Journal
SIAM JOURNAL ON IMAGING SCIENCES
Volume 9, Issue 1, Pages 412-444Publisher
SIAM PUBLICATIONS
DOI: 10.1137/15M1016552
Keywords
super-resolution; single-molecule imaging; diffraction limit; Rayleigh regularity; sparsity; stable reconstruction via convex optimization
Categories
Funding
- Swiss National Science Foundation [PA00P2_139678]
- Simons Foundation
- NSF [CCF-0963835]
- Swiss National Science Foundation (SNF) [PA00P2_139678] Funding Source: Swiss National Science Foundation (SNF)
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In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by f(c), which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about 1/f(c). The theoretical performance guarantee is complemented with a converse result showing that our simple convex program is nearly optimal. Finally, numerical experiments illustrate our methods.
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