Metasurface calculation imaging algorithm based on regularization

Abstract

The invention discloses a regularization-based metasurface calculation imaging algorithm, and relates to the technical field of metasurface calculation imaging, and the algorithm comprises the following steps: constructing a noisy observation model, and building a linear relation between an observation signal and a target signal; designing a Tikhonov regularization objective function, and balancing the signal reconstruction precision and the noise suppression capability; according to the method, a Tikhonov regularization objective function is constructed and a filtering factor is designed in combination with singular value decomposition, so that noise interference under underdetermined observation is effectively inhibited; the SVD closed-form solution and generalized cross validation are used for automatic parameter adjustment, and the reconstruction time of the 1024-dimensional sparse signal is greatly shortened to 0.2296 seconds from 11.335 seconds of the OMP algorithm; self-adaptive optimization of regularization parameters is achieved through the GCV criterion, different noise environments can be adapted without manual intervention, hardware complexity is remarkably reduced based on the low sampling rate characteristic of compressed sensing, and the method is suitable for scenes such as remote sensing spectrum analysis and industrial sparse imaging and has high precision, high speed, self-adaptability and engineering universality.

Description

Technical Field

[0001] The present invention relates to the technical field of metasurface computational imaging, and specifically to a metasurface computational imaging algorithm based on regularization. Background Art

[0002] Traditional full-channel imaging systems (such as high-speed cameras, CT scans, remote sensing satellites) rely on dense sensor arrays, and have problems such as complex structure, high cost, and large data processing volume. For this reason, single-pixel imaging technology has been proposed. By combining a metasurface aperture with compressive sensing, it uses a small amount of observation data to reconstruct the target signal, significantly reducing the hardware complexity.

[0003] Defects of the prior art:

[0004] Ill-conditioning of the observation matrix and noise sensitivity: The metasurface observation matrix is usually an underdetermined matrix (the observation dimension is much lower than the signal dimension), and direct solution is vulnerable to noise interference. The reconstruction error of the traditional least squares (LSQ) algorithm increases significantly at low signal-to-noise ratios.

[0005] Low reconstruction efficiency: Iterative algorithms such as orthogonal matching pursuit (OMP) need to iteratively select signal components multiple times, with high computational complexity and difficult to meet the requirements of real-time imaging.

[0006] Parameter dependence and manual parameter tuning: Traditional regularization methods rely on manually setting regularization parameters and cannot adapt to different noise environments, resulting in unstable reconstruction accuracy.

[0007] Although existing single-pixel image reconstruction methods achieve real-time reconstruction, the system memory occupancy is relatively high; at the same time, existing Fourier single-pixel imaging methods (FSI-GAN) have problems such as low imaging quality and slow speed. These methods have not effectively solved the problems of noise suppression and fast reconstruction under underdetermined observations.

[0008] In view of this, a metasurface computational imaging algorithm based on regularization is provided to overcome the above problems. Summary of the Invention

[0009] The purpose of the present invention is to provide a metasurface computational imaging algorithm based on regularization to solve the problems raised in the above background art.

[0010] To solve the above technical problems, the metasurface computational imaging algorithm based on regularization provided by the present invention includes the following steps:

[0011] Construct a noisy observation model and establish a linear relationship between the observation signal and the target signal;

[0012] Design a Tikhonov regularization objective function to balance the signal reconstruction accuracy and noise suppression ability;

[0013] Perform singular value decomposition (SVD) on the observation matrix and decompose it into a linear combination of orthogonal basis vectors and singular values;

[0014] Design a regularization filtering factor based on the singular values to weight the singular value components of the observation matrix and suppress noise;

[0015] Automatically optimize the regularization parameter through the Generalized Cross-Validation (GCV) criterion to achieve data-driven adaptive selection of parameters;

[0016] Reconstruct the one-dimensional sparse signal using the parameters and decomposition results of the above steps.

[0017] Furthermore, the noisy observation model is the observation equation:

[0018] g = Hσ + n (Equation 1);

[0019] In the equation, g is the observation signal vector, which is composed of the noisy data collected by the metasurface aperture; H is the metasurface observation matrix, which characterizes the modulation characteristics of the metasurface unit on the target signal; σ is the one-dimensional sparse signal to be reconstructed; and n is the additive Gaussian white noise.

[0020] Furthermore, the Tikhonov regularization objective function is:

[0021]

[0022] The first term is the data fitting term; the second term is the L2-norm regularization term, and λ is the regularization parameter, which is used to adjust the noise suppression intensity.

[0023] Furthermore, the expression of the singular value decomposition (SVD) is:

[0024]

[0025] r is the matrix rank; β i are the singular values arranged in descending order u i and V i * are the left and right singular vectors respectively.

[0026] Furthermore, the matrix form of the least squares (LSQ) solution of the observation matrix is: 0]

[0027]

[0028] is the projection of the observation signal in the left singular vector space, and β i are the corresponding singular values.

[0029] Furthermore, the regularization filtering factor is:

[0030] Introduce a filtering factor to suppress noise:

[0031]

[0032] f i Weight the singular value components: large singular values correspond to the dominant components of the signal, and their contributions are retained; small singular values correspond to noise, which is suppressed through the denominator λ 2 Suppress;

[0033] Solution after regularization:

[0034]

[0035] Through the frequency-domain filtering mechanism, noise is automatically suppressed during signal reconstruction, improving the reconstruction accuracy under low signal-to-noise ratios.

[0036] Furthermore, the expression of the generalized cross-validation GCV criterion is:

[0037]

[0038] S(λ) = H(H * H + λ 2 I N ) -1 H * where H is the projection matrix; trace(S(λ)) represents the effective degrees of freedom of the model, reflecting the risk of overfitting.

[0039] Compared with the prior art, the beneficial effects of the present invention are:

[0040] 1. Significantly improved noise robustness: By introducing the L2 norm constraint through the Tikhonov regularization objective function (Equation 2) and designing the filtering factor (Equation 7) in combination with singular value decomposition (SVD), the noise components corresponding to small singular values are attenuated, and the dominant signal components of large singular values are retained; effectively suppressing the influence of additive white Gaussian noise on the reconstruction accuracy.

[0041] 2. Greatly improved computational efficiency: Using SVD to decompose the observation matrix into orthogonal bases (Equation 3), avoiding iterative calculations through the closed-form solution (Equation 8), and automatically optimizing the regularization parameter (Equation 9) in combination with generalized cross-validation (GCV); the reconstruction time for a 1024-dimensional sparse signal is reduced from 11.335 seconds of the OMP algorithm to 0.2296 seconds, with a speed increase of more than 50 times, meeting the requirements of real-time imaging systems for low latency.

[0042] 3. Parameter Adaptive Optimization: Automatically select the optimal regularization parameter λ through the GCV criterion (Equation 9) without manual intervention. GCV balances the model complexity (characterized by the trace of the projection matrix trace(S(λ))) and the reconstruction error by minimizing the unbiased prediction error, adapting to different noise levels and the characteristics of the observation matrix. In scenarios with unknown noise power, the algorithm can still automatically select the optimal parameter, avoiding the cumbersome and subjective manual parameter tuning and enhancing the generalization ability of the algorithm.

[0043] 4. Low Complexity and High Universality: Based on the compressive sensing theory, only a small amount of observation data (sampling rate ≥ 50%) is required to reconstruct one-dimensional sparse signals, reducing the requirements for hardware acquisition devices. The combination of SVD and GCV does not depend on a specific metasurface structure and is applicable to metasurface apertures of different materials and sizes.

[0044] Application Value: It can be directly applied to scenarios such as remote sensing satellite spectrum analysis and industrial CT sparse imaging. While maintaining high precision, it significantly simplifies the hardware system design and reduces costs.

[0045] Through the combination of Tikhonov regularization and SVD, the present invention systematically solves the problems of noise sensitivity, low efficiency, and parameter dependence in one-dimensional signal reconstruction in metasurface computational imaging, achieving a comprehensive improvement in noise robustness, computational efficiency, and adaptive ability, providing key support for the practical engineering application of single-pixel imaging technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is the schematic diagram of the metasurface computational imaging algorithm based on regularization of the present invention;

[0047] Figure 2 is the experimental result diagram of the metasurface computational imaging algorithm based on regularization of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0048] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0049] Please refer to Figure 1 - Figure 2 , the present invention provides a technical solution:

[0050] In metasurface computational imaging, the reconstruction of one-dimensional signals (such as spectral signals, sparse target scattering coefficients) faces the following challenges:

[0051] Ill-conditioning of the observation matrix: The hypersurface observation matrix H is often an underdetermined matrix (the number of rows M is much smaller than the number of columns N). Solving directly is vulnerable to noise interference, resulting in large reconstruction errors.

[0052] Noise sensitivity: The traditional least squares (LSQ) algorithm does not consider noise suppression, and the reconstruction effect significantly degrades under low signal-to-noise ratios.

[0053] Parameter dependence problem: The regularization parameter λ needs to be manually tuned and it is difficult to adapt to different observation environments.

[0054] This embodiment systematically solves the above problems by performing SVD decomposition on the observation matrix, designing a Tikhonov regularization filtering factor, and automatically optimizing parameters using generalized cross-validation (GCV).

[0055] Refer to Figure 1 - Figure 2 As shown in the following, an embodiment of a hypersurface computational imaging algorithm based on regularization:

[0056] I. Observation model and objective function:

[0057] Observation equation:

[0058] g = Hσ + n (Equation 1);

[0059] Where g is the observation signal vector, composed of noisy data collected by the hypersurface aperture; H is the hypersurface observation matrix, characterizing the modulation characteristics of the hypersurface units on the target signal; σ is the one-dimensional sparse signal to be reconstructed (such as the target spectrum); n is additive white Gaussian noise.

[0060] Tikhonov regularization objective function:

[0061]

[0062] By minimizing the data fitting term (the first term) and the L2 norm regularization term (the second term), the signal reconstruction accuracy and noise suppression ability are balanced;

[0063] is the regularization parameter, controlling the noise suppression intensity. The larger λ is, the stronger the noise suppression, but it may introduce signal distortion.

[0064] By constructing an optimization objective with noise robustness, the ill-conditioning problem of directly solving the underdetermined equation is avoided.

[0065] II. Singular value decomposition (SVD) of the observation matrix:

[0066] Perform orthogonal decomposition on H:

[0067]

[0068] r is the matrix rank; β iAre the singular values arranged in descending order

[0069] u i and V i * are the left and right singular vectors respectively.

[0070] Least squares (LSQ) solution expansion:

[0071]

[0072] Equation 4 is the matrix form of the LSQ solution, and Equation 5 is the SVD expansion form, which projects the observed signal onto the singular vector space; is the projection of the observed signal onto the left singular vector space, and β i is the corresponding singular value;

[0073] The observation matrix is decomposed into orthogonal bases by SVD, reducing the computational complexity and providing a basis for subsequent regularization filtering.

[0074] III. Regularization filtering factor:

[0075] Introduce a filtering factor to suppress noise:

[0076]

[0077] f i Weight the singular value components: large singular values correspond to the dominant components of the signal, and their contributions are retained; small singular values correspond to noise and are suppressed by the denominator λ 2 ;

[0078] Solution after regularization:

[0079]

[0080] Through the frequency-domain filtering mechanism, noise is automatically suppressed in signal reconstruction, improving the reconstruction accuracy at low signal-to-noise ratios.

[0081] IV. Generalized cross-validation (GCV) to optimize parameters:

[0082] Automatically select the optimal by the GCV criterion:

[0083]

[0084] S(λ) = H(H * H + λ 2 I N ) -1 H * is the projection matrix;

[0085] trace(S(λ)) represents the effective degrees of freedom of the model and reflects the risk of overfitting;

[0086] The numerator is the normalized reconstruction error, and the denominator penalizes the error by the degrees of freedom to avoid an overly complex model.

[0087] Implementation method:

[0088] 1. Set the search interval of λ as [λ min , λ max (e.g., 10 -4 ≤ λ ≤ 10 2 ), and adopt logarithmic interval sampling;

[0089] 2. For each λ, calculate (quickly solve using the SVD result);

[0090] 3. Select the λ that minimizes GCV * as the optimal parameter.

[0091] Avoid manual parameter tuning, adaptively match the noise environment, and improve the generalization ability of the algorithm.

[0092] V. Result verification and output (refer to Figure 2 as shown):

[0093] Algorithm Recovery time (s) Recovery effect OMP algorithm 11.335 Poor effect Regularization algorithm 0.2296 Good effect

[0094] A physical observation model is constructed through Equation 1, and the preprocessing operation eliminates the signal deviation, providing a standardized input for subsequent reconstruction;

[0095] The observation matrix is decomposed into orthogonal bases through Equation 3, reducing the complexity of matrix operations. Equations 4 and 5 give the solutions without regularization, which can be used as a comparison benchmark for subsequent regularization processing and also provide a theoretical basis for regularization processing;

[0096] Data-driven parameter optimization is achieved through Equation 9, avoiding the cumbersome and uncertain manual parameter tuning. Equation 6 is used to calculate the filtered signal, providing the necessary intermediate result for GCV calculation, enabling the algorithm to adapt to different noise environments and reducing the risk of model overfitting;

[0097] Equation 2 provides the goal orientation for regularization processing, improving the accuracy and robustness of reconstruction by balancing data fitting and signal sparsity. Equations 7 and 8 reconstruct the signal using the singular value components after regularization filtering, effectively suppressing noise in the orthogonal basis space and improving the reconstruction accuracy at low signal-to-noise ratios;

[0098] By evaluating the error of the reconstruction result, it is ensured that the reconstructed signal meets the accuracy requirements, achieving closed-loop quality control. The entire process is based on the previous equations, forming a complete and operable technical solution from the establishment of the observation model to signal reconstruction and result verification.

[0099] VI. Summary:

[0100] Formula linkage:

[0101] Equation 1 constructs a physical observation model, Equation 2 provides a regularization objective, Equations 3 - 5 decompose the observation matrix and establish a reference solution, Equations 6 - 8 achieve noise suppression, and Equation 9 optimizes key parameters, forming a complete technical chain.

[0102] Technical advantages:

[0103] Noise robustness: Suppress noise through the regularization filtering factor and improve the reconstruction accuracy of low signal - to - noise ratio;

[0104] Computational efficiency: The closed - form solution of SVD and the fast parameter tuning of GCV meet the requirements of real - time imaging;

[0105] Adaptability: Data - driven parameter optimization, without prior knowledge, and adaptable to different observation environments.

[0106] Engineering value:

[0107] For the one - dimensional signal reconstruction in metasurface computational imaging, a reproducible, efficient and robust technical solution is formed, providing key algorithm support for scenarios such as single - pixel spectral imaging and sparse target localization.

[0108] This embodiment systematically solves the one - dimensional signal reconstruction problem in metasurface imaging through the combination of mathematical formulas and engineering implementation, and has clear practical application value.

Claims

1. A regularization-based metasurface computational imaging algorithm, characterized in that It includes the following steps: Construct a noisy observation model and establish a linear relationship between the observation signal and the target signal; Design a Tikhonov regularization objective function to balance the signal reconstruction accuracy and the noise suppression ability; Perform singular value decomposition (SVD) on the observation matrix and decompose it into a linear combination of orthogonal basis vectors and singular values; Design a regularization filtering factor based on the singular values and weight the singular value components of the observation matrix to suppress noise; Automatically optimize the regularization parameter through the Generalized Cross-Validation (GCV) criterion to achieve data-driven adaptive selection of parameters; Reconstruct a one-dimensional sparse signal using the parameters and decomposition results of the above steps.

2. The regularization-based metasurface computational imaging algorithm according to claim 1, wherein: The noisy observation model is the observation equation: g = Hσ + n (Equation 1); where g is the observation signal vector, composed of the noisy data collected by the metasurface aperture; H is the metasurface observation matrix, characterizing the modulation characteristics of the metasurface elements on the target signal; σ is the one-dimensional sparse signal to be reconstructed; and n is additive white Gaussian noise.

3. The regularization-based metasurface computational imaging algorithm according to claim 1, characterized in that: The Tikhonov regularization objective function is: The first term is the data fitting term; the second term is the L2-norm regularization term, and λ is the regularization parameter used to adjust the noise suppression intensity.

4. The regularization-based metasurface computational imaging algorithm according to claim 1, characterized in that: The expression of singular value decomposition (SVD) is: r is the rank of the matrix; β i are the singular values arranged in descending order u i and V i * are the left and right singular vectors respectively.

5. The regularization-based metasurface computational imaging algorithm according to claim 1, wherein: The matrix form of the least squares (LSQ) solution of the observation matrix is: is the projection of the observed signal in the left singular vector space, and β i is the corresponding singular value.

6. The regularization-based metasurface computational imaging algorithm according to claim 1, wherein: The regularization filtering factor is: Introduce the filtering factor to suppress noise: f i Weight the singular value components: large singular values correspond to the dominant components of the signal, and their contributions are retained; Small singular values correspond to noise and are suppressed by the denominator λ 2 Suppress; The solution after regularization: Through the frequency-domain filtering mechanism, automatically suppress noise in signal reconstruction and improve the reconstruction accuracy at low signal-to-noise ratios.

7. The regularization-based metasurface computational imaging algorithm according to claim 1, characterized in that: The expression of the Generalized Cross-Validation (GCV) criterion is: S(λ) = H(H * H + λ 2 I N ) -1 H * where H is the projection matrix; trace(S(λ)) represents the effective degrees of freedom of the model and reflects the overfitting risk.

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