Juri Ranieri

Near-Optimal Sensor Placement for Linear Inverse Problems

A classic problem is the estimation of a set of parameters from measurements collected by only a few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is near-optimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods.

Euclidean Distance Matrices: Essential theory, algorithms, and applications

Euclidean distance matrices (EDMs) are matrices of the squared distances between points. The definition is deceivingly simple; thanks to their many useful properties, they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness, and show how the various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes, and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. The code for all of the described algorithms and to generate the figures in the article is available online at http://lcav.epfl.ch/ivan.dokmanic. Finally, we suggest directions for further research.

Sampling and reconstructing diffusion fields with localized sources

We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results.

EigenMaps: algorithms for optimal thermal maps extraction and sensor placement on multicore processors

Chip designers place on-chip sensors to measure local temperatures, thus preventing thermal runaway situations in multicore processing architectures. However, thermal characterization is directly dependent on the number of placed sensors, which should be minimized, while guaranteeing full detection of all hot-spots and worst case temperature gradient. In this paper, we present EigenMaps: a new set of algorithms to recover precisely the overall thermal map from a minimal number of sensors and a near-optimal sensor allocation algorithm. The proposed methods are stable with respect to possible temperature sensor calibration inaccuracies, and achieve significant improvements compared to the state-of-the-art. In particular, we estimate an entire thermal map for an industrial 8-core industrial design within 1°C of accuracy with just four sensors. Moreover, when the measurements are corrupted by noise (SNR of 15 dB), we can achieve the same precision only with 16 sensors.

Sensor networks for diffusion fields: Detection of sources in space and time

We consider the problem of reconstructing a diffusion field, such as temperature, from samples collected by a sensor network. Motivated by the fast decay of the eigenvalues of the diffusion equation, we approximate the field by a truncated series. We show that the approximation error decays rapidly with time. On the other hand, the information content in the field also decays with time, suggesting the need for a proper choice of the sampling strategy. We propose two algorithms for sampling and reconstruction of the field. The first one reconstructs the distribution of point sources appearing at known times using the finite rate of innovation (FRI) framework. The second algorithm addresses a more difficult problem of estimating the unknown times at which the point sources appear, in addition to their locations and magnitudes. It relies on the assumption that the sources appear at distinct times. We verify that the algorithms are capable of reconstructing the field accurately through a set of numerical experiments. Specifically, we show that the second algorithm successfully recovers an arbitrary number of sources with unknown release times, satisfying the assumption. For simplicity, we develop the 1-D theory, noting the possibility of extending the framework to more general domains.

Compressive sensing of localized signals: Application to Analog-to-Information conversion

Compressed sensing hinges on the sparsity of signals to allow their reconstruction starting from a limited number of measures. When reconstruction is possible, the SNR of the reconstructed signal depends on the energy collected in the acquisition. Hence, if the sparse signal to be acquired is known to concentrate its energy along a known subspace, an additional “rakeness” criterion arises for the design and optimization of the measurement basis. Formal and qualitative discussion of such a criterion is reported within the framework of a well-known Analog-to-Information conversion architecture and for signals localized in the frequency domain. Non-negligible inprovements are shown by simulation.

The Fukushima inverse problem

Knowing what amount of radioactive material was released from Fukushima in March 2011 is crucial to understand the scope of the consequences. Moreover, it could be used in forward simulations to obtain accurate maps of deposition. But these data are often not publicly available, or are of questionable quality. We propose to estimate the emission waveforms by solving an inverse problem. Previous approaches rely on a detailed expert guess of how the releases appeared, and they produce a solution strongly biased by this guess. If we plant a nonexistent peak in the guess, the solution also exhibits a nonexistent peak. We propose a method based on sparse regularization that solves the Fukushima inverse problem blindly. Together with the atmospheric dispersion models and worldwide radioactivity measurements our method correctly reconstructs the times of major events during the accident, and gives plausible estimates of the released quantities of Xenon.

Sampling and reconstruction of time-varying atmospheric emissions

We study the spatio-temporal sampling of physical fields representing the dispersion of a substance in the atmosphere. We consider the following setup: N sensors are deployed at ground level and measure the concentration of a particular substance, while M smokestacks are located in the same area and emit a time-varying amount of the substance. To recover the emission rates of the smokestacks with a limited number of spatio-temporal samples, we consider time varying emissions rates lying in two specific low-dimensional subspaces. We propose efficient algorithms and sufficient conditions to recover the emission rates of the smokestacks from the local measurements collected by the sensor network.

DASS: Distributed Adaptive Sparse Sensing

Wireless sensor networks are often designed to perform two tasks: sensing a physical field and transmitting the data to end-users. A crucial design aspect of a WSN is the minimization of the overall energy consumption. Previous researchers aim at optimizing the energy spent for the communication, while mostly ignoring the energy cost of sensing. Recently, it has been shown that considering the sensing energy cost can be beneficial for further improving the overall energy efficiency. More precisely, sparse sensing techniques were proposed to reduce the amount of collected samples and recover the missing data using data statistics. While the majority of these techniques use fixed or random sampling patterns, we propose adaptively learning the signal model from the measurements and using the model to schedule when and where to sample the physical field. The proposed method requires minimal on-board computation, no inter-node communications, and achieves appealing reconstruction performance. With experiments on real-world datasets, we demonstrate significant improvements over both traditional sensing schemes and the state-of-the-art sparse sensing schemes, particularly when the measured data is characterized by a strong intra-sensor (temporal) or inter-sensors (spatial) correlation.

Sampling and reconstructing diffusion fields in presence of aliasing

The reconstruction of a diffusion field, such as temperature, from samples collected by a sensor network is a classical inverse problem and it is known to be ill-conditioned. Previous work considered source models, such as sparse sources, to regularize the solution. Here, we consider uniform spatial sampling and reconstruction by classical interpolation techniques for those scenarios where the spatial sparsity of the sources is not realistic. We show that even if the spatial bandwidth of the field is infinite, we can exploit the natural low-pass filter given by the diffusion phenomenon to bound the aliasing error.