# Inverse Problems

Chapter

First Online:

## Abstract

An understanding of forward and inverse problems [12, 301] lies at the heart of any large estimation problem.
Abstractly, most physical systems can be defined or parametrized in terms of a set of attributes, or unknowns, from which other attributes, or measurements, can be inferred. In other words, the quantities \(\underline{m}\) which we measure are some mathematical function of other, more basic, underlying quantities \(\underline{z},\) where for the deterministic or stochastic cases, respectively. Normally \(\underline{z},\) is an ideal, complete representation of the system: detailed, noise-free, and regularly structured (e.g.,pixellated), whereas the measurements \(\underline{m},\) are incomplete and approximate: possibly noise-corrupted, irregularly structured, limited in number, or somehow limited by the physics of the measuring device.

$$\underline{m} = f(\underline{Z})$$

(2.1)

*f*may be deterministic or stochastic. In the special case when*f*is linear, a case of considerable interest to us, then (2.1) may be expressed as$$\underline{m} = C\underline{Z} \quad {\rm or} \quad \underline{m} = C\underline{Z} + \underline{v}$$

(2.2)

## Keywords

Inverse Problem Condition Number Data Fusion Prior Model Forward Problem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer New York 2011