Robust recovery of stream of pulses using convex optimization

Abstract

This paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses. This problem is motivated by a variety of applications, such as ultrasound and radar. We show that for univariate and bivariate stream of pulses, one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem, provided that a pulse-dependent separation condition is satisfied. The main result of this paper states that the recovery is robust to additive noise or model mismatch.

Super-Resolution on the Sphere Using Convex Optimization

“…A consecutive paper [12] showed that the recovery is robust to noisy measurements. Similar results are given for support detection from low Fourier coefficients [4], [23], recovery of non-uniform splines from their projection onto spaces of algebraic polynomials [8], [18] and recovery of streams of pulses [6], [9]. (see also [17]).…”

Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees

“…Our techniques are inspired by the deterministic certificate used to establish exact recovery of an atomic measure from low-pass measurements in [14]. Similarly, if the whole convolution K is assumed to be known, as in [5,6,54], a certificate can be built by interpolating the sign pattern with the convolution kernel in the same way. This polynomial is constructed via interpolation using a lowpass interpolation kernel with fast decay.…”

“…6 To define piecewise bounds that are valid on each U j , we maximize over the sets in the partition of OE .S; T /; .S; T / 2 , which contain points that satisfy the sample-separation condition in Definition 2.2. These bounds can be computed by interval arithmetic using the fact that the functions B .i / and W .i / can be expressed in terms of exponentials and polynomials.…”

“…A series of recent papers analyze TV norm minimization for recovering point sources from low-pass measurements [14,23,33] and derive stability guarantees [4,13,30,32,46,68]. This framework has been extended to obtain both exact-recovery and robustness guarantees for deconvolution via convex programming for bumplike kernels including the Gaussian in [5,6] and specifically Ricker kernels in [54] under the assumption that the complete convolution between a train of spikes and the kernel of interest is known (i.e., without sampling).…”

“…The crucial difference with our setting is that in [14] the interpolating function just needs to be low-pass, which makes it possible to center the shifted copies of the interpolation kernel at the elements of the signal support (T in our notation). Similarly, if the whole convolution K is assumed to be known, as in [5,6,54], a certificate can be built by interpolating the sign pattern with the convolution kernel in the same way. In contrast, if we incorporate sampling into the measurement process, we are constrained to use shifted kernels centered at the sample locations s 1 ; s 2 ; : : : ; s n .…”