3D reconstruction of wave-propagated point sources from boundary measurements using joint sparsity and finite rate of innovation

Abstract

Reconstruction of point sources from boundary measurements is a challenging problem in many applications. Recently, we proposed a new sensing and non-iterative reconstruction scheme for systems governed by the three-dimensional wave equation. The points sources are described by their magnitudes and positions. The core of the method relies on the principles of finite-rate-of-innovation, and allows retrieving the parameters in the continuous domain without discretization.Here we extend the method when the source configuration shows joint sparsity for different temporal frequencies; i.e., the sources have same positions for different frequencies, not necessarily the same magnitudes. We demonstrate that joint sparsity improves upon the robustness of the estimation results. In addition, we propose a modified multi-source version of Dijkstra's algorithm to recover the Z parameters. We illustrate the feasibility of our method to reconstruct multiple sources in a 3-D spherical geometry.

Real-time temperature field measurement based on acoustic tomography

“…Besides, in all three cases the reconstruction error of OIRTF is much smaller than OIRTT. One of possible reasons is that the smooth filter can be considered as frequency domain projection, and then the iterative calculation defined in equation (13) can be considered as a projected Landweber with preconditioning which helps to accelerate convergence. It follows from many numerical simulations and practical applications that this method can provide much better estimations than the usual linear regularisation methods [28].…”

“…To improve the accuracy of non-iterative method, many researchers tried to iteratively calculate the optimal inversion operator beforehand for non-iterative online reconstruction, for instance, Offline Iteration Online Reconstruction (OIOR) [25] based on Landweber iteration, and Direct Landweber (DLW) based on modified Landweber [26]. In order to build a real-time acoustic tomography system, the offline iteration method is applied based on the SIRT method defined in equation (12) and equation (13), named offline iteration reconstruction technique using Tikhonov regularisation (OIRTT) and smooth filter (OIRTF). Consequently, the processing time of online reconstruction can be reduced to the same level as non-iterative method like LBP.…”

“…However, according to Green's research on the acoustic tomography measurement error for a temperature field in a combustion gas [11], in the worst case, refraction effects cause about a 2% reconstruction error. Therefore, a straight ray model, where the acoustic signal travels along a straight line between the transmitter and receiver, is often used in acoustic tomography for simplicity [12][13][14].…”

A Sampling Framework for Solving Physics-Driven Inverse Source Problems

“…In [37], [43] it was shown that for the case of the diffusion field, the generalised measurements can be found by imposing that Ψ k (x) be analytic. A similar strategy has been used in [26], [44] for the wave and Poisson equation. The disadvantage of these approaches is that (i) they cannot be easily extended to d > 2 and (ii) the constraint that Ψ k is analytic leads in some cases to less stable reconstruction algorithms.…”

Spatio-temporal sampling and reconstruction of diffusion fields induced by point sources

“…Given that we have access only to sparse field measurements, we adapt the reciprocity gap method to operate properly within this new context. In particular, contrary to common practice [11,14], we utilize sensor measurements both along and inside an arbitrary domain boundary to perform source localization using measurements taken over a short time interval. This allows estimation of intensities, locations and activation times of active sources with high accuracy, even in the presence of noise.…”