The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its @math consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is @math and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than @math apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT (Estimation of Signal Parameters via Rotation Invariance Techniques) is an efficient method that does not depend on the sign of the measure and this paper provides a : we prove an explicit upper bound on the recovery error of ESPRIT in terms of the minimum singular value of Vandermonde matrices. Using prior estimates for the minimum singular value explains its resolution limit -- when the support of @math consists of multiple well-separated clumps, the noise level that ESPRIT can tolerate scales like @math , where the super-resolution factor @math governs the difficulty of the problem and @math is the cardinality of the largest clump. Our theory is validated by numerical experiments.
Prior works @cite_3 @cite_10 addressed super-resolution from an information theoretic view. They considered the situation where the atoms are located on a grid on @math with spacing @math and the given information consists of noisy continuous Fourier measurements. Both papers derived lower and upper bounds for a min-max error. These results showed that @math minimization is optimal for this discrete model, but it is not computationally feasible. ESPRIT is a polynomial-time algorithm that can be used for the more general off-the-grid-model". The investigation of the optimality of ESPRIT is an interesting future research direction.
@cite_10 @cite_3
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In this paper Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) is developed for spectral estimation with single-snapshot measurement. Stability and resolution analysis with performance guarantee for Single-Snapshot ESPRIT (SS-ESPRIT) is the main focus. In the noise-free case, exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise and under the assumption that the true frequencies are separated by at least two times Rayleigh's Resolution Length, an explicit error bound for frequency reconstruction is given in terms of the dynamic range and the separation of the frequencies. The separation and sparsity constraint compares favorably with those of the leading approaches to compressed sensing in the continuum. Super-resolution refers to the process of recovering the locations and amplitudes of a collection of point sources, represented as a discrete measure, given @math of its noisy low-frequency Fourier coefficients. The recovery process is highly sensitive to noise whenever the distance @math between the two closest point sources is less than @math . This paper studies the fundamental difficulty of super-resolution and the performance guarantees of a subspace method called MUSIC in the regime that @math . The most important quantity in our theory is the minimum singular value of the Vandermonde matrix whose nodes are specified by the source locations. Under the assumption that the nodes are closely spaced within several well-separated clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our estimate is given as a weighted @math sum, where each term only depends on the configuration of each individual clump. This implies that, as the noise increases, the super-resolution capability of MUSIC degrades according to a power law where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the resolution limit of MUSIC. When there are @math point sources located on a grid with spacing @math , the fundamental difficulty of super-resolution can be quantitatively characterized by a min-max error, which is the reconstruction error incurred by the best possible algorithm in the worst-case scenario. We show that the min-max error is closely related to the minimum singular value of Vandermonde matrices, and we provide a non-asymptotic and sharp estimate for the min-max error, where the dominant term is @math .