A sparse recovery method with fast convergence, device and storage medium

Abstract

The application discloses a fast-converging sparse recovery method, which comprises the following steps: S1, acquiring an observation signal, a measurement matrix, a sparsity of a to-be-solved signal and an iteration error threshold; S2, initializing a first estimated signal and a support set of a sequence of vector product positions of a transpose of the measurement matrix and the observation signal; S3, acquiring an iteration step length based on the observation signal, the measurement matrix, the support set and the first estimated signal; S4, bringing the iteration step length into an iteration formula to acquire a second estimated signal; S5, if an error between the second estimated signal and the first estimated signal is not greater than the iteration error threshold, terminating iteration, and the target estimated signal is the second estimated signal; otherwise, taking the second estimated signal as the first estimated signal, and repeating steps S3-S5. The application further discloses a corresponding device and a storage medium for fast-converging sparse recovery. The fast-converging sparse recovery method provided by the application can improve the convergence speed and the estimation accuracy.

Description

Technical Field

[0001] This application relates to the field of signal processing technology, and more specifically, to a fast-converging sparse recovery method, apparatus, and storage medium. Background Technology

[0002] In image transmission and processing, compressed sensing algorithms can accurately reconstruct high-dimensional sparse signals from low-dimensional linear measurements. This problem can be expressed in the following form:

[0003]

[0004] This problem is a nondeterministic polynomial problem. To solve it, existing methods typically involve constructing a new objective function by introducing a penalty term and a regularization parameter. Examples include using l0-iterative thresholding algorithms, l1-iterative thresholding algorithms, and l... 1 / 2 Methods such as iterative thresholding algorithms.

[0005] However, the existing l 1 / 2 The iterative thresholding algorithm suffers from the problem of unreasonable iteration step size selection when converging large singular value matrices, and the convergence speed and accuracy cannot meet the requirements of the working scenario. Summary of the Invention

[0006] In response to at least one defect or improvement requirement of the prior art, the present invention provides a fast-converging sparse recovery method, device and storage medium that can solve at least one of the problems existing in the prior art.

[0007] To achieve the above objectives, according to a first aspect of the present invention, a fast-converging sparse recovery method is provided, the method comprising:

[0008] S1 acquires the observed signal, measurement matrix, sparsity of the target signal, and iteration error threshold;

[0009] S2 initializes the support set of the position sequence of the vector product of the first estimated signal and the transpose of the measurement matrix with the observed signal, which is obtained based on the sparsity of the observed signal, the measurement matrix and the target signal;

[0010] S3 obtains the iteration step size based on the support set of the position sequence of the vector product of the observed signal, the measurement matrix, the transpose of the measurement matrix and the observed signal, and the first estimated signal;

[0011] S4 substitutes the iteration step size into the iteration formula to obtain the second estimated signal;

[0012] S5 If the error between the second estimated signal and the first estimated signal is not greater than the iteration error threshold, then the iteration is terminated, and the target estimated signal is the second estimated signal; if the error between the second estimated signal and the first estimated signal is greater than the iteration error threshold, then the second estimated signal is used as the first estimated signal, and steps S3 to S5 are repeated.

[0013] Furthermore, in the above-mentioned fast-converging sparse recovery method, after substituting the iteration step size into the iteration formula to obtain the second estimated signal, the support set of the second estimated signal is calculated;

[0014] If the support set of the second estimated signal is the same as the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, then the second estimated signal is retained.

[0015] If the support set of the second estimated signal is different from the support set of the position sequence of the vector product of the transpose of the measurement matrix and the observed signal, and the iteration step size is not less than the minimum iteration step size, then the second estimated signal is retained.

[0016] If the support set of the second estimated signal is different from the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step size is less than the minimum iteration step size, update the iteration step size and repeat step S4 until the iteration step size is not less than the minimum iteration step size, then retain the second estimated signal.

[0017] Furthermore, the above-mentioned fast-converging sparse recovery method, specifically including the acquisition of the sparsity of the observed signal, measurement matrix, target signal, and iteration error threshold, includes:

[0018] Obtain the observed signal y∈R M Measurement matrix The sparsity k of the target signal, the iteration error threshold η, and the constants c = 0.01 and κ = 1.1.

