This book summarizes Grenander’s lifelong efforts to represent empirical knowledge of complex real world processes in a mathematical form. The authors characterize the book’s aim as “the formalization of a small set of ideas for constructing the representation of the pattern themselves, which accommodate variability and structure simultaneously”. After a short introduction, the basic paradigm of Bayesian setup of the estimation on the posterior distribution is explained in chapter 2. The next four chapters analyze the role of pattern representations in conditioning structure. Chapters 7 and 8 examine groups of geometric transformations suitable for the representation of geometric objects. Chapter 9 is devoted to random processes and fields on the background spaces – the continuum limits of the finite graphs (e.g. Gaussian processes representing physical processes in the world). Chapters 10 and 11 develop metric space structure of shapes and further extensions of this approach are treated in the next chapter.

Chapters 13 to 15 pass from pure representations of shape to their Bayes estimations and parametric representations. The next two chapters are devoted to infinite dimensional shape covered by a new field of computational anatomy, an emerging discipline introduced by Miller and Grenander at the end of the last century with applications in the biological variability of human anatomy. The last two chapters conclude the inference under the assumption that the posterior distributions describing patterns contain all the information about the underlying regular structure. Markov processes are introduced as models for changes of this structure and random samples are generated via their simulation. The book is designed for a broad public; it contains numerous exercises, examples of applications and an extended bibliography. Appendices outlining some theorems, their proofs and solutions of selected problems are on the website: www.oup.com/uk/booksites/content/0198505701/appendices.