisition terminal devices with relatively limited hardware computing resources.

[0065] Its main uses include:

[0066] This method supports the reconstruction of signals with arbitrary sparsity. By employing the Chebyshev iteration method, it reduces the computational complexity of the ADMM algorithm by simplifying high-complexity matrix multiplication and inversion operations into low-complexity multiplication and addition operations. Furthermore, it reduces the order of the Chebyshev iterative solution by decomposing the data dependencies in the iterative calculation, thereby increasing parallelism and computational speed. This parallel Chebyshev iteration-based ADMM compressed sensing fast reconstruction method holds promise for widespread application in IoT information acquisition terminal devices where hardware computing resources are relatively scarce. Attached Figure Description

[0067] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0068] Figure 1 This is a flowchart of the PCIADMM method provided in an embodiment of the present invention.

[0069] Figure 2 This is a comparison chart of the reconstruction results of embodiments of the present invention. Detailed Implementation

[0070] To make the features and advantages of this patent more apparent and understandable, specific embodiments are provided below for detailed explanation:

[0071] It should be noted that the following detailed descriptions are illustrative and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0072] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0073] The compressed sensing fast reconstruction method provided by this invention includes three steps: 1. Acquire the original signal and preprocess it; 2. Map the preprocessed input signal into a low-dimensional measurement signal based on compressed sensing theory; 3. Use the parallel Chebyshev iterative ADMM fast reconstruction method to quickly reconstruct the measurement signal, and then obtain the reconstructed signal.

[0074] This embodiment provides a detailed explanation of the above three steps.

[0075] 1. Acquire the raw signal and perform preprocessing.

[0076] 2. The preprocessed input signal is mapped into a low-dimensional measurement signal based on compressed sensing theory;

[0077] Let x be a signal of length N and sparsity K. This signal is defined as:

[0078] x=Ψα (1)

[0079] In the formula, Ψ is an N×N sparse transformation basis matrix, and α is an N×1 sparse transformation coefficient vector with exactly K non-zero coefficients.

[0080] Using an M×N (M << N) dimensional observation matrix Φ, the signal x is compressed into an M-dimensional observation vector y, i.e.:

[0081] y=Φx=ΦΨα=Θα (2)

[0082] In the formula, Θ=ΦΨ is an M×N dimensional sensing matrix.

[0083] 3. The measured signal is reconstructed using the parallel Chebyshev iterative ADMM reconstruction method, and the reconstructed signal is obtained further. For example... Figure 1 As shown, it specifically includes:

[0084] 3.1 Initialize all variables and parameters

[0085]

[0086]

[0087] Where I is a unit vector, η and β are parameters related to the eigenvalues ​​of matrix A, and λ max , λ min These are the maximum and minimum eigenvalues ​​of the matrix, ω1 and ω2, respectively. Here, z represents the Chebyshev parameter, u represents the dual variable, the superscript indicates the number of ADMM iterations, the subscript indicates the number of Chebyshev iterations, and ρ represents the penalty coefficient.

[0088] 3.2 Compressed Sensing ADMM Reconstruction Solution Algorithm Based on L1 Regularization Model:

[0089]

[0090] Equation (3-a) shows that the ADMM algorithm involves large-scale matrix multiplication and inversion operations, making it a typical computationally intensive algorithm. The signal α for the (t+1)th iteration is obtained through Chebyshev iteration, and then the final estimated value of signal α is obtained by iteratively solving the problem in the order α→z→u.

[0091] 3.2.1 The signal α for the (t+1)th iteration is obtained through Chebyshev iteration, including:

[0092] 3.2.1.1 Calculation of updated Chebyshev equation parameters:

[0093] Let c t+1 =Θ T y+ρ(z t -u t )

[0094] 3.2.1.2 Chebyshev Iteration Initial Value Solution:

[0095] Update signal Calculation:

[0096] Update residual vector Calculation:

[0097] Update correction vector Calculation:

[0098] 3.2.1.3 Parallel Chebyshev Iterative ADMM Accelerated Computation:

[0099] 3.2.1.3.1 Calculation of Chebyshev parameters

[0100]

[0101]

[0102]

[0103]

[0104] 3.2.1.3.2 Parallel computation of modified variables:

[0105]

[0106]

[0107] 3.2.1.3.3 Calculate the residual vector for the j-th iteration:

[0108]

[0109] 3.2.1.3.4 Update the iteration count j = j + 2. If the iteration count j ≤ J, then execute step 3.2.1.3; otherwise, proceed to the next step.

[0110] 3.2.2 Calculate the reconstructed signal

[0111]

[0112] 3.2.3 Calculate the auxiliary variable: z t+1 =S(α) t+1 +u t ,1 / ρ);

[0113] 3.2.4 Update the dual variable: u t+1 =u t +(α t+1 -z t+1 );

[0114] 3.2.5 Update the iteration count t = t + 1. If the iteration count t ≤ T, then execute step 3.2.1; otherwise, end the calculation.

