In graduate school and for the first few years as an assistant professor I concentrated on pure mathematics, mainly topology and functional analysis. Around 1979 I was drawn, largely by accident, into signal processing, collaborating with friends at the Naval Research Laboratory who were working on SONAR. I quickly found out that the intersection of the mathematics that I knew and that they knew was nearly empty. For the last 25 years I have been trying to remedy that situation. In writing this book I have tried to gather together in one place the mathematics I wish I had known in 1979 but did not, in the hope that it will be helpful to others undertaking a similar journey.

The situations of interest to us here can be summarized as follows: the data has been obtained through some form of sensing; physical models, often simplified, describe how the data we have obtained relates to the information we seek; there usually isn t enough data and what we have is corrupted by noise and other distortions. Although applications differ from one another in their details they often make use of a common core of mathematical ideas; for example, the Fourier transform and its variants play an important role in many areas of signal and image processing, as do the language and theory of matrix analysis, iterative optimization and approximation techniques, and the basics of probability and statistics. This common core provides the subject matter for this text. Applications of the core material...