In this thesis, we study the skin surface as a new paradigm for the deformable surfaces. The skin surface handles deformation and topology changes robustly, supported by the un-derlying structure of Delaunay triangulations and alpha shapes. The surface serves as a deformable manifold in various disciplines, such as computer graphics, molecular modeling, and mechanical engineering. We develop an algorithm and software for the construction and visualization of the skin surface in 3D in various ways, namely, a parametric representation, static and dynamic triangulations. The triangulation algorithm is guaranteed to terminate with a high quality triangle mesh. In our investigation, