We consider simultaneously identifying the membership and locations of point sources that are convolved with different band-limited point spread functions, from the observation of their superpositions. This problem arises in three-dimensional super-resolution single-molecule imaging, neural spike sorting, multi-user channel identification, among other applications. We propose a novel algorithm, based on convex programming, and establish its near-optimal performance guarantee for exact recovery in the noise-free setting by exploiting the spectral sparsity of the point source model as well as the incoherence between point spread functions. Furthermore, robustness of the recovery algorithm in the presence of bounded noise is also established. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.
In this paper, we study the problem of super-resolving the mixture model when there are two modalities, i.e. @math . The methodology in this paper can be extended straightforwardly to the analysis of the case @math and is left for future work. We start by recognizing that in the Fourier domain, the observed signal can be regarded as a linear combination of two spectrally-sparse signals, each composed of a small number of distinct complex sinusoids. The atomic norm @cite_19 @cite_23 of spectrally-sparse signals is developed and proposed as an efficient convex optimization framework to motivate parsimonious structures @cite_26 @cite_20 @cite_23 @cite_8 @cite_5 recently, which can be computed efficiently via semidefinite programming. We then separate and recover the two signals by motivating their spectral structures using atomic norm minimization, in addition to satisfying the observation constraints. The proposed algorithm, denoted by AtomicDemix , is reminiscent of the algorithms for sparse error correction @cite_32 , robust principal component analysis @cite_36 , demixing of sines and spikes @cite_21 @cite_24 , and source separation @cite_38 , where one aims to separate two low-dimensional signals with incoherent structures via convex optimization.
@cite_38 @cite_26 @cite_8 @cite_36 @cite_21 @cite_32 @cite_24 @cite_19 @cite_23 @cite_5 @cite_20
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Suppose a discrete-time signal S(t), 0 spl les t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1 sub t= spl tau and sinusoids exp 2 spl pi iwt N spl radic N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l sup 1 norm of the coefficients among all decompositions. Here "highly sparse" means that N sub t +N sub w < spl radic N 2 where N sub t is the number of time atoms, N sub w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l sup 1 uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l sup 1 norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2 sup j 2 . This paper studies the problem of recovering a sparse signal x ∈ ℝn from highly corrupted linear measurements y = Ax + e ∈ ℝm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any sufficiently sparse signal x can be recovered by solving an l1 -minimization problem min ||x||1 + ||e||1 subject to y = Ax + e. More precisely, if the fraction of the support of the error e is bounded away from one and the support of a: is a very small fraction of the dimension m, then as m becomes large the above l1 -minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of independent identically distributed (i.i.d.) Gaussian vectors with nonzero mean and small variance, dubbed the "cross-and-bouquet" (CAB) model. Simulations and experiments corroborate the findings, and suggest extensions to the result. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. The scope of this work is the separation of N sources from M linear mixtures when the underlying system is underdetermined, that is, when Mi N. If the input distribution is sparse the mixing matrix can be estimated either by external optimization or by clustering and, given the mixing matrix, a minimal l1 norm representation of the sources can be obtained by solving a low-dimensional linear programming problem for each of the data points. Yet, when the signals per se do not satisfy this assumption, sparsity can still be achieved by realizing the separation in a sparser transformed domain. The approach is illustrated here for M = 2. In this case we estimate both the number of sources and the mixing matrix by the maxima of a potential function along the circle of unit length, and we obtain the minimal l1 norm representation of each data point by a linear combination of the pair of basis vectors that enclose it. Several experiments with music and speech signals show that their time-domain representation is not sparse enough. Yet, excellent results were obtained using their short-time Fourier transform, including the separation of up to six sources from two mixtures. ? 2001 Elsevier Science B.V. All rights reserved.