Robust Uncertainty Principles:
Exact Signal Reconstruction from Highly Incomplete
Frequency Information
Emmanuel Candes†, Justin Romberg†, and Terence Tao]
† Applied and Computational Mathematics, Caltech, Pasadena, CA 91125
] Department of Mathematics, University of California, Los Angeles, CA 90095
June 2004; Revised August 2005
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. 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. Suppose that f is a superposition of |T |
spikes f(t) =
∑
τ∈T f(τ) δ(t− τ) obeying
|T | ≤ CM · (logN)
−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 `1 minimization problem
min
g
N−1∑
t=0
|g(t)|, s.t. gˆ(ω) = fˆ(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 | · logN). 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 | ·
logN .
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 func-
tionals such as the total variation of f .
Keywords. Random matrices, free probability, sparsity, trigonometric expansions, uncertainty
principle, convex optimization, duality in optimization, total-variation minimization, image recon-
struction, linear programming.
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