Chapter 1 MULTIRESOLUTION MODELING OF THREE - DIMENSIONAL SHAPES

Manipulation of models of complex three-dimensional shapes is common in computer graphics and virtual reality, computer-aided-design, geographic information systems, medicine, etc. In this chapter, we deal with geometric models based on spatial decompositions, also known as geometric meshes. A mesh is an aggregate of a finite number of basic entities, called cells. When a mesh is used to describe the shape of an object, each cell represents a portion of the object. The resolution of a mesh is related to the density of its cells, and it is a fundamental parameter for the accuracy of the object representation. Accurate representations need high resolutions, and, thus, a high number of cells, which leads to high costs for managing the mesh. Three-dimensional models are either fully synthetic or obtained from sampling physical objects. In both cases, the models produced by modern tools for either 3D acquisition and reconstruction, or computer-aided shape design, become more and more accurate, with larger and larger sizes. Models at high resolution can provide very accurate object representations, but they can easily become too large to be effectively used in tasks such as rendering, recognition, classification, collision detection, and manipulation planning. This problem cannot be simply solved by using more powerful machines, since, while hardware and software technology improves, the resolution of geometric models that we are able to construct increases at an even faster rate. However, most applications tasks do not need a high resolution in each parts of an object, at all times. Thus, we can reduce memory and computational costs