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Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information
, 2006
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this pa ..."
Abstract

Cited by 1416 (45 self)
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. 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. Suppose that is a superposition of spikes @ Aa @ A @ A obeying @�� � A I for some constant H. We do not know the locations of the spikes nor their amplitudes. Then with probability at least I @ A, can be reconstructed exactly as the solution to the I minimization problem I aH @ A s.t. ” @ Aa ” @ A for all
Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information
, 2004
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal f ∈ C N 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 res ..."
Abstract

Cited by 1 (0 self)
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(Show Context)
This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal f ∈ C N 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 · (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 g(t), s.t. ˆg(ω) = ˆ f(ω) for all ω ∈ Ω.