[0115] The following table further outlines the specific execution flow of the above algorithms:

[0116] Table 2. PCIADMM Reconstruction Algorithm Based on Chebyshev Iterative Parallel Optimization

[0117] Table 2PCI ADMM reconstruction algorithm

[0118]

[0119]

[0120] To evaluate the performance of the parallel Chebyshev iterative optimization PCIADMM algorithm, this embodiment verifies and simulates the proposed algorithm using Python in the PyCharm environment.

[0121] In this embodiment, a Python program on a PC is used to divide the 256×256 Lena image into 64 image sequences of length 32×32=1024 as the original data, and the parallel Chebyshev iterative optimization PCIADMM algorithm is verified.

[0122] The peak signal-to-noise ratio (PSNR) of the reconstruction algorithm used in this invention is:

[0123]

[0124] MAX I MAX represents the highest pixel value of the image. If the pixel value is represented in binary, then MAX is the highest pixel value. I =2 C -1. MSE is the mean square value of the pixel difference between the original image and the reconstructed image.

[0125] like Figure 2 As shown, experimental results demonstrate that the Python-implemented PCI ADMM algorithm suffers only a 1.01 dB loss compared to the ADMM algorithm. Therefore, the PCI ADMM algorithm improves reconstruction speed with a smaller loss in peak signal-to-noise ratio, proving the performance of the parallel Chebyshev iterative optimization PCI ADMM method.

[0126] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0127] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0128] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0129] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0130] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

[0131] This patent is not limited to the above-described preferred embodiment. Anyone can derive other forms of parallel Chebyshev iterative ADMM compressed sensing fast reconstruction method under the guidance of this patent. All equivalent changes and modifications made within the scope of this patent application shall fall within the scope of this patent.

Claims

1. A parallel Chebyshev iterative ADMM compressed sensing fast reconstruction method, characterized in that, Includes the following steps: Step S1: Acquire the image sequence as the raw signal and perform preprocessing; Step S2: Based on compressed sensing theory, the preprocessed input signal is mapped into a low-dimensional measurement signal; Step S3: The measured signal is rapidly reconstructed using the parallel Chebyshev iterative ADMM fast reconstruction method, and the reconstructed signal is obtained further. The parallelism of the ADMM fast reconstruction method based on parallel Chebyshev iteration is 2. It reduces the high-complexity matrix multiplication and inversion operations to low-complexity multiplication and addition operations by using the Chebyshev iteration method, and reduces the order of the Chebyshev iteration solution by decomposing the data dependencies of the iterative calculation. In step S2: Let the length be sparsity is signal Defined as: (1) In the formula yes 3D sparse transformation basis matrix, for A dimensional transformation sparse coefficient vector, in which the vector contains one and only One non-zero coefficient; use 3D observation matrix , will signal Compress into one dimensional observation vector ,Right now: (2) In the formula yes 3D sensing matrix; Step S3 specifically includes the following steps: Step S31: Initialize all variables and parameters: in, Unit vector Is with matrix Eigenvalue-related parameters, , These are the maximum and minimum eigenvalues ​​of the matrix, respectively. , Here, z represents the Chebyshev parameters, u represents the dual variable, the superscript indicates the number of ADMM iterations, and the subscript indicates the number of Chebyshev iterations. Indicates the penalty coefficient; Step S32: Execute based on Compressed Sensing ADMM Reconstruction Solution Algorithm with Regularized Model: The first step is obtained through Chebyshev iteration. Signal of the next iteration Then by The solution is obtained by iteratively solving the sequence. The final estimate; Step S32 specifically includes the following steps: Step S321: Calculate the first step using Chebyshev iteration. Signal of the next iteration ,include: Step S3211: Update the calculation of Chebyshev equation parameters: make ; Step S3212: Solving for initial values ​​using Chebyshev iteration: Update signal Calculation: ; Update residual vector Calculation: ; Update correction vector Calculation: ; Step S3213: Parallel Chebyshev Iterative ADMM Accelerated Computation: Step S32131: Calculation of Chebyshev parameters: Step S32132: Parallel computation of the corrected variables: Step S32133: Calculate the first... The residual vector of the order: Step S32134: Update the number of iterations Determine if the number of iterations is... If yes, return to step S3213; otherwise, proceed to the next step. Step S322: Calculate the reconstructed signal : ； Step S323: Calculate auxiliary variables: ; Step S324: Update the dual variable: ; Step S325: Update the number of iterations Determine if the number of iterations is... If the condition is met, return to step S321; otherwise, end the calculation.

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