ABSTRACT
The goal of shape modeling is to seek mathematical representations to not only capture the intrinsic morphologies of various shapes, but to also account for their variability. Formally, part of pattern theory, whose formalism is to a large extent due to Grenander in the 1970s (Grenander, 1976, 1978), seeks to unravel and to quantify the structure of patterns present in an image. A considerable research activity has yielded a wealth of perspectives and approaches, each with its advantages and limitations. The earliest approach to modeling structure was based on rigid models, and fell short of accounting for the inherent variability, as evidenced by biological and anatomical shapes, e.g., a beating heart or a stomach shape. More flexible and adapted models, namely, deformable templates, were later proposed by Grenander (1996), which addressed the statistical variability of shapes. These models specifically looked upon object boundaries as a set of sites (carefully selected landmarks) joined by arcs (or segments) whose spatial attributes were captured by probability distributions. This mathematically elegant and conceptually sound approach led to inferences on infinite dimensional spaces with an often prohibitive computational demand.
