17 research outputs found
Towards Generalized FRI Sampling with an Application to Source Resolution in Radioastronomy
It is a classic problem to estimate continuous-time sparse signals, like point sources in a direction-of-arrival problem, or pulses in a time-of-flight measurement. The earliest occurrence is the estimation of sinusoids in time series using Prony's method. This is at the root of a substantial line of work on high resolution spectral estimation. The estimation of continuous-time sparse signals from discrete-time samples is the goal of the sampling theory for finite rate of innovation (FRI) signals. Both spectral estimation and FRI sampling usually assume uniform sampling. But not all measurements are obtained uniformly, as exemplified by a concrete radioastronomy problem we set out to solve. Thus, we develop the theory and algorithm to reconstruct sparse signals, typically sum of sinusoids, from non-uniform samples. We achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements. These uniform samples are known to satisfy the annihilation equations. A valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint. Thanks to this new approach, we unify a variety of FRI-based methods. We demonstrate the versatility and robustness of the proposed approach with five FRI reconstruction problems, namely Dirac reconstructions with irregular time or Fourier domain samples, FRI curve reconstructions, Dirac reconstructions on the sphere and point source reconstructions in radioastronomy. The proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled Fourier measurements under severe noise conditions
Sparse Recovery of Strong Reflectors With an Application to Non-Destructive Evaluation
In this paper we show that it is sufficient to recover the locations of K strong reflectors within an insonified medium from three receive elements and 2K+1 samples per element. The proposed approach leverages advances in sampling signals with a finite rate of innovation along each element and rank properties from the Euclidean distance matrix construction across elements. With the proposed approach, it is not necessary to construct an image in order to identify strong reflective sources, which is why much fewer receive elements are needed. However, the assumed transmit scheme still uses a standard linear array in order to excite the entire medium with sufficient energy. The approach is validated with simulated data and a measurement that emulates a scenario in non-destructive evaluation
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
On the Uniqueness of Inverse Problems with Fourier-domain Measurements and Generalized TV Regularization
We study the super-resolution problem of recovering a periodic
continuous-domain function from its low-frequency information. This means that
we only have access to possibly corrupted versions of its Fourier samples up to
a maximum cut-off frequency. The reconstruction task is specified as an
optimization problem with generalized total-variation regularization involving
a pseudo-differential operator. Our special emphasis is on the uniqueness of
solutions. We show that, for elliptic regularization operators (e.g., the
derivatives of any order), uniqueness is always guaranteed. To achieve this
goal, we provide a new analysis of constrained optimization problems over Radon
measures. We demonstrate that either the solutions are always made of Radon
measures of constant sign, or the solution is unique. Doing so, we identify a
general sufficient condition for the uniqueness of the solution of a
constrained optimization problem with TV-regularization, expressed in terms of
the Fourier samples.Comment: 20 page
Neuromorphic Sampling of Sparse Signals
Neuromorphic sampling is a bioinspired and opportunistic analog-to-digital
conversion technique, where the measurements are recorded only when there is a
significant change in the signal amplitude. Neuromorphic sampling has paved the
way for a new class of vision sensors called event cameras or dynamic vision
sensors (DVS), which consume low power, accommodate a high-dynamic range, and
provide sparse measurements with high temporal resolution making it convenient
for downstream inference tasks. In this paper, we consider neuromorphic sensing
of signals with a finite rate of innovation (FRI), including a stream of Dirac
impulses, sum of weighted and time-shifted pulses, and piecewise-polynomial
functions. We consider a sampling-theoretic approach and leverage the close
connection between neuromorphic sensing and time-based sampling, where the
measurements are encoded temporally. Using Fourier-domain analysis, we show
that perfect signal reconstruction is possible via parameter estimation using
high-resolution spectral estimation methods. We develop a kernel-based sampling
approach, which allows for perfect reconstruction with a sample complexity
equal to the rate of innovation of the signal. We provide sufficient conditions
on the parameters of the neuromorphic encoder for perfect reconstruction.
Furthermore, we extend the analysis to multichannel neuromorphic sampling of
FRI signals, in the single-input multi-output (SIMO) and multi-input
multi-output (MIMO) configurations. We show that the signal parameters can be
jointly estimated using multichannel measurements. Experimental results are
provided to substantiate the theoretical claims
Sampling and Reconstruction of Bandlimited Signals with Multi-Channel Time Encoding
Sampling is classically performed by recording the amplitude of the input at given time instants; however, sampling and reconstructing a signal using multiple devices in parallel becomes a more difficult problem to solve when the devices have an unknown shift in their clocks. Alternatively, one can record the times at which a signal (or its integral) crosses given thresholds. This can model integrate-and-fire neurons, for example, and has been studied by Lazar and Tóth under the name of "Time Encoding Machines". This sampling method is closer to what is found in nature. In this paper, we show that, when using time encoding machines, reconstruction from multiple channels has a more intuitive solution, and does not require the knowledge of the shifts between machines. We show that, if single-channel time encoding can sample and perfectly reconstruct a 2Ω-bandlimited signal, then M-channel time encoding can sample and perfectly reconstruct a signal with M times the bandwidth. Furthermore, we present an algorithm to perform this reconstruction and prove that it converges to the correct unique solution, in the noiseless case, without knowledge of the relative shifts between the machines. This is quite unlike classical multi-channel sampling, where unknown shifts between sampling devices pose a problem for perfect reconstruction