Total Focusing in the Virtual Wave Domain: 3D Defect Reconstruction Using Spatially Structured Laser Heating ^†

Abstract

Classical active thermographic testing of industrial goods has mostly been limited to generating 2D defect maps. While for surface or near-surface defect detection, this is a desired result, for deeply buried defects, a 3D reconstruction of the defect geometry is coveted. This general trend can also be well observed in widely used NDT methods (radiography, ultrasonic testing), where the progression from 2D to 3D reconstruction methods has already made profound progress (CT, UT phased array transducers). Achieving a fully 3D defect reconstruction in active thermographic testing suffers from the diffusive nature of thermal processes. One possible solution to deal with thermal diffusion is the application of the virtual-wave concept, which, by solving an inverse problem, allows the diffusiveness to be extracted from the thermographic data in the post-processing stage. What is left follows propagating-wave physics, enabling the usage of well-known algorithms from ultrasonic testing. In this work, we present our progress in the 3D reconstruction of deeply buried defects using spatially structured laser heating in conjunction with applying the well-known total focusing method (TFM) in the virtual-wave domain.

1. Introduction

Active thermographic testing, as a non-destructive testing method, being non-contact and covering large areas at a reasonable speed, offers a unique set of advantages over other non-destructive testing methods. However, one of its main drawbacks is the diffusive nature of thermal processes, which makes it difficult to detect and reconstruct deeply buried defects. Furthermore, in its application highly ill-posed inverse problems arise, as usually the object under test (OuT) can only be excited at its surfaces, and the temperature response is measured at the surface as well. Achieving a fully 3D tomographic reconstruction of defects—a much sought-after result—is therefore a challenging task for active thermographic testing. In this work, we present our progress towards achieving this goal by combining multiple spatially structured laser heatings with the total focusing method (TFM), known from ultrasonic testing in the virtual wave domain.

2. Methodology

Burgholzer et al. [1] developed a virtual wave concept that connects the diffusive heat equation with the propagating wave equation for ultrasound waves. The result is a Fredholm integral relationship that allows the computation of the virtual wave field ${\displaystyle T_{\mathrm{virt}}}$ based on a known temperature distribution ${\displaystyle T}$. The formal equation is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T(\mathbf{r},t)& ={\int }_{-\infty }^{\infty }{T}_{\mathrm{virt}}(\mathbf{r},t^{\prime })K(t,t^{\prime })\mathrm{d}{t}^{\prime },\hfill \\ \hfill K(t,t^{\prime })& ={\displaystyle \frac{c}{\sqrt{\pi \alpha t}}}\exp \left(-{\displaystyle \frac{c^2{t}^{\prime 2}}{4\alpha t}}\right)\mathrm{for}t>0,\hfill \end{array}\end{array}$$

(1)

with ${\displaystyle \mathbf{r}}$ being the position vector and ${\displaystyle t}$ and ${\displaystyle t^{\prime }}$ being the different time scales in the time and virtual wave domains. The kernel ${\displaystyle K(t,t^{\prime })}$ transforms between both domains and contains the thermal diffusivity ${\displaystyle \alpha }$ and the virtual speed of sound ${\displaystyle c}$ as parameters. The corresponding discretized version of Equation (1) for two spatial dimensions ${\displaystyle x}$ and ${\displaystyle y}$ is ${\displaystyle T=\mathbf{K}{T}_{\mathrm{virt}}}$, where ${\displaystyle T\in {\mathbb{R}}^{N_t\times N_x\times N_y}}$ is the observed temperature data, ${\displaystyle \mathbf{K}\in {\mathbb{R}}^{N_t\times N_{t^{\prime }}}}$ is the discretized kernel, and ${\displaystyle T_{\mathrm{virt}}\in {\mathbb{R}}^{N_{t^{\prime }}\times N_x\times N_y}}$ is the virtual wave field. The entries of the discrete kernel ${\displaystyle \mathbf{K}=\left[\begin{array}{c}{K}_{i,j}\end{array}\right]\in {\mathbb{R}}^{N_t\times N_{t^{\prime }}}}$ can be calculated with

$$K_{i,j}={\displaystyle \frac{\tilde{c}}{\sqrt{\pi \tilde{\alpha }i}}}\exp \left(-{\displaystyle \frac{{\tilde{c}}^2{(j-1)}^2}{4\tilde{\alpha }i}}\right),\mathrm{where}\tilde{c}=c{\displaystyle \frac{{\mathsf{\Delta }}_t}{{\mathsf{\Delta }}_z}}\mathrm{and}\tilde{\alpha }=\alpha {\displaystyle \frac{{\mathsf{\Delta }}_{t^{\prime }}}{{\mathsf{\Delta }}_z^2}}.$$

(2)

Here ${\displaystyle {\mathsf{\Delta }}_t,{\mathsf{\Delta }}_{t^{\prime }}}$, and ${\displaystyle {\mathsf{\Delta }}_z}$ are the increments of the time, virtual time, and depth vector. The kernel ${\displaystyle \mathbf{K}}$ is highly rank deficient, which makes solving for ${\displaystyle T_{\mathrm{virt}}}$ non-trivial. To solve this ill-posed inverse problem, we apply the alternating direction method of multipliers (ADMM) [2]. The minimization problem version of the discretized problem is given by [3] (Chapter 6.4) as

$$\underset{T_{\mathrm{virt}}}{min}{\displaystyle \frac{1}{2}}{∥\mathbf{K}{T}_{\mathrm{virt}}-T∥}_2^2+\lambda {∥T_{\mathrm{virt}}∥}_1.$$

(3)

Since an ultrasound field is naturally very sparse, it makes sense to incorporate sparsity as an additional constraint into the minimization problem. The sparsity of the solution is achieved by the ${\displaystyle \ell 1}$-norm of ${\displaystyle T_{\mathrm{virt}}}$ and controlled by the parameter ${\displaystyle \lambda }$, which has to be determined empirically.

3. Total Focusing Method

The total focusing method (TFM) is a well-established image reconstruction technique for ultrasonic testing [4]. Combined with an full matrix capture (FMC) experimental approach, the method relies on taking multiple measurements of the same OuT with different transmitter positions for each measurement while recording the response using multiple receivers. The transfer to an application within thermographic testing is very natural, as an infrared camera usually records a large region of interest (ROI) at all times, providing a high phase-coherent receiver count. Hence, only a change in the structure of the excitation is necessary for TFM application. This can be facilitated by step-scanning with a single laser spot of any shape [5] or using a full-scale light modulator [6]. Enough cooling time between excitations is necessary to guarantee sufficient independence of all measurements. Given ${\displaystyle N_{\mathrm{meas}}}$ two-dimensional virtual wave fields ${\displaystyle T_{\mathrm{virt}}\in {\mathbb{R}}^{N_{t^{\prime }}\times N_x\times N_y\times N_{\mathrm{meas}}}}$, we can compute the TFM reconstruction result using the following equation:

$$T_{\mathrm{rec}}\left(P\right)=\sum _{n=1}^{N_{\mathrm{meas}}}\sum _{P_{tx}}\sum _{P_{rx}}{T}_{\mathrm{virt}}(t_{tx,rx}(P_{tx,j},P,P_{rx,i}),P_{rx,i},n),$$

(4)

where ${\displaystyle P=(x,y,z)\in {\mathbb{R}}^3}$ is the location of a volume element within the discretized OuT, ${\displaystyle P_{rx,i}=(x_{rx,i},y_{rx,i})\in {\mathbb{R}}^2}$ is the location of the ${\displaystyle i}$-th receiver, and ${\displaystyle P_{tx,j}=(x_{tx,j},y_{tx,j})\in {\mathbb{R}}^2}$ is the location of the ${\displaystyle j}$-th transmitter. ${\displaystyle t_{tx,rx}}$ describes the travel time of the virtual wave from the transmitter ${\displaystyle P_{tx,j}}$ via the volume element at ${\displaystyle P}$ to the receiver ${\displaystyle P_{rx,i}}$.

Using a Euclidean distance function between two points ${\displaystyle d(P_1,P_2)={∥P_2-P_1∥}_2}$ and the virtual wave speed ${\displaystyle c}$, the travel time ${\displaystyle t_{tx,rx}}$ of the virtual wave can be determined as ${\displaystyle t_{tx,rx}(P_{tx},P,P_{rx})=\left(d(P_{tx},P)+d(P,P_{rx})\right)/c.}$

4. Experimental Setup

The experimental setup is shown in Figure 1. Here, an OuT is examined that has been additively manufactured from stainless steel (316L, 1.4404, ${\displaystyle \alpha =3.5 {\mathrm{m}\mathrm{m}}^2 {\mathrm{s}}^{-1}}$) and features a height and width of ${\displaystyle 58.5 \mathrm{m}\mathrm{m}}$ and a thickness of ${\displaystyle 4.5}$ ${\displaystyle \mathrm{m}}$${\displaystyle \mathrm{m}}$. It contains four defect pairs ( ${\displaystyle 2 \mathrm{m}\mathrm{m}\times 2 \mathrm{m}\mathrm{m}}$ rectangular channels) where the gaps between the two defects of each pair each double from ${\displaystyle 0.5 \mathrm{m}\mathrm{m} \mathrm{to} 4 \mathrm{m}\mathrm{m}}$. The OuT is symmetric in the ${\displaystyle y}$-direction as the channels run through the whole width. On this OuT, ${\displaystyle N_{\mathrm{meas}}=121}$ line excitations for ${\displaystyle t_{\mathrm{pulse}}=300 \mathrm{m}\mathrm{s}}$ at ${\displaystyle P=90 \mathrm{W}}$ have been performed, using a laser line with a width of ${\displaystyle 0.38}$ ${\displaystyle \mathrm{m}}$${\displaystyle \mathrm{m}}$ and a step-over of ${\displaystyle 0.4}$ ${\displaystyle \mathrm{m}}$${\displaystyle \mathrm{m}}$ in the ${\displaystyle x}$-direction. The resulting temperature response has been measured using an MWIR cooled infrared camera with a pixel resolution of ${\displaystyle \mathsf{\Delta }x,\mathsf{\Delta }y=53 \mathsf{\mu }\mathrm{m}}$ in reflection configuration. Because of the symmetry of the OuT, the ROI is limited to a ${\displaystyle 64}$ ${\displaystyle \mathrm{pixel}}$-wide strip across the ${\displaystyle y}$-direction.

A dichroic mirror is used to compact the setup and avoid perspective distortion. The OuT is allowed to cool for ${\displaystyle 40}$ ${\displaystyle \mathrm{s}}$ between consecutive measurements.

5. Results and Discussion

Calculating the resulting virtual wave fields for all ${\displaystyle N_{\mathrm{meas}}=121}$ measurements, using a virtual speed of sound of ${\displaystyle c=1}$, ${\displaystyle \lambda =0.01}$ and ${\displaystyle {\rho }_{\mathrm{ADMM}}=0.0039}$ at a depth discretization of ${\displaystyle {\mathsf{\Delta }}_z=4.5 \mathsf{\mu }\mathrm{m}}$, leads to the result shown in Figure 2a.

Here, the sum of all virtual wave fields is displayed. The white line indicates the back wall of the OuT at ${\displaystyle z=4.5 \mathrm{m}\mathrm{m}}$. In total, depths up to two times the thickness of the OuT have been reconstructed. The result already shows a clear indication of the defects and the wakes in the virtual wave field introduced by them. This mostly affects the area below the defects and directly above them.

Applying the afore-described TFM procedure to the virtual wave fields, making full use of all three spatial dimensions, leads to the result shown in Figure 2b. Here, a single slice through the middle of the ROI is shown. Compared to the result of the virtual wave transformation, the defect contrast has clearly improved, and defects are clearly visible. The defect wakes are somewhat suppressed, but the remains are still visible. Even the smallest defect pair, with a gap of ${\displaystyle 0.5 \mathrm{m}\mathrm{m}}$ at a depth of ${\displaystyle 2}$ ${\displaystyle \mathrm{m}}$${\displaystyle \mathrm{m}}$, can be differentiated using this technique.

6. Outlook

The presented results are very promising and show that the combination of multiple measurements with spatially structured laser heating and the application of the TFM in the virtual wave domain is a feasible method for 3D defect reconstruction. Additional measurements using round-laser-spot step-scanning on an OuT with a more complex defect geometry also reinforce this assumption. Due to the limited scope of this manuscript, a more detailed overview is to be published in a follow-up publication.