Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information

Abstract

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f ∈ CN and a randomly chosen set of frequencies Ω. It is possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set Ω? A typical result of this paper is as follows. Suppose that f is a superposition of T spikes f(t) = ∑τ∈T f(τ)δ(t-τ) obeying T ≤ CM · (log N)-1 · Ω for some constant CM > 0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1 - O(N-M, f can be reconstructed exactly as the solution to the ℓ1 minimization problem ming∑t=oN-1 g(t) , s.t.ĝ(ω) = f̂(ω) for all ω ∈ Ω. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for CM which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of T spikes may be recovered by convex programming from almost every set of frequencies of size O( T · log N). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1 - O(N-M) would in general require a number of frequency samples at least proportional to T · log N. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f. © 2006 IEEE.