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This paper studies stable recovery of a collection of point sources from its noisy M+1 low-frequency Fourier coefficients. We focus on the super-resolution regime where the minimum separation of the point sources is below 1/M. We propose a separated clumps model where point sources are clustered in far apart sets, and prove an accurate lower bound of the Fourier matrix with nodes restricted to the source locations. This estimate gives rise to a theoretical analysis on the super-resolution limit of the MUSIC algorithm. 

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M 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 is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like SRF2λ-2×Noise, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and λ is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of M in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments. Keywords: Super-resolution, subspace methods, ESPRIT, stability, uncertainty principle. 

Unmanned vehicles, equipped with radiation detection sensors, can serve as a valuable aid to personnel responding to radiological incidents. The use of tele-operated ground vehicles avoids human exposure to hazardous environments, which in addition to radioactive contamination, might present other risks to personnel. Autonomous unmanned vehicles using algorithms for radioisotope classification, source localization, and efficient exploration allow these vehicles to conduct surveys with reduced human supervision allowing teams to address larger areas in less time. This work presents systems for autonomous radiation search with results presented in several proof-of-concept demonstrations.