A Markov Random Field Approach to Multi-scale Shape Analysis

Abstract

With a mind towards achieving means of image comprehension by computer, we intend to convey the benefits of (1) characterizing the geometry of object complexes in the real world as contrasted with the geometric conformation of their images, and (2) describing populations of object complexes probabilistically. We show how a multi-scale description of inter-scale residues of geometric features provides a set of efficiently trainable probability distributions via a Markov random field approach, and specifies the location and scale of geometric differences between populations. These ideas and methods are illustrated using medial representations for 3D objects, depending on their properties (1) that local descriptors have an associated coordinate frame and distance metric, and (2) that continuous geometric random variables can be used to describe all members of a population of object complexes with a common structure and the variation among those members. We demonstrate with respect to the following object-complex-relative discrete scale levels: a whole object complex, individual objects, various object parts and sections, and fine boundary details. Using this illustrative framework, we show how to build Markov random field (MRF) models on the geometry scale space based on the statistics of shape residues across scales and between neighboring geometric entities at the level of locality given by its scale. In this paper, we present how to design and estimate MRF models on two scale levels, namely boundary displacement and object sections.