Reconstructing diffusion fields sampled with a network of arbitrarily distributed sensors

Sensor networks are becoming increasingly prevalent for monitoring physical phenomena of interest. For such wireless sensor network applications, knowledge of node location is important. Although a uniform sensor distribution is common in the literature, it is normally difficult to achieve in reality. Thus we propose a robust algorithm for reconstructing two-dimensional diffusion fields, sampled with a network of arbitrarily placed sensors. The two-step method proposed here is based on source parameter estimation: in the first step, by properly combining the field sensed through well-chosen test functions, we show how Prony's method can reveal locations and intensities of the sources inducing the field. The second step then uses a modification of the Cauchy-Schwarz inequality to estimate the activation time in the single source field. We combine these steps to give a multi-source field estimation algorithm and carry out extensive numerical simulations to evaluate its performance

Guaranteed performance in the FRI setting

Finite Rate of Innovation (FRI) sampling theory has shown that it is possible to sample and perfectly reconstruct classes of non-bandlimited signals such as streams of Diracs. In the case of noisy measurements, FRI methods achieve the optimal performance given by the Cramér-Rao bound up to a certain PSNR and breaks down for smaller PSNRs. To the best of our knowledge, the precise anticipation of the breakdown event in FRI settings is still an open problem. In this letter, we address this issue by investigating the subspace swap event which has been broadly recognised as the reason for performance breakdown in SVD-based parameter estimation algorithms. We work out at which noise level the absence of subspace swap is guaranteed and this gives us an accurate prediction of the breakdown PSNR which we also relate to the sampling rate and the distance between adjacent Diracs. Simulation results validate the reliability of our analysis

Rumour source detection in social networks using partial observations

The spread of information on graphs has been extensively studied in engineering, biology, and economics. Re- cently, however, several authors have started to address the more challenging inverse problem, of localizing the origin of an epidemic, given observed traces of infection. In this paper, we introduce a novel technique to estimate the location of a source of multiple epidemics on a general graph, assuming knowledge of the start times of rumours, and using observations from a small number of monitors

Sparse Signal Recovery Using Structured Total Maximum Likelihood

In this paper, we consider the sparse signal recovery problem when the dictionary is a Fourier frame. Based on the annihilation relation, the sparse signal recovery from noisy observations is posed as a structured total maximum likelihood (STML) problem. The recent structured total least squares (STLS) approach for finite rate of innovation signal recovery can be viewed as a particular version of our method. We transform the STML problem which has an additional logdet term into a form similar to the STLS problem. It can be effectively tackled using an iterative quadratic maximum likelihood like algorithm. From simulation results, our proposed STML approach outperforms the STLS based algorithm and the state-of-the-art sparse recovery algorithms

Estimating localized sources of diffusion fields using spatiotemporal sensor measurements

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

Spatiotemporal Sampling Trade-off for Inverse Diffusion Source Problems

We consider the spatiotemporal sampling of diffusion fields induced by M point sources, and study the associated inverse problem of recovering the initial parameters of the unknown sources. In particular, we focus on characterising qualitatively the error of the obtained source estimates. To achieve this, we obtain an expression with which we can trade the sensor density for performance accuracy. In other words, by evaluating the optimal sampling instant for a given sensor density-and using the corresponding field samples at that instant-we can expect to obtain an improvement in the estimation performance when compared to an arbitrary sampling instant. Finally, several numerical simulations are presented, to support the theoretical results obtained