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

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f N and a randomly chosen set of frequencies Ω of mean size τN. Is it 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: for each M > 0, suppose that f obeys # \t, f (t) 0 \ α (M) (N) ^-1 # Ω, then with probability at least 1-O (N^-M), f can be reconstructed exactly as the solution to the ₁ minimization problem g ₓ = ₀^N-1 |g (t) |, s. t. g (ω) = f (ω) for all ω Ω. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for α 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.