[0019] Furthermore, in the aforementioned fast-converging sparse recovery method, the initialization of the support set of the vector product position sequence of the first estimated signal and the transpose of the measurement matrix and the observed signal specifically includes:

[0020] Get x n , Γ n =supp[H k (Φ T y)], where n represents the number of iterations, supp(a) represents the support set of vector a, H k (m) means keeping only the k elements with the largest absolute value in vector m, and setting the other elements in vector m to 0;

[0021] Initialize x n , Γ n =supp[H k (ΦT When n = 1, x1 = 0, Γ1 = supp[H k (Φ T y)).

[0022] Furthermore, the above-mentioned fast-converging sparse recovery method, specifically including the acquisition of the iteration step size based on the support set of the position sequence of the vector product of the observed signal, the measurement matrix, the transpose of the measurement matrix and the observed signal, and the first estimated signal, includes:

[0023] Get

[0024] g n =Φ T (y-Φx n )

[0025] Based on g n Obtain the iteration step size α n

[0026]

[0027] Furthermore, in the aforementioned fast-converging sparse recovery method, the step of substituting the iteration step size into the iteration formula to obtain the second estimated signal specifically includes:

[0028] Get

[0029]

[0030] Get

[0031]

[0032] Get

[0033]

[0034] Obtain the i-th element of the second estimated signal respectively i∈(1,k+1)

[0035]

[0036] Obtain the second estimated signal in,

[0037]

[0038] Furthermore, in the aforementioned fast-converging sparse recovery method, the step of calculating the support set of the second estimated signal after obtaining the second estimated signal specifically includes:

[0039] Get

[0040]

[0041] If Γ n+1 =Γ n ,but

[0042] If Γ n+1 ≠Γ n ,when At that time,

[0043] If Γ n+1 ≠Γ n ,when At that time, α n =κα n Repeat step S4 until... at this time

[0044] Furthermore, in the above-described fast-converging sparse recovery method, step S5 specifically includes:

[0045] like Then the iteration terminates, and the target estimated signal is x. n+1 ;

[0046] like Then x n =x n+1 , n = n + 1, repeat steps S3 to S5.

[0047] According to a second aspect of the invention, a fast-converging sparse recovery device is also provided, comprising at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program that, when executed by the processing unit, causes the processing unit to perform the steps of any of the methods described above.

[0048] According to a third aspect of the invention, a storage medium is also provided, which stores a computer program executable by a fast-converging sparse recovery device, which, when run on the fast-converging sparse recovery device, causes the fast-converging sparse recovery device to perform the steps of any of the methods described above.

[0049] In summary, compared with the prior art, the above-described technical solutions conceived by this invention can achieve the following beneficial effects:

[0050] The sparse recovery method with fast convergence provided in this application, under the premise of adhering to the principle of minimizing the step size parameter, can improve the convergence speed and estimation accuracy by adaptively adjusting the step size parameter during the iteration process, and has the advantages of fast convergence speed and small reconstruction error. Attached Figure Description

[0051] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0052] Figure 1 A flowchart illustrating a fast-converging sparse recovery method provided in this application embodiment;

[0053] Figure 2 A schematic diagram illustrating the entire process of a fast-converging sparse recovery method provided in this application embodiment;

[0054] Figure 3 A graph showing the variation of the iteration error of a fast-converging sparse recovery method provided in this application embodiment as a function of the number of iterations;

[0055] Figure 4 The graph shows the mean square error of a fast-converging sparse recovery method provided in this application embodiment as a function of signal-to-noise ratio. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0057] The terms "first," "second," "third," etc., in the specification, claims, and accompanying drawings of this application are used to distinguish different objects, not to describe a specific order. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or apparatus that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to these processes, methods, products, or apparatuses.

[0058] Figure 1 This is a flowchart illustrating a fast-converging sparse recovery method provided in an embodiment of this application, as shown below. Figure 1 As shown, this application provides a fast-converging sparse recovery method, comprising:

[0059] S1 acquires the observed signal, measurement matrix, sparsity of the target signal, and iteration error threshold.

[0060] Specifically, compressed sensing can be used in image transmission and processing. It leverages the redundancy of data, collecting only a small number of samples to reconstruct the original data. By compressing the data during acquisition, it ensures the original data is recovered without distortion. The signal obtained through compressed sensing can be represented as a sparse signal, where sparsity k represents the number of non-zero values ​​in the sparse representation.

