Finite time solver design method for sparse signal reconstruction problem

Abstract

The invention provides a finite time solver design method for a sparse signal reconstruction problem, and the method comprises the steps: designing a solver based on a classical neurodynamics algorithm in combination with a sliding mode controller and a projection operator; according to the method, the optimal solution of the sparse signal reconstruction problem can be converged in finite time, the convergence speed is higher, and compared with a traditional solver, the calculation speed is greatly improved.

Description

Technical Field

[0001] The present invention relates to a solver design method, in particular to a finite time solver design method for sparse signal reconstruction problems. Background Art

[0002] With the continuous development of information technology and the advent of the big data era, the requirements for signal processing are becoming increasingly stringent. Information sampling, transmission, and storage have become major challenges. Traditional Nyquist sampling, especially for signals with relatively wide bandwidths, simply cannot meet the sampling rate and cannot reconstruct the original signal without distortion. Through the tireless efforts of researchers, the theory of compressed sensing, a theory with significant significance and practical value, has been proposed. Since its introduction, compressed sensing has attracted widespread attention in academia and industry, garnering significant attention in fields such as information theory, image processing, pattern recognition, and biomedical engineering. It was even recognized as one of the top ten scientific and technological advances of 2007 in the United States. This demonstrates the profound significance and practical value of research on the theory and application of compressed sensing.

[0003] Compressed sensing, a new signal processing theory, can recover desired information with fewer measurements than the Nyquist sampling theorem. Compressed sensing discards redundant information when acquiring information, directly obtaining compressed samples from the continuous-time signal transform. These samples are then processed using an optimization algorithm during signal processing. The concept can be understood as follows: a signal becomes sparse after a certain transformation. The transformed high-dimensional signal can then be projected onto a low-dimensional space using a measurement matrix. Finally, by solving an optimization problem, the original signal can be reconstructed with a high probability. The measurement matrix is independent of the transformation basis.

[0004] In compressed sensing theory, solving compressed sensing problems can be divided into the following steps:

[0005] (1) Assume that an original signal x of dimension n is K-sparse on a sparse basis Ψ;

[0006] (2) Find a measurement matrix Φ that is uncorrelated with the sparse basis Ψ;

[0007] (3) Using the measurement matrix Φ to observe the original signal x, we obtain the measurement vector y with dimension m, where K < m < < n;

[0008] (4) Reconstruct x with high probability from the measurement vector y using an optimization algorithm;

[0009] That is, assume an original signal \(x\) of dimension \(n\) (this signal is assumed to be a sparse signal) with sparsity \(K\), \(\varPhi\) is an \(m\times n\) matrix (\(m < n\)), and \(y=\varPhi x\) is the measurement vector with dimension \(m\). The compressive sensing problem is to solve the underdetermined system of equations \(y = \varPhi x\) based on the observation matrix \(\varPhi\) and the measurement vector \(y\) using an optimization algorithm to obtain the original signal \(x\). Each row of \(\varPhi\) can be regarded as a sensor, which then multiplies with the signal to obtain a part of the information in the signal, and finally the original signal is reconstructed with high probability using an optimization algorithm.

[0010] However, generally signals are not sparse and often need to be represented sparsely, that is, to make a transformation on a certain sparse basis. Let \(x=\varPsi s\), where \(\varPsi\) is the sparse basis and \(s\) is the sparse coefficient. The original problem is transformed into: \(y = \varPhi\varPsi s\). Given \(y\), \(\varPhi\), and \(\varPsi\), solve for \(s\). Let \(\varTheta=\varPhi\varPsi\), where \(\varTheta\) is the observation matrix, then the problem is equivalent to: \(y=\varTheta s\). Given \(y\) and \(\varTheta\), solve for \(s\). Finally, the original signal \(x\) is reconstructed from \(x = \varPsi s\). Since the equation \(y=\varTheta s\) is an underdetermined system of equations, it is difficult to solve. Summary of the Invention

