This monograph represents a systematic introduction to the general theory of the statistics of manifolds, with special emphasis on the manifolds of shapes. The statistics of shapes is a relatively new area of research, with only 30 to 40 years of history. Most of the current literature on the topic is focused on parametric models, with little attention given to nonparametric models. In this book, however, the emphasis is on nonparametric analysis of shape distributions on a given manifold, a minimizer of the Fréchet function. The book is primarily suitable for graduate students in computer science who do research in computer vision, but the last two chapters could also be useful for physicists and researchers dealing with modeling.

Following an introductory chapter, chapter 2 establishes six core examples of various entities, which are further analyzed in subsequent chapters. As a natural addition to the “studied objects” defined in chapter 2, chapter 3 forms the conceptual fabric of the book with the main definitions and concepts used in the rest of the text. For example, this chapter provides a clear definition of the Fréchet function of probability distribution as an integral on the expected square distance from a point on a manifold. That definition is used in the analysis on shapes described in chapters 7 to 12.

Chapter 4 deals with extrinsic inferences on differentiable manifolds. It is well known that the extrinsic mean is a projection of the Euclidean mean over a manifold. However, in this chapter, the authors expand this definition to show that there is a unique mean if and only if there is a unique projection.

Chapter 5 steps back to complete the Fréchet analysis on Riemannian manifolds, using the metric of the geodesic distance. Chapter 6 introduces some extra shapes used later in the book, including similarity shapes, reflection similarity shapes, affine shapes, and projective shapes. Chapters 7 to 12 discuss the geometry of each of these shapes, and analyze them using the methods outlined in chapter 5. Chapters 8 to 11 discuss the planar shape space in detail, including Kendall type shape spaces and affine shape spaces. Chapter 12 presents methodologies for the analysis of projective spaces applicable to machine vision algorithms.

The most interesting material in the book is found in the last two chapters, 13 and 14, which deal with functional inference. I found these most valuable because they go beyond the scope of machine vision, introducing density estimation, classification, and regression, all based on nonparametric Bayes methodology.

In the end, I have to say that this is an excellent text that will benefit many students in computer science, mathematics, and physics. However, I must stress that a proper background in differential geometry and differential calculus is needed to fully understand the material, as well as some graduate learning in advanced statistics. A significant plus of the book is the library of MATLAB codes and datasets available for download from the authors’ site.