Browse Theses

This thesis presents a complete framework for the description of implicit surfaces and solids. We propose piecewise representations that are based on decomposition models and constructed using methods adapted to the geometry of objects. These models are general, in the sense that they can represent arbitrary shapes, and support variable precision, permitting approximations at any desirable resolution.

The contribution of our work consists of: (i) an original characterization of the implicit description of shapes, (ii) a new method for generating a smooth implicit function corresponding to a solid object, and (iii) two new piecewise representation schemes based on a multiscale decomposition of the implicit function and on an adapted simplicial decomposition of the domain of the implicit function.

We characterize the implicit model through an analysis of the implicit function. We show that the skeleton of a shape and the tubular neighborhood of its boundary are dual structures that relate an object with the space in which it is embedded.

We develop a method that allows the generation of smooth implicit functions from the characteristic function of an object. It employs multiresolution edge detection and reconstruction using dyadic wavelets.

We introduce a functional decomposition model based on B-spline scaling functions that generate nested multiscale approximating spaces. The Laplacian transform is employed to compute a pyramid in terms of these B-spline bases.

We introduce a spatial decomposition model based on adapted simplicial subdivision. Physics based deformation adapts to the boundary of the object a mesh derived from this simplicial complex.

Some of the applications of these methods include: approximate conversion between volumetric, implicit and parametric representations; surface rendering; volume visualization; and animation of implicit objects.

In summary, the relevance of this thesis is twofold: it provides a conceptual, as well as a practical scheme for piecewise shape description. The piecewise implicit representations that we have developed are effective and efficient. They capture the spatial features of objects using composite structures that are constructed from simple elements.