Noise-resistant signal reconstruction method for compressed sensing system based on resolution reduction constraint

Abstract

The invention relates to the field of target signal observation by the compressed sensing observation system, and solves the problems that the existing underdetermined compressed sensing system is sensitive to noise and cannot realize effective signal reconstruction under noise interference. The invention overcomes the problem that the compressed sensing system is sensitive to noise interference and cannot realize effective signal reconstruction. The response of the reconstruction system under variable resolution is subjected to pathological analysis and selection, and the low-resolution high-precision target signal is solved first. The reconstruction of the compressed sensing high-precision target signal is constrained by the low-resolution reconstructed signal, which increases the anti-interference ability of the compressed sensing signal reconstruction system. Without introducing additional environmental conditions, the compressed sensing signal reconstruction that is resistant to noise interference is realized by the system itself.

Description

Technical Field

[0001] The present invention relates to the problem of observing target signals by a compressed sensing observation system and reconstructing them under noise interference. By introducing reduced-resolution reconstruction results, high-resolution signal solution is constrained, the system's anti-interference performance is improved, and effective reconstruction is achieved. Background Art

[0002] When a signal acquisition (observation) system acquires a target signal, it is usually required that the system signal acquisition frequency or the number of observations be greater than or equal to the required target signal data volume; however, due to practical factors such as time, space, and system integration, the system is unable to achieve the required number of observations, that is, the target signal is undersampled. In response to the problem of undersampling of target signals, researchers have proposed a solution based on the principle of compressed sensing. The main idea of ​​compressed sensing is to combine the observation and compression steps during the data acquisition process, usually acquiring signals at a sub-Nyquist rate (for example, one-fifth of the Nyquist rate), and accurately reconstructing the world from these samples with a certain probability. In the theory and application of compressed sensing, sparsity is a prerequisite for reconstruction, so signals that meet any of the following two conditions can be effectively reconstructed:

[0003] 1. The signal itself is sparse and is defined as a vector whose most elements are zero. That is, the number of non-zero elements k in the signal is much smaller than the number of zero elements in the signal. It is called a k-sparse signal.

[0004] 2. A signal with k elements can represent the overall energy of the system and is also called a k-sparse signal.

[0005] Compressed sensing systems are mostly used in situations where the output signal is a mixture of the target signal, that is, there is a clear linear relationship between the output signal and the target signal. Therefore, compressed sensing systems are often represented by discrete linearity. After introducing sparse transformation (dictionary) and adding sparsity constraints, sparse vectors are calculated first, and then the sparse vectors are combined with the dictionary to achieve signal reconstruction.

[0006] The signal acquisition process is inevitably affected by input interference, electronic readout noise, and thermal noise. To complete signal solution and reconstruction, the compressed sensing system must be calibrated to obtain the system response. However, the calibration results of the system response contain uncertainty. Therefore, numerous interference factors are inevitably introduced into the reconstruction calculation, affecting reconstruction accuracy. Based on system type, the order of solution stability affected by noise is, from high to low, overdetermined systems, then positively determined systems, and finally underdetermined systems. Given that compressed sensing systems are underdetermined, noise interference will significantly impact signal reconstruction accuracy, potentially making it impossible to effectively reconstruct the target signal. Summary of the Invention

[0007] The present invention aims to solve the problems of existing compressed sensing underdetermined systems being sensitive to noise and unable to achieve effective signal reconstruction under noise interference, and to provide a noise-resistant reconstruction method for compressed sensing system signals based on resolution reduction constraints.

[0008] A noise-resistant reconstruction method for compressed sensing system signals based on resolution reduction constraints is implemented by the following steps:

[0009] Step 1: Build a signal acquisition model for the compressed sensing system, linearly discretize the model, and determine the data range and resolution of the observed target signal;

[0010] Step 2: Perform high-resolution compressed sensing system response calibration to obtain the basic data source;

[0011] Step 3: Determine the data range and resolution of the observed target signal according to step 1 and obtain the basic data source in step 2, and perform resolution reduction calculation of the compressed sensing system response;

[0012] Step 4: Perform pathological analysis on the system response after different levels of resolution reduction. Combined with the condition that the amount of system response data after resolution reduction is less than or equal to the number of observations of the compressed sensing system, select the reduced-resolution system response; and draw a curve of the system response condition number changing with the resolution.

[0013] Step 5: Use the resolution selected in step 4 to perform a resolution reduction operation on the system response. After obtaining the low-resolution system response, use the regularization method to reconstruct the low-resolution target signal and analyze the mathematical relationship between the low-resolution target signal and the high-resolution target signal.

[0014] Step 6: Introduce the mathematical relationship between the low-resolution target signal and the high-resolution target signal in step 5 into the reconstruction calculation of the high-resolution signal, constrain the sparse reconstruction iterative process, and reduce the high-resolution signal reconstruction deviation.

