Feature Adaptive Ray Tracing of Subdivision Surfaces

abstract: Subdivision surfaces have gained more and more traction since it became the standard surface representation in the movie industry for many years. And Catmull-Clark subdivision scheme is the most popular one for handling polygonal meshes. After its introduction, Catmull-Clark surfaces have been extended to several eminent ways, including the handling of boundaries, infinitely sharp creases, semi-sharp creases, and hierarchically defined detail. For ray tracing of subdivision surfaces, a common way is to construct spatial bounding volume hierarchies on top of input control mesh. However, a high-level refined subdivision surface not only requires a substantial amount of memory storage, but also causes slow and inefficient ray tracing. In this thesis, it presents a new way to improve the efficiency of ray tracing of subdivision surfaces, while the quality is not as good as general methods.Dissertation/ThesisMasters Thesis Computer Science 201

Semi-sharp creases on subdivision curves and surfaces

We explore a method for generalising Pixar semi-sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high-degree univariate examples and explore analogous methods for subdivision surfacesThis work was supported by the Engineering and Physical Sciences Research Council [EP/H030115/1].This is the author accepted manuscript and will be under embargo until the 23rd of August 2015. The final version has been published in Computer Graphics Forum here: http://onlinelibrary.wiley.com/doi/10.1111/cgf.12447/abstract

Subdivision surfaces with creases and truncated multiple knot lines

We deal with subdivision schemes based on arbitrary degree B-splines. We focus on extraordinary knots which exhibit various levels of complexity in terms of both valency and multiplicity of knot lines emanating from such knots. The purpose of truncated multiple knot lines is to model creases which fair out. Our construction supports any degree and any knot line multiplicity and provides a modelling framework familiar to users used to B-splines and NURBS systems

Compression for Smooth Shape Analysis

Most 3D shape analysis methods use triangular meshes to discretize both the shape and functions on it as piecewise linear functions. With this representation, shape analysis requires fine meshes to represent smooth shapes and geometric operators like normals, curvatures, or Laplace-Beltrami eigenfunctions at large computational and memory costs. We avoid this bottleneck with a compression technique that represents a smooth shape as subdivision surfaces and exploits the subdivision scheme to parametrize smooth functions on that shape with a few control parameters. This compression does not affect the accuracy of the Laplace-Beltrami operator and its eigenfunctions and allow us to compute shape descriptors and shape matchings at an accuracy comparable to triangular meshes but a fraction of the computational cost. Our framework can also compress surfaces represented by point clouds to do shape analysis of 3D scanning data

Subdivision surface fitting to a dense mesh using ridges and umbilics

Fitting a sparse surface to approximate vast dense data is of interest for many applications: reverse engineering, recognition and compression, etc. The present work provides an approach to fit a Loop subdivision surface to a dense triangular mesh of arbitrary topology, whilst preserving and aligning the original features. The natural ridge-joined connectivity of umbilics and ridge-crossings is used as the connectivity of the control mesh for subdivision, so that the edges follow salient features on the surface. Furthermore, the chosen features and connectivity characterise the overall shape of the original mesh, since ridges capture extreme principal curvatures and ridges start and end at umbilics. A metric of Hausdorff distance including curvature vectors is proposed and implemented in a distance transform algorithm to construct the connectivity. Ridge-colour matching is introduced as a criterion for edge flipping to improve feature alignment. Several examples are provided to demonstrate the feature-preserving capability of the proposed approach

