Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing

Complex geometric features such as oriented points, lines or 3D frames are increasingly used in image processing and computer vision. However, processing these geometric features is far more difficult than processing points, and a number of paradoxes can arise. We establish in this article the basic mathematical framework required to avoid them and analyze more specifically three basic problems: (1) what is a random distribution of features, (2) how to define a distance between features, (3) and what is the “mean feature” of a number of feature measurements? We insist on the importance of an invariance hypothesis for these definitions relative to a group of transformations that models the different possible data acquisitions. We develop general methods to solve these three problems and illustrate them with 3D frame features under rigid transformations. The first problem has a direct application in the computation of the prior probability of a false match in classical model-based object recognition algorithms. We also present experimental results of the two other problems for the statistical analysis of anatomical features automatically extracted from 24 three-dimensional images of a single patient's head. These experiments successfully confirm the importance of the rigorous requirements presented in this article.