[0011] To solve at least one technical problem in the prior art, an embodiment of the present invention provides a method for designing a finite-time solver for a sparse signal reconstruction problem, which can converge to the optimal solution of the sparse signal reconstruction problem within a finite time and has a faster convergence speed. Compared with traditional solvers, the computing speed is greatly improved. To achieve the above technical objectives, the technical solution adopted in the embodiment of the present invention is:

[0012] An embodiment of the present invention provides a method for designing a finite-time solver for a sparse signal reconstruction problem, including the following steps:

[0013] Step S10, the L1-norm minimization problem model in the sparse signal reconstruction problem is shown as formula (1):

[0014]

[0015] where \(\|\cdot\|_1\) is the L1 norm, \(x=(x_1,x_2,\cdots,x

[0016] ,

[0018] ,

[0017] , , , \cdots,x n ) T \in\mathbb{R} n is the original signal to be reconstructed with dimension \(n\), \(\varPhi\in\mathbb{R} m×n is the observation matrix, \(y\in\mathbb{R} m is the measurement vector with dimension \(m\);

[0016] Step S20, design a solver by combining a sliding mode controller and a projection operator;

[0017] The solver is shown as formula (2);

[0018]

[0019] Among them, the sliding mode controller[·] α =sgn(·)|·| α ,α∈(0,1);z=(z1,z2,…z i …,z n ) T ∈R n is the decision variable, P = Φ T (ΦΦ T ) -1 Φ is the projection matrix, q = Φ T (ΦΦ T ) -1 y, g(·) is the projection operator;

[0020] Specifically,

[0021]

[0022] Among them, ||·||2 is the L2 norm; When Ω is a box constraint, then Ω={z∈R n :-1≤z i ≤1,i=1,2,…,n},g(z)=[g(z1),g(z2),…g(z i )…,g(z n )] T ;in,

[0023]

[0024] The technical solution provided by the embodiment of the present invention has the following beneficial effects:

[0025] 1) This application is different from the traditional neural dynamics method. It combines the sliding mode controller and the projection operator to design a solver, which can solve the optimal solution of the sparse reconstruction problem in a limited time, has a faster convergence speed, and greatly improves the calculation speed.

[0026] 2) The present application can adjust the convergence speed of the solver by adjusting the sliding mode parameter α. When α=1, the solver degenerates from finite-time convergence to exponential convergence; therefore, the solver has a strong generalization ability. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 This is a numerical simulation experiment diagram in an embodiment of the present invention.

[0028] Figure 2 Schematic diagram of sparse signal reconstruction in an embodiment of the present invention.

[0029] Figure 3 Schematic diagram of convergence trajectory in an embodiment of the present invention.

[0030] Figure 4 Schematic diagram of error comparison in an embodiment of the present invention. DETAILED DESCRIPTION

[0031] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present 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 only used to explain the present invention and are not intended to limit the present invention.

[0032] In the description of the embodiments of the present invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings and are intended solely to facilitate the description of the present invention and simplify the description. They do not indicate or imply that the devices or components referred to must have a specific orientation, be constructed, or operate in a specific orientation. Therefore, they should not be construed as limitations on the present invention. Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0033] In the description of the embodiments of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "installed," "connected," and "connected" should be understood in a broad sense. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to mechanical connections or electrical connections; they can refer to direct connections or indirect connections through an intermediate medium; they can refer to internal connections between two components; and they can refer to wireless connections or wired connections. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0034] In addition, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0035] The embodiment of the present invention proposes a finite-time solver design method for the sparse signal reconstruction problem, comprising the following steps:

[0036] In step S10, the L1 norm minimization problem model in the sparse signal reconstruction problem is shown in formula (1):

[0037]

[0038] Among them, ||·||1 is the L1 norm, x=(x1,x2,…x i …,x n ) T ∈R n is the original signal to be reconstructed, with dimension n, Φ∈R m×nis the observation matrix, y∈R m is the measurement vector with dimension m;

