Conferencia Luis Santalo
 |
 IX Escuela Santaló - CIMPA 2017 |
|---|---|
|
Departamento de Matemática
|
Conferencia Luis SantalóSuper-resolution by means of Beurling minimal extrapolationJohn Benedetto University of Maryland, USA jjb@math.umd.edu
We address the super-resolution question: Given spectral data defined on a finite set of d-dimensional
multi-integers; of all complex Radon measures on the d-dimensional torus, whose Fourier transform equals
this data, does there exist exactly one with minimal total variation? We first note that this is a mathematical
formulation of a large class of super-resolution problems that arises in image processing, that it
generalizes some fundamental problems in compressed sensing, and that it has wide ranging applications
in other fields.
We prove a theorem that has quantitative implications about the possibility and impossibility of constructing
such a unique measure. Our method introduces the notion of an admissibility range that fundamentally
connects Beurling’s theory of minimal extrapolation with the Candes and Fernandez-Granda theory of
super-resolution. The method is also well -suite for the construction of explicit examples.
This is part of an on-going collaboration with Weilin Li.
|
