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Stochastic geometry : modern research frontiers
Coupier D., Springer International Publishing, New York, NY, 2019. 248 pp. Type: Book (978-3-030135-46-1)
Date Reviewed: Apr 13 2021

In the era BC (“before computers”), the standard way to reason about geometric objects, dating back to Euclid, was a logical proof system based on axioms. This approach is symbolic, dealing with entities such as points and lines. It leverages human spatial intuitions, but makes the computerization of geometric reasoning challenging. Symbolic computation is certainly possible (and was a cornerstone of “good old fashioned artificial intelligence,” or GOFAI), but it is inefficient compared with quantitative computations.

AI has largely turned from GOFAI to quantitative methods (such as deep learning). This book is a summary of methods that can similarly be used to do geometric reasoning without the need for a symbolic representation of the problem. The basic idea is that geometric structures constrain the distribution of results obtained when those structures are sampled randomly, thus the title, Stochastic geometry.

The book is a survey of research frontiers, not an introduction. The first chapter provides a review of classical results, but explores extensions beyond the original problems, and the rest of the book assumes that readers are familiar with existing work in the area. The chapters in the book, by different authors, are introductory lectures from the annual meeting of a French working group focused on stochastic geometry.

Chapter 1 surveys four classic problems in stochastic geometry. Buffon’s needle relates the fundamental geometric constant pi to the probability that a randomly dropped needle intersects one of a set of parallel lines. Bertrand’s paradox highlights the difficulty of defining a geometric problem probabilistically by exhibiting three different ways of defining a random line in the plane that give different answers. Sylvester’s problem seeks the probability that four randomly chosen points constrained to a convex polygon form the vertices of a convex quadrilateral. The bicycle wheel problem quantifies the probability of detecting an anomaly on a circle based on the number of partial random samples of the circle. In each case, the discussion goes beyond the initial problem to show the current research to which it leads.

Chapter 2 develops the intensity approach to evaluating the spatial density of events of interest (such as trees in a forest). Intensity in this case is the local probability of observing an event, which may be conditioned on other events (for example, what is the probability of a tree at (x, y), given that there is one at (3,2)?).

Chapter 3 deals with stochastic approaches to the analysis of images and recognition of geometric structures. It focuses on two areas: comparison of an image with random noise (the a contrario method), and the problem of characterizing texture versus discrete objects over a range of scales.

Chapter 4 develops, in more detail, methods for dealing with textures that exhibit scale invariance, using random fields.

Chapter 5 returns to the theme of point processes (treated in chapter 2) with special attention to a Gibbs point process, which examines geometry-dependent interactions (attraction or repulsion) among the points, compared with the non-interactive baseline of a Poisson point process.

Each chapter has its own references, and there is no overall index. The volume will be of interest to active researchers in stochastic geometry who want a concise summary of current frontiers in the areas that it covers.

Reviewer:  H. Van Dyke Parunak Review #: CR147238 (2108-0194)
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