Sparsity Based Defect Imaging in Pipes using Guided-Waves Andrew Golato*, Sridhar Santhanam, Fauzia Ahmad, and Moeness G. Amin Center for Advanced Communications, College of Engineering, Villanova University, 800 E. Lancaster Ave., Villanova, PA 19085, USA. ABSTRACT Pipes are used for the transport of fluids and gases in urban and industrial settings, such as buried pipelines to transport water, oil, and other resources. To ensure reliable operation, it is essential that an inspection system be in place to identify and localize damage/defects in the pipes. Unfortunately, many of the typical nondestructive evaluation techniques are inadequate due to limited pipe access; often, only the beginning and end sections of the pipe are physically accessible. As such, this problem is well suited to the use of ultrasonic guided-wave based structural health monitoring. With a limited number of transducers, ultrasonic guided waves can be used to interrogate long lengths of pipes. In this paper, we propose a damage detection and localization scheme that relies upon the inherent sparsity of defects in the pipes. A sparse array of transducers, deployed in accessible areas of the pipes, is utilized in pitch-catch mode to record signals scattered by defects in the pipe. Both the direct path scattering off the defect, and the helical modes, which are paths that spiral around the circumference of the pipe before or after interaction with the defect, are recorded. A Lamb wave based signal model is formulated that accounts for this multipath approach. The signal model is then inverted via group sparse reconstruction, in order to produce an image of the scene. The model accounts for the specificities of Lamb wave propagation through the pipe. Performance validation of the proposed approach is provided using simulated data for an aluminum pipe. Keywords: Sparse reconstruction, pipe inspection, Lamb waves, compressed sensing, structural health monitoring

1. INTRODUCTION Structural Health Monitoring (SHM) has recently emerged as a primary technology in the assessment of the integrity of a variety of structures.1-4 In particular, the emergence of SHM is due in part to its self-sensing capabilities in structures. Guided-wave SHM (GWSHM) is utilized for real-time and in situ imaging of defects in thin-walled plate-like and shelllike structures.5-6 Pipe inspection is of particular importance as pipes are a major component of a developed country’s infrastructure, including water, gas, and sewage transportation systems. Aging of the infrastructure is leading to increasing pipe failures, which are often very expensive to repair. Therefore, there is much motivation for a non-invasive pipe inspection technique that has the ability to detect defects in pipe systems prior to catastrophic failure. GWSHM has this potential, and therefore, there has been an increased investment in its application to pipe inspection.5 In GWSHM, Lamb waves have become the preferred wave mode for thin plate and shell structures due to their ability to travel large distances without experiencing significant attenuation. Lamb waves also provide rich interactions with defects.7-9 Lamb waves are multimodal in nature and can be separated into symmetric (S) and anti-symmetric (A) modes.10-11 While inherently an infinite number of symmetric and anti-symmetric modes exist, for a simple thin platelike structure, the number present in the given structure can be regulated by the frequency of the generated signal. Higher frequencies provide an overabundance of wave modes, while lower frequencies excite only the fundamental symmetric and anti-symmetric (S0 and A0) modes. These propagation characteristics are well understood for plate inspection such that the complexities of Lamb waves can be reduced via frequency selection. For ultrasonic wave propagation in a pipe or pipe-like structure, the wave modes propagating are even more complex, though frequency selection can similarly reduce the complexity.12-13 The added complexity of wave propagation in pipes is due to the presence of three distinct wave modes: Longitudinal, Torsional and Flexural.11,14-15 In addition to these three sets of waves unique to propagation in a pipe, in many scenarios these modes can intermingle with the aforementioned Lamb wave modes.16 The propagation complexity is further *[email protected]; http://www1.villanova.edu/villanova/engineering/research/centers/cac/facilities/aul.html

enhanced by dispersion. That is, the phase and group velocities of each individual mode are frequency-dependent, which causes the wave shape to change during propagation. The complexity of pipe propagation can be simplified via excitation approach and frequency selection. In this work, surface transducers are employed in such a way that only the Lamb modes are excited, and none of the three pipe modes are induced. The number of induced modes can be further simplified using the appropriate selection of frequency, which allows for only a single or the first few of the desired wave modes to propagate.17-18 Recent efforts have taken advantage of this simplification in order to provide efficient inspection techniques for pipes. In the context of GWSHM, most of these works have focused on time reversal and beamforming techniques.17-19 While successful in single-defect detection in a long pipe, these techniques are limited in their robustness, as they require a large number of transducers and often lose their accuracy in multi-defect scenarios. Additional complications in guided-wave pipe inspection involve the helical modes or spiral paths, whereby the wave propagation path wraps around the circumference of the pipe en route to the receiver. Often times, these spiral paths are eliminated by simply limiting the length of the observation window; however, depending on the topology of the transducer arrays and the region of interest, this approach is not always possible. Therefore, in contrast to helical mode mitigation, some work has been recently performed to exploit these helical modes.20-21 The work in ref. [20] employs a tomography-based approach and uses large Electromagnetic Acoustic Transducer (EMAT) arrays. This approach relies on the helical modes to increase the effective array aperture, thereby improving ray coverage, which in turn permits more accurate wall thickness estimation. The work in ref. [21] is limited only to exploiting the helical spirals in a lineof-sight approach. In this paper, we present the pipe inspection problem as a sparse reconstruction problem assuming use of only few transducers. The number of defects is assumed to be much less than the number of potential defect locations; thus, the spatial sparsity assumption holds. Higher order helical modes as well as the direct scattered A0 wave modes are built into the signal model. The scene recovery problem is cast within a block sparse framework22-25 with the grouping applied across the direct scattered and helical modes in a similar fashion to the multipath exploitation approach for inspection of a thin plate.26 The remainder of the paper is organized as follows: Section 2 describes the multipath (direct and spiral path) signal propagation model accounting for the helical modes, and presents the sparse reconstruction algorithm used for multipath exploitation. Section 3 provides the supporting simulation results. Section 4 presents the concluding remarks.

Figure 1. 3D image of the pipe. The ROI, denoted by the blue mesh-grid, is centered along the length of the pipe and circumferentially wraps around the entirety of the pipe surface. The transducers represented by red circles are adhered to the surface of the pipe and aligned in two circumferential rings on either side, each equidistant from the ROI.

2. HELICAL MODE SIGNAL MODEL AND SPARSE RECONSTRUCTION 2.1

Signal model

Consider a network of 𝐽 piezoelectric (PZT) transducers adhered to the surface of the pipe. The transducer network is organized into two uniformly-spaced circumferential arrays (A and B) of 𝐽/2 transducers each as shown in Fig. 1. The arrays are considered as uniform linear arrays along the circumference of the “unwrapped” pipe, as shown in Fig. 2. The 𝐽 transducers are operated in pitch-catch mode providing a total of 𝐿 = 𝐽!/(𝐽 − 2)! transmit-receive combinations. These

Figure 2. The unrolled pipe showing both physical pipe region, ℜ, and the virtual region,  𝒱. Prime(’) denotes virtual transducers. The three considered spiral paths are: the direct scattering 𝑚 = 0 (red solid line), the second half spiral 𝑚 = 1 (blue dash-dotted line) and the first half spiral 𝑚 = 2 (orange dashed line).

include both forward propagation between the two arrays and backscatter towards the array from which the signal was transmitted. Further, the self-receiver versions are ignored, but the distinction between spiral propagations between pairs of transducers is taken into account (see Fig. 2). Incorporation of these paths into the pipe inspection problem presented herein is similar to the multipath exploitation approach taken in Ref. [22], whereby a system of virtual transducers is implemented as denoted by A’ and B’. As shown in Fig. 2, the model utilized in the proposed approach includes the first order helical modes wherein the propagating packet wraps around the circumference once before being received by the receiving transducer. This spiraling can occur on the second half of the path after interaction with the defect as shown by path 𝑚 = 1, or the spiraling can occur on the first half of the path before interaction with the defect as shown by path 𝑚 = 2 in Fig. 2. The transmitter and receiver locations corresponding to the lth pair are denoted by 𝐭 ! and 𝐫! , respectively, while 𝐭′! and 𝐫′! represent the respective virtual locations. The excited waveform utilized herein is a windowed sinusoidal pulse with Fourier transform 𝐻 𝑓 . We assume access to baseline signals collected with no defect present.27 Consider 𝑃 structural defects present in the pipe. Let the pth defect be located at 𝐬! . Divide the region of interest (ROI) into 𝑁 ≫ 𝑃 pixels; hence, sparsity of the defects is an applicable assumption. Let 𝐳!    be the    𝐾×1 vector obtained by sampling the 𝑙th received signal, with the baseline already subtracted out, at time instants, 𝑡! , 𝑘 = 0,1, … 𝐾 − 1. Declare 𝐱 ! , 𝑚 = 0,1,2, as the 𝑁×1 image vector, which corresponds to the mth propagation path (see Fig. 2). Only 𝑃 entries of 𝐱 ! , corresponding to the actual defect locations, are nonzero. The received signal vector 𝐳!    is expressed as 𝒛! = 𝚿!,! 𝐱 ! + 𝚿!,! 𝐱! + 𝚿!,! 𝐱 ! .

(1)

The (k, n)th elements of the 𝐾×𝑁 dictionary matrices 𝚿!,! , 𝚿!,! , and 𝚿!,! are, respectively, given by 𝚿!,! 𝚿!,!

!,!

!,!

= ℱ !!  𝐻 𝑓 exp = ℱ !!  𝐻 𝑓 exp

!!!" 𝒕! !𝒔! ! ! 𝒓! !𝒔! ! ! !! ! !!!"

𝒕!! !𝒔! ! ! 𝒓! !𝒔! ! ! !! !

(2) !!!!

(3) !!!!

𝚿!,!

!,!

= ℱ !!  𝐻 𝑓 exp

!!!"

𝒕! !𝒔! ! ! 𝒓!𝒍 !𝒔! ! ! !! !

 

(4)

!!!!

where ℱ !! denotes the inverse Fourier transform, 𝑐!!  is the frequency-dependent phase speed of the A0 mode, 𝐬! is the location of the nth pixel, 𝒕!!  and 𝒓!! represent locations of the lth virtual transmitter-receiver pair, and ∙ ! denotes the 𝑙! norm. The virtual transducers are obtained by replicating the spiral path through the unrolled pipe boundary as shown in Fig. 2. Utilization of the virtual geometry (denoted by the region 𝒱 in Fig. 2 as opposed to physical region ℜ) permits easier computation of the propagation delays associated with the helical modes. Next, stack the received signal 𝒛!! 𝒛!! … 𝒛!!!! !  such that

𝐳! , 𝑙 = 0,1, ⋯ , 𝐿 − 1 ,

vectors

to

obtain

the

𝐾𝐿×1

𝒛 = 𝚿! 𝐱 ! + 𝚿! 𝐱! + 𝚿! 𝐱 ! ,

vector

𝒛= (5)

where superscript ‘T’ represents matrix transpose. Each 𝐾𝐿×𝑁 dictionary matrix, 𝚿! , is constructed as !

! ! ! 𝚿! = 𝚿!,!    𝚿!,!  ⋯    𝚿!!!,! ,    for  𝑚 = 0,1,2.      

(6)

Before proceeding to sparse reconstruction, two qualifications must be established. First, the provided multipath model, which accounts for first-order helical modes, does not require any of the spiral path signals to be resolvable. Second, while the model provided herein only involves first-order spiral paths, extension to higher order spiral paths follows naturally such that they can be easily incorporated. 2.2 Group sparse reconstruction While the direct scattering and the spiral scatterings have different defect reflectivities, their corresponding reflectivity vectors share a common support or sparsity pattern. More simply, when a particular element of 𝐱 ! is nonzero, then so must be the corresponding elements of 𝐱! , and 𝐱 ! because the elements are representing the same physical location in the ROI. This calls for a group sparse reconstruction, which is applied in the following manner. The reflectivity vectors 𝐱 ! , 𝐱! , and 𝐱 ! are stacked to form a single 3𝑁×1 tall vector 𝐱 =   𝐱 !! 𝐱!! 𝐱 !! ! . Then, the measurement vector, 𝒛, can be expressed most compactly as 𝒛 =  𝚿𝐱 (7) where 𝚿 = 𝚿!    𝚿!      𝚿!  is the composite dictionary matrix of dimension 𝐾𝐿×3𝑁 and the vector  𝐱 exhibits a group sparse structure, with the group extending across the three considered paths for each pixel location. The vector  𝐱 can be recovered from the measurement vector 𝒛 through either a mixed 𝑙! /𝑙! norm optimization, 𝐱 = arg  min 𝐱

! !

𝒛 − 𝚿𝐱

! !

+𝜆 𝐱

!,! ,

(8)

! where 𝐱 !,! =   !!! !!! 𝑥!,! 𝑥!,! 𝑥!,! ! and 𝜆 is a regularization parameter, or a block version of the Orthogonal 23 Matching Pursuit algorithm (BOMP). For this work, BOMP is utilized for scene recovery.

Having obtained the recovered vector 𝐱, the individual reflectivity vectors 𝐱 ! , 𝐱! , and 𝐱 ! , which are contained in 𝐱, can be combined to obtain the single composite scene representation 𝐱. This is achieved simply through calculating the 𝑙! norm across the elements of these vectors as 𝐱 ! = 𝑥! = 𝑥!,! 𝑥!,! 𝑥!,! ! ! . (9)

3. SIMULATION RESULTS 3.1 Simulation parameters and setup Consider a simulation environment corresponding to a 1.824 m long aluminum pipe with a 0.1524 m diameter and a 3.125 mm wall thickness. The employed network of transducers consists of two uniformly spaced linear arrays of 3 transducers each. Table 1 lists the transducer locations relative to the coordinate system employed in the schematic shown in Fig. 2. These 𝐽 = 6 transducers imply a total of 𝐿 = 30 combinations. These transducers are excited via a fivecycle burst of a 50 kHz sinusoidal signal. This frequency is chosen relative to our pipe thickness so that only the fundamental anti-symmetric (A0) wave mode propagates from the transmitter. Relative to the coordinate system in Fig. 2, the ROI ranges from x = 0.7125 m to x = 1.111 m along the length of the pipe and wraps fully around the

circumference as shown in both Fig. 1 and Fig. 2. The ROI is divided into a 41×49 pixel grid providing a total of 𝑁 = 2009 pixels. The signal is sampled at 1 MHz over a time window of 1024 microseconds leading to 𝐾 = 1024 samples; hence, the measurement vector, 𝒛, has a length of 𝐾𝐿 = 30720. Each individual dictionary is of size 30720×2009; thus the composite dictionary, 𝚿, is of dimension 𝐾𝐿×3𝑁 = 30720×6027. Array A x (m) y (m) 0.608 0 0.608 0.159 0.608 0.319

Array B x (m) y (m) 1.216 0.079 1.216 0.239 1.216 0.399

Array A' x (m) y (m) 0.608 -0.479 0.608 -0.319 0.608 -0.159

Array B' x (m) y (m) 1.216 -0.399 1.216 -0.239 1.216 -0.079

Table 1. List of coordinates for the real and virtual transducers relative to the coordinate system portrayed in Fig. 2.

Two simulations are presented. The first scenario is that of a single defect modeled as a point-scatterer. The second scenario considers two defects both modeled as point scatterers. For both simulations, the simulated signal vector is corrupted by additive white Gaussian noise whereby the resultant signal-to-noise ratio (SNR) is 0 dB and 100 Monte Carlo runs are performed. The sparsity estimate provided to BOMP is overdetermined to be three times the actual number of defects present (either 1 or 2 depending on the simulation). This overestimate of the sparsity allows for simultaneous reconstruction, one using the multipath model that accounts for spirals and one using only the direct scatterer model. For both reconstructions, the received signal vector is the same such that the spiral contributions are present in the signal regardless of whether or not they are accounted for in the signal model. This allows us to visualize the performance gain of the proposed approach in highlighting the accuracy of the helical propagation based model and the failure inherent in ignoring the spiral components. 3.2 Simulation results For the first simulation, the single defect is located at (0.192, 0.189) m. This location is chosen such that for all transmitter-receiver pairs, the first order spirals arrive within a time window that corresponds to the arrival time of a direct path scatterer; hence, no appropriate time windowing can be applied to successfully ignore the spiral paths in using the single direct path scattering model. As expected, the reconstructed images, averaged over 100 trails, in Fig. 3 support this claim. As seen in Fig. 3(a), use of only the direct path scattering model that ignores the spirals results in high intensity false selections. These false defects can be attributed to equivalent direct path scattering arrivals from pixels with high correlation to the unaccounted for spiral paths. Contrastingly, Fig. 3(b) shows the results when BOMP is used with sparsity level specified as 3. However, since the spirals are correctly accounted for in the signal model, the two additional false selections are placed at different locations for each of the 100 runs, and assigned a weak amplitude each time; therefore, the average image over the 100 Monte Carlo runs provides essentially a clutter-free reconstruction for the decibel range considered herein. The second simulation follows the same procedure as for the first; however, two defects are implemented. The first defect is still located at (0.192, 0.189) m, whereas the second defect is at (0.712, 0.479) m. These locations provide a cluttered signal with significant time overlap of the direct scattered signals and the spiral scattered signals from both defects. The resultant reconstruction, averaged over 100 runs, using the proposed BOMP approach is shown in Fig. 4(b), with the sparsity level specified as 6. For comparison, Fig. 4(a) depicts the averaged reconstruction in which OMP with sparsity of 6 is utilized and the spiral contributions are unaccounted for in the model. Similar to the single defect case, the proposed approach provides both an intensity boost in the correct selections, as well as offers a significant clutter reduction. For Fig. 4(a), the unaccounted spirals are manifested in the selection of false locations whose direct path scatterings have corresponding arrivals. These selections tend to be consistent throughout the Monte Carlo runs and are reconstructed with non-negligible amplitudes, leading to a cluttered image.

4. CONCLUSION This paper provided a sparse reconstruction approach to pipe inspection problem within Structural Health Monitoring. The approach exploited the spiral-path propagation as well as the direct path scatterings for detecting defects in thinwalled pipes. The fundamental anti-symmetric Lamb mode was utilized and a block sparse structure was imposed for scene reconstruction. Model-based dictionaries accounting for the associated dispersion and attenuation through the

medium were constructed for the three modes (direct path, and two first order spirals). Simulation results for an aluminum pipe validated the proposed approach by exposing the shortcomings of direct path only reconstructions and highlighting the superior performance of the proposed spiral path block reconstruction approach.

a)

b)

Figure 3. Single defect simulation results. Shown are the averaged images over 100 Monte Carlo Runs at SNR of 0 dB. a) The direct scattering path reconstruction using OMP with sparsity set to 3. b) The BOMP based results employing the proposed model with a sparsity level of 3. In both figures, the true defect location is indicated by the black circle.

a)

b)

Figure 4. Two defect simulation results. Shown are the averaged images over 100 Monte Carlo Runs at an SNR of 0 dB. a) The direct scattering path reconstruction using OMP with sparsity set to 6. b) The BOMP results employing the proposed model with sparsity level of 6. In both figures, the true locations of the defects are indicated by the black circles.

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