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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)=στ∈Tf(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. 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 functionals such as the total variation of f.