Fitting smooth surfaces to dense polygon meshes

Recent progress in acquiring shape from range data permits the acquisition of seamless million-polygon meshes from physical models. While dense polygon meshes are an adequate representation for some applications, many users prefer smooth surface representations for reasons of compactness, control, manufacturability, or appearance. In this thesis, we present algorithms and an end-to-end software system for converting dense irregular polygon meshes of arbitrary topology into tensor product B-spline surface patches with accompanying displacement maps. This choice of representation yields a coarse but efficient model suitable for interactive modification and animation and a fine but more expensive model suitable for rendering. The first step in our process consists of interactively painting patch boundaries onto the polygonal surface. In many applications, the placement of patch boundaries is considered part of the creative process and is not amenable to automation. We present efficient techniques for representing, creating and editing curves on dense polygonal surfaces. The second step in our process consists of finding a gridded resampling of each bounded section of the mesh. Our resampling algorithm lays a grid of springs across the polygon mesh, then iterates between relaxing this grid and subdividing it. This grid provides a parameterization for the mesh section, which is initially unparameterized. Our parameterization algorithm is automatic, efficient, and robust, even for complex polygonal surfaces. Prior algorithms have lacked one or more of these properties, making them unusable for dense meshes. Our strategy also provides the user a flexible method to design parameterizations--an ability that previous literature in surface approximation does not address. The third and final step of our process consists of fitting a hybrid of B-spline surfaces and displacement maps to our gridded re-sampling. The displacement map is an image representation of the error between the fitted B-spline surfaces and our spring grid. Since displacement maps are just images our hybrid representation facilitates the use of image processing operators for manipulating the geometric detail of an object. Our resampling and fitting steps are fast enough to surface a million polygon mesh in under 10 minutes--important for an interactive system.