This chapter introduces some of the fundamental notions from stochastic geometry. Background information from convex geometry is provided as far as this is required for the applications to stochastic geometry. First, the necessary definitions and concepts related to geometric point processes and from convex geometry are provided. These include Grassmann spaces and invariant measures, Hausdorff ...

Given independent random points X1, . . . , Xn ∈ Rd with common probability distribution ν, and a positive distance r = r(n) > 0, we construct a random geometric graph Gn with vertex set {1, . . . , n} where distinct i and j are adjacent when ‖Xi − Xj‖ ≤ r. Here ‖.‖may be any norm on Rd, and ν may be any probability distribution on Rd with a bounded density function. We consider the chromatic n...

of Thesis presented to COPPE/UFRJ as a partial fulfillment of the requirements for the degree of Doctor of Science (D.Sc.) DISTANCES IN RANDOM GEOMETRIC GRAPHS AND THEIR APPLICATIONS TO THE LOCALIZATION PROBLEM IN SENSOR NETWORKS Rodrigo Simões Camara Leão

We describe a method for modelling and locating deformable objects using a combination of global and local shape models. An object is represented as a set of patches together with a geometric model of their relative positions. The geometry is modelled with a global pose and linear shape model, together with a Markov Random Field (MRF) model of local displacements from the global model. Matching...