A method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging

Abstract

The present invention discloses a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging, which relates to the technical field of ultrasonic Lamb wave imaging. The method specifically comprises the following steps: first, based on a full-focusing data acquisition method, an array transducer is used for point-by-point excitation and reception, complete full-matrix data is acquired, and a multimodal Lamb wave propagation model is established; second, based on partial derivative information of each channel and in combination with the observed noise variance, a corresponding Fisher information matrix is ​​constructed; and third, based on the derivation result of the Fisher information matrix, a MATLAB platform is used for numerical calculation of CRLB. The present invention introduces a minimum variance expression for deriving defect positioning in Lamb wave imaging from statistical estimation theory, theoretically establishes a physical relationship between defect imaging accuracy and wavelength and modal characteristics, and provides a quantitative basis for multimodal selection and system design.

Description

Technical Field

[0001] The present invention belongs to the technical field of ultrasonic Lamb wave imaging, and in particular relates to a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging. Background Art

[0002] With the development of structural health monitoring (SHM) technology, Lamb waves have gained widespread application in nondestructive testing of plate structures in aerospace, rail transportation, civil engineering, and other fields due to their long propagation distance, high sensitivity, and sensitivity to surface and near-surface defects. However, Lamb wave propagation is accompanied by significant dispersion and modal diversity, and its propagation behavior is significantly affected by factors such as excitation frequency, structure thickness, and material parameters. In complex environments where multiple modalities coexist, signal interpretation and imaging quality are highly dependent on the effective differentiation and rational utilization of each modality.

[0003] Current methods for imaging resolution analysis primarily focus on empirical parameter optimization, image quality index assessment, or numerical simulation techniques, such as transverse intensity curve (TIC) measurements and simulation comparison experiments. These methods suffer from subjectivity or strong reliance on specific models, lack theoretically universal accuracy assessment metrics, and are difficult to generalize to complex structures, arbitrary modal combinations, or diverse imaging system design scenarios.

[0004] In the field of statistical signal processing, the Cramer-Rao lower bound (CRLB), as a theoretical lower limit for measuring the minimum variance of parameter estimates, has long been used for performance analysis in scenarios such as radar, multi-antenna arrays, and sound source localization. Its advantage lies in its ability to calculate the minimum variance achievable by any unbiased estimator based on a given signal model and parameter relationship using the Fisher Information Matrix (FIM). Therefore, a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging is urgently needed. By decoupling from specific algorithmic dependencies, a theoretical limit for spatial resolution can be obtained, establishing a physical connection between modal characteristics and imaging capabilities. Summary of the Invention

[0005] The object of the present invention is to provide a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging, which is used to solve the technical problem of the lack of theoretical evaluation standards for spatial resolution in the prior art.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A method for quantifying the spatial resolution of multimodal ultrasound Lamb wave imaging is proposed. CRLB stands for Cramer-Rao lower bound, which includes:

[0008] Step 1: Based on the full focusing method, an array transducer is used for point-by-point excitation and reception to collect complete matrix data and establish a multi-modal Lamb wave propagation model.

[0009] Step 2: Based on the partial derivative information of each channel and the observed noise variance, the corresponding Fisher information matrix is ​​constructed;

[0010] Step 3: Based on the derivation results of the Fisher information matrix, the MATLAB platform is used to perform numerical calculations of CRLB.

[0011] Furthermore, an array transducer is used for point-by-point excitation and reception to collect complete matrix data. The specific method is as follows:

[0012] Record the time domain signals of all transmit-receive channels, perform a three-dimensional fast Fourier transform on the full matrix data, convert the time domain signals to the frequency-wavenumber domain, obtain the energy distribution characteristics of the signal in three-dimensional space, identify the wavenumber ranges and energy concentration areas of different Lamb wave modes through frequency-wavenumber spectrum analysis, use the Hilbert-Huang transform, extract the intrinsic mode function of the signal through empirical mode decomposition, combine with instantaneous frequency analysis, separate the time-frequency components of the aliased mode, design a matching window function or implement a bandpass filtering operation for the wavenumber distribution range of different modes, use the Hilbert envelope detection or short-time energy integration method to extract the amplitude information of each mode, determine the Lamb wave dispersion curve, and determine the wavenumber based on the excitation frequency and group velocity in the Lamb wave dispersion curve.

[0013] Furthermore, a multi-modal Lamb wave propagation model is established. The specific method is as follows:

[0014] The time domain signal expression of the multimodal Lamb wave is obtained by linearly superposing each modal component, and a signal model is constructed. The signal component of each mode is a function of the defect position, and the formula is used. represents the signal expression, where S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, and M represents the total number of Lamb wave modes. represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving array element i, represents the wave number of the j-th Lamb wave mode, represents the spatial coordinates of receiving array element i, Represents the spatial coordinates of the defect point, represents the phase of the j-th Lamb wave mode, The variance is Gaussian white noise.

[0015] Furthermore, based on the partial derivative information of each channel and combined with the observed noise variance, the corresponding Fisher information matrix is ​​constructed. The specific method is:

[0016] The partial derivatives of the observation signals of each array element channel with respect to the spatial position parameters of the defect are obtained respectively. According to the chain rule, the partial derivatives of the signal intensity with respect to the position parameters are expanded into the joint contribution of the amplitude term and the phase term. Based on the partial derivative information of each channel and combined with the observation noise variance, the corresponding Fisher information matrix is ​​constructed. Specifically, each element of the Fisher information matrix is ​​composed of the outer product of the partial derivatives of each channel, where the diagonal elements measure the sensitivity to a single position parameter, and the non-diagonal elements reflect the correlation between the position parameters. The theoretical lower limit of the defect position estimation error under given imaging conditions is derived through the Fisher information matrix. The Fisher information matrix provides a quantitative reference standard for the optimization of different modal combinations, array configurations, and excitation frequency selection. Through systematic analysis of FIM, the differences in the spatial resolution contributions of different Lamb wave modes are revealed.

[0017] Furthermore, the partial derivative of the signal strength with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term. The specific method is:

[0018] Using the formula With the formula , expand the partial derivative of the signal intensity with respect to the location parameter into the joint contribution of the amplitude term and the phase term, where It represents the signal strength of a Lamb wave mode at time t, where t represents time, A represents the amplitude of the corresponding Lamb wave mode, and f represents the excitation frequency, i.e. the frequency of the transmitted signal. represents the phase of the corresponding Lamb wave mode, x represents the horizontal coordinate, and y represents the vertical coordinate.

[0019] Furthermore, the corresponding Fisher information matrix is ​​constructed by combining the observation noise variance. The specific method is as follows:

[0020] There are M modes of Lamb waves, and the size of the Fisher information matrix (FIM) is (2M + 3) × (2M + 3). represents the Fisher information matrix, where represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode, and represents the partial differential of the amplitude and phase respectively, Represents the spatial coordinates of the defect point source, represents the observation noise variance, Represents the partial derivative of the signal model with respect to the j-th modal auxiliary variable.

[0021] Furthermore, the partial derivative of the signal model with respect to the j-th modal auxiliary variable is as follows:

[0022] Using the formula represents the partial derivative of the signal model with respect to the j-th modal auxiliary variable, where represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode, M represents the total number of Lamb wave modes, Express the characteristic factor, using the formula express.

[0023] Furthermore, based on the derivation of the Fisher information matrix, the numerical calculation of CRLB was performed using the MATLAB platform. The specific method is as follows:

[0024] The MATLAB platform is used for the numerical calculation of CRLB. The physical parameters are set according to the actual detection scenario, including the plate geometric parameters, excitation frequency range and step size, modal wave velocity, wave number, and amplitude attenuation law. A complete input data set is constructed, and the inverse matrix of the Fisher information matrix is ​​obtained through matrix operations. The diagonal elements corresponding to the spatial position parameters of the defect are extracted to obtain its theoretical minimum variance, i.e., the CRLB value.

[0025] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0026] 1. The present invention adopts a full-focusing data acquisition method, using a two-dimensional array transducer for point-by-point excitation and reception to collect complete full-matrix data. This comprehensive data acquisition method can obtain richer information, covering the time domain signals of all transmit-receive channels. By performing a three-dimensional fast Fourier transform on the full-matrix data, the time domain signal is converted to the frequency-wavenumber-space domain, and the multi-dimensional energy distribution characteristics of the signal are obtained, which can more intuitively understand the propagation of Lamb waves in different dimensions.

[0027] 2. By decomposing the partial derivative of signal intensity with respect to position parameters into the joint contribution of amplitude and phase terms, the present invention can more carefully analyze the factors affecting the change of signal intensity with defect position, providing a more precise theoretical basis for accurately evaluating the defect position. The Fisher information matrix constructed based on the partial derivative information of each channel and the observation noise variance can derive the theoretical lower limit of the defect position estimation error under given imaging conditions. This lower limit provides an objective standard for evaluating the performance of the detection system, allowing researchers and engineers to understand the minimum error range that can be achieved in defect position estimation under current detection conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0029] Figure 1 A step diagram of a method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging is shown;

[0030] Figure 2 A diagram showing the steps of constructing the Fisher information matrix is ​​shown;

[0031] Figure 3 This is a comparison chart of the results of TFM imaging implemented in the present invention;

[0032] Figure 4 A comparison chart of the results of implementing 3D-FFT in the present invention;

[0033] Figure 5 This is a schematic diagram of the CRLB results of the present invention;

[0034] Figure 6 This is a quantitative comparison diagram of TIC results after TFM imaging in the present invention. DETAILED DESCRIPTION

[0035] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0036] like Figure 1 、 Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 and Figure 6 As shown, a method for quantifying the spatial resolution of multimodal ultrasonic Lamb wave imaging specifically includes the following steps:

[0037] Step 1: Based on the full-focusing method, an array transducer is used for point-by-point excitation and reception to collect complete full-matrix data and establish a multi-modal Lamb wave propagation model.

[0038] A two-dimensional array transducer is used, with an element spacing of half the Lamb wave wavelength. A*A array of elements covers the data detection area. A laser rangefinder is used to measure and record the actual spatial coordinates of each element. Each element is activated sequentially as a transmitter, while the remaining elements receive the echo signal synchronously. The array transducer is excited and received point by point, acquiring complete full-matrix-convolution (FMC) data. This includes recording the time-domain signals of all transmit and receive channels. A three-dimensional fast Fourier transform (3D-FFT) is then performed on the FMC data to convert the time-domain signals into the frequency-wavenumber-space domain, thereby obtaining the multidimensional energy distribution characteristics of the signal. Frequency-wavenumber spectrum analysis, combined with predefined Lamb wave dispersion curves, identifies the wavenumber-frequency mapping relationships and energy concentration regions of different modes (such as S0, A0, and A1). For aliased modes in the frequency-wavenumber domain, the Hilbert-Huang transform (HHT) is further employed to extract the signal's intrinsic mode functions (IMFs) through empirical mode decomposition (EMD). This is then combined with transient frequency analysis to separate transient interference components. Based on the dynamically generated wavenumber-frequency relationship, adaptive window functions (such as Gaussian windows or Hanning windows) or bandpass filters are designed. For example, low-pass filtering is performed on the S0 mode (the low-wavenumber region at the current frequency) and band-pass filtering is performed on the A0 mode (the medium- and high-wavenumber region) to suppress intermodal crosstalk. Finally, Hilbert envelope detection or short-time energy integration methods are used to extract the amplitude information of each mode. Combined with the known group velocity and excitation frequency from the dispersion curve, the wavenumber calculation results are verified and corrected.

[0039] The time domain signal expression of the multimodal Lamb wave is obtained by linearly superposing each modal component, and a signal model is constructed. The signal component of each mode is a function of the defect position. The specific signal expression formula is as follows:

[0040] ;

[0041] Where S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, and M represents the total number of Lamb wave modes. represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving array element i, represents the wave number of the j-th Lamb wave mode, Represents the spatial coordinates of receiving array element i, defining the array layout and signal acquisition position, Represents the spatial coordinates of the defect point, represents the phase of the j-th Lamb wave mode, The variance is Gaussian white noise.

[0042] Step 2: Based on the partial derivative information of each channel and combined with the observation noise variance, construct the corresponding Fisher information matrix.

[0043] Based on the existing measurement model, the partial derivative of the observation signal of each array element channel with respect to the defect spatial position parameter is calculated and expanded using the chain rule to obtain the sensitivity expression of signal intensity and position change. The existing measurement model refers to a measurement model based on the total focusing method (TFM) data acquisition principle and combined with the propagation characteristics of multimodal Lamb waves. According to the chain rule, the partial derivative of the signal intensity with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term. The specific formula is as follows:

[0044] ;

[0045] ;

[0046] in, It represents the signal strength of a Lamb wave mode (such as S0, A0) at time t, where t represents time and A represents the amplitude of the corresponding Lamb wave mode. It is obtained by extracting the amplitudes of different modes through wave number analysis and modal separation. f represents the excitation frequency, that is, the frequency of the transmitted signal. It represents the phase of the corresponding Lamb wave mode, which is determined by the position of the sensor in the array and the aperture ratio, and can be obtained by geometric calculation of the wavefront propagation path. x represents the abscissa and y represents the ordinate.

[0047] Subsequently, based on the partial derivative information of each channel and combined with the observed noise variance, the corresponding Fisher information matrix is ​​constructed. Specifically, each element of the Fisher information matrix is ​​composed of the outer product of the partial derivatives of each channel. The diagonal elements measure the sensitivity to a single position parameter, and the off-diagonal elements reflect the correlation between position parameters. The noise variance is usually set to a constant based on the Gaussian white noise assumption. If the Lamb wave has M modes, the size of the Fisher information matrix (FIM) is (2M + 3) × (2M + 3). The Fisher information matrix is ​​shown as follows:

[0048] ;

[0049] in, represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode (representing the partial differential of amplitude and phase respectively), Represents the spatial coordinates of the defect point source, represents the observation noise variance.

[0050] Furthermore, the specific calculation formula of the signal model for the j-th modal auxiliary variable is as follows:

[0051] ;

[0052] in, represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode (representing the partial differential of amplitude and phase respectively), M represents the total number of Lamb wave modes, Express the characteristic factor, using the formula It is stated that the theoretical lower limit of the defect position estimation error under given imaging conditions is derived through the Fisher information matrix. The Fisher information matrix provides a quantitative reference standard for the optimization of different modal combinations, array configurations, and excitation frequency selection. Through systematic analysis of FIM, the differences in the spatial resolution contributions of different Lamb wave modes are revealed.

[0053] Step 3: Based on the derivation results of the Fisher information matrix, the MATLAB platform is used to perform numerical calculations of CRLB.

[0054] Based on the derivation of the Fisher information matrix mentioned above, the MATLAB platform was used to perform numerical calculations of the CRLB. The physical parameters were set according to the actual detection scenario, including the plate geometric parameters (thickness, material density, elastic modulus), excitation frequency range and step size, modal wave velocity, wave number, and amplitude attenuation law. A complete input data set was constructed, and the inverse matrix of the Fisher information matrix was obtained through matrix operations. The diagonal elements corresponding to the spatial position parameters of the defect were extracted to obtain its theoretical minimum variance, i.e., the CRLB value.

[0055] In the actual calculation process, in order to accurately reflect the physical characteristics of the imaging system, the propagation velocity and wavenumber parameters need to be updated in real time according to the dispersion relation of each Lamb wave mode at different excitation frequencies, thereby ensuring that the CRLB results are highly consistent with the actual detection conditions. By changing the frequency, mode type, and array aperture input parameters, the system can generate the minimum variance change curve under the corresponding imaging conditions.

[0056] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

[0057] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to specific embodiments. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging, characterized in that: include: Step 1: Based on the full focusing method, an array transducer is used for point-by-point excitation and reception to collect complete matrix data and establish a multi-modal Lamb wave propagation model. Step 2: Based on the partial derivative information of each channel and the observed noise variance, the corresponding Fisher information matrix is ​​constructed; The partial derivative of the observation signal of each array element channel with respect to the spatial position parameter of the defect is calculated separately. According to the chain rule, the partial derivative of the signal intensity with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term. Based on the partial derivative information of each channel and the observation noise variance, the corresponding Fisher information matrix is ​​constructed. Specifically, each element of the Fisher information matrix is ​​composed of the outer product of the partial derivatives of each channel. The diagonal elements measure the sensitivity to a single position parameter, and the off-diagonal elements reflect the correlation between position parameters. The theoretical lower limit of the defect position estimation error under given imaging conditions is derived from the Fisher information matrix. By systematically analyzing the Fisher information matrix, the differences in the spatial resolution contributions of different Lamb wave modes are revealed. Step 3: Based on the derivation results of the Fisher information matrix, the MATLAB platform is used to perform numerical calculations of CRLB.

2. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, wherein: An array transducer is used for point-by-point excitation and reception to collect complete matrix data. The specific method is as follows: Record the time domain signals of all transmit-receive channels, perform a three-dimensional fast Fourier transform on the full matrix data, convert the time domain signals to the frequency-wavenumber domain, obtain the energy distribution characteristics of the signal in three-dimensional space, identify the wavenumber ranges and energy concentration areas of different Lamb wave modes through frequency-wavenumber spectrum analysis, use the Hilbert-Huang transform, extract the intrinsic mode function of the signal through empirical mode decomposition, combine with instantaneous frequency analysis, separate the time-frequency components of the aliased mode, design a matching window function or implement a bandpass filtering operation for the wavenumber distribution range of different modes, use the Hilbert envelope detection or short-time energy integration method to extract the amplitude information of each mode, determine the Lamb wave dispersion curve, and determine the wavenumber based on the excitation frequency and group velocity in the Lamb wave dispersion curve.

3. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, wherein: Establish a multi-modal Lamb wave propagation model. The specific method is as follows: The time domain signal expression of the multimodal Lamb wave is obtained by linearly superposing each modal component, and a signal model is constructed. The signal component of each mode is a function of the defect position, and the formula is used. represents the signal expression, where S represents the signal model, i represents the i-th receiving array, j represents the j-th Lamb wave mode, N represents the total number of receiving arrays, and M represents the total number of Lamb wave modes. represents the out-of-plane amplitude of the j-th Lamb wave mode at the receiving array element i, represents the wave number of the j-th Lamb wave mode, represents the spatial coordinates of receiving array element i, Represents the spatial coordinates of the defect point, represents the phase of the j-th Lamb wave mode, The variance is Gaussian white noise.

4. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, wherein: The partial derivative of the signal strength with respect to the position parameter is expanded into the joint contribution of the amplitude term and the phase term. The specific method is: Using the formula With the formula , expand the partial derivative of the signal intensity with respect to the location parameter into the joint contribution of the amplitude term and the phase term, where It represents the signal strength of a Lamb wave mode at time t, where t represents time, A represents the amplitude of the corresponding Lamb wave mode, and f represents the excitation frequency, i.e. the frequency of the transmitted signal. represents the phase of the corresponding Lamb wave mode, x represents the horizontal coordinate, and y represents the vertical coordinate.

5. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, wherein: Combined with the observation noise variance, the corresponding Fisher information matrix is ​​constructed. The specific method is: There are M modes of Lamb waves, and the size of the Fisher information matrix is ​​(2M + 3) × (2M + 3). represents the Fisher information matrix, where represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode, and represents the partial differential of the amplitude and phase respectively, Represents the spatial coordinates of the defect point source, represents the observation noise variance, Represents the partial derivative of the signal model with respect to the j-th modal auxiliary variable.

6. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 5, characterized in that: The partial derivative of the signal model with respect to the j-th modal auxiliary variable is as follows: Using the formula represents the partial derivative of the signal model with respect to the j-th modal auxiliary variable, where represents the signal model, j represents the jth mode, represents the auxiliary variable of the jth mode, M represents the total number of Lamb wave modes, Express the characteristic factor, using the formula express.

7. The method for quantifying spatial resolution of multimodal ultrasonic Lamb wave imaging according to claim 1, wherein: Based on the derivation of Fisher information matrix, the numerical calculation of CRLB is performed using MATLAB platform. The specific method is as follows: The MATLAB platform is used for numerical calculation of CRLB. The physical parameters are set according to the actual detection scenario, including the plate geometric parameters, excitation frequency range and step size, modal wave velocity, wave number and amplitude attenuation law. A complete input data set is constructed, and the inverse matrix of the Fisher information matrix is ​​obtained through matrix operations. The diagonal elements corresponding to the spatial position parameters of the defect are extracted to obtain its theoretical minimum variance, i.e., the CRLB value.

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