Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing

  • Authors:

  • Xavier Pennec;Nicholas Ayache

  • Affiliations:

  • INRIA, B.P. 93, 2004 route des Lucioles, 06902 Sophia Antipolis Cedex, France. E-mail: Xavier.Pennec@sophia.inria.fr;INRIA, B.P. 93, 2004 route des Lucioles, 06902 Sophia Antipolis Cedex, France. E-mail: Xavier.Pennec@sophia.inria.fr

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 1998

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Abstract

Complex geometric features such as oriented points, lines or 3Dframes are increasingly used in image processing and computer vision.However, processing these geometric features is far more difficult thanprocessing points, and a number of paradoxes can arise. We establishin this article the basic mathematical framework required to avoidthem 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 featuremeasurements?We insist on the importance of an invariance hypothesisfor these definitions relative to a group of transformations thatmodels the different possible data acquisitions. We developgeneral methods to solve these three problems and illustrate themwith 3D frame features under rigid transformations.The first problem has a direct application in the computation of theprior probability of a false match in classical model-based objectrecognition algorithms. We also present experimental results of the twoother problems for the statistical analysis ofanatomical features automatically extracted from 24three-dimensional images of a single patient‘s head. Theseexperiments successfully confirm the importance of the rigorousrequirements presented in this article.