[0015] Beneficial effects of the present invention:

[0016] The present invention overcomes the problem that the compressed sensing system is sensitive to noise interference and cannot achieve effective signal reconstruction. It performs pathological analysis and selection on the reconstruction system response under variable resolution, prioritizes solving low-resolution and high-precision target signals, and uses low-resolution reconstructed signals to constrain the reconstruction of compressed sensing-high-resolution target signals, thereby increasing the anti-interference ability of the compressed sensing signal reconstruction system. Without introducing additional environmental conditions, the compressed sensing signal reconstruction that is resistant to noise interference is achieved by the system itself. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1This is a schematic diagram of a compressed sensing signal acquisition system in the compressed sensing system signal noise-resistant reconstruction method based on resolution reduction constraint according to the present invention;

[0018] Figure 2 Schematic diagram of the compressed sensing signal reconstruction structure based on resolution reduction constraint of the present invention;

[0019] Figure 3 A diagram of the response selection method of the reduced-resolution compressed sensing system of the present invention. DETAILED DESCRIPTION

[0020] 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.

[0021] like Figure 1 As shown in the figure, compressed sensing (underdetermined) systems are one of the effective methods for obtaining high-resolution signals with a limited number of observations. However, the systems themselves lack the ability to resist interference. Positively determined or overdetermined systems, on the other hand, have the ability to resist interference but cannot obtain high-resolution signals with a limited number of observations. By analyzing the relationship between the reconstruction solutions of positively determined or overdetermined systems and compressed sensing reconstruction solutions, and introducing this relationship into compressed sensing signal reconstruction, we improve the anti-interference capability of the compressed sensing-high-resolution signal reconstruction system, that is, compressed sensing signal reconstruction based on reduced resolution constraints, further ensuring the accuracy of signal reconstruction.

[0022] refer to Figure 2 A noise-resistant reconstruction method for compressed sensing system signals based on resolution reduction constraints includes:

[0023] The observation value g obtained by the compressed sensing system for the mth observation of the target signal m It can be expressed as:

[0024] ∫ x Ω m (x)φ(x)dx=g m

[0025] Using rectangular mean integral for discretization, the formula changes to:

[0026]

[0027] Where Δx represents the signal resolution of the current compressed sensing system. If the compressed sensing system observes the target signal M times, it can be expressed in matrix form as follows:

[0028] Ωφ=g

[0029] in After the compressed sensing system is calibrated, the measurement matrix is ​​obtained, where M is the number of observations of the compressed sensing system, N is the length of the discrete signal data, is the discretization result of the target signal, is the observation value output by the target signal after the compressed sensing system. The noise-resistant reconstruction method of the compressed sensing system signal based on the resolution reduction constraint specifically includes:

[0030] 1. Determine the data range where the target signal φ is located [x start , x end ] and resolution Δx0,x start is the starting point of the target signal, x end is the end point of the target signal, then the resolution of the required compressed sensing system response is the same as the target signal resolution, and the system response at this time is expressed as And M<N.

[0031] 2. Calibrate the compressed sensing system to obtain the system response. The resolution obtained by the system calibration is required to be Δx c , satisfying Δx0=Δx c / t,t∈Z + , that is, calibrate the system with a higher resolution to ensure the accuracy of the reduced resolution calculation. The system response obtained after calibration is:

[0032] 3. Carry out the resolution reduction calculation of the compressed sensing system, and let the low resolution Δx k =kΔx0, k∈Z + , where the amount of data after resolution reduction is constrained to be less than or equal to the number of compressed sensing observations, that is, (x end -x start ) / Δx k ≤M. The current system changes from solving an underdetermined system with high resolution and large amount of data to a positive or overdetermined system with low resolution and small amount of data. The low resolution is Δx k System response The calculation method is

[0033]

[0034] Where x j with x i are the discrete points obtained at different resolutions;

[0035] in j∈Z, and similarly we can get x i Calculation method.

[0036] IV. Conduct the ill - conditioning analysis of the variable - resolution system response. Based on the down - resolution calculation, perform multiple down - resolution calculations and construct the corresponding down - resolution system response. Calculate the condition number of the system response after down - resolution, and plot the curve of the change in the condition number of the system response with respect to the resolution.

[0037] V. A smaller condition number means that the current reconstructed system has a stronger anti - noise interference ability. Therefore, referring to Figure 3 , select the resolution Δx at the low - value inflection point of the curve s to perform the down - resolution operation and obtain the corresponding system response Ω s as the environment for implementing the constraint, and use the Tikhonov regularization method to reconstruct the signal at the current low resolution. Therefore, the objective function for data reconstruction after down - resolution is:

[0038]

[0039] The solution for reconstructing the low - resolution signal is:

[0040]

[0041] In the formula, is the observed value output by the target signal after passing through the compressive sensing system; I is the identity matrix, μ is the weight coefficient, which is solved by the L - curve or generalized cross - validation method.

[0042] VI. For the reconstruction of the compressive sensing target signal, since the amount of target signal data is greater than the number of observations, use the sparse transform (i.e., dictionary) method to add constraints to the current under - determined system. Select the dictionary Let

[0043] ΩDy = g subject to: Dy = φ

[0044] Where is a k - sparse vector (k << L). The role of the dictionary D is to map the original signal data to a new data domain, making the mapped data y sparse. After calculating y, the target signal φ can be calculated according to the selected dictionary D.

