# Residue and Sampling Techniques in Deconvolution

• Stephen Casey
• David Walnut
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

In this chapter we will study techniques for solving the following general problem, which we will refer to as the Multisensor Deconvolution Problem or MDP: Given a collection of compactly supported distributions, $$\{ {\mu _i}\} _{i = 1}^m \subseteq \mathcal{E}\prime ({\mathbb{R}^d})$$ how can we recover an arbitrary function $$f \in {C^\infty }({\mathbb{R}^d})$$ from the data $$\left\{ {{S_i}} \right\}_{i = 1}^m = \left\{ {f*{\mu _i}} \right\}_{i = 1}^m?$$ The solution technique that we consider involves the construction of deconvolvers. These come in a variety of types but are essentially a collection of distributions which (1) depend only on the convolvers $$\left\{ {{\mu _i}} \right\}_{i = 1}^m$$ and (2) allow for the solution of the MDP with only simple linear operations on the data $$\left\{ {{\mu _i}} \right\}_{i = 1}^m$$

### Keywords

Convolution Dition Remote Sensor Deconvolution