[0039] Step S20, designing a solver by combining the sliding mode controller and the projection operator;

[0040] The solver is shown in formula (2);

[0041]

[0042] Among them, the sliding mode controller[·] α =sgn(·)|·| α ,α∈(0,1);z=(z1,z2,…z i …,z n ) T ∈R n is the decision variable, P = Φ T (ΦΦ T ) -1 Φ is the projection matrix, q = Φ T (ΦΦ T ) -1 y, g(·) is the projection operator;

[0043] Specifically,

[0044]

[0045] Among them, ||·||2 is the L2 norm; When Ω is a box constraint, then Ω={z∈R n :-1≤z i ≤1,i=1,2,…,n},g(z)=[g(z1),g(z2),…g(z i )…,g(z n )] T ;in,

[0046]

[0047] Embodiment 1;

[0048] The L1 norm minimization problem model in the sparse signal reconstruction problem is shown in formula (1):

[0049]

[0050] in,

[0051] in y=(2,3,1) T .

[0052] Assuming that the projection matrix P is a positive definite matrix, the solutions of the solver proposed in this application and the other two solvers are (0,-0.3384,0.3240,0.3921,0,0), (0,-0.3381,0.3239,0.3920,0,0) and (0,-0.3376,0.3251,0.3924,0,0) respectively. Figure 1 The numerical simulation process of the solver finding the solution to the problem is shown in Figure 2. Figure 1 In the figure, it can be seen intuitively that the solver proposed in this application and the other two solvers converge to the optimal solution of the problem, and the solver proposed in this application converges significantly faster than the other two solvers.

[0053] Embodiment 2;

[0054] Further consider the high-dimensional example and apply it to sparse signal reconstruction; use the "Φ=rand(m,n)" command in Matlab to randomly generate the observation matrix, with the initial value x0=zeros(n,1), m=100, n=50; sparsity K=6; then perform experimental simulation. Figure 2 Schematic diagram of sparse signal reconstruction. Figure 3 is a schematic diagram of the convergence trajectory, Figure 4 Schematic diagram of error comparison;

[0055] Figure 2 In the example, the signal reconstructed by the solver of this application has a sparsity of 6, which is consistent with the original signal and the signals reconstructed by other solvers; Figure 3 In the example, it can be clearly seen that the solver of this application can converge to the optimal solution of the problem faster; Figure 4 The above further illustrates that the solver of the present application can reach stability faster, i.e., has a faster convergence speed. Therefore, the solver of the present application has a faster solution speed and higher efficiency than the traditional solver.

[0056] Finally, it should be noted that the above specific implementation methods are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that the technical solutions of the present invention can be modified or replaced by equivalents without departing from the spirit and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A finite-time solver design method for sparse signal reconstruction problem, characterized in that: The following steps are involved: In step S10, the L1 norm minimization problem model in the sparse signal reconstruction problem is shown in formula (1): Among them, ||·||1 is the L1 norm, x=(x1,x2,…x i …,x n ) T ∈R n is the original signal to be reconstructed, with dimension n, Φ∈R m×n is the observation matrix, y∈R m is the measurement vector with dimension m; Step S20: Design a solver by combining the sliding mode controller and the projection operator.

2. The finite-time solver design method for the sparse signal reconstruction problem according to claim 1, wherein: The solver is shown in formula (2); Among them, the sliding mode controller[·] α =sgn(·)|·| α ,α∈(0,1);z=(z1,z2,…z i …,z n ) T ∈R n is the decision variable, P = Φ T (ΦΦ T ) -1 Φ is the projection matrix, q = Φ T (ΦΦ T ) -1 y, g(·) is the projection operator.

3. The finite-time solver design method for the sparse signal reconstruction problem according to claim 2, characterized in that: In the solver, Among them, ||·||2 is the L2 norm; When Ω is a box constraint, then Ω={z∈R n :-1≤z i ≤1,i=1,2,…,n},g(z)=[g(z1),g(z2),…g(z i )…,g(z n )] T ;in,

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