Analysis and Parameterization of Triangulated Surfaces

This dissertation deals with the analysis and parameterization of surfaces represented by triangle meshes, that is, piecewise linear surfaces which enable a simple representation of 3D models commonly used in mathematics and computer science. Providing equivalent and high-level representations of a 3D triangle mesh M is of basic importance for approaching different computational problems and applications in the research fields of Computational Geometry, Computer Graphics, Geometry Processing, and Shape Modeling. The aim of the thesis is to show how high-level representations of a given surface M can be used to find other high-level or equivalent descriptions of M and vice versa. Furthermore, this analysis is related to the study of local and global properties of triangle meshes depending on the information that we want to capture and needed by the application context. The local analysis of an arbitrary triangle mesh M is based on a multi-scale segmentation of M together with the induced local parameterization, where we replace the common hypothesis of decomposing M into a family of disc-like patches (i.e., 0-genus and one boundary component) with a feature-based segmentation of M into regions of 0-genus without constraining the number of boundary components of each patch. This choice and extension is motivated by the necessity of identifying surface patches with features, of reducing the parameterization distortion, and of better supporting standard applications of the parameterization such as remeshing or more generally surface approximation, texture mapping, and compression. The global analysis, characterization, and abstraction of M take into account its topological and geometric aspects represented by the combinatorial structure of M (i.e., the mesh connectivity) with the associated embedding in R^3. Duality and dual Laplacian smoothing are the first characterizations of M presented with the final aim of a better understanding of the relations between mesh connectivity and geometry, as discussed by several works in this research area, and extended in the thesis to the case of 3D parameterization. The global analysis of M has been also approached by defining a real function on M which induces a Reeb graph invariant with respect to affine transformations and best suited for applications such as shape matching and comparison. Morse theory and the Reeb graph were also used for supporting a new and simple method for solving the global parameterization problem, that is, the search of a cut graph of an arbitrary triangle mesh M. The main characteristics of the proposed approach with respect to previous work are its capability of defining a family of cut graphs, instead of just one cut, of bordered and closed surfaces which are treated with a unique approach. Furthermore, each cut graph is smooth and the way it is built is based on the cutting procedure of 0-genus surfaces that was used for the local parameterization of M. As discussed in the thesis, defining a family of cut graphs provides a great flexibility and effective simplifications of the analysis, modeling, and visualization of (time-depending) scalar and vector fields; in fact, the global parameterization of M enables to reduce th

Completing unknown portions of 3D scenes by 3D visual propagation

Institute of Perception, Action and BehaviourAs the requirement for more realistic 3D environments is pushed forward by the computer {graphics | movie | simulation | games} industry, attention turns away from the creation of purely synthetic, artist derived environments towards the use of real world captures from the 3D world in which we live. However, common 3D acquisition techniques, such as laser scanning and stereo capture, are realistically only 2.5D in nature - such that the backs and occluded portions of objects cannot be realised from a single uni-directional viewpoint. Although multi-directional capture has existed for sometime, this incurs additional temporal and computational cost with no existing guarantee that the resulting acquisition will be free of minor holes, missing surfaces and alike. Drawing inspiration from the study of human abilities in 3D visual completion, we consider the automated completion of these hidden or missing portions in 3D scenes originally acquired from 2.5D (or 3D) capture. We propose an approach based on the visual propagation of available scene knowledge from the known (visible) scene areas to these unknown (invisible) 3D regions (i.e. the completion of unknown volumes via visual propagation - the concept of volume completion). Our proposed approach uses a combination of global surface fitting, to derive an initial underlying geometric surface completion, together with a 3D extension of nonparametric texture synthesis in order to provide the propagation of localised structural 3D surface detail (i.e. surface relief). We further extend our technique both to the combined completion of 3D surface relief and colour and additionally to hierarchical surface completion that offers both improved structural results and computational efficiency gains over our initial non-hierarchical technique. To validate the success of these approaches we present the completion and extension of numerous 2.5D (and 3D) surface examples with relief ranging in natural, man-made, stochastic, regular and irregular forms. These results are evaluated both subjectively within our definition of plausible completion and quantitatively by statistical analysis in the geometric and colour domains