7 - Features

Abstract

This chapter discusses the features of forward models and provides the basics of functional analysis, which are useful in many of the presented formulations. Since an image consists of a number of discrete pixels or voxels within which the physical attributes are estimated, it is necessary to discretize the forward models. The discrete forms of equations should be definite, and repeatable in their structure for the same measurement configuration. As such, their inverse is likely to exist. For a linear system, the uniqueness of solution can be ascertained by the argument that if the system has two solutions, corresponding to the same measurement, then both must satisfy the basic equation. A well-conditioned problem has a condition number of nearly unity, i.e. with no significant magnification of measurement error through the inversion process. An ill-conditioned problem, on the other hand, has a large condition number, leading to magnification of error propagation during the solution of the inverse problem. The forward problem maps a set of physical attributes into the measurement space. The measurement functions on the limit can approach an infinite-dimensional space. The measurement space encompasses all possible measurement vectors. Both the physical parameters and measurements are real numbers, and the mapping processes are operators on the functions. The branch of mathematics that deals with these functions and their mapping is called functional analysis.