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

Unifying Geometry and Mesh Adaptive Refinement Using Loop Subdivision

RÉSUMÉ Cette thèse présente une nouvelle approche pour le raffinement de trois types de maillages : courbes, surfaces triangulaires et maillages tétraédriques tridimensionnels. Cette approche utilise des représentations par subdivisions afin de définir, modifier, analyser et visualiser des modèles géométriques de topologie arbitraire pour les applications de simulation numérique. Les représentations par subdivisions sont générées à l’aide des subdivisions de Loop. Après avoir étudié les inconvénients du manque de flexibilité dans le contrôle des niveaux de détails et du manque de précision dans les représentations de modèles géométriques utilisant les subdivisions itératives, approximatives et non-uniformes pour se rapprocher des modèles simulés, nous introduisons une nouvelle méthode de subdivision adaptative pour le raffinement de maillages. Cette méthode de raffinement à un seul niveau a été développée afin de supporter les subdivisions adaptatives pour les trois types de maillages. Cette méthode évite le stockage par hiérarchie et les problèmes d’assemblage rencontrés durant la génération des maillages multi-résolutions par subdivisions, surtout pour les maillages tétraédriques. La mise en œuvre de subdivisions pour les maillages adaptatifs tétraédriques amène deux innovations : la configuration de forme de fractionnement des tétraèdres et l’amélioration de la paramétrisation des surfaces de subdivision. La combinaison naturelle de ces deux innovations permet la génération par subdivision de maillages multi-résolutions tétraédriques dont les surfaces frontières sont exactement sur les limites de subdivision. Notre recherche contient cinq parties. Premièrement, nous développons un schéma de Loop pour la subdivision des solides, lequel permet d’intégrer le fractionnement topologique des arêtes avec le lissage géométrique des surfaces frontières. Deuxièmement, nous fusionnons les raffinements adaptatifs avec les techniques de subdivision, ce qui permet la subdivision adaptive complète du maillage tout en ayant les surfaces frontières projetées sur les limites de subdivision. Troisièmement, nous étudions et comparons des techniques existantes de paramétrisation des surfaces de subdivision, ce qui permet d’obtenir directement la limite de subdivision de toutes positions arbitraires sur les surfaces de subdivision de Loop. Quatrièmement, nous construisons les règles de création des sommets fixes et des arêtes vives du schéma de subdivision de Loop pour les modèles solides, ce qui permet de préserver les caractéristiques anguleuses des surfaces frontières des maillages tétraédriques. Finalement, nous utilisons un critère de qualité des maillages pour valider nos résultats et nous présentons la performance des calculs en ce qui a trait à la modélisation des solides.----------ABSTRACT In this thesis, we present a new refinement approach on three types of meshes: curves, triangular surfaces and 3D tetrahedral meshes. This approach utilizes subdivision-based representations to create, modify, analyze and visualize geometric models with arbitrary topology for numerical simulation applications. The subdivision-based representations are generated by utilizing Loop subdivisions. After studying the disadvantage of lack of flexibility in controlling LODs (Level Of Details) and accuracy in representing geometric models by using the non-uniform approximating subdivision iterations to approach simulated models, we introduce adaptive subdivisions in our refinement work. We develop a single-level refinement method to support adaptive subdivisions on the three types of meshes. This single-level method eliminates the hierarchy storage and the stitching issues encountered during the generation of multi-resolution subdivision meshes, especially 3D tetrahedral meshes. The implementation of adaptive tetrahedral mesh subdivisions brings up two innovations: the configuration of tetrahedron split patterns and the improvement in subdivision surface parameterizations. The natural combination of these two innovations fulfills generating multi-resolution subdivision tetrahedral meshes, whose boundary surfaces lie exactly on their subdivision limits. Our research work includes five parts. Firstly, we develop the Loop-based solid subdivision scheme, which permits integrating edge-based topological splits with geometrical smoothing on boundary surfaces. Secondly, we merge subdivision techniques with adaptive refinements with, which permits whole meshes to be adaptively subdivided and boundary meshes to be projected to their subdivision limits. Thirdly, we study and compare the existing subdivision surface parameterization techniques, which eventually permits obtaining the limit subdivision of any arbitrary position on Loop subdivision surfaces. Fourthly, we complete vertex and edge crease creation rules of the Loop-based solid subdivision scheme, which permits preserving sharp features on boundary surfaces of 3D tetrahedral meshes. Finally, we use a mesh quality evaluator to validate our results and we evaluate system performance in the context of solid modeling