Rapidly convergent sparse recovery method and device, and storage medium

Abstract

The invention discloses a fast convergence sparse recovery method. The method comprises the following steps: S1, obtaining an observation signal, a measurement matrix, the sparseness of a signal to be solved and an iteration error threshold; s2, initializing a support set of the transpose of the first estimation signal and the measurement matrix and a vector product position sequence of the observation signal; s3, obtaining an iteration step length based on the observation signal, the measurement matrix, the support set and the first estimation signal; s4, substituting the iteration step length into an iteration formula to obtain a second estimation signal; s5, if the error between the second estimation signal and the first estimation signal is not greater than an iteration error threshold value, iteration is terminated, and the target estimation signal is the second estimation signal; and otherwise, taking the second estimation signal as the first estimation signal, and repeating the steps S3-S5. The invention also discloses a device and a storage medium corresponding to fast convergence sparse recovery. The fast convergence sparse recovery method provided by the invention can improve the convergence speed and the estimation precision.

Description

Technical Field

[0001] The present application relates to the field of signal processing technology, and more specifically, to a fast-converging sparse recovery method, device, and storage medium. Background Art

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

[0003]

[0004] This problem is a non-deterministic polynomial problem. To solve this problem, existing methods usually introduce penalty terms and regularization parameters to construct a new objective function to solve the problem, such as using the l0 iterative threshold algorithm, l1 iterative threshold algorithm, and l 1 / 2 Iterative thresholding algorithm and other methods.

[0005] However, the existing 1 / 2 The iterative threshold algorithm has the problem of unreasonable selection of iteration step size for large singular value matrix convergence, 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 need of the prior art, the present invention provides a fast-converging sparse recovery method, device and storage medium, which can solve at least one of the problems existing in the above-mentioned prior art.

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

[0008] S1 obtains the observation signal, measurement matrix, sparsity of the target signal and the iterative error threshold;

[0009] S2 initializes the support set of the vector product position sequence of the transpose of the first estimation signal and the measurement matrix and the observation signal, wherein the support set is obtained based on the sparsity of the observation signal, the measurement matrix and the target signal;

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

[0011] S4 brings the iteration step into the iteration formula to obtain a second estimation signal;

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

[0013] Furthermore, in the above-mentioned fast-converging sparse recovery method, after the iteration step length is brought 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 and the transpose of the measurement matrix are the same as the support set of 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 transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step is not less than the minimum iteration step, the second estimated signal is retained;

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

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

[0018] Get the observation signal y∈R M , the measurement matrix The sparsity k of the target signal, the iterative error threshold η, the constants c = 0.01 and κ = 1.1.

[0019] Furthermore, in the above-mentioned fast-converging sparse recovery method, the support set of the position sequence of the vector product of the transpose of the initialization first estimation signal and the measurement matrix and the observation 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, and H k (m) means that only the k elements with the largest absolute values ​​in vector m are retained, and the other elements in vector m are changed to 0;

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

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

[0023] Get

[0024] g n =Φ T (y-Φx n )

[0025] Based on g n Get the iteration step α n

[0026]

[0027] Furthermore, the above-mentioned fast-convergence sparse recovery method, wherein the iteration step length is brought into the iteration formula to obtain the second estimation signal, specifically includes:

[0028] Get

[0029]

[0030] Get

[0031]

[0032] Get

[0033]

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

[0035]

[0036] Get the second estimated signal in,

[0037]

[0038] Furthermore, in the above-mentioned fast-converging sparse recovery method, after obtaining the second estimated signal, calculating the support set of 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 When

[0043] If Γ n+1 ≠Γ n ,when When α n =κα n , repeat step S4 until at this time

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

[0045] like Then the iteration is terminated, 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 the second aspect of the present invention, there is also provided a fast-converging sparse recovery device, which includes at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the steps of any one of the above-mentioned methods.

[0048] According to the third aspect of the present invention, a storage medium is also provided, which stores a computer program executable by a fast-converging sparse recovery device. When the computer program runs on the fast-converging sparse recovery device, the fast-converging sparse recovery device executes the steps of any one of the above-mentioned methods.

[0049] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0050] The embodiment of the present application provides a fast-converging sparse recovery method, which, under the premise of following the principle of keeping the step size parameter as small as possible, 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. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0052] Figure 1 A schematic diagram of a process flow of a fast-converging sparse recovery method provided in an embodiment of the present application;

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

[0054] Figure 3 A curve diagram showing the variation of the iteration error of a fast-converging sparse recovery method with the number of iterations provided in an embodiment of the present application;

[0055] Figure 4 A graph showing the variation of the mean square error with the signal-to-noise ratio of a fast-converging sparse recovery method provided in an embodiment of the present application. DETAILED DESCRIPTION

[0056] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present 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 and claims of this application and the above-mentioned drawings are used to distinguish different objects, rather than to describe a specific order. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not limited to the listed steps or units, but optionally includes steps or units that are not listed, or optionally includes other steps or units inherent to these processes, methods, products or devices.

[0058] Figure 1 is a flowchart of a fast-converging sparse recovery method provided by an embodiment of the present application, such as Figure 1 As shown, the present application provides a fast-converging sparse recovery method, comprising:

[0059] S1 obtains the observation signal, measurement matrix, sparsity of the target signal and the iterative error threshold.

[0060] Specifically, compressed sensing can be used in the process of image transmission and processing. It can take advantage of the redundant characteristics of data and collect only a small number of samples to restore the original data. In the process of collecting data, the data is compressed to ensure that the original data is restored without distortion. The compressed sensing signal can be represented as a sparse signal, and the sparsity k means that there are k values ​​that are not 0 in the result of the sparse representation of the signal.

[0061] The present invention is based on the observation signal y∈R M , the measurement matrix and the known conditions such as the sparsity k of the signal to be sought, around the objective function A compressed sensing algorithm based on logarithmic sum is proposed.

[0062] Figure 2 FIG. 1 is a schematic diagram of the entire process of a fast-converging sparse recovery method provided by an embodiment of the present application. Figure 2 As shown, first obtain the observation signal y∈R M , the measurement matrix The sparsity k of the target signal, the iterative error threshold η, the constants c = 0.01 and κ = 1.1.

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

[0064] S2 initializes the support set of the vector product position sequence of the transpose of the first estimation signal and the measurement matrix and the observation signal, wherein the support set is obtained based on the sparsity of the observation signal, the measurement matrix and the target signal.

[0065] Specifically, 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, and H k (m) means that only the k elements with the largest absolute values ​​in vector m are retained, and the other elements in vector m are changed to 0;

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

[0067] The support set of a is the number of non-zero elements in a if a subset of a satisfies the elements in this subset. k (m) means that only the k elements with the largest absolute value in vector m are retained, and the other elements in vector m are changed to 0. T represents the transpose of Φ, then Γ n =supp[H k (Φ T y)] represents Φ T The position sequence of the k elements with the largest absolute value in the vector product of y.

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

[0069] Specifically, obtain

[0070] g n =Φ T (y-Φx n )

[0071] Based on g n Get the iteration step α n

[0072]

[0073] g n Represents Φ T (y-Φx n ) is the gradient of g n Get the iteration step α n .

[0074] S4 brings the iteration step into the iteration formula to obtain the second estimation signal.

[0075] Specifically, obtain

[0076]

[0077] Get

[0078]

[0079] Get

[0080]

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

[0082]

[0083] Get the second estimated signal in,

[0084]

[0085] For each element of the second estimated signal, calculate Get the second estimated signal The sign function represents a function for extracting the sign of the number of pieces (positive or negative).

[0086] After the iteration step length is substituted into the iteration formula and the second estimation signal is obtained, the support set of the second estimation signal is calculated.

[0087] If the support set of the second estimated signal and the transpose of the measurement matrix are the same as the support set of 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 transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step is not less than the minimum iteration step, the second estimated signal is retained;

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

[0090] Specifically, obtain

[0091]

[0092] If Γ n+1 =Γ n ,but

[0093] If Γ n+1 ≠Γ n ,when When

[0094] If Γ n+1 ≠Γ n ,when When α n =κα n , repeat step S4 until at this time

[0095] That is, when the support set of the second estimated signal no longer changes from before the iteration, it can be considered that the sparse signal has completed convergence, and the second estimated signal at this time can be regarded as the first estimated signal; if the support set of the second estimated signal changes from before the iteration, if the iteration step size is not less than the minimum iteration step size, the second estimated signal at this time can be regarded as the first estimated signal; if the support set of the second estimated signal changes from before the iteration, if the iteration step size is less than the minimum iteration step size, then change the iteration step size and iterate again.

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

[0097] Specifically, if Then the iteration is terminated, 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, when the error between the estimated signal after iteration and the estimated signal before iteration is not greater than the iteration error threshold, it is considered that the convergence of the sparse signal has 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 number of iterations is increased by one, and the iteration is continued 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 is a curve diagram showing the variation of the iteration error of a fast-converging sparse recovery method with the number of iterations provided in an embodiment of the present application, such as 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 the improved l are compared. It can be seen that the iterative error decreases monotonically with the number of iterations. When the number of iterations n = 75, the improved l 1 / 2 The regularized iterative threshold algorithm has reached convergence.

[0101] Figure 4 is a curve diagram showing the variation of the mean square error of a fast-converging sparse recovery method with the signal-to-noise ratio provided in an embodiment of the present application, such as Figure 4 As shown, when the signal-to-noise ratio is greater than SNR=10 4 When the improved l1 / 2 The performance of the regularized iterative threshold algorithm is significantly better than the original algorithm (step size parameter α = max(eig(Φ T Φ))+0.01), the mean square error can be reduced to MSE=10 -5 .

[0102] The embodiment of the present application provides a fast-converging sparse recovery method, which, under the premise of following the principle of keeping the step size parameter as small as possible, 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] The present application also provides a computer-readable storage medium on which a computer program is stored, and when the program is executed by a processor, the steps of the above method are implemented. The computer-readable storage medium may include, but is not limited to, any type of disk, including a floppy disk, an optical disk, a DVD, a CD-ROM, a micro drive, and a magneto-optical disk, a ROM, a RAM, an EPROM, an EEPROM, a DRAM, a VRAM, a flash memory device, a magnetic card or an optical card, a nanosystem (including a molecular memory IC), or any type of medium or device suitable for storing instructions and / or data.

[0104] It should be noted that, for the aforementioned method embodiments, for the sake of simplicity, they are all expressed as a series of action combinations, but those skilled in the art should be aware that the present application is not limited by the described order of actions, because according to the present application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily required by the present application.

[0105] In the above embodiments, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0106] In the several embodiments provided in the present application, it should be understood that the disclosed devices can be implemented in other ways. For example, the device embodiments described above are only schematic, such as the division of the units, which is only a logical function division. There may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some service interfaces, and the indirect coupling or communication connection of devices or units can be electrical or other forms.

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

[0108] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0109] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable memory. Based on this understanding, the technical solution of the present application is essentially 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, and the computer software product is stored in a memory, including a number of instructions to enable a computer device (which can be a personal computer, server or network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present application. The aforementioned memory includes: U disk, read-only memory (ROM), random access memory (RAM), mobile hard disk, disk or optical disk and other media that can store program code.

[0110] A person of ordinary skill in the art can understand that all or part of the steps in the various methods of the above embodiments can be completed by instructing related hardware through a program, and the program can be stored in a computer-readable memory, which can include: a flash drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk or an optical disk, etc.

[0111] The above is only an exemplary embodiment of the present disclosure, and the scope of the present disclosure cannot be limited thereto. That is, any equivalent changes and modifications made according to the teachings of the present disclosure are still within the scope of the present disclosure. After considering the specification and practicing the disclosure here, those skilled in the art will easily think of the implementation scheme of the present disclosure. This application is intended to cover any modification, use or adaptation of the present disclosure, which follows the general principles of the present disclosure and includes common knowledge or customary technical means in the technical field not recorded in the present disclosure. The description and examples are regarded as exemplary only, and the scope and spirit of the present disclosure are defined by the claims.

[0112] The technical features of the above embodiments may be arbitrarily combined. To make the description concise, 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] It will be easily understood by those skilled in the art that the above description is only 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 in the protection scope of the present invention.

Claims

1. A fast-converging sparse recovery method, characterized in that: include: S1 obtains the observation signal, measurement matrix, sparsity of the target signal and the iterative error threshold; S2 initializes the support set of the vector product position sequence of the transpose of the first estimation signal and the measurement matrix and the observation signal, wherein the support set is obtained based on the sparsity of the observation signal, the measurement matrix and the target signal; S3 obtains an iteration step size based on the observation signal, the measurement matrix, the support set of the vector product position sequence of the transpose of the measurement matrix and the observation signal, and the first estimated signal; S4 brings the iteration step into the iteration formula to obtain a second estimation signal; S5: if the error between the second estimation signal and the first estimation signal is not greater than the iteration error threshold, the iteration is terminated, and the target estimation signal is the second estimation signal; If the error between the second estimation signal and the first estimation signal is greater than the iteration error threshold, the second estimation signal is used as the first estimation signal, and steps S3 to S5 are repeated.

2. A fast-converging sparse recovery method as claimed in claim 1, characterized in that: After the iteration step length is brought into the iteration formula to obtain the second estimated signal, a support set of the second estimated signal is calculated; If the support set of the second estimated signal and the transpose of the measurement matrix are the same as the support set of 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 transpose of the measurement matrix and the vector product position sequence of the observed signal, and the iteration step is not less than the minimum iteration step, the second estimated signal is retained; If the support set of the second estimated signal is different from the support set of the vector product position sequence of the transpose of the measurement matrix and the observed signal, and the iteration step is less than the minimum iteration step, update the iteration step and repeat step S4 until the iteration step is not less than the minimum iteration step, then retain the second estimated signal.

3. A fast-converging sparse recovery method as claimed in claim 2, characterized in that: The step of obtaining the observation signal, the measurement matrix, the sparsity of the target signal and the iterative error threshold specifically includes: Get the observation signal y∈R M , the measurement matrix The sparsity k of the target signal, the iterative error threshold η, the constants c = 0.01 and κ = 1.

1.

4. A fast-converging sparse recovery method as claimed in claim 3, characterized in that: The support set of the position sequence of the vector product of the transpose of the initialization first estimation signal and the measurement matrix and the observation signal specifically includes: 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, and H k (m) means that only the k elements with the largest absolute values ​​in vector m are retained, and the other elements in vector m are changed to 0; Initialize x n , Γ n =supp[H k (Φ T y)], when n = 1, x1 = 0, Γ1 = supp[H k (Φ T y)].

5. A fast-converging sparse recovery method as claimed in claim 4, characterized in that: The step of obtaining the iteration step based on the support set of the vector product position sequence of the observation signal, the measurement matrix, the transpose of the measurement matrix and the observation signal and the first estimation signal specifically includes: Get g n =Φ T (y-Φx n ) Based on g n Get the iteration step α n 6. A fast-converging sparse recovery method as claimed in claim 5, characterized in that: Substituting the iteration step length into the iteration formula to obtain the second estimation signal specifically includes: Get Get Get Get the i-th element of the second estimated signal respectively Get the second estimated signal in, 7. A fast-converging sparse recovery method as claimed in claim 6, characterized in that: After obtaining the second estimated signal, calculating the support set of the second estimated signal specifically includes: Get If Γ n+1 =Γ n ,but If Γ n+1 ≠Γ n ,when When If Γ n+1 ≠Γ n ,when When α n =κα n , repeat step S4 until at this time 8. A fast-converging sparse recovery method as claimed in claim 7, characterized in that: The step S5 specifically includes: like Then the iteration is terminated, and the target estimated signal is x n+1 ; like Then x n =x n+1 , n=n+1, repeat steps S3 to S5.

9. A fast-converging sparse recovery device, characterized in that: The method comprises at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the steps of the method according to any one of claims 1 to 8.

10. A storage medium, characterized in that: It stores a computer program executable by a fast-converging sparse recovery device. When the computer program runs on the fast-converging sparse recovery device, the fast-converging sparse recovery device executes the steps of the method described in any one of claims 1 to 8.

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