Estimating Localized Sources of Diffusion Fields Using Spatiotemporal Sensor Measurements

Abstract

We consider diffusion fields induced by a finite number of spatially localized sources and address the problem of estimating these sources using spatiotemporal samples of the field obtained with a sensor network. Within this framework, we consider two different time evolutions: the case where the sources are instantaneous, as well as, the case where the sources decay exponentially in time after activation. We first derive novel exact inversion formulas, for both source distributions, through the use of Green's second theorem and a family of sensing functions to compute generalized field samples. These generalized samples can then be inverted using variations of existing algebraic methods such as Prony's method. Next, we develop a novel and robust reconstruction method for diffusion fields by properly extending these formulas to operate on the spatiotemporal samples of the field. Finally, we present numerical results using both synthetic and real data to verify the algorithms proposed herein.Index Terms-Spatiotemporal sampling, diffusion fields, finite rate of innovation (FRI), Prony's method, sensor networks.

Physics-driven quantized consensus for distributed diffusion source estimation using sensor networks

“…In what follows, we provide an overview for the centralized recovery of diffusion fields as proposed in [36], wherein we demonstrate through the use of Green's second theorem that given access to some generalized measurements of the form…”

“…, L} N n=1 . A way to compute good approximations of (15) using standard quadrature techniques is discussed in [36]. When using these quadrature techniques to obtain an approximation of the generalized measurements {R(k)} k , the performance of the centralized algorithm is near-optimal, in that it comes very close to achieving the Cramer-Rao bound (CRB) as shown in Fig.…”

“…To further improve the noise robustness of our source estimation approach, we adapt the sequential local line search technique introduced [36]. This modification is based on searching for a time interval over where only a single diffusion source is active.…”

Solving physics-driven inverse problems via structured least squares

“…Given these estimates the activation time τ 1 is then recovered by performing a line search. For multiple sources we can recover each source sequentially as described in [7], for example. The results are summarized in Figure 3, as expected the unknown source location, intensity and activation times are recovered reliably.…”

“…This class of inverse problems continues to receive considerable research interests from a range of communities, including the signal processing community, due to their ubiquity across many applications involving, for example, sound/wave source localization [4], brain source localization [5], and plume/leakage detection [6,7]. Our proposed This work is supported by the European Research Council (ERC) starting investigator award Nr.…”