Inverse Problems

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

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

(2.1)

of other, more basic, underlying quantities $\underline{z},$ where 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)

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.