Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

TL;DR

This paper demonstrates that exact reconstruction of signals from incomplete frequency data is possible using convex optimization, under certain sparsity conditions, with high probability, extending to higher dimensions and other convex functionals.

Contribution

It establishes sharp conditions under which sparse signals can be exactly recovered from partial Fourier samples via convex optimization, with probabilistic guarantees.

Findings

Exact recovery with high probability using $ ext{l}_1$ minimization.

Support size condition is nearly sharp, up to a logarithmic factor.

Extension to higher dimensions and piecewise constant objects.

Abstract

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $f \in \C^N$ and a randomly chosen set of frequencies $\Omega$ of mean size $\tau N$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $\Omega$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \{t, f(t) \neq 0 \} \le \alpha(M) \cdot (\log N)^{-1} \cdot # \Omega, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$ \min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(\omega) = \hat f(\omega) \text{for all} \omega \in \Omega. $$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $\alpha$ which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp. The methodology extends to a variety of other setups 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 $f$.