Fieguth Statistical Image Processing and Multidimensional Modeling

1 Introduction Images are all around us! Inexpensive digital cameras, video cameras, computer web-cams, satellite imagery, and images off the Internet give us access to spatial imagery of all sorts. The vast majority of these images will be of scenes at human scales — pictures of animals / houses / people / faces and so on — relatively complex images which are not well described statistically or mathematically. Many algorithms have been developed to process / denoise / compress / segment such images, described in innumerable textbooks on image processing [36, 54, 143, 174, 210], and briefly reviewed in Appendix C. Somewhat less common, but of great research interest, are images which do allow some sort of mathematical characterization, and to which standard image-processing algorithms may not apply. In most cases we do not necessarily have images here, per se, but rather spatial datasets, with one or more measurements taken over a two-or higher-dimensional space. There are many important problems falling into this latter group of scientific images , and where this text seeks to make a contribution. Although a great deal of research has been applied to scientific images, in most cases the resulting methods are not well documented in common textbooks, such that many experienced researchers will be unfamiliar with the use of the FFT method (Section 8.3) or of posterior sampling (Chapter 11), for example. The goal, then, of this text is to address methods for solving multidimensional inverse problems. In particular, the text seeks to avoid the pitfall of being entirely mathematical / theoretical at one extreme, or primarily applied / algorithmic on the other, by deliberately developing the basic theory (Part I), the mathematical mod-2 1 Introduction