# Dispersion aware matching pursuit decomposition of Lamb wave signals for structural health monitoring of thin plates structures

Sebastian Rodriguez, Shweta Paunikar, Pierre Margerit, Eric Monteiro, Francisco Chinesta, Nazih Mechbal, Marc Rébillat

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## ► To cite this version:

Sebastian Rodriguez, Shweta Paunikar, Pierre Margerit, Eric Monteiro, Francisco Chinesta, et al. Dispersion aware matching pursuit decomposition of Lamb wave signals for structural health monitoring of thin plates structures. 11th European Workshop on Structural Health Monitoring (EWSHM 2024), Jun 2024, Potsdam, Germany. 10.58286/29712. hal-05299179

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Submitted on 6 Oct 2025

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# Dispersion aware matching pursuit decomposition of Lamb wave signals for structural health monitoring of thin plates structures

Sebastian RODRIGUEZ¹, Shweta PAUNIKAR¹, Pierre MARGERIT¹, Eric MONTEIRO¹, Francisco CHINESTA¹,², Nazih MECHBAL¹, Marc REBILLAT¹

¹PIMM, Arts et M'etiers ParisTech, CNRS, CNAM, HESAM University, Paris, France, sebastian.rodriguez_iturra@ensam.eu; shweta.paunikar@ensam.eu; pierre.margerit@ensam.eu; eric.monteiro@ensam.eu; francisco.chinesta@ensam.eu; nazih.mechbal@ensam.eu; marc.rebillat@ensam.eu;

²CNRS @ CREATE LTD, 1 Create Way, 08-01 CREATE Tower, 138602, Singapore, francisco.chinesta@ensam.eu

Abstract. Structural Health Monitoring (SHM) techniques are key to monitor in real time the health state of engineering structures, where damage type, location and severity are to be estimated by applying signal processing and deep learning techniques to temporal signals measured by sensors coming from the structure under study. Among the most widely used signal processing techniques in the SHM community is the Matching Pursuit Method (MPM), which allows to extract key features from a measured signal. MPM simply consists in approximating temporal data as a sum of different terms composed of a delayed and scaled signal called atom. This signal decomposition approach is however limited to non-dispersive signals which is not the case when dealing with Lamb waves signals widely used in the SHM of thin structures. In this case, in addition of delays and attenuations, wave packed also endure dispersion caused by the fact that all the frequencies do not propagate at the same speed within the structures under study. In this context, an improved version of Matching Pursuit is proposed in the present work, which address the dispersion phenomena by decomposing a measured signal as delayed and dispersed impulse response of an atom, obtaining better features for Lamb wave measurements. The performance of the developed technique is tested for the approximation of real measured signals, showing its better performance compared to classical MPM, especially when dealing with dispersive signals. This improved signal decomposition can be very useful for SHM purposes.

**Keywords:** Lamb Waves, Convolutional Matching Pursuit, Matching Pursuit, Structural Health Monitoring, Single Atom Dictionary.
Introduction

Matching Pursuit Method (MPM) [1] consists in approximating a given signal s(t) as follows:

$$s(t) \approx \sum_{i=1}^{m} \alpha_{i} \Psi_{i}(t)$$

The features extracted by MPM corresponds to the atoms $\Psi_i(t)$ and their amplitudes $\alpha_i$. These atoms are selected on an over-complete learning dictionary “D” that needs to be provided a priori [2, 3, 4], where each atom is selected in a greedy process in the sparsest way possible.

However, MPM is limited only to non-dispersive signals, preventing its application for thin structures in SHM scenarios where dispersion is classically present during Lamb Waves (LW) propagation. In this case, in addition to delays and attenuation, wave packets also endure dispersion caused by the fact that all the frequencies do not propagate at the same speed within the structures under study. Furthermore, the learning dictionary D needs to be a priori defined which limits the practical use of MPM as not all delayed, attenuated and dispersed atoms can be precomputed in advance.

Despite these drawbacks, MPM has however already been applied in Lamb Waves based SHM context in combination with other methods for damage monitoring. Recently, Li et al. [5] developed a method using orthogonal matching pursuits and model updating to locate damages in hinge joints of a fully functional hollow slab bridge using data collected from a single location. To decompose and reconstruct the various wave packets in a signal, Gao et al. [6] used orthogonal matching pursuit algorithm based on dictionary matrix, while Kim and Yuan [7] used it for imaging damage in an aluminum plate. Mu et al. [8] also used orthogonal MP in conjunction with dispersion removal for identifying LW packets in shells. MPM is employed by Hong et al. [9] to analyze guided wave signals obtained from FE simulations and experiments in covered region of the metallic messenger cable in electrified railway catenary and successfully detect damage larger than 2.5 mm in the cable. An optimized dictionary based MPM was employed to study and size axial defects in in-service or corroded pipelines by Tse and Wang [10]. It is noteworthy that all of these methods are based on predefined overcomplete dictionaries for employing MPM.

To overcome the limitations in the approximation of complex signals that present dispersion, two techniques are proposed in this work. The first one consists of an approximation very similar to the MPM techniques but whose main difference lies in the use of a single atom. In this sense, the need for the use of a dictionary is eliminated. This method is called here the Single Atom Matching Pursuit Method. Additionally, the SAMPM method is extended to be able to approximate more complex signals where dispersion is present, this method is called Single Atom Convolutional Matching Pursuit Method (SACMPM).

The present paper is structured as follows: Sec. 2 introduces the SAMPM. Then Sec. 3 introduce the extension of the SAMPM to its convolutional counterpart the SACMPM. Following Sec. 4 provide numerical results of the proposed algorithms when they approximate signals. Finally, Sec. 5 provides conclusions and perspectives.
# 1. Single Atom Matching Pursuit Method

The main idea consists in approximating a given signal $s(t)$ by the following decomposition:

$$s(t) \approx \sum_{i=1}^{m} \alpha_i \Psi(t - \tau_i) \qquad (1)$$

where the principal unknowns correspond to the amplitude of each term and its temporal delay $\alpha_i$ and $\tau_i$ respectively. Let us notice that the considered approximation considers a single atom that is delayed and whose amplitude is changed.

Each term of the decomposition is determined one after the other in an incremental way, so called Greedy process. In this sense, if one seeks to determine the term “m”, one has:

$$\{\alpha_m, \tau_m\} = \arg \min_{\{\alpha_m, \tau_m\}} \|\alpha_m \Psi(t - \tau_m) - s_{res}^{m-1}(t)\|_{I_t}^2 \qquad (2)$$

with $s_{res}^{m-1}(t) = s(t) - \sum_{i=1}^{m-1} \alpha_i \Psi(t - \tau_i)$ and $I_t = [0, T]$.

In parallel one can define the above temporal norm in the frequency domain as well due to Parseval’s theorem as follows:

$$\{\alpha_m, \tau_m\} = \arg \min_{\{\alpha_m, \tau_m\}} \|\alpha_m \hat{\Psi}(\omega) e^{-j\omega\tau_m} - \hat{s}_{res}^{m-1}(\omega)\|_{I_\omega}^2 \qquad (3)$$

with $\hat{\Psi}(\omega) = \text{FFT}(\cdot)$. Where the norms are defined as follows:

$$\| \cdot \|_{I_t}^2 = \int_{I_t} (\cdot)^2 dt \qquad (4)$$

and

$$\| \cdot \|_{I_\omega}^2 = \int_{I_\omega} (\cdot)^*(\cdot) d\omega \qquad (5)$$

Since two unknowns must be determined (its amplitude and temporal delay), therefore it consists on a nonlinear problem. In this context, a fixed-point iterative strategy is considered for its resolution. The iterative resolution is applied in the frequency domain, this is, by minimizing (3). Each term of the decomposition is computed one after the other in a Greedy way.

# 2. Single Atom Convolutional Matching Pursuit Method

In order to be able to better approximate complex signals, here one propose the following decomposition:

$$s(t) \approx \sum_{i=1}^{m} [\alpha_i(t) * \Psi(t - \tau_i)](t) \qquad (6)$$

where * corresponds to the convolution operator.
Let's remark that here the amplitude now corresponds to a temporal function that needs to be determined. For its determination, different techniques can be applied, here the temporal function is approximated by means of a Finite Element Method (FEM) [11] in time. In consequence we have:

$$ \alpha_m(t) \approx \sum_{i=1}^{N} \beta_i N_i(t) \qquad (7) $$

with $\beta_i$ and $N_i(t)$ the corresponding nodal unknowns and FEM shape functions respectively. For this case, the approximation is constructed by solving:

$$ \{\alpha_m(t), \tau_m\} = \arg \min_{\{\alpha_m(t), \tau_m\}} \left\| [\alpha_m(t) * \Psi(t - \tau_m)](t) - s_{res}^{m-1}(t) \right\|_{I_t}^2 \qquad (8) $$

here $s_{res}^{m-1}(t) = s(t) - \sum_{i=1}^{m-1}[\alpha_i(t) * \Psi(t - \tau_i)](t)$.

In this case, as presented for the SAMPM of Sec. 2, a fixed-point strategy is also considered for the resolution of the time function and its time delay. Each term of the decomposition is computed one after the other in a Greedy way.

### 3. Numerical examples

The following subsections provide numerical results for the SAMPM case presented in Sec. 1 and the SACMPM method as presented in Sec. 2.

#### 3.1 Application of SAMPM

To illustrate how SAMPM works when approximating a given signal, first lets consider the atom represented in figure 1, which will be used to approximate the reference signal of figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Input signal considered (atom).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Reference signal.</figcaption></figure>

In this situation, the SAMPM allows to approximate the reference signal with an error below 10 % only with 4 terms as the graph of error vs terms represents in figure 3. The final approximation is illustrated in figure 4.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Error vs number of terms.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Reference vs approximated signal.</figcaption></figure>

Finally, figures 5 and 6 illustrate the first 2 terms of the decomposition.

<figure><img src="image_7.png" /><figcaption>Fig. 5. First term.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 6. Second term.</figcaption></figure>

## 3.2 Application of SACMPM

Here, the SACMPM is used for the approximation of a complex signal, this signal is illustrated in figure 8. The atom used is represented in figure 7.

<figure><img src="image_9.png" /><figcaption>Fig. 7. Input signal considered (atom).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Reference signal.</figcaption></figure>

The SACMPM allows to approximate the reference signal with an error below 10 % only using 13 terms as shown in figure 9. The final approximation is given in figure 10.
<figure><img src="image_11.png" /><figcaption>Fig. 9. Error vs number of terms.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Reference vs approximated signal.</figcaption></figure>

Finally, the first 2 terms are illustrated in figures 11 and 12 for the first and second terms respectively.

<figure><img src="image_13.png" /><figcaption>Fig. 11. First term.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Second term.</figcaption></figure>
## 4. Conclusions

In the present work, two techniques are presented, which are inspired by the classical Matching Pursuit technique. These methods consist of the SAMPM and SACMPM. The SAMPM consists in approximating a signal as the sum of time-phased atoms whose amplitude is modified by a scalar. The construction of this approximation is performed completely in the frequency domain, which allows its construction in an optimal way. On the other hand, the SACMPM extends the idea of the previous method by considering that the atom is convolved by a time-dependent function. This approach allows to better capture the time evolution of the signal to be approximated. The methods proposed in this manuscript allow to obtain important characteristics of the approximated signal, which can be used later in different applications and domains, such as its use in SHM techniques, where the characteristics obtained for each signal can be used for the detection of damage in structures. The latter application is considered as a perspective and will be addressed in future releases.

## References

[1] Stéphane G Mallat and Zhifeng Zhang. Matching pursuits with time-frequency dictionaries. IEEE Transactions on signal processing, 41(12):3397-3415, 1993.

[2] Buli Xu, Victor Giurgiutiu, and Lingyu Yu. Lamb waves decomposition and mode identification using matching pursuit method. In Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems 2009, volume 7292, pages 161-172. SPIE, 2009.

[3] Debejyo Chakraborty, Narayan Kovvali, Jun Wei, Antonia Papandreou-Suppappola, Douglas Cochran, and Aditi Chattopadhyay. Damage classification structural health monitoring in 7 bolted structures using time-frequency techniques. Journal of Intelligent Material Systems and Structures, 20(11):1289-1305, 2009.

[4] Weilei Mu, Yuqing Gao, and Guijie Liu. Ultrasound defect localization in shell structures with lamb waves using spare sensor array and orthogonal matching pursuit decomposition. Sensors, 21(23):8127, 2021.

[5] Shengli Li, Haoxiang Yang, Pan Guo, Duochang Ren, Bin Xu, and Zhenzhen Liang. Damage identification of hinge joint in hollow slab bridge based on model updating and orthogonal matching pursuit algorithm. Measurement, 224:113867, January 2024.

[6] Yuqing Gao, Weilei Mu, Fuh-Gwo Yuan, and Guijie Liu. A defect localization method based on self-sensing and orthogonal matching pursuit. Ultrasonics, 128:106889, February 2023.

[7] Howuk Kim and Fuh-Gwo Yuan. Adaptive signal decomposition and dispersion removal based on the matching pursuit algorithm using dispersion-based dictionary for enhancing damage imaging. Ultrasonics, 103:106087, April 2020.

[8] Weilei Mu, Yuqing Gao, and Guijie Liu. Ultrasound Defect Localization in Shell Structures with Lamb Waves Using Spare Sensor Array and Orthogonal Matching Pursuit Decomposition. Sensors, 21(23):8127, December 2021.

[9] Xiaobin Hong, Jianxi Zhou, and Yongkui He. Damage detection of anchored region on the messenger cable based on matching pursuit algorithm. Mechanical Systems and Signal Processing, 130:221-247, September 2019.

[10] Peter W. Tse and Xiaojuan Wang. Characterization of pipeline defect in guided-waves based inspection through matching pursuit with the optimized dictionary. NDT & E International, 54:171-182, March 2013.

[11] Olek C Zienkiewicz, Robert L Taylor, and Jian Z Zhu. The finite element method: its basis and fundamentals. Elsevier, 2005.