Superresolution without separation
Abstract
This article provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of $M$ shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from $2M+1$ observations. This recovery can be achieved by solving a weighted version of basis pursuit over a continuous dictionary. Our analysis shows that $\ell_1$-based methods enjoy the same separation-free recovery guarantees as polynomial root finding techniques, such as de Prony’s method or Vetterli’s method for signals of finite rate of innovation. Our proof techniques combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- May 29, 2017
- Source ID
- 10.1093/imaiai/iax006
Entities
People
- Benjamin Recht
- Elina Robeva
- Geoffrey Schiebinger
Organizations
- Air Force Office of Scientific Research
- Defense Advanced Research Projects Agency
- Lawrence Berkeley National Laboratory
- National Science Foundation
- Office of Naval Research
- University of California