Subdivision Surface based One-Piece Representation

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject\u27s topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications

AMM: Adaptive Multilinear Meshes

We present Adaptive Multilinear Meshes (AMM), a new framework that significantly reduces the memory footprint compared to existing data structures. AMM uses a hierarchy of cuboidal cells to create continuous, piecewise multilinear representation of uniformly sampled data. Furthermore, AMM can selectively relax or enforce constraints on conformity, continuity, and coverage, creating a highly adaptive and flexible representation to support a wide range of use cases. AMM supports incremental updates in both spatial resolution and numerical precision establishing the first practical data structure that can seamlessly explore the tradeoff between resolution and precision. We use tensor products of linear B-spline wavelets to create an adaptive representation and illustrate the advantages of our framework. AMM provides a simple interface for evaluating the function defined on the adaptive mesh, efficiently traversing the mesh, and manipulating the mesh, including incremental, partial updates. Our framework is easy to adopt for standard visualization and analysis tasks. As an example, we provide a VTK interface, through efficient on-demand conversion, which can be used directly by corresponding tools, such as VisIt, disseminating the advantages of faster processing and a smaller memory footprint to a wider audience. We demonstrate the advantages of our approach for simplifying scalar-valued data for commonly used visualization and analysis tasks using incremental construction, according to mixed resolution and precision data streams

Review of the mathematical foundations of data fusion techniques in surface metrology

The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed