Deformable models: Software implementation based on mathematical fundamentals

Abstract

This paper is devoted to the mathematical approaches and software implementations in the domain of deformable models. Based on a general result of calculus of variations, we deduce the Euler-Lagrange-Poisson (ELP) Equation associated to the 2D deformable models, with relevant applications in medical imaging; the optimal solution of these models is found among the solutions (named extremals) of ELP Equation. Then, we perform a discretization of ELP Equation, using the method of finite differences, in order to obtain an approximation of the optimal solution for static and dynamic deformable models; this discrete solution is given by a matriceal system, which involves the so-called stiffness matrix. Finally, based on the good quality of the results obtained with this method, providing the formal definitions required by the image processing standardization, we extend the set of standard image processing operators with the "snake operator", taking into account its performances in the domain of feature extraction and object recognition in medical imaging.