Paper
Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information
Authors
Emmanuel Candes, Justin Romberg, Terence Tao
Abstract
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal and a randomly chosen set of frequencies of mean size . Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set ?
A typical result of this paper is as follows: for each , suppose that obeys then with probability at least , can be reconstructed exactly as the solution to the minimization problem In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for 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 .