Subspace estimation adaptive matrix sensing method for compressed sensing imaging

Abstract

The invention discloses a subspace estimation adaptive matrix sensing method for compressed sensing imaging. A Gaussian random linear operator A is used as a stimulation mode to be applied to a compressed sensing imaging system so as to carry out p1-time measurement on a scene to be measured, and a corresponding observed quantity y1 is collected. And obtaining first matrix estimation of an unknown matrix X of the scene to be measured through a matrix low-rank recovery algorithm by using y1 and A. And designing a second measurement operator AH based on specific structure information estimated by the first matrix. AH is used as a stimulation mode to be applied to the compressed sensing imaging system for measurement, new observed quantity y2 is collected, and the unknown matrix X is recovered by combining the first matrix estimation, the coefficient matrix estimation and the column space base. According to the method, the total number of times of measurement of the unknown matrix representing the image can be reduced, and the method has great value in invisible light wave band imaging, machine learning and big data, especially in scenes extremely sensitive to measurement cost.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of image compressive sensing, in particular to a subspace estimation adaptive matrix sensing method for compressive sensing imaging. BACKGROUND

[0002] Matrix sensing technology is an extension of compressive sensing theory in the field of matrix, which aims to reconstruct a low-rank matrix from partial observations or noisy data. It is applied in various fields of science and engineering, such as non-visible light band imaging, such as terahertz imaging, infrared thermal imaging, microwave imaging, or extreme conditions such as ultra-high-speed imaging, low-light imaging. It can also be applied in the field of computer vision, for example, video repair or hyperspectral image reconstruction. In these applications, it is difficult to obtain a complete image directly (here, the conventional image and video are collectively referred to as image), because it is extremely difficult and expensive, or even impossible, to manufacture a high-performance, high-resolution planar array detector, so it is difficult to obtain the image directly through the device. We know that traditional imaging is to take a video with a high-speed planar array camera (such as CMOS or CCD), and each pixel on the camera sensor array directly records the light intensity at the corresponding position, so that the image can be obtained directly. In contrast, for non-visible light bands (such as terahertz imaging, infrared thermal imaging, microwave imaging) or extreme conditions (such as ultra-high-speed imaging, low-light imaging), it is physically impossible or very difficult to obtain a complete image, i.e. matrix X, so the compressive sensing method is used to obtain the image. In order to determine the matrix X, the prior art provides some methods, which are mainly divided into two categories: convex relaxation methods (such as nuclear norm minimization) and non-convex optimization methods (such as alternating least squares). These methods have their own advantages, but they all have problems such as too high measurement number, high computational complexity, slow convergence speed, or sensitivity to initial values.

[0003] The prior art also proposes a GNMR (Gauss-Newton Matrix Recovery) algorithm, which has the following significant advantages: the Gauss-Newton method is used to achieve local quadratic convergence, which is theoretically guaranteed; the update strategy is flexibly adjusted by the parameter a, which is robust to ill-conditioned matrices; and the least squares solution framework has good numerical stability.

[0004] The GNMR algorithm needs to follow the RIP (Restricted Isometry Property) condition:

[0005] A linear mapping A: satisfies the rank-r restricted isometry property (r-RIP), for a constant δr∈[0,1), if and only if for all matrices of rank not exceeding r The following inequality holds:

[0006]

[0007] in,

[0008] Linear mapping: A is a mapping of matrix space Mapping to vector space Linear operators.

[0009] Rank-r restricted isometry (r-RIP): This property requires that A approximately maintains its norm (the relationship between the Frobenius norm and the l2 norm) on all matrices whose rank does not exceed r.

[0010] The constant δr measures the degree of equidistant approximation of the mapping. The smaller δr is, the closer A is to the equidistant mapping.

[0011] Frobenius norm (||X||) F ): The square root of the sum of the squares of all elements of a matrix.

[0012] l2 norm (||A(X)||2): vector The Euclidean norm.

[0013] Like existing matrix sensing techniques, GNMR uses non-adaptive measurements, meaning the linear operator A is fixed before any data is collected; such linear operators are called non-adaptive. Because the choice of linear operator A is fixed (e.g., Gaussian measurement or uniform sampling), a large number of measurements are required to recover the matrix, resulting in excessively high measurement costs.

[0014] Therefore, it is necessary to design an adaptive matrix sensing method that can reduce the number of measurements. Summary of the Invention

[0015] To address the problems existing in the prior art, this invention proposes a subspace estimation adaptive matrix sensing method for compressed sensing imaging.

[0016] The subspace estimation adaptive matrix sensing method for compressed sensing imaging of the present invention includes the following steps:

[0017] A Gaussian random measurement matrix A_i is applied as a stimulus pattern to a compressed sensing imaging system that measures the scene under test, and the observed values ​​y1_i of the compressed sensing imaging system are recorded.

[0018] Where i = 1, 2, 3, ..., p1, p1 is the first measurement number, y1 is the observation vector, y1 = A(X) + e1, and y1 = (y1_1, y1_2, y1_3, ..., y1_p1) T Let A be a linear operator composed of A_i, and let X be an unknown matrix representing the unknown image corresponding to the scene to be tested, X∈R. n1×n2 y1∈R p1×1n1 and n2 are the number of rows and columns of the unknown matrix X, respectively, e1 is additive noise, A(X) = (<A_1,X> ,<A_2,X> ,…,<A_p1,X> ) T A_i and the unknown matrix X have the same dimension;

[0019] The estimate of the unknown matrix X determined from (y1, A) is used as the first matrix estimate. And extract column space basis and It is an orthonormal basis, i.e. r is the rank of the unknown matrix X;

[0020] Estimate the first matrix Convert to column space basis The product of the unknown coefficient matrix C, i.e.

[0021] Utilizing column space Generate the adaptive measurement matrix A using a Gaussian random matrix H_j. H_j j = 1, 2, 3, ..., p2, where H = (H_1,H_2,H_3,…,H_j),A H_j Composition of adaptive measurement operator A H , p2 represents the second measurement number;

[0022] The adaptive measurement matrix A H_j A compressed sensing imaging system is used to apply a stimulus pattern to the scene under test and to measure it. The observed values ​​y2_j of the compressed sensing imaging system are recorded. The observation vector y2 is obtained based on the observed values ​​y2_j, where y2 = A. H (X)+e2, where e2 is additive noise;

[0023] use and Let y2 = A H (X)+e2 is converted to y2=H(C)+e2;

[0024] according to Estimation of the deterministic coefficient matrix using the GNMR algorithm As the coefficient matrix C;

[0025] The recovery matrix is ​​determined as the unknown matrix based on the first matrix estimate, the coefficient matrix estimate, and the column space basis.

[0026] The technical solution of this invention adopts a two-stage approach: first, initial measurement is performed; then, the norm index of the matrix is ​​calculated; next, the measurement operator is adjusted based on the index; and finally, measurement is performed again to recover the unknown matrix. The first stage solves the problem of "finding the main direction of the signal" from a high-dimensional (n1×n2) unknown matrix, obtaining the column space of the high-dimensional matrix X. The unknown matrix is ​​then represented using a column space basis and an unknown coefficient matrix, thus transforming the unknown matrix to be determined into a definite coefficient matrix, and converting the high-dimensional problem into a low-dimensional (n2×r) problem. The second stage operates in a low-dimensional (n2×r) space, and the measurement operator used is an adaptive operator redesigned based on the column space basis obtained in the first stage. This fully utilizes the measurement information from the first stage, resulting in extremely high measurement efficiency in this stage.

[0027] In the method of this invention, the measurement energy is concentrated on the subspace where the signal actually exists, avoiding wasteful measurements in noise-dominated, ineffective directions. On the one hand, the two-stage measurement method of this invention can significantly reduce the total number of measurements compared to conventional GNMR algorithms; the total number of measurements p1+p2 is typically much smaller than the number required for linear operator A in the prior art. On the other hand, it can also utilize the information obtained after the first-stage measurement to construct a measurement operator that better matches the unknown matrix, thereby better recovering the unknown matrix.

[0028] In this invention, the first stage does not require determining the full picture of the unknown matrix, and the dimension of the coefficient matrix to be determined in the second stage is greatly reduced. Therefore, the total number of measurements required can be reduced. This two-stage adaptive strategy can break through some sampling rate limits of non-adaptive matrix sensing. Attached Figure Description

[0029] Figure 1 This is a flowchart of an adaptive matrix sensing method for subspace estimation in compressed sensing imaging, according to an embodiment of the present invention. Detailed Implementation

[0030] In this invention, conventional images and videos are collectively referred to as images. The unknown matrix X represents the complete image of the scene to be tested.

[0031] As is well known, for non-visible light bands such as terahertz imaging, infrared thermal imaging, and microwave imaging, or under extreme conditions such as ultra-high-speed imaging and low-light imaging, it is extremely difficult, expensive, and even impossible to manufacture high-performance, high-resolution area array detectors. For example, there is no "terahertz camera" capable of directly capturing 1-megapixel terahertz images. Therefore, it is physically impossible or very difficult to directly obtain a complete image of the scene under test, i.e., the unknown matrix X. However, it is feasible to manufacture a single high-performance detector. Technicians have discovered that although the unknown matrix X cannot be obtained directly, it is possible to obtain measurement values ​​by applying a stimulus pattern to a compressed sensing imaging system consisting of a light source, a spatial light modulator (SLM), and a single-pixel detector, thereby physically performing linear measurements. The complete image X of the scene under test can then be reconstructed using the stimulus pattern and the measurement values. The structure and working principle of the compressed sensing imaging system are known and will not be elaborated here.

[0032] In this invention, a single linear measurement y i =<A_h,X> This is done as follows: A Gaussian random measurement matrix A_h is generated, h = 1, 2, 3, ..., p. The measurement matrix A_h represents the stimulus pattern to be applied to the compressed sensing imaging system, for example, a beam of terahertz waves of a specific shape in terahertz imaging. <,> represents the integration process of a single-pixel detector, y i This is the total energy detected. The result of p measurements, y i Combining these, we obtain the observation vector y. The p measurement matrices A_k are combined to form the linear operator A, therefore the linear measurement y ≡ A(X) + ξ. Similarly, the p1th and p2th linear measurements mentioned below involve the same physical quantities as those in the aforementioned linear measurement term, and will not be described in detail again. The use of the Gaussian random measurement matrix allows it to comply with the RIP (Restricted Isometry) condition.

[0033] This invention performs two measurements on the scene under test. After the second measurement, the matrix is ​​directly recovered using a known GNMR algorithm. To this end, a known GNMR algorithm applied to compressed sensing imaging is first described. For linear measurements y≡A(X)+ξ, A:R n1×n2 →R m ξ represents additive noise. In the GNMR algorithm, the low-rank unknown matrix X is factored, X = UV. T ,U∈R n1×r ,V∈R n2×r Let y be the observation vector and r be the rank of matrix X. The algorithm is given an estimate (U0, V0) and an update (ΔU, ΔV) such that (U1, V1) = (U0 + ΔU, V0 + ΔV).

[0034] The GNMR algorithm employs the convex least squares method, and solves for the minimum norm solution of (ΔU, ΔV) according to the following formula, i.e., it requires... Minimum,

[0035]

[0036] Furthermore, the GNMR algorithm is defined. α is an adjustment factor, thus transforming formulas (1) and (2) into:

[0037]

[0038]

[0039] According to formulas (3) and (4), under the aforementioned condition of minimum norm, the final solution can be obtained. Then according to X=UV T The unknown matrix is ​​obtained.

[0040] The GNMR algorithm uses a fixed linear operator A, which limits its application. This is because, in the GNMR algorithm, to recover the unknown matrix X, p measurements of the scene to be measured are required, where p > β0r(n1+n2)log. 2 (n1+n2). Typically, the coefficient β0 takes values ​​between 4 and 10. It is evident that for a fixed-measurement linear operator, the number of measurements is very large, and the linear operator A is a very large matrix. Due to the large number of measurements required, existing algorithms, including GNMR, are limited in their application in compressed sensing imaging, especially in scenarios where measurement costs are extremely sensitive, such as certain medical imaging (MRI) or high-energy physics experiments, and channel detection in line communication. Reducing the number of measurements directly translates to shorter scan times, lower radiation doses, or cost savings.

[0041] To address the drawback of existing technologies requiring a large number of measurements, this invention proposes an optimized design for the non-adaptive fixed linear operator used in existing technologies. This optimization utilizes feedback information from existing observations to dynamically optimize the sensing strategy, thereby reconstructing the target signal more efficiently. See also... Figure 1 This illustrates the main steps of the adaptive matrix sensing method based on subspace estimation of the present invention. See also Figure 1 First, an initial Gaussian random linear operator A is applied as the stimulus pattern to the compressed sensing imaging system to perform p1 measurements on the scene under test, and the corresponding observations y1 are collected. Second, using y1 and A, a preliminary, coarse first matrix estimate of the unknown matrix X is obtained through a low-rank recovery algorithm. Next, based on this first matrix estimation Given specific structural information (e.g., its row / column space basis), represent the unknown matrix using this structural information and a low-order coefficient matrix, and design a second measurement operator A. H Finally, use A. H The stimulus pattern is applied to the compressed sensing imaging system to measure the scene under test and collect new observations y2. Then, the first matrix estimation and coefficient matrix estimation in two stages, as well as the column space basis, are combined to finally recover the unknown matrix X.

[0042] The adaptive matrix sensing method based on subspace estimation of the present invention is described in detail below.

[0043] In this invention, the linear operator A is directly constructed from p1 measurement matrices {A_1, A_2, ..., A_p1}. Its definition is as follows:

[0044] A = (A_1, A_2, ..., A_p1)

[0045] This equation means that when you apply a linear operator A to a matrix X, its operational rules are uniquely determined:

[0046] A(X)=(<A_1,X> ,<A_2,X> ,...,<A_p1,X> ) T .

[0047] Similarly, the adaptive operator A used in the p2 measurements below H It also has a similar meaning to the linear operator A.

[0048] In embodiments of the present invention, a Gaussian random linear operator A is applied to an unknown matrix X to obtain an observation vector y1. Specifically, a Gaussian random measurement matrix A_i is applied as a stimulus pattern to a compressed sensing imaging system measuring the scene under test, and the observation values ​​y1_i of the compressed sensing imaging system are recorded, where i = 1, 2, 3, ..., p1, p1 is the first measurement number, y1 is the observation vector, y1 = A(X) + e1, and y1 = (y1_1, y1_2, y1_3, ..., y1_p1). T Let A be a linear operator composed of A_i, and let X be an unknown matrix representing the unknown image corresponding to the scene to be tested, X∈R. n1×n2 y1∈R p1×1 n1 and n2 are the number of rows and columns of the unknown matrix X, respectively, e1 is additive noise, A(X) = (<A_1,X> ,<A_2,X> ,…,<A_p1,X> ) TA_i and the unknown matrix X have the same dimension. In this invention, using the Gaussian random linear operator A on the unknown matrix X means that p1 independent n1×n2 Gaussian random measurement matrices are generated to construct the Gaussian random linear operator A. The method for generating Gaussian random measurement matrices is well known and will not be described in detail here.

[0049] Next, the estimate of the unknown matrix X is obtained from (y1,A) and used as the first matrix estimate. And extract column space basis r is the rank of the unknown matrix X. In this invention, since the goal of the first stage is not to directly recover the unknown matrix X with high quality, but rather to estimate its principal direction (subspace) at low cost, the estimate of the unknown matrix X obtained from (y1, A) is used as the first matrix estimate. And extract column space basis There are multiple ways to implement the steps.

[0050] According to an embodiment of the present invention, the task objective of the first stage is downgraded from "perfectly recovering the entire high-dimensional matrix" to "roughly estimating a low-dimensional subspace". The easier task of "roughly estimating a low-dimensional subspace" reduces the number of measurements p1 compared to conventional matrix-aware recovery algorithms.

[0051] According to one embodiment of the present invention, a coarse estimation matrix M is first constructed. Since if A_i is a Gaussian random matrix with zero mean, the coarse estimation matrix M is an unbiased but high-variance estimate of the unknown matrix X (mathematical expectation E[M] ∝ X). Its principal eigenvector directions contain column space information of the unknown matrix X. Next, an economical singular value decomposition (SVD) is performed on the coarse estimation matrix M, retaining only the first r principal components. The corresponding column vectors of the matrix M corresponding to the r principal components are used as the column space basis. Where p1 = β*r(n1+n2), β is a coefficient. The coefficient β ranges from 4 to 10. For scenarios with high signal-to-noise ratio and less stringent requirements for initialization quality, β is generally chosen to be 4; for low signal-to-noise ratio environments or scenarios with very high requirements for the quality of the first matrix estimation, β is chosen to be 10. Generally, choosing β to be 6 is sufficient to provide a good subspace estimate in most cases. Therefore, in the embodiments of this invention, since it is only necessary to stabilize the direction of the principal eigenvector of the coarse estimation matrix M, the number of measurements p1 is much less than the number of measurements p required for the GNMR algorithm to fully recover the unknown matrix X.

[0052] In another embodiment of the present invention, the estimate of the unknown matrix X obtained from (y1,A) is used as the first matrix estimate. And extract column space basis Column space estimation based on matrix sampling can be implemented as follows.

[0053] First, randomly select s columns from the n² columns of the unknown matrix X to form a column index set Ω, where Ω = {k1, k2, ..., k s}, s is greater than the rank r of the unknown matrix; for each k-th column, k = k1, k2, ..., k s .

[0054] According to the formula Solve to recover the estimate of the k-th column of the unknown matrix X. λ3 is a regularization parameter, and its value is determined according to a known method in this technical field; where Φ k For compressed sensing matrix;

[0055] For the Ath j The k-th column of the operator, X k Let represent the k-th column of the unknown matrix X, where j = 1, 2, ..., p1.

[0056] Estimate the s columns The matrix consisting of n1×s And perform singular value decomposition on the matrix. The left singular vectors obtained by performing singular value decomposition serve as the basis of the column space. The estimate.

[0057] In this embodiment, a certain number of global random measurements, p1, are first performed. Then, calculations are used to focus on specific columns, which is equivalent to sampling the matrix. This obtains both global information and column information for estimating the column space basis. On the other hand, since the measurement matrix A... j The randomness of the perceptual matrix Φ constructed for each column in the index set Ω. k It also satisfies desirable properties such as RIP, making it possible to restore a single column.

[0058] Next, the first matrix is ​​estimated. Using column space basis The unknown coefficient matrix C represents, where, Those skilled in the art will understand that column space basis It is usually obtained through singular value decomposition, therefore it is an orthonormal basis, i.e. I r It is an r×r identity matrix. Therefore, we can obtain... Clearly, perceptual recovery of the unknown matrix X is transformed into perceptual recovery of the coefficient matrix C. Since n2×r is necessarily much smaller than n1×n2, the embodiments of the present invention can transform the high-dimensional problem of the original unknown matrix into a low-dimensional problem, significantly reducing the number of measurements required to determine the coefficient matrix C.

[0059] Those skilled in the art will understand that other methods can be designed to obtain an estimate of the unknown matrix X from (y1,A) as the first matrix estimate. And extract column space basis For example, if there are few restrictions on the number of measurements p1, a strategy of minimizing the nuclear norm can be used to obtain the first matrix estimate. Specifically, according to Solution to the first matrix estimate Where ||·||_* is the nuclear norm, Z is a candidate matrix of the position matrix X, and λ4 is the regularization parameter, the value of which is determined in a manner known in this technical field. Then, matrix estimation is performed. Perform singular value decomposition, and take the first r left singular vectors to form the column space of the matrix estimate.

[0060] The next step, in order to determine the coefficient matrix C, is to utilize the column space. Generate the adaptive measurement matrix A using a Gaussian random matrix H_j. H_j j = 1, 2, 3, ..., p2, where H = (H_1,H_2,H_3,…,H_j),A H_j Composition of adaptive measurement operator A H , p2 represents the second measurement count. Next, the adaptive measurement matrix A... H_j The compressed sensing imaging system is used as a stimulus mode to measure the scene under test, and the observed value y2_j of the compressed sensing imaging system is recorded. The observation vector y2 is obtained based on the observed value y2_j, where y2 = (y21, y22, ..., y2_p2). T And y2 = A H (X) + e2, where e2 represents noise. In embodiments of the present invention, since the effective information of the low-rank matrix X is mainly concentrated in its row and column spaces, the adaptive measurement operator A... H It is designed based on the column space basis obtained from the first matrix estimation. It utilizes preliminary matrix-aware information and the designed adaptive operator can better determine the coefficient matrix C with fewer measurements. It overcomes the defect of fixed linear operators in conventional GNMR algorithms.

[0061] The linear measurement y2 = A is obtained. HAfter (X)+e2, using and Let y2 = A H (X)+e² is transformed into y²=H(C)+e². This shows that, although physically it utilizes the adaptive measurement operator A H This is applied to a compressed sensing imaging system to measure the scene under test, but the resulting observation vector y2 is mathematically equivalent to directly measuring the coefficient matrix C, and the measurement linear operator used is H. Then, according to... The GNMR algorithm is used to obtain the estimated restored coefficient matrix. The coefficient matrix C is used as the initial matrix. Finally, the recovery matrix is ​​constructed as the unknown matrix based on the first matrix estimate, the coefficient matrix estimate, and the column space basis.

[0062] In this invention, the first stage uses a relatively small number of measurements p1 for global, non-adaptive exploration. The goal is not to obtain a perfect complete image, but to find the direction of the strongest signal (principal component) of the unknown matrix. This is a low-precision, high-efficiency task. The second stage is high-precision focusing: most of the measurement budget p2 is concentrated on fine, adaptive measurements within the discovered subspace. Since the search space is reduced from n1×n2 dimensions to n2×r dimensions, the measurement efficiency is extremely high. According to this invention, not only are the first and second measurement counts p1 and p2 less than the number of measurements p required by directly using conventional GNMR algorithms, but the total number of measurements p is also less. total =p1+p2 can be performed in fewer ways than the number of times p is required for a one-time global recovery in conventional GNMR methods, because the first stage does not require obtaining complete information about the unknown matrix, and the adaptive measurement in the second stage avoids waste in the "no signal" or "weak signal" direction.

[0063] In one embodiment of the present invention, a recovery matrix is ​​constructed as the unknown matrix based on a first matrix estimate, a coefficient matrix estimate, and a column space basis in the following manner. First, based on... Calculate the reconstruction matrix Next, according to the formula Obtain the recovery matrix The unknown matrix is ​​represented by α, a weighting coefficient between 0 and 1, which can be set based on the signal-to-noise ratio or reconstruction error during the first and second stages (p1 and p2 measurements) and the recovery process of the unknown matrix. For example, if the first-stage measurement is more reliable, α is larger; if the second-stage measurement is more accurate, (1-α) is larger. Obtaining the unknown matrix in this way is simple and computationally fast, although this method does not truly fuse the data but rather performs post-processing on the two recovery results.

[0064] According to another embodiment of the present invention, a better strategy is proposed to construct the recovery matrix as an implementation of the unknown matrix based on the first matrix estimate, the coefficient matrix estimate, and the column space basis. Specifically, according to the formula... and The optimized matrix Z is obtained as the unknown matrix, where ||·||_* is the nuclear norm, Z is a candidate matrix of the unknown matrix X, and λ1 is the regularization parameter, the value of which is determined according to a known method in this technical field. The specific solution process can be efficiently solved using the well-known Alternating Direction Multiplier Method (ADMM), which will not be elaborated here. In this embodiment, prior information about the subspace is explicitly utilized. This reduces the problem dimension from n1×n2 to n2×r, significantly decreasing computational complexity and the number of measurements required. (Constraints) Crucially, it couples the two variables, forcing the final solution Z to lie in the subspace estimated in the first stage. This significantly reduces the degrees of freedom in solving the problem. This embodiment truly integrates all data, rather than simply merging results; moreover, it maintains low-rank constraints, ensures global optimality through kernel norm terms, and addresses the potentially imperfect subspace estimation in the first stage. It has a certain degree of robustness.

[0065] In another embodiment of the present invention, the observations from the two stages are directly spliced ​​together to form a larger integrated observation system, thereby obtaining the recovery of the unknown matrix. First, according to formula y combined =[y1; y2] yields a joint observation vector y of (p1+p2)×1. combined Secondly, according to formula A combined (Z)=[A(Z;A H (Z)] yields the joint measurement operator A combined Finally, according to the formula Obtain the recovery matrix As an unknown matrix; where ||·||_* is the nuclear norm, Z is a candidate matrix of the position matrix X, and λ2 is the regularization parameter whose value is determined in a manner known in this technical field. Where A H It is the design that acts on the subspace of, that is The original intention is to measure C; however, in the implementation, it can be mapped back to the measurement of Z based on its physical meaning, because the m-th measurement

[0066] According to the present invention, in some cases, only the method described above may be used. Calculated reconstruction matrix It is directly treated as the unknown matrix X. This approach simplifies the calculation.

[0067] This invention addresses the pain point of conventional non-adaptive, i.e., fixed linear operator compressed sensing / matrix sensing—insufficient measurement efficiency. It employs a two-stage approach to recover the matrix, essentially introducing a feedback mechanism to more intelligently allocate measurement resources, thereby reducing the total number of measurements. This has significant value in scenarios where measurement costs are extremely sensitive, such as certain medical imaging (MRI) or high-energy physics experiments, and channel detection in line communication. Reducing the number of measurements directly translates to shorter scan times, lower radiation doses, or cost savings. The technical solution of this invention can also be applied to machine learning, artificial intelligence, and big data.

Claims

1. A subspace estimation adaptive matrix sensing method for compressed sensing imaging, comprising the following steps: A Gaussian random measurement matrix A_i is applied as a stimulus pattern to a compressed sensing imaging system that measures the scene under test, and the observed values ​​y1_i of the compressed sensing imaging system are recorded. in, i = 1, 2, 3, ..., p1, where p1 is the first measurement number, y1 is the observation vector, y1 = A(X) + e1, and y1 = (y1_1, y1_2, y1_3, ..., y1_p1) T A is a linear operator composed of A_i, and X is an unknown matrix representing the unknown image corresponding to the scene to be tested, X∈R. n1×n2 y1∈R p1×1 n1 and n2 are the number of rows and columns of the unknown matrix X, respectively, e1 is additive noise, A(X) = (<A_1,X> ,<A_2,X> ,…,<A_p1,X> ) T A_i and the unknown matrix X have the same dimension; The estimate of the unknown matrix X determined from (y1, A) is used as the first matrix estimate. And extract the first matrix estimate column space basis Among them, column space basis It is an orthonormal basis, i.e. r is the rank of the unknown matrix X; Estimate the first matrix Convert to column space basis The product of the unknown coefficient matrix C, where, C∈R n2×r ; Utilizing column space Generate the adaptive measurement matrix A using a Gaussian random matrix H_j. H _j, j = 1, 2, 3, ..., p2, where H_j∈R n2×r ,H=(H_1,H_2,H_3,…,H_j),A H _j forms the adaptive measurement operator A H , p2 represents the second measurement number; The adaptive measurement matrix A H Stimulus _j is applied as a stimulus pattern to a compressed sensing imaging system that measures the scene under test, and the observed value y2_j of the compressed sensing imaging system is recorded. Based on the observed value y2_j, the observation vector y2 is obtained, where y2 = A. H (X)+e2, where e2 is additive noise, and y2=(y2_1,y2_2,y2_3,…,y2_p2) T ; use and Let y2 = A H (X)+e2 is converted to y2=H(C)+e2; according to Using the GNMR algorithm to determine the coefficient matrix estimate As the coefficient matrix C; The recovery matrix is ​​determined as the unknown matrix based on the first matrix estimate, the coefficient matrix estimate, and the column space basis.

2. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 1, wherein, The recovery matrix is ​​determined based on the first matrix estimate, the coefficient matrix estimate, and the column space basis, including the following: according to Calculate the reconstruction matrix According to the formula Obtain the recovery matrix As an unknown matrix, α is a weighting coefficient between 0 and 1.

3. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 2, wherein, The weighting coefficient α is set based on the signal-to-noise ratio in measurements p1 and p2.

4. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 1, wherein, The recovery matrix is ​​determined based on the first matrix estimate, the coefficient matrix estimate, and the column space basis, including the following: According to the formula and The optimized matrix Z is obtained as the unknown matrix; where ||·||_* is the nuclear norm, Z is a candidate matrix of the unknown matrix X, and λ1 is the regularization parameter.

5. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 1, wherein, The recovery matrix is ​​determined as an unknown matrix based on the first matrix estimate, the coefficient matrix estimate, and the column space, including: According to the formula y combined =[y1; y2] yields a joint observation vector y of (p1+p2)×1. combined ; According to formula A combined (Z) = [A(Z); AJ(Z)] yields the joint measurement operator A. combined ; According to the formula Obtain the recovery matrix As the unknown matrix; where ||·||_* is the nuclear norm, Z is a candidate matrix of the unknown matrix X, and λ2 is the regularization parameter.

6. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 1, wherein, The estimate of the unknown matrix X determined from (y1, A) is used as the first matrix estimate. And extract column space basis The steps include: Construct a rough estimation matrix M. Singular value decomposition (SVD) is performed on the coarse estimation matrix M, and only the first r principal components are retained. The corresponding column vectors of the coarse estimation matrix M corresponding to the r principal components are used as the basis of the column space. Where p1 = β*r(n1+n2), β is a coefficient.

7. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 6, wherein, The coefficient β ranges from 4 to 10.

8. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as claimed in claim 1, wherein, The estimate of the unknown matrix X determined from (y1, A) is used as the first matrix estimate. And extract column space basis The steps include: Randomly select s columns from the n² columns of the unknown matrix X to form a column index set Ω, Ω = {k1, k2, ..., kn}. s }, s is greater than the rank r of the unknown matrix; For each k-th column in the index set, k = k1, k2, ..., k s , According to the formula Solve to recover the estimate of the k-th column of the unknown matrix X. λ3 is the regularization parameter; in, Estimate the s columns The matrix consisting of n1×s And perform singular value decomposition on the matrix. The left singular vectors obtained by performing singular value decomposition serve as the basis of the column space. The estimate.

9. The subspace estimation adaptive matrix sensing method for compressed sensing imaging as described in claim 1, wherein, The estimate of the unknown matrix X determined from (y1, A) is used as the first matrix estimate. And extract column space basis The steps include: according to Solve the matrix estimate Where ||·||_* is the nuclear norm, Z is a candidate matrix of the position matrix X, and λ4 is the regularization parameter; Matrix estimation Perform singular value decomposition, and take the first r left singular vectors to form the column space of the matrix estimate.

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