Fusing electrical and elasticity imaging

We will consider here a two-dimensional bounded domain $\Omega\subset\mathbb{R}^{2}$, but note that an extension to three dimensions is straight-forward. For conducting electrical measurements we place a set of $M$ disjoint electrodes on the boundary $E_{m}\subset\partial\Omega$ for $1\leq m\leq M$. For the measurement process, we apply currents $I_{m}$ at the boundary and measure the resulting voltage at each electrode $U_{m}$, given a conductivity distribution inside the domain $\gamma(x)$. The full model is then given by the established complete electrode model [43]:

Here $z_{m}$ denotes the contact impedance at each electrode and $\bar{n}$ the outer unit normal on the boundary. We note that due to conservation of charge all injected currents sum to zero, similarly a ground level for the voltage potential can be fixed by assuming that the measured voltages sum to zero as well. The forward model is then given by the mapping from conductivity $\gamma$ to a set of measured voltages $V$, we write $\gamma\mapsto U_{\mathrm{EIT}}(\gamma)$.

Given the set of measured voltages $V$, we can compute a reconstruction of the conductivity $\gamma$ as the minimiser of a penalty functional

where $\mathcal{R}$ denotes a suitable regulariser and $\alpha>0$ is a weighting parameter balancing the data fidelity term and the influence of prior information contained in the regulariser.

Specifically, the regulariser does not only stabilise the inversion process, but also offers the possibility to flexibly incorporate prior information into the reconstruction task. Such prior information can come in large variety, from simple assumptions on continuity [48] or piece-wise constant reconstructions, incorporated as a total variation penalty [18], to more informative priors using structural information [26] and geometric assumptions [29]. In particular, as the reconstruction task in EIT is non-linear and highly ill-posed such prior information is essential to improve reconstruction quality. This positive effect can be even clearly demonstrated for direct reconstructions with the D-bar method [1, 2, 3] and recent advances in utilising deep learning techniques to infer prior information from large datasets [22, 21].

In this work, rather than using specially designed priors, we consider the possibility to supply auxiliary information from a secondary modality providing parallel measurements and thus improving the reconstruction quality. In particular, a complementary modality would primarily need to compensate for the diffusive nature of EIT. That means, in EIT measurements are only obtained at the boundary from diffusive electric currents and thus loss of contrast as well as distortions are stronger within the target, but a higher accuracy at the boundary can be obtained. Consequently, we will discuss next the synergy with a full-field modality to improve reconstruction quality over the whole domain.

In contrast to boundary measurements, in QSEI the inverse problem is to compute the distributed elasticity modulus $E$ from a measured displacement field $u_{m}$ inside the domain. To be more precise, the elastic forward model is defined herein for the same two-dimensional geometry $\Omega\subset\mathbb{R}^{2}$ with boundary $\partial\Omega$ as above. On part of the boundary $\partial\Omega_{1}$ a prescribed boundary traction $f(x):\partial\Omega_{1}\to\mathbb{R}$ is applied, referred to as imposed boundary conditions. Concurrently, we denote the (unknown) displacements on $\partial\Omega_{2}$ by $\hat{u}$. Neglecting body forces, the isotropic QSEI forward problem is then written as follows

In particular, the recovery of $E$ from a measured displacement field $u_{m}$ is (as well) a non-linear inverse problem, i.e. the forward model $E\mapsto U_{\mathrm{QSEI}}(E)$ for the simulated displacement field is nonlinear in $E$. Generally, one aims to minimise the following penalty functional to reconstruct $E$ from displacement measurements $u_{m}$

It is worth mentioning here what advantage QSEI may provide in the joint imaging framework. Since QSEI is a full-field imaging modality, it is highly sensitive to localized parameter changes in $\Omega$. In contrast, EIT is a diffusive modality and is therefore less sensitive to localized parameter changes. Further, the noted sensitivity decreases proportionally with the distance from the electrodes. As such, it is surmised that including QSEI in the joint imaging framework may compensate for the lack of EIT sensitivity far from the electrodes and in return joint imaging may reduce noise sensitivity for both modalities – a hypothesis to be confirmed in the results section.

