An Overview of Some Mathematical Methods for Medical Imaging

Abstract

This chapter based on a series of lecture notes proposes an overview of mathematical concepts commonly used in medical physics and in medical imaging. One goal is to summarize basic and, in principle, well-known tools such as the continuous and discrete Fourier transforms, Shannon’s sampling theorem, or the singular value decomposition of a matrix or operator. Tomographic reconstruction is covered in another chapter. Due to differences in vocabulary, notations and context, it is often difficult to grasp the links between inverse problems as different as image reconstruction in emission tomography,¹⁸ photoacoustic tomography,²⁵ image registration, ¹² or simply data fitting in the analysis of time sequences. The second goal of this chapter is therefore to give a synthetic and coherent view of the structure of the inverse problems encountered in this field, by rigorously defining concepts such as ill-posedness, stability, and regularization. This will allow a better understanding of the rationale behind data processing methods described in the literature.