[0045] VII. The mathematical relationship between the reconstructed down - resolution signal and the required high - resolution signal is:

[0046]

[0047] Introduce this relationship into the reconstruction calculation of the sparse representation of the target signal φ. At this time, the objective function of the system is written as:

[0048] min||y||1

[0049]

[0050] Where ε0 is the l2 norm calculated after system error analysis; ε s is the absolute value of the maximum relative error discretized at different resolutions; for the above objective function, methods including but not limited to: convex optimization and constrained matching pursuit can be used to solve and calculate, so as to obtain a more accurate approximate solution.

[0051] The technical features of the above-mentioned embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above-mentioned 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.

[0052] The above-described embodiments merely illustrate several implementations of the present invention. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent. It should be noted that a person skilled in the art would be able to make numerous variations and improvements without departing from the spirit of the present invention, all of which fall within the scope of protection of the present invention. Therefore, the scope of protection of the patent for this invention shall be determined by the appended claims.

Claims

1. A noise-resistant reconstruction method for compressed sensing system signals based on resolution reduction constraints, characterized by: The method is implemented by the following steps: Step 1: Build a signal acquisition model for the compressed sensing system, linearly discretize the model, and determine the data range and resolution of the observed target signal; Step 2: Perform high-resolution compressed sensing system response calibration to obtain the basic data source; Step 3: Determine the data range and resolution of the observed target signal according to step 1 and obtain the basic data source in step 2, and perform resolution reduction calculation of the compressed sensing system response; Step 4: Perform pathological analysis on the system response after different levels of resolution reduction, and select the system response after resolution reduction based on the condition that the amount of data in the system response after resolution reduction is less than or equal to the number of observations of the compressed sensing system; Draw the curve of the system response condition number changing with the resolution; Step 5: Use the resolution selected in step 4 to perform a resolution reduction operation on the system response. After obtaining the low-resolution system response, use the regularization method to reconstruct the low-resolution target signal and analyze the mathematical relationship between the low-resolution target signal and the high-resolution target signal. Step 6: Introduce the mathematical relationship between the low-resolution target signal and the high-resolution target signal in step 5 into the reconstruction calculation of the high-resolution signal, constrain the sparse reconstruction iterative process, and reduce the high-resolution signal reconstruction deviation; The mathematical relationship between the low-resolution target signal and the required resolution target signal is introduced as a constraint into the reconstruction of the required resolution target signal. The resulting constraint formula is: min||y||1 Where Δx0 is the resolution, Ω is the system response, D is the dictionary, y is the k sparse vector, g is the observation value output by the target signal after the compressed sensing system, φ is the observed target signal, and φ is the target signal. s is the low-resolution signal reconstruction solution, ε0 is the l2 norm calculated after system error analysis; ε s is the absolute value of the maximum relative error discretized at different resolutions; the constraint formula is used as the objective function to achieve noise-resistant compressed sensing high-resolution target signal reconstruction.

2. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 1, characterized in that: In step 1, determine the data range [x start ,x end ] and resolution Δx0, then the resolution of the required compressed sensing system response is the same as the resolution of the observed target signal, and the system response at this time is And M<N; M is the number of observations of the compressed sensing system, and N is the length of the discretized signal data.

3. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 2, characterized in that: In step 2, obtaining the basic data source includes requiring the compressed sensing system response calibration to obtain a resolution of Δx c , satisfying Δx0=Δx c / t,t∈Z + , and the system response obtained after calibration is:

4. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 3, characterized in that: The specific process of step three is: Set low resolution Δx k =kΔx0, k∈Z + , where the amount of data after resolution reduction is constrained to be less than or equal to the number of compressed sensing observations M, that is, (x end -x start ) / Δx k ≤M; Low resolution is Δx k System response The calculation method is: Where x j with x i are the discrete points obtained at different resolutions.

5. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 4, characterized in that: In step 4, the curve of the system response condition number changing with the resolution is drawn. It is required that the amount of data after the resolution reduction operation is less than or equal to the number of compressed sensing observations. Therefore, the resolution Δx at the low value inflection point in the curve is selected. s And the corresponding system response Ω s .

6. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 5, characterized in that: Using resolution Δx s Perform resolution reduction calculation on the system response to obtain the corresponding system response Ω s As the low-resolution target signal reconstruction condition, the low-resolution target signal is reconstructed using the Tikhonov regularization method. The objective function of the reconstructed target signal after resolution reduction is: The low-resolution signal reconstruction solution is: Where, is the observation value output by the target signal after the compressed sensing system; I is the identity matrix; μ is the weight coefficient, which is solved by L-curve or generalized cross-validation method.

7. The method for noise-resistant reconstruction of compressed sensing system signals based on resolution reduction constraints according to claim 1, characterized in that: The objective function is solved and calculated by a method based on convex optimization or constrained matching pursuit.

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