Formulating a joint EIT-QSEI imaging framework aims to take advantage of two mutually-beneficial factors (a) the sensitivity of QSEI and EIT to changes far from and near to the boundaries, respectively and (b) the assumption of anomaly sharpness within the imaged target $\Omega$. To capitalise on (a), full data sets from both EIT and QSEI will be utilised and for (b), a bespoke form of the joint total variation (JTV) will be formulated that is capable to penalise spatial smoothness, while benefiting from shared information and maintaining the correct quantitative values of both modalities. With these factors identified, instead of computing reconstructions for each problem separately in Eqs. 2 and 4, we aim to recover both reconstruction jointly by minimising the following joint functional

Nevertheless, we need to carefully adjust the joint total variation regulariser to the problem at hand, since a primary concern in the joint recovery for EIT-QSEI imaging is to preserve the quantitative values for both modalities, while benefiting from shared information on regularity and edge alignment. To achieve this, we first write the JTV functional $J$ in the discrete form as follows

It is important to note the the JTV functional described in Eq. 7 is familiar to joint TV functionals used in previous works (e.g. [13, 11, 49]). However, unlike similar works using either a level-set or global normalisation scheme (e.g., the optimisation parameter is scaled from 0 to 1), we choose to scale one of the joint parameters with respect to the other for simplicity and to prevent over regularisation of one modality while reducing necessary regularisation parameters. To do this, the parameter $\kappa$ is used to scale $E$ proportionally to $\gamma$ for computing the regulariser. For this, we choose the rational scaling of $\kappa=\frac{\gamma_{\mathrm{exp}}}{E_{\mathrm{exp}}}$, where $\gamma_{\mathrm{exp}}$ and $E_{\mathrm{exp}}$ are the expected (scalar) values computed as the best homogeneous estimates. The homogeneous estimates are computed following [39]. Finally to compute the reconstructions by minimising Eq. 5, a Gauss-Newton scheme equipped with a linesearch was adopted where iterations were terminated when the change in the cost functional was less than $10^{-2}$.

In the following we will evaluate the effectiveness of the proposed joint reconstruction framework by comparing it to the respective single-modality reconstructions. In doing this, we first simulate a realistic crack, shown in Fig. 1(c), within a (left-end) clamped $1\times 1$m square domain representative of a 5mm thick plate subjected to a 10kN/m distributed end load to be non-destructively evaluated using QSEI and EIT. For this study, the crack was considered non-conductive ($\sigma\approx 0$) and non-elastic ($E\approx 0$), consistent with a structural through crack [41]. Following, using the same crack characteristics, we simulate a double crack case as shown in Fig. 1(d). Lastly, an edge through hole simulating puncture or impact was generated as depicted in Fig. 1(e). To prevent inverse crime, separate data generation and inversion meshes are used which are shown in Figs. 1(a-b). Note that the same data generation and coarse inversion discretisations are used for EIT and QSEI.

In generating the EIT data, 16 boundary electrodes were used in conjunction with a 1mA opposite electrode current injection and adjacent electrode measurement protocol. Correspondingly, the electrode contact impedances were set to 0.01$\Omega\mathrm{cm}^{2}$. This EIT measurement strategy resulted in 192 measurements. Meanwhile, QSEI measurements were collected from the fine mesh nodes and interpolated to the coarse mesh nodes using linear interpolation, thus resulting in 1674 total displacement measurements (including $x$- and $y$- displacement fields). After collecting EIT and QSEI measurements, 1% noise standard deviation was added.

All reconstructions were carried out on a quad-core Intel i7-7700HQ machine equipped with 16GB of RAM. Generally, JTV computations required 20-30 iterations to reach stopping criteria with each iteration taking approximately 3-5 minutes (primarily in computing the Jacobians). Single modality reconstructions each required approximately 10-50% of the former computing time depending on the modality.

In this study, both joint and single-modality inversion frameworks are tested and compared. Importantly, two parameters, $\alpha$ and $\lambda$, are used for JTV/TV and the $L^{1}$ regularisation term, respectively, in addition to the JTV/TV stabilisation parameter $\beta$. We emphasise here, that also the single-modality framework uses a TV and $L^{1}$ penalty for fair comparisons. In order to rationally select the regularisation parameters in the experiments, we first utilise the methodology proposed in [18], where the TV weighting parameter is selected using $\alpha=\frac{\ln(1-p/100)}{\sigma_{\mathrm{exp}}/d}$. Namely, $d$ is the element width and $p$ is the probability (%) that the estimated parameter is predicted to lie within the expected range – herein we use $p=99$. Recalling that $\kappa$ is used to normalise gradients of $E$, the expected range is accurately captured by $\sigma_{\mathrm{exp}}$ (since $\sigma$ estimates are to expected lie between $\sigma_{\mathrm{exp}}$ and zero). For both $\lambda$ and $\beta$, empirical testing was used to select appropriate values, for which $\beta=10^{-3}$ and $\lambda=10^{-5}$ were used throughout the study.

In this subsection, we report the reconstructions using JTV and single-modality (TV regularised) frameworks. These images are shown in Fig. 2, where EIT and QSEI reconstructions are shown in the left and right columns, respectively. As a whole, all reconstructions localise the crack(s) well, while the single-modality images generally show more artefacts in the background and notably less continuity along the crack length(s). The latter is particularly true in the case of the double crack, where the continuity of cracks is far less distinguishable when single modality reconstruction is used. These visual observations are quantitatively supported by computing the mean square error (MSE, normalised with respect to the ground-truth for each modality), as reported in Table 1. Indeed, image MSEs reported in Table 1 are notable lower in cases where JTV is used.

What remains to be discussed, however, is the influence of measurement noise on reconstruction quality. To more closely investigate this, we compare previous through hole reconstructions (1% added noise) to substantially corrupted through hole reconstructions (5% added noise) in Fig. 3. Visually, it is immediately apparent that the backgrounds in images reconstructed using JTV presents with less artefacts while the through hole is localized in all cases. Interestingly, we also note an improvement in sensitivity to the (near zero) conductivity and elasticity values at the through hole location using JTV. This observation demonstrates a relative improvement in effectiveness in capturing damage processes nearing distinguishability limit by leveraging joint information and regularisation – in comparison to using a single modality. Further, we note the quantitative improvements in using JTV, with respect to increasing noise levels, reported in Table 1.

Based on the visual and quantitative observations, it can be concluded that the JTV reconstructions drastically improved image quality and reduced reconstruction errors. This may be attributed to a number of key factors, including (a) the increase in spatial information from dual data sets, (b) joint regularisation/penalisation of the EIT and QSEI problems, and (c) the inclusion of simultaneous spatial information in solving the joint problem. It is interesting to note that, while the JTV problem includes twice as many unknown parameters as the single-modality problems (and is thus, more non-unique/ill-posed), the leveraging of information from items (a-c) was sufficient in significantly improving reconstructions. This underscores a critical strength of joint inversion techniques, whereby additional prior information can be advantageously included in solving the inverse problem.

In the context of this work, where EIT in relatively insensitive to changes far from the electrodes and QSEI has more uniform sensitivity throughout the domain, EIT reconstruction quality can be drastically improved by mutually-beneficial QSEI measurements. This is supported by a visual improvement of the crack resolution that is especially evident in the EIT reconstruction. Further yet, joint QSEI image quality is also improved via utilisation of JTV penalisation.

As a whole, however, a primary advantage in using the JTV framework is that we double the amount of information available for assessing structural condition via the simultaneous reconstruction of $\gamma$ and $E$. This leverage will be especially important in the context of missing measurement data, for instance due to detached electrodes, or inaccurate information in the geometry. We expect that a joint framework will be especially beneficial in overcoming the sensitivity of these nonlinear inverse problems to incomplete model information.