Snake pedals: Active geometric models for shape modeling and recovery

Modeling shapes, be it in 2D or 3D, is a fundamental constitutent of computer vision and computer graphics. Shape modeling techniques have been widely used in shape synthesis, shape reconstruction, recognition, motion tracking, and so on. In this dissertation, we introduce a compact and versatile geometric shape modeling scheme that can model a large class of shapes and is amenable to stable and efficient numerical implementations. Geometric models are traditionally well suited for representing global shapes but not the local detail. However, in this thesis we propose a powerful geometric shape modeling scheme which allows for the representation of global shapes with local detail and permits model shaping as well as topological changes via physics-based control. The proposed geometric models are a blend of geometric and physics-based models and are in spirit "similar" to the now popular deformable superquadric models but differ from them considerably in the expressiveness and numerical stability leading to greater applicability.; The proposed modeling scheme consists of representing shapes by pedal curves and surfaces--pedal curves/surfaces are the loci of the foot of perpendiculars to the tangents of a fixed curve/surface from a fixed point called the pedal point. By varying the location of the pedal point, one can synthesize a large class of shapes which exhibit both local and global deformations. We introduce physics-based control for shaping these geometric models by letting the control point vary and use a dynamic spline, that is, a snake, to represent the position of this varying pedal point. The model dubbed as a "snake pedal" allows for interactive manipulation via forces applied to the snake. We extend the model to automatically cope with topological changes by introducing the concept of hybrid geometric active contour/surface models wherein the traditional snake in the snake pedal is replaced with a geometric active contour which is realized via a level-set implementation.; Efficient numerical algorithms for shape recovery from image data using the proposed snake pedal model are presented. These algorithms involve novel mathematical derivations that lead to efficient numerical solutions of the model fitting problem. We demonstrate the applicability of this modeling scheme via shape synthesis examples and shape estimation examples from image data.