[0061] This invention is based on the observed signal y∈R M Measurement matrix Given conditions such as the sparsity k of the signal to be determined, and focusing on the objective function... A compressed sensing algorithm based on logarithmic summation is proposed.

[0062] Figure 2 The diagram shown is a complete flowchart of a fast-converging sparse recovery method provided in an embodiment of this application. Figure 2 As shown, the observed signal y∈R is first obtained. M Measurement matrix The sparsity k of the target signal, the iteration error threshold η, and the constants c = 0.01 and κ = 1.1.

[0063] The observed signal y has a length of M, and the observation matrix Φ is a matrix of size M×N representing the compression process. It is a random subsampling of the time-domain signal, which can compress the high-dimensional signal x. n Projecting into a low-dimensional space, the sparsity k of the target signal is the number of non-zero values ​​in the sparse representation of the signal, where c = 0.01 and κ = 1.1 are constants.

[0064] S2 initializes the support set of the position sequence of the vector product of the first estimated signal and the transpose of the measurement matrix with the observed signal, which is obtained based on the sparsity of the observed signal, the measurement matrix, and the target signal.

[0065] Specifically, obtain x n , Γ n =supp[H k (Φ T y)], where n represents the number of iterations, supp(a) represents the support set of vector a, H k (m) means keeping only the k elements with the largest absolute value in vector m, and setting the other elements in vector m to 0;

[0066] Initialize x n , Γ n =supp[H k (Φ T When n = 1, x1 = 0, Γ1 = supp[H k (Φ T y)).

[0067] The support set of a is defined as follows: if a subset of a satisfies the condition that all elements in that subset are non-zero elements of a, then the support set of a is the number of such non-zero elements. The function H... k (m) indicates that only the k elements with the largest absolute values ​​in vector m are retained, and the other elements in vector m are set to 0. Φ T Let Γ be the transpose of Φ. n =supp[H k (Φ T y)] represents Φ T The sequence of the positions of the k elements with the largest absolute value in the product of vectors y.

[0068] S3 obtains the iteration step size based on the support set of the position sequence of the vector product of the observed signal, the measurement matrix, the transpose of the measurement matrix and the observed signal, and the first estimated signal.

[0069] Specifically, to obtain

[0070] g n =Φ T (y-Φx n )

[0071] Based on g n Obtain the iteration step size α n

[0072]

[0073] g n Φ T (y-Φx n The gradient of ) based on g n Obtain the iteration step size α n .

[0074] S4 substitutes the iteration step size into the iteration formula to obtain the second estimated signal.

[0075] Specifically, to obtain

[0076]

[0077] Get

[0078]

[0079] Get

[0080]

[0081] Obtain the i-th element of the second estimated signal respectively i∈(1,k+1)

[0082]

[0083] Obtain the second estimated signal in,

[0084]

[0085] Calculate each element of the second estimated signal separately. Obtain the second estimated signal The sign function extracts the sign (positive or negative) of the number of items.

[0086] After substituting the iteration step size into the iteration formula to obtain the second estimated signal, the support set of the second estimated signal is calculated.

[0087] If the support set of the second estimated signal is the same as the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, then the second estimated signal is retained.

[0088] If the support set of the second estimated signal is different from the support set of the vector product position sequence of the measurement matrix and the transpose of the measurement matrix, and the iteration step size is not less than the minimum iteration step size, then the second estimated signal is retained.

[0089] If the support set of the second estimated signal is different from the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step size is less than the minimum iteration step size, update the iteration step size and repeat step S4 until the iteration step size is not less than the minimum iteration step size, then retain the second estimated signal.

[0090] Specifically, to obtain

[0091]

[0092] If Γ n+1 =Γ n ,but

[0093] If Γ n+1 ≠Γ n ,when At that time,

[0094] If Γ n+1 ≠Γ n ,when At that time, α n =κα n Repeat step S4 until... at this time

[0095] When the support set of the second estimated signal remains unchanged from the previous iteration, the sparse signal can be considered to have converged, and the second estimated signal at this point can be regarded as the first estimated signal. If the support set of the second estimated signal changes from the previous iteration, and the iteration step size is not less than the minimum iteration step size, the second estimated signal at this point can be regarded as the first estimated signal. If the support set of the second estimated signal changes from the previous iteration, and the iteration step size is less than the minimum iteration step size, the iteration step size is changed, and the iteration is restarted.

[0096] S5 If the error between the second estimated signal and the first estimated signal is not greater than the iteration error threshold, then the iteration is terminated, and the target estimated signal is the second estimated signal; if the error between the second estimated signal and the first estimated signal is greater than the iteration error threshold, then the second estimated signal is used as the first estimated signal, and steps S3 to S5 are repeated.

[0097] Specifically, if Then the iteration terminates, and the target estimated signal is x. n+1 ;

[0098] like Then x n =x n+1 , n = n + 1, repeat steps S3 to S5.

[0099] That is, if the error between the estimated signal after iteration and the estimated signal before iteration is not greater than the iteration error threshold, the convergence of the sparse signal is considered to have met the requirements, and the iteration can be terminated at this time; if the error between the estimated signal after iteration and the estimated signal before iteration is greater than the iteration error threshold, the iteration number is incremented by one, and the iteration continues until the error between the estimated signal after iteration and the estimated signal before iteration is not greater than the iteration error threshold.

[0100] Figure 3 This is a graph showing the variation of the iteration error of a fast-converging sparse recovery method provided in this application with the number of iterations, as illustrated in the embodiments. Figure 3 As shown, the improved l 1 / 2 Regularized iterative threshold algorithm and original algorithm (step size parameter α = max(eig(Φ)) T The convergence characteristics of Φ))+0.01) were compared. It can be seen that the iteration error monotonically decreases with the number of iterations. When the number of iterations n=75, the improved l 1 / 2 The regularized iterative threshold algorithm then converged.

[0101] Figure 4 This is a graph showing the mean square error of a fast-converging sparse recovery method provided in this application as a function of the signal-to-noise ratio, as illustrated in the embodiments of this application. Figure 4 As shown, when the signal-to-noise ratio is greater than SNR=10 4 At that time, the improved l1 / 2 The regularized iterative threshold algorithm significantly outperforms the original algorithm (step size parameter α = max(eig(Φ)). T With Φ))+0.01), the mean square error can be reduced to MSE=10. -5 .

[0102] The sparse recovery method with fast convergence provided in this application, under the premise of adhering to the principle of minimizing the step size parameter, can improve the convergence speed and estimation accuracy by adaptively adjusting the step size parameter during the iteration process, and has the advantages of fast convergence speed and small reconstruction error.

[0103] This application also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the above-described method. The computer-readable storage medium may include, but is not limited to, any type of disk, including floppy disks, optical disks, DVDs, CD-ROMs, microdrives, as well as magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, DRAMs, VRAMs, flash memory devices, magnetic cards or optical cards, nanosystems (including molecular memory ICs), or any type of medium or device suitable for storing instructions and / or data.

[0104] It should be noted that, for the sake of simplicity, the foregoing method embodiments are all described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, as some steps may be performed in other orders or simultaneously according to this application. Furthermore, those skilled in the art should also understand that the embodiments described in the specification are preferred embodiments, and the actions and modules involved are not necessarily essential to this application.

[0105] In the above embodiments, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0106] In the several embodiments provided in this application, it should be understood that the disclosed apparatus can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some service interface; the indirect coupling or communication connection between devices or units may be electrical or other forms.

[0107] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0108] Furthermore, the functional units in the various embodiments of this application 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.

[0109] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage device (CMD). Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a memory and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this application. The aforementioned memory includes various media capable of storing program code, such as USB flash drives, read-only memory (ROM), random access memory (RAM), portable hard drives, magnetic disks, or optical disks.

[0110] Those skilled in the art will understand that all or part of the steps in the various methods of the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, which may include: a flash drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk, or an optical disk, etc.

[0111] The foregoing description is merely an exemplary embodiment of this disclosure and should not be construed as limiting the scope of this disclosure. Any equivalent changes and modifications made in accordance with the teachings of this disclosure shall still fall within the scope of this disclosure. Those skilled in the art will readily conceive of embodiments of this disclosure upon considering the specification and practicing the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not described herein. The specification and embodiments are to be considered exemplary only, and the scope and spirit of this disclosure are defined by the claims.

[0112] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0113] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A fast-converging sparse recovery method, applied in image transmission and processing, characterized in that, include: S1 acquires the observed signal, measurement matrix, sparsity of the target signal, and iteration error threshold; S2 initializes the support set of the position sequence of the vector product of the first estimated signal and the transpose of the measurement matrix with the observed signal, which is obtained based on the sparsity of the observed signal, the measurement matrix and the target signal; S3 obtains the iteration step size based on the support set of the position sequence of the vector product of the observed signal, the measurement matrix, the transpose of the measurement matrix and the observed signal, and the first estimated signal; S4 substitutes the iteration step size into the iteration formula to obtain the second estimated signal; S5 If the error between the second estimated signal and the first estimated signal is not greater than the iteration error threshold, then the iteration is terminated, and the target estimated signal is the second estimated signal; If the error between the second estimated signal and the first estimated signal is greater than the iteration error threshold, then the second estimated signal is used as the first estimated signal, and steps S3 to S5 are repeated. After substituting the iteration step size into the iteration formula to obtain the second estimated signal, the support set of the second estimated signal is calculated. If the support set of the second estimated signal is the same as the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, then the second estimated signal is retained. If the support set of the second estimated signal is different from the support set of the position sequence of the vector product of the transpose of the measurement matrix and the observed signal, and the iteration step size is not less than the minimum iteration step size, then the second estimated signal is retained. If the support set of the second estimated signal is different from the support set of the transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step size is less than the minimum iteration step size, update the iteration step size and repeat step S4 until the iteration step size is not less than the minimum iteration step size, then retain the second estimated signal.

2. The sparse recovery method with fast convergence as described in claim 1, characterized in that, The acquisition of the observed signal, measurement matrix, sparsity of the target signal, and iteration error threshold specifically includes: Obtain the observed signal y∈R M Measurement matrix Φ∈ M×N sparsity of the target signal k Iteration error threshold η .

3. The sparse recovery method with fast convergence as described in claim 2, characterized in that, The support set for the position sequence of the vector product of the first estimated signal and the transpose of the measurement matrix with the observed signal specifically includes: Obtain the first estimated signal x n Г n =supp[ H k (Φ T y)], where n Let supp(a) represent the number of iterations, and let sup(a) represent the support set of vector a. This means keeping only the element with the largest absolute value in vector m. k The element is set to 0, and all other elements in vector m are changed to 0. Initialize x n Г n =supp[ H k (Φ T y)], when At that time, x1=0, Г1=supp[ H k (Φ T y)).

4. The fast-converging sparse recovery method as described in claim 3, characterized in that, The method for obtaining the iteration step size based on the support set of the position sequence of the vector product of the observed signal, the measurement matrix, the transpose of the measurement matrix and the observed signal, and the first estimated signal specifically includes: Get Based on g n Get the iteration step size α n 。 5. The fast-converging sparse recovery method as described in claim 4, characterized in that, The step of substituting the iteration step size into the iteration formula to obtain the second estimated signal specifically includes: Get Get Get The second estimated signal is obtained respectively. i element , i ∈(1, k +1) Obtain the second estimated signal ,in, 。 6. The fast-converging sparse recovery method as described in claim 5, characterized in that, After obtaining the second estimated signal, the support set of the second estimated signal is calculated, specifically including: Get If Г n+1 =Г n ,but ; If Г n+1 ≠Г n ,when At that time, ; If Г n+1 ≠Г n ,when hour, Repeat step S4 until... ,at this time , , where constant c =0.

01.

7. The sparse recovery method with fast convergence as described in claim 6, characterized in that, Step S5 specifically includes: like If the iteration terminates, the target estimation signal is obtained. ; like ,but , n = n +1, repeat steps S3~S5.

8. A sparse recovery device with fast convergence, characterized in that, The method includes at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program that, when executed by the processing unit, causes the processing unit to perform the steps of the method according to any one of claims 1 to 7.

9. A storage medium, characterized in that, It stores a computer program executable by a fast-converging sparse recovery device, which, when run on the fast-converging sparse recovery device, causes the fast-converging sparse recovery device to perform the steps of the method according to any one of claims 1 to 7.

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