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<p><b>Abstract</b>—In this paper, we introduce a novel geometric shape modeling scheme which allows for representation of global and local shape characteristics of an object. Geometric models are traditionally well-suited for representing global shapes without local detail. However, 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 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 pedal point vary and use 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 develop a fast numerical iterative algorithm for shape recovery from image data using this geometric shape modeling scheme. The algorithm involves the Levenberg-Marquardt (LM) method in the outer loop for solving the global parameters and the Alternating Direction Implicit (ADI) method in the inner loop for solving the local parameters of the model. The combination of the global and local scheme leads to an efficient numerical solution to the model fitting problem. We demonstrate the applicability of this modeling scheme via examples of shape synthesis and shape estimation from real image data.</p>
Geometric models, snakes, pedal curves/surfaces, alternating direction implicit method, Levenberg-Marquardt method.

Y. Guo and B. C. Vemuri, "Snake Pedals: Compact and Versatile Geometric Models with Physics-Based Control," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 22, no. , pp. 445-459, 2000.
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