Composite material terahertz thickness measuring method based on physical interpretable sparse expansion network

Abstract

The invention discloses a composite material terahertz thickness measuring method based on a physical interpretable sparse expansion network, and relates to the crossing field of terahertz nondestructive testing technology and artificial intelligence. Comprising the steps of obtaining a terahertz time domain echo signal of a to-be-detected composite material, establishing a sparse inverse problem model, constructing a physically-guided interpretable sparse expansion network, designing a dynamic sparse weighted focus loss function and calculating the thickness of each layer. The problems of dispersion interference, signal aliasing and noise submerging of the composite material are effectively solved, and the method has the advantages of high measurement precision, high processing efficiency, high physical interpretability and the like.

Description

Technical Field

[0001] This invention relates to the intersection of terahertz nondestructive testing technology and artificial intelligence, and more specifically, to a terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks. Background Technology

[0002] Composite materials, due to their superior properties such as high specific strength and corrosion resistance, are widely used in aerospace, wind power generation, and other fields. However, defects such as thickness variations and delamination can easily occur during manufacturing and service, seriously affecting structural safety. Terahertz time-domain spectroscopy, with its advantages of being non-ionizing and having good penetration into non-polar materials, has become an important tool for non-destructive testing of composite materials. By analyzing the flight time of terahertz pulses propagating within the material, it is possible to measure the thickness of structural layers and the thickness and depth of defects.

[0003] However, achieving high-precision terahertz thickness measurement of composite materials in practical applications faces extremely severe challenges. First, composite materials typically possess heterogeneous and anisotropic structural characteristics, causing terahertz waves to experience strong frequency-dependent attenuation and dispersion effects during propagation. This results in severe broadening, asymmetric distortion, and peak shift in the reflected echo pulse in the time domain, rendering traditional pulse peak-based localization methods ineffective. Second, when inspecting thin-layer or multi-layered bonded structures, the small thickness of each layer leads to severe temporal overlap (signal aliasing) of reflected echoes from the front and rear interfaces due to pulse broadening, forming complex composite waveforms that make it impossible to directly distinguish individual reflecting interfaces. Furthermore, the complex industrial inspection environment often obscures weak interlayer reflection signals due to scattering noise and system noise, and the low signal-to-noise ratio further complicates high-precision time-of-flight extraction.

[0004] To address the aforementioned signal processing challenges, researchers have proposed various signal restoration and parameter estimation methods, which can be mainly categorized into physical model-based deconvolution methods, sparse representation-based optimization algorithms, and data-driven deep learning methods. The first category comprises physical model-based deconvolution methods, such as frequency domain deconvolution and Wiener filtering. These methods attempt to recover ideal pulse sequences by eliminating the influence of the system response function, and can compress pulse width and improve resolution to some extent. However, their limitations lie in their extreme sensitivity to high-frequency noise, easily leading to noise amplification and Gibbs ringing effects. Furthermore, they typically assume the system is linear and time-invariant and neglect the complex dispersion characteristics of materials, making it difficult to accurately handle waveform distortion caused by frequency-dependent attenuation. The second category consists of sparse representation-based optimization algorithms, which is currently the mainstream direction for improving terahertz resolution. This type of method utilizes the sparsity of terahertz interlayer reflection signals in the time domain, modeling them as convolutions of dictionary atoms and sparse coefficients. Representative algorithms include Basis Pursuit, Matching Pursuit, and the Iterative Shrinkage-Thresholding Algorithm (ISTA). These methods have the advantage of a solid mathematical foundation, enabling them to recover sparse reflection interfaces from undersampled or aliased signals. However, they also have significant drawbacks: first, low computational efficiency; traditional iterative algorithms like ISTA require hundreds or thousands of iterations to converge, making it difficult to meet the real-time requirements of industrial online inspection; second, poor parameter adaptability; regularization parameters (such as sparsity weights) and step sizes usually require manual tuning, making it difficult to adapt to changing inspection environments; and third, dictionary mismatch; pre-defined analytical dictionaries (such as Gaussian wavelets) often cannot perfectly represent the complex dispersive echoes in actual composite materials, limiting reconstruction accuracy. The third category is deep learning methods that have emerged in recent years. Models such as Convolutional Neural Networks (CNNs) have demonstrated superior performance in terahertz signal processing due to their powerful feature extraction capabilities, enabling rapid end-to-end inference of thickness parameters. However, in the field of nondestructive testing with high reliability requirements, conventional deep learning methods suffer from "black box" characteristics. Their network structure design lacks physical guidance, their internal decision-making processes are difficult to interpret, and they are highly dependent on training data, resulting in limited generalization ability.

[0005] In summary, existing technologies for terahertz thickness measurement of composite materials with severe dispersion interference and high signal aliasing are still limited by the trade-off between model accuracy, computational efficiency, and physical interpretability. Therefore, there is an urgent need for a new method that can address waveform distortion by incorporating physical mechanisms and leverage deep learning to improve efficiency and adaptability. Summary of the Invention

[0006] The purpose of this invention is to solve the problems mentioned in the background art, and to propose a terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks.

[0007] A terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks includes the following steps: S1. Obtain the terahertz time-domain echo signal of the composite material under test; S2. Based on the terahertz wave propagation mechanism and convolutional sparse coding theory, a sparse inverse problem model is established. S3. Construct a physically guided interpretable sparse unfolding network, embedding the physical mechanism into a deep neural network through the optimization process of an iterative shrinking threshold algorithm; the interpretable sparse unfolding network includes a physically constrained relaxed dictionary learning module, a hybrid scale dense connection strategy, a dynamic threshold mechanism, and an effective compression-excitation module. S4. Design a dynamic sparse weighted focus loss function to solve the problem of weak signal missed detection caused by the extreme sparseness of interlayer reflection signals; S5. Calculate the thickness of each layer based on the flight time series output by the network.

[0008] Furthermore, in step S1, the step of acquiring the terahertz time-domain echo signal of the composite material under test includes: S11. Prepare composite material samples including single-layer, multi-layer and damaged structures; S12. Using a terahertz time-domain spectroscopy system in reflection mode, terahertz data is acquired through an ultrafast femtosecond laser source, an optical delay line, a central control unit, a terahertz transmitter-receiver, and an xy motion platform.

[0009] Furthermore, in step S2, the step of establishing the sparse inverse problem model includes: S21. Use Fourier transform to convert the time-domain waveforms of the incident and reflected signals to the frequency domain; S22. Analyze the reflection and transmission characteristics of terahertz waves at the interface of a medium using Fresnel's formula; S23. Model the signal as a sparse convolution combination of multiple dictionary atoms and solve the L1 regularization problem to recover the sparse vector.

[0010] Furthermore, in step S3, the step of constructing the physically guided interpretable sparse unfolded network includes: S31. A physical constraint relaxation dictionary learning strategy is adopted to construct a parameterized physical kernel function to simulate an ideal terahertz echo, and a relaxation term is introduced to adaptively fine-tune waveform details. S32. Design a deep sparse unfolding framework, where each layer corresponds to one iteration step of the optimization algorithm, including physical initialization, cascaded iterative update, and feature fusion based on attention mechanism. S33. Introduce a hybrid scale dense connection strategy and a dynamic attention threshold mechanism in each layer to improve the network's ability to fit sparse vectors.

[0011] Furthermore, in step S31, a parameterized physical kernel function is constructed based on the propagation mechanism of terahertz waves. The ideal physical echo is simulated by using the linear superposition of two Gaussian derivative functions: ; in, This represents a set of learnable physical parameters that control the amplitude ratio, pulse width, and time delay, respectively. An additional relaxation term was introduced on top of the physical kernel. dictionary atoms The definition is as follows: .

[0012] Furthermore, in step S32, during physical initialization, the sparse code is initialized using a relaxed dictionary of physical constraints, achieved through transposed convolution: .

[0013] Furthermore, in step S32, a compression excitation module is introduced at the end of the physically guided interpretable sparse unfolded network: ; in, This represents the Sigmoid function. Indicates average pooling. This represents the weights of the fully connected layer.

[0014] Furthermore, in step S33, the input feature map The dynamic threshold module outputs a value that matches the input image. Thresholds with the same dimensions : ; in It is the basic threshold parameter; It is a lightweight attention module that contains two convolutional layers.

[0015] Furthermore, in step S4, the step of designing the dynamic sparse weighted focus loss function includes: S41. Introduce a focus loss mechanism to reduce the dominance of a large number of simple negative samples in gradient updates; S42. Design dynamic weighting coefficients to dynamically adjust the loss weights based on the proportion of positive samples in the real labels, preventing weak signals from being missed.

[0016] Furthermore, the dynamic sparse weighted focus loss function is: ; in Indicates network output The result obtained by applying the Sigmoid function; It is a positive sample indicator function; It is a dynamic weighting coefficient; ; in This is the sensitivity scaling factor; Define focus item for: ; in To predict probabilities, For focusing parameters.

[0017] Furthermore, in step S5, when the terahertz wave is incident perpendicularly, the thickness is defined as: ; in, To measure thickness, At the speed of light, This represents the time difference between adjacent pulses in the flight time sequence output by the network. The value is the refractive index of the sample.

[0018] As can be seen from the above, the composite material terahertz thickness measurement method based on physically interpretable sparse unfolded network provided in this application includes acquiring terahertz time-domain echo signals, establishing a sparse inverse problem model, constructing a physically guided interpretable sparse unfolded network, designing a dynamic sparse weighted focus loss function, and calculating the thickness of each layer. By embedding the physical mechanism into a deep neural network and optimizing the signal processing flow, it effectively overcomes the problems of composite material dispersion interference, signal aliasing, and noise overwhelmance, and has the advantages of high measurement accuracy, high processing efficiency, and strong physical interpretability. Attached Figure Description

[0019] Figure 1 This is the overall flowchart of the present invention; Figure 2 This is a schematic diagram of the terahertz time-domain spectral system in the reflection mode of the present invention; Figure 3 The flight time prediction results for single-layer composite laminates of different thicknesses according to the present invention; Figure 4This is a schematic diagram of the multilayer composite material laminate of the present invention; Figure 5 The flight time prediction results for the multilayer composite laminate of the present invention are shown. Figure 6 This is a schematic diagram of the composite laminate containing damage according to the present invention; Figure 7 The time-of-flight prediction results for the damaged composite laminate of the present invention are shown below. Figure 8 This is a schematic diagram of terahertz wave propagation in the double-layer structure of the present invention; Figure 9 This invention describes the convolution process between the terahertz echo signal and the physical kernel. Figure 10 This is a schematic diagram of the deep sparse unfolded network architecture of the present invention; Figure 11 This is a schematic diagram of the structural architecture of the eSE module of the present invention. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] A method for measuring the thickness of terahertz composite materials based on physically interpretable sparse unfolded networks includes the following steps: S1. Obtain the terahertz time-domain echo signal of the composite material under test; First, composite material samples including single-layer, multi-layer, and damaged structures were prepared. For single-layer composite samples, glass fiber reinforced polymer (GFRP) samples with thicknesses of 0.1 mm, 0.2 mm, 0.4 mm, 0.6 mm, 0.8 mm, and 1.0 mm were used. Each sample had a transverse dimension of 50 mm × 50 mm and a refractive index calibration of 2.24. For single-layer composite samples, a three-layer sandwich structure was prepared, such as... Figure 4 As shown, the transverse dimensions are 50 mm × 50 mm. The surface layer is a GFRP laminate with thicknesses of 0.35 mm and 0.50 mm, and its refractive index is calibrated to 2.24. The intermediate layer is an extruded polystyrene (XPS) laminate with a thickness of 1.00 mm and a refractive index of 1.02. For composite materials with damaged structures, such as... Figure 6As shown, a GFRP laminate with overall dimensions of 50 mm × 50 mm × 2 mm and a refractive index of 1.65 was used. The sample was divided into a non-damaged region and a damaged region. A square air gap with dimensions of 20 mm × 20 mm × 0.6 mm was prefabricated at the bottom of the sample to simulate void defects.

[0022] Figure 2 This is a schematic diagram of the terahertz time-domain spectroscopy system in reflection mode used in the experiment. The system includes an ultrafast femtosecond laser source, an optical delay line, a central control unit, a terahertz transmitter-receiver, and an XY motion platform. During detection, the femtosecond laser beam emitted by the ultrafast laser is split into two beams by a polarization beam splitter. The high-energy beam serves as a pump pulse, passing through the delay module and exciting the terahertz transmitter to generate broadband terahertz radiation. The remaining beam serves as a probe pulse, guided to the receiver for coherent detection of the terahertz echo carrying the spectral characteristics of the sample. During detection, the transmitter, composed of a photoconductive antenna, is excited by the femtosecond laser, generating a terahertz pulse with a bandwidth ranging from 10 GHz to 4.5 THz. After penetrating the sample, the terahertz wave is reflected by a smooth metal mirror and guided into the receiver. Subsequently, the optical delay of the probe pulse is corrected by adjusting the length of the optical delay line, thereby achieving sampling of the entire time-domain terahertz signal. Finally, a stepper motor on the XY motion platform performs grating scanning to acquire terahertz data from all sampling points within the entire detection area. Before measuring the samples, the transmitter first emits a terahertz wave that reaches a smooth metal mirror. After reflection, the wave enters the receiver to obtain a terahertz reference signal. Then, the samples to be measured are sequentially placed on the smooth metal mirror for measurement. To ensure measurement stability, the experimental environment is strictly controlled, with the temperature maintained at 24℃ and the relative humidity between 25% and 27%. The data acquisition parameters are configured as follows: sampling interval of 0.02 ps, signal length of 2048 points, and spatial scan step size of 0.2 mm.

[0023] S2. Based on the terahertz wave propagation mechanism and the theory of convolutional sparse coding (CSC), a sparse inverse problem model is established. When a terahertz wave passes through a test sample, the dielectric discontinuity between adjacent layers will cause reflection at the interface. Figure 8The propagation path of the reflected terahertz signal from a test sample with a double-layer structure and its corresponding time-domain waveform are shown. It is noteworthy that, as a high-frequency electromagnetic wave, the terahertz signal is subject to attenuation and dispersion effects, including dissipation, absorption, and multiple reflections. These phenomena inevitably lead to distortion of the reflected echo relative to the incident wave, severely affecting the accuracy of time-of-flight estimation, which is a key parameter for accurately calculating thickness. To comprehensively evaluate the impact of these propagation effects on time-of-flight estimation, this study utilizes Fourier transform to convert the time-domain waveforms of the incident and reflected signals to the frequency domain: ; in , and These are the Fourier transforms of the reflected wave, the incident wave, and the system response, respectively.

[0024] As a special type of electromagnetic wave, terahertz waves can be further analyzed using Fresnel's formula: ; in , and These represent the reflection coefficient, transmission coefficient, and propagation coefficient, respectively. Represents the complex refractive index, including the real refractive index. and extinction coefficient ; This represents the propagation distance of terahertz waves in a continuous medium, while and These correspond to the propagation components of the refractive index and extinction coefficient, respectively. It is important to note that due to cumulative losses from transmission and reflection, the amplitudes of higher-order multiple reflections within the sample decay rapidly. Therefore, these components are usually negligible. Assuming the terahertz wave is incident perpendicularly, the system response function can be expressed as: ; Where 1, 2, and 3 correspond to the air layer, the first layer of the sample, and the second layer, respectively. This formula shows that a complete terahertz signal is essentially a linear combination of a finite number of basic waveforms, each with a specific time delay and amplitude. Therefore, the matrix form of formula (5) can be expressed as: ; in This represents the dispersion matrix that causes distortion in the reflected echo, while It is a sparse pulse vector related to the time delay (i.e., flight time). Therefore, the matrix form of equation (1) can be expressed as: ; in This represents the convolution dictionary, which consists of a series of dispersive terahertz reference echoes that exhibit temporal pulse broadening and amplitude attenuation. Indicates noise.

[0025] As shown in Equation (7), the flight time is closely related to the sample thickness. Therefore, the main goal of this study is to extract the accurate flight time from complex terahertz signals, which can be mathematically expressed as an inverse problem. The traditional matrix multiplication form can be more naturally expressed as a CSC model, which is more advantageous for processing time-series signals. Under the CSC framework, the signal is modeled as a sparse convolution combination of multiple dictionary atoms: ; in, It is the observed terahertz time-domain signal. It is the first in the dictionary Each atom represents a reflected echo with a specific time delay and dispersion pattern. This represents the convolution operation. It corresponds to the first A sparse feature map of atoms, where the positions of non-zero elements indicate the precise moment of interface reflection. Indicates noise. This represents the total number of atoms in the dictionary. The model can be concisely written as: ; in It is a convolutional dictionary. It is a concatenated sparse vector. To recover the sparse vector... The following L1 regularization problem needs to be solved: ; in, It is a regularization parameter that controls the trade-off between sparsity and reconstruction error. Let L1 norm be denoted. ISTA is a classic method for solving this type of problem, which approximates the optimal solution through two iterative steps: First, it updates the current solution along the negative gradient direction of the reconstruction error to minimize the data fidelity error; second, it applies a soft thresholding function to the result of the gradient step to enhance sparsity. Therefore, the th... The update rules for this iteration are as follows: ; in, It is the iteration step size related to the Lipschitz constant. It is a soft threshold operator, defined as follows: , for The transpose of .

[0026] S3. Construct a physics-guided interpretable sparse unfolding network that embeds physical mechanisms into a deep neural network through the optimization process of unfolding ISTA. The network includes a relaxed dictionary learning module for physical constraints, a hybrid scale dense connection strategy, a dynamic thresholding mechanism, and an efficient compression-excitation module. Furthermore, to transform the optimization process of CSC into an end-to-end inference process, an interpretable sparse unfolding network is constructed based on formula (11), unfolding the ISTA algorithm into a deep neural network. This network consists of four core components: physical constraint relaxation dictionary learning, deep sparse unfolding framework, hybrid scale dense connection strategy, and dynamic threshold module. The components are described in detail below: S31. In CSC theory, the quality of the global dictionary directly determines the accuracy of the sparse representation and the uniqueness of the solution. Using a general analytical dictionary (e.g., a wavelet basis) or a completely unconstrained learning dictionary often fails to accurately capture the unique bipolar pulse shape and frequency-dependent attenuation characteristics of terahertz echoes, leading to severe reconstruction errors or artifacts. Therefore, a physically constrained relaxed dictionary learning strategy is proposed. To give the dictionary a clear physical meaning, a parameterized physical kernel function is constructed based on the propagation mechanism of terahertz waves. Considering the dispersion and frequency-selective attenuation experienced by terahertz pulses during propagation, a linear superposition of two Gaussian derivative functions is used to simulate an ideal physical echo: ; in, This represents a set of learnable physical parameters that control the amplitude ratio, pulse width, and time delay, respectively. Unlike traditional networks that use random initialization, this parameterized design forces the convolutional kernels to lie on terahertz physical waves from the initialization stage, making the feature maps more interpretable. Detailed description follows... Figure 9 As shown. However, ideal physical models are often oversimplified and fail to fully characterize the complex waveform distortions caused by system jitter, material inhomogeneities, or non-perpendicular incidence in actual detection. To compensate for this deficiency, an additional relaxation term is introduced on top of the physical kernel. The final dictionary atoms The definition is as follows: ; The relaxation term is initialized with a very small initial value and adaptively fine-tunes waveform details in a data-driven manner during training. This design has two implications: on the one hand, physical constraints provide strong regularization, preventing the network from getting stuck in local minima or overfitting noise in the early stages of training; on the other hand, the relaxation term gives the model the freedom to fine-tune within the physical framework, significantly enhancing the dictionary's ability to represent complex real-world signals.

[0027] S32. Next, we introduce the deep sparse unfolding framework. Unlike simple layer stacking, each layer of this network strictly corresponds to one iteration step of the optimization algorithm, thus giving the network structure a clear mathematical meaning. For example... Figure 10 As shown, the overall architecture aims to establish an end-to-end mapping from the original terahertz signal to a sparse coefficient map (i.e., the estimated time-of-flight series). The framework comprises three key stages: physical initialization, cascaded iterative updates, and attention-based feature fusion.

[0028] The first step is physical initialization. The network input is the raw terahertz signal. Before entering the iterative phase, to accelerate convergence, the sparse code needs to be initialized using a relaxation dictionary of physical constraints. This can be achieved through transposed convolution: ; This operation is similar to matched filtering, enabling rapid location of potential reflected echoes and providing a reliable initial state for subsequent optimization. The signal is then sequentially unwound through cascaded multi-scale convolutional sparse coding (MS-CSC) layers. For example... Figure 10 As shown in the upper part, the MS-CSC unfolded layer is the core component of the network. The ISTA iteration steps are mapped into a deep module containing physical constraints and learnable parameters. Updating a layer involves the following steps: (1) Physical gradient calculation: First, using a shared physical dictionary Calculate reconstruction error This error contains the portion of the signal that has not yet been fitted. The gradient is then obtained by convolving the reconstruction error with the convolution kernel. This step identifies which components of the reconstruction error are closest to the terahertz waveform, ensuring that network updates strictly follow the physical constraints of terahertz wave propagation, thereby meeting data fidelity requirements.

[0029] (2) Nonlinear correction: Traditional ISTA only updates linearly along the gradient direction. To further improve the solution accuracy, the current state is... With gradient The data is concatenated and fed into a hybrid-scale dense update module (see S33) to learn the optimal descent direction. This ensures that the network can adaptively adjust and optimize the step size, achieving faster convergence than traditional mathematical iterations.

[0030] (3) Dynamic thresholding: At the end of MS-CSC, a dynamic attention thresholding module (see S33) is applied to eliminate noise artifacts, thereby obtaining an updated sparse feature map. .

[0031] go through Layer iterations result in a multidimensional sparse feature map containing atomic channels. Since atoms with different physical parameters may have varying sensitivities to noise or signal details at different frequency bands, direct summation can introduce interference. To address this, an effective Squeeze-and-Excitation (eSE) module is employed at the network's end. ; in, This represents the Sigmoid function. Indicates average pooling. The weights of the fully connected layer are represented by [value]. The eSE module adaptively recalibrates the channel feature responses, enhancing atomic channels containing the main echo information while suppressing noise-dominant channels. Finally, the channel fusion layer (implemented via a 1×1 convolutional layer) linearly combines the weighted multi-channel feature maps into the final single-channel time-of-flight prediction sequence. The detailed structure of the eSE module is shown below. Figure 11 As shown.

[0032] S33. Because high-frequency components of terahertz waves attenuate rapidly with depth when propagating in multilayer media, the pulse width of deep echoes is significantly larger than that of shallow echoes. Single-scale convolutional kernels struggle to simultaneously match these time-varying frequency characteristics. Furthermore, the fixed threshold in iterative algorithms cannot effectively distinguish between truly weak signals and structural noise. Therefore, a Mixed-Scale Dense (MSD) connection strategy and a dynamic attention threshold mechanism are introduced into each unfolded layer to improve the network's ability to fit sparse vectors. First, to endow the network with multi-resolution feature capture capabilities, multi-scale dilated convolution is introduced in the nonlinear update module. Specifically, this update module contains three parallel dilated convolution branches with a dilation rate of... The branch with the smallest dilatation rate has the smallest receptive field, focusing on capturing high-frequency, sharp interface reflection peaks. Branches with larger dilatation rates focus on capturing low-frequency, broadened dispersive waveforms and long-duration interlayer reflections. Simultaneously, a dense connection strategy is introduced to concatenate the input of the previous layer with the outputs of all branches along the channel dimension to maximize gradient flow and preserve signal details. The mathematical formula for calculating the update vector is as follows: ; in, It is a combination feature of the current sparse graph and gradient. Indicates the expansion rate One-dimensional convolution is used for cross-channel information fusion and dimensionality reduction. This represents a 1×1 fused convolutional layer. (Updated feature map of the MS-CSC network) It is determined by both the linear gradient descent term and the nonlinear correction term: ; Furthermore, real-world terahertz signals are often accompanied by non-stationary scattering noise. In this case, using a dynamic thresholding module instead of a fixed soft thresholding function is a relatively better approach. For the input feature map... This module outputs a threshold with the same dimensions as the input. : ; in It is a learnable basic threshold parameter. It is a lightweight attention module containing two convolutional layers; It's the Sigmoid function, which restricts the attention weights to... Within the range. The final activation function is transformed into a pointwise adaptive soft thresholding operation: ; The physical meaning of this mechanism is that in areas of strong reflection, the attention network outputs a smaller scaling factor, thereby reducing the threshold to preserve signal details; in areas dominated by noise, it outputs a larger scaling factor to suppress spurious peaks.

[0033] S4. Design a dynamic sparse weighted focus loss function to solve the problem of weak signal missed detection caused by the extreme sparseness of interlayer reflection signals; In terahertz time-domain waveforms containing thousands of sampling points, only a very small percentage (typically less than 0.1%) of the time points correspond to the actual interlayer interfaces (positive samples), while the vast majority of the remaining points are background (negative samples). This extreme class imbalance causes standard MSE or cross-entropy loss functions to fail: the network tends to predict all outputs as zero to obtain extremely low average errors, but completely loses crucial physical interface information. To address this "missed detection" problem, a dynamic sparse weighted focus loss is proposed. Total loss function. It is a composite objective function, consisting of a weighted classification loss and an L1 sparse regularization term: ; in, Indicates network output The results obtained by applying the Sigmoid function. First, to reduce the dominance of a large number of simple negative samples (background noise) in gradient updates, we introduce a focus loss mechanism. The focus term is defined as: ; in To predict probabilities, To focus on the parameters, and to prevent missed detections under extremely sparse constraints, a dynamic weighting coefficient was designed. This amplifies the loss weights for positive samples. For each training batch, the true labels are calculated. Average proportion of positive samples The formula for calculating dynamic weights is: ; in This is the sensitivity scaling factor. The formula indicates that the sparser the data, the greater the penalty weight. The larger the value, the better. The final classification loss is defined as: ; in It is a positive sample indicator function (when the result is greater than 0). ;otherwise, When the actual reflection peak appears, its corresponding gradient will be amplified. This effectively offsets the gradient dilution effect caused by the large number of negative samples, ensuring that the model can accurately capture every interlayer reflection signal.

[0034] S5. Calculate the thickness of each layer based on the flight time series output by the network.

[0035] Terahertz waves generate reflected echoes at different interfaces, and the time difference between adjacent pulses during their flight time reflects the distance information between adjacent interfaces (i.e., material thickness). Specifically, when a terahertz wave is incident perpendicularly, the thickness is defined as: ; in, To measure thickness, At the speed of light, This represents the time difference between adjacent pulses in the flight time sequence output by the network. The value is the refractive index of the sample.

[0036] To verify the effectiveness of the terahertz composite material thickness measurement method based on physically interpretable sparse unfolded networks described in this invention, this embodiment designs and conducts experimental and numerical simulation studies on terahertz composite material thickness measurement based on physically interpretable sparse unfolded networks.

[0037] like Figure 2As shown, the terahertz time-domain spectroscopy system in reflection mode used in the experiment acquired terahertz signals from single-layer, multi-layer, and damaged composite material samples. The system includes an ultrafast femtosecond laser source, an optical delay line, a central control unit, a terahertz transmitter-receiver, and an xy-motion platform. During detection, the femtosecond laser beam emitted by the ultrafast laser is split into two beams by a polarization beam splitter. The high-energy beam serves as a pump pulse, passing through the delay module and exciting the terahertz transmitter to generate broadband terahertz radiation. The remaining beam serves as a probe pulse, guided to the receiver for coherent detection of the terahertz echo carrying the spectral characteristics of the sample. During detection, the transmitter, composed of a photoconductive antenna, is excited by the femtosecond laser, generating a terahertz pulse with a bandwidth ranging from 10 GHz to 4.5 THz. After penetrating the sample, the terahertz wave is reflected by a smooth metal mirror and guided into the receiver. Subsequently, the optical delay of the probe pulse is corrected by adjusting the length of the optical delay line, thereby achieving sampling of the entire time-domain terahertz signal. Finally, grating scanning is performed using a stepper motor on the xy motion platform to acquire terahertz data from all sampling points within the entire detection area. Before measuring the sample, the transmitter first emits a terahertz wave that reaches a smooth metal mirror, is reflected, and then enters the receiver to obtain the terahertz reference signal. The Levenberg-Marquardt (LM) algorithm is used to determine the optimal parameters of the physical kernel that best match the experimental reference waveform. Subsequently, the samples to be tested were placed sequentially on a smooth metal mirror for measurement. To ensure measurement stability, the experimental environment was strictly controlled, with the temperature maintained at 24℃ and the relative humidity between 25% and 27%. The data acquisition parameters were configured as follows: sampling interval of 0.02 ps, signal length of 2048 points, and spatial scan step size of 0.2 mm.

[0038] For single-layer composite laminates, GFRP samples with thicknesses of 0.1 mm, 0.2 mm, 0.4 mm, 0.6 mm, 0.8 mm, and 1.0 mm were fabricated. Each sample had a transverse dimension of 50 mm × 50 mm and a refractive index calibration of 2.24. To reduce system random errors, five signal acquisitions were performed at each sampling point, and the average value was taken as the acquired terahertz signal. A total of 3000 signals were acquired for each thickness category. 80% of the signals were randomly selected as the network training set, and the remaining signals were used as the test set. Finally, the predicted time of flight was substituted into the thickness calculation formula to obtain the predicted thickness of the single-layer sample.

[0039] like Figure 3As shown, the time-of-flight prediction results for single-layer composite laminates of different thicknesses are presented. In the thinner 0.1 mm sample, severe temporal overlap of the terahertz signal generates numerous interference pulses. The amplitudes of these interference pulses are comparable to the true reflection peaks, easily leading to misjudgments. Conversely, in thicker samples such as 1.0 mm, terahertz waves undergo multiple internal reflections between interfaces, generating intermediate echoes between the main wave packets. This phenomenon significantly increases the difficulty of identifying the corresponding wave packets at the upper and lower interfaces. Despite these challenges, experimental results show that for the 0.1 mm sample, the network successfully decouples the overlap features, accurately locates the time of flight of the two interfaces, and effectively suppresses the non-triggered events caused by sidelobe interference. Similarly, in the 1.0 mm sample, the network accurately distinguishes the main reflection pulse, indicating that it has learned to use physical constraints to differentiate between the main reflection and multiple reflection artifacts. Table 1 lists the predicted thickness and accuracy results. The results show that for all single-layer composite samples, the prediction accuracy is higher than 99%. This superior performance is attributed to the powerful nonlinear fitting capability of the deep unfolded network, which effectively compensates for distortions caused by system response, noise, and material dispersion. Therefore, this method enables high-precision measurements at subwavelength scales, thus validating the effectiveness of the proposed method in single-layer thickness estimation.

[0040] Table 1. Predicted thickness and accuracy results for single-layer composite laminates of different thicknesses. ; like Figure 4 The diagram shows a schematic of the multilayer composite laminate used in the experiment. For the single-layer composite sample, a three-layer sandwich structure with a transverse dimension of 50 mm × 50 mm was prepared. The outer layer was a GFRP laminate with thicknesses of 0.35 mm and 0.50 mm, and its refractive index was calibrated to 2.24. The middle layer was an XPS laminate with a thickness of 1.00 mm and a refractive index of 1.02. Similar to the previous experimental scheme, the signal at each sampling point was averaged after five samplings to minimize random errors. A total of 3000 signals were collected, of which 80% were randomly selected as the training dataset, and the remainder as the test dataset.

[0041] like Figure 5As shown, the time-of-flight prediction results for multilayer composite laminates are presented. Clearly, the introduction of multiple closely adjacent interfaces with significant differences in refractive index leads to unprecedented complexity in the terahertz signal. According to Fresnel's law, when a wave travels from a high-refractive-index medium to a low-refractive-index medium, the reflected wave undergoes a 180° phase reversal. This manifests in the time domain as a negative pulse in the second reflected wave packet, a stark contrast to the conventional positive pulse reflection, posing a fundamental challenge to traditional algorithms based on positive peak detection. Due to the large refractive index difference, most of the energy is reflected at the first interface, drastically reducing the energy transmitted into the sample, causing the fourth wave packet to be almost completely submerged in noise. Within a limited time window, reflected wave packets from the four interfaces, multiple reflections within each layer, and the system's dispersion effects superimpose and interfere with each other, forming an extremely complex mixed waveform, making manual or rule-based time-of-flight extraction virtually impossible. However, the network-output time of flight corresponds one-to-one with the actual time of flight, further demonstrating the method's powerful signal extraction capabilities. Table 2 lists the prediction results and accuracy for each layer thickness. For the intermediate XPS layer, its measured thickness is 0.99 mm, while the proposed method predicts a thickness of 0.993 mm, achieving a measurement accuracy of 99.69%. The thickness extraction of this layer relies on accurate identification... Figure 5 The timing of the second and third wave packets is crucial. One of these packets has a negative phase, and the other has an extremely low amplitude; they are also very close to each other. Traditional methods are prone to producing large errors or even failing. The high accuracy achieved by this method directly verifies its robustness in processing complex interface signals.

[0042] Table 2. Predicted thickness and accuracy results for multilayer composite laminates ; like Figure 6 The diagram shows a schematic of the damaged composite laminate used in the experiment. For the composite material with the damaged structure, a GFRP laminate with overall dimensions of 50 mm × 50 mm × 2 mm and a refractive index of 1.65 was used. The sample was divided into undamaged and damaged regions. A square air gap of 20 mm × 20 mm × 0.6 mm was prefabricated at the bottom of the sample to simulate void defects. Similarly, the average of five samplings at the damaged region was used as the terahertz signal acquired at that sampling point. A total of 3000 terahertz signals were collected, with 80% randomly selected as the training dataset and the remaining signals used as the test dataset.

[0043] like Figure 7As shown, the time-of-flight prediction results for the damaged composite laminate are presented. Observing the second wave packet, phase reversal also occurs, and the latter two wave packets severely overlap in the time domain, almost merging into a wide and distorted waveform. Notably, the amplitude of the second wave packet is smaller than that of the third wave packet. This is because when terahertz waves reach the air layer, the vast majority of the wave is transmitted into the air layer, and a very small portion is reflected. However, the proposed network again demonstrates its excellent waveform interpretation and feature resolution capabilities. The time of flight predicted by the network is precisely aligned with the actual time of flight of all four interfaces. Table 3 lists the thickness and accuracy results predicted using the proposed method. Among them, the thickness measurement of the damaged region itself is a key indicator for evaluating the performance of the method. The results show that the actual thickness of the damaged region is 0.654 mm, and the predicted thickness of the proposed method is 0.642 mm, with a measurement accuracy of up to 98.12%. Considering that the measurement of this thickness depends entirely on the time of flight extracted from the weakly reflected signal, this accuracy has significant engineering value. It proves that this method can not only measure the thickness of conventional samples, but also achieve stable and high-precision measurement capabilities of the defect's own size and the thickness of the remaining structure.

[0044] Table 3. Predicted thickness and accuracy results for composite laminates containing damage. ; The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of protection claimed by the present invention. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks, characterized in that, Includes the following steps: S1. Obtain the terahertz time-domain echo signal of the composite material under test; S2. Based on the terahertz wave propagation mechanism and convolutional sparse coding theory, a sparse inverse problem model is established. S3. Construct a physics-guided interpretable sparse unfolding network and embed the physical mechanism into a deep neural network through the optimization process of the unfolding iterative shrinking threshold algorithm. The interpretable sparse unfolded network includes a physically constrained relaxed dictionary learning module, a hybrid-scale dense connection strategy, a dynamic thresholding mechanism, and an effective compression-excitation module. S4. Design a dynamic sparse weighted focus loss function to solve the problem of weak signal missed detection caused by the extreme sparseness of interlayer reflection signals; S5. Calculate the thickness of each layer based on the flight time series output by the network.

2. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 1, characterized in that, In step S1, the step of acquiring the terahertz time-domain echo signal of the composite material under test includes: S11. Prepare composite material samples including single-layer, multi-layer and damaged structures; S12. Using a terahertz time-domain spectroscopy system in reflection mode, terahertz data is acquired through an ultrafast femtosecond laser source, an optical delay line, a central control unit, a terahertz transmitter-receiver, and an xy motion platform.

3. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 1, characterized in that, In step S2, the step of establishing the sparse inverse problem model includes: S21. Use Fourier transform to convert the time-domain waveforms of the incident and reflected signals to the frequency domain; S22. Analyze the reflection and transmission characteristics of terahertz waves at the interface of a medium using Fresnel's formula; S23. Model the signal as a sparse convolution combination of multiple dictionary atoms and solve the L1 regularization problem to recover the sparse vector.

4. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 1, characterized in that, In step S3, the step of constructing the physically guided interpretable sparse unfolded network includes: S31. A physical constraint relaxation dictionary learning strategy is adopted to construct a parameterized physical kernel function to simulate an ideal terahertz echo, and a relaxation term is introduced to adaptively fine-tune waveform details. S32. Design a deep sparse unfolding framework, where each layer corresponds to one iteration step of the optimization algorithm, including physical initialization, cascaded iterative update, and feature fusion based on attention mechanism. S33. Introduce a hybrid scale dense connection strategy and a dynamic attention threshold mechanism in each layer to improve the network's ability to fit sparse vectors.

5. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 4, characterized in that, In step S31, a parameterized physical kernel function is constructed based on the propagation mechanism of terahertz waves. The ideal physical echo is simulated by using the linear superposition of two Gaussian derivative functions: ; in, This represents a set of learnable physical parameters that control the amplitude ratio, pulse width, and time delay, respectively. An additional relaxation term was introduced on top of the physical kernel. dictionary atoms The definition is as follows: 。 6. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 4, characterized in that, In step S32, during physical initialization, the sparse code is initialized using a relaxed dictionary of physical constraints, achieved through transposed convolution: 。 7. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 4, characterized in that, In step S32, a compression excitation module is introduced at the end of the physically guided interpretable sparse unfolded network: ; in, This represents the Sigmoid function. Indicates average pooling. This represents the weights of the fully connected layer.

8. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 4, characterized in that, In step S33, the input feature map is... The dynamic threshold module outputs a value that matches the input image. Thresholds with the same dimensions : ; in It is the basic threshold parameter; It is a lightweight attention module that contains two convolutional layers.

9. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 4, characterized in that, In step S4, the step of designing the dynamic sparse weighted focus loss function includes: S41. Introduce a focus loss mechanism to reduce the dominance of a large number of simple negative samples in gradient updates; S42. Design dynamic weighting coefficients to dynamically adjust the loss weights based on the proportion of positive samples in the real labels to prevent weak signals from being missed. The dynamic sparse weighted focus loss function is: ; in Indicates network output The result obtained by applying the Sigmoid function; It is a positive sample indicator function; These are dynamic weighting coefficients; ; in This is the sensitivity scaling factor; Define focus item for: ; in To predict probabilities, For focusing parameters.

10. The terahertz thickness measurement method for composite materials based on physically interpretable sparse unfolded networks according to claim 1, characterized in that, In step S5, when the terahertz wave is incident perpendicularly, the thickness is defined as: ; in, To measure thickness, At the speed of light, This represents the time difference between adjacent pulses in the flight time sequence output by the network. The value is the refractive index of the sample.

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