Implicit shapes: Reconstruction and explicit transformation

In this thesis, I present solutions to two specific problems associated with implicit representations composed of radial basis functions: Automated 3D content creation. Constructing an implicit function to represent a desired geometric shape is a non-trivial task. I present a method for generating an implicit surface from discrete, image-based range data that is noisy and non-uniform. Explicit mappings for implicit transformations. A shape morph describes the transformation that changes one shape into another. In an implicit morph, an implicit function describes the changing shape. Explicit information about how points on the surface move as the shape transforms is not maintained in an implicit morph. I present a method for tracking points on the changing surface in an implicit morph.; In reconstructing an implicit function from image-based range data, I apply a technique called volumetric regularization to obtain a smooth three dimensional implicit function. Regularization operates by minimizing an energy functional that consists of a data fitness term that measures how far the observed data values are from the reconstructed function and a smoothness assumption that describes how smooth the reconstructed function should be. An implicit function formulated as the summation of radial basis functions minimizes such an energy functional. In this thesis, I address a number of practical issues regarding the construction of the implicit function from image-based range data. These issues include the use of an energy-minimizing radial basis function, sampling of the image-based data set, validation metrics to guide reconstruction parameters, and development of an anisotropic basis function to preserve sharp edges.; Reconstruction of time-varying data sets is an important area of application. A morph describes how one shape geometrically changes into another, and can thus represent time-varying data. Implicit representations are especially popular for shape morphing because they gracefully handle changes in topology, but explicit correspondences between the shapes are lost. I construct an explicit mapping between the shapes in an implicit morph by solving two partial differential equations (PDE) defined on the morphing surface. Solving the first PDE generates characteristic paths across the morphing sequence, while solving the second PDE explicitly marks the paths. The collection of paths form the explicit mapping function. By solving PDEs on the morphing surface, the resulting explicit correspondences are not arbitrary, but intrinsic, to the morphing surface.