About the book
Images are all around us! The proliferation of low-cost, high-quality imaging devices has led to an explosion in acquired images. Where these images are acquired from a microscope, telescope, satellite, or medical imaging device there is a statistical image processing task: the inference of something --- an artery, a road, a DNA marker, an oil spill --- from imagery, possibly noisy, blurry, or incomplete.
A great many textbooks have been written on image processing, however this book does not so much focus on {\em images}, per se, rather on spatial data sets, with one or more measurements taken over a two or higher dimensional space, and to which standard image-processing algorithms may not apply.
There are many important data analysis methods, developed in this text, for such statistical image problems. Examples abound throughout remote sensing (satellite data mapping, data assimilation, climate-change studies, land use), medical imaging (organ segmentation, anomaly detection), computer vision (image classification, segmentation), and other 2D / 3D problems (biological imaging, porous media).
The goal, then, of this text is to address methods for solving multidimensional statistical problems. The text strikes a balance between mathematics and theory on the one hand, versus applications and algorithms on the other, by deliberately developing the basic theory (Part I), the mathematical modeling (Part II), and the algorithmic and numerical methods (Part III) of solving a given problem. The particular emphases of the book include inverse problems, multidimensional modeling, random fields, and hierarchical methods.
This text has as its origins the graduate course in Stochastic Image Processing which I taught for the first time as a young professor in 1997. Looking back on my rough notes from that time, the course must have been a near impenetrable disaster for the graduate students enrolled, with a long list of errors, confusions, and bad notation.
With every repetition the course improved, with significant changes to notation, content, and flow. However, at the same time that a cohesive, large-scale form of the course took shape, the absence of any textbook covering this material became increasingly apparent. There are countless texts on the subjects of image processing, Kalman filtering, and signal processing, however precious little for random fields or spatial statistics. The few texts that do cover Gibbs models or Markov random fields tend to be highly mathematical research monographs, not well suited as a textbook for a graduate course.
More than just a graduate course textbook, this text was developed with the goal of being a useful reference for graduate students working in the areas of image processing, spatial statistics, and random fields. In particular, there are many concepts which are known and documented in the research literature, which are useful for students to understand, but which do not appear in many textbooks. This perception is driven by my own experience as a PhD student, which would have been considerably simplified if I had had a text accessible to me addressing some of the following gaps:
- FFT-based estimation
- A nice, simple, clear description of multigrid
- The inference of dynamic models from cross-statistics
- A clear distinction and relationship between squared and unsquared kernels
- A graphical summary relating Gibbs and Markov models
To facilitate the use of this textbook and the methods described within it, this web site makes available much of the code which I developed for this text, allowing interested readers to reproduce a number of the figures and examples, and also giving readers a code base which they can edit to use in other projects.