Nonlinear Inverse Problems in Imaging

Chapter 1

Introduction

We consider a physical system where variables and parameters interact in a domain of interest. Variables are physical quantities that are observable or measurable, and their values change with position and time to form signals. We may express system structures and properties as parameters, which may also change with position and time. For a given system, we understand its dynamics based on underlying physical principles describing the interactions among the variables and parameters. We adopt mathematical tools to express the interactions in a manageable way.

A physical excitation to the system is an input and its response is an output. The response is always accompanied by some form of energy transfer. The input can be applied to the system externally or internally. For the internal input, we may also use the term “source”. Observations or measurements can be done at the outside, on the boundary and also on the inside of the system. For the simplicity of descriptions, we will consider boundary measurements as external measurements. Using the concept of the generalized system, we will introduce the forward and inverse problems of a physical system.

1.1 Forward Problem

The generalized system H in Figure 1.1 has a system parameter p, input x and output y. We first need to understand how they are entangled in the system by understanding underlying physical principles. A mathematical representation of the system dynamics is the forward problem formulation. We ...