Thursday,
October 27, 4:30 p.m. PSA 118
Ben Adcock
Simon Fraser University
Generalized sampling: a new framework for image and signal reconstruction
Abstract
An important problem in many areas of science and engineering is the recovery of an object - e.g. an image or signal - from a collection of its measurements. For example, in Magnetic Resonance Imaging (MRI) one seeks to reconstruct an image from pointwise samples of its Fourier transform. The purpose of this talk is to introduce a new framework for this problem, known as generalized sampling. Unlike more common approaches, this framework is both numerically stable, and therefore robust in the presence of noise or other types of errors, and possesses guaranteed error bounds. As a consequence, if the given image or signal is known a priori to have a good representation in terms of a particular system of functions (e.g. splines, wavelets, piecewise polynomials,…), then one can always compute a good approximation of the image/signal in this system, regardless of the type of measurements.
The standard setup for generalized sampling for the MRI problem assumes that measurements are taken uniformly. However, in practice one often is faced with nonuniform sampling pattens. Nonuniform sampling presents both significant theoretical and practical difficulties. In the final part of this talk I will describe the first steps towards extending the generalized sampling framework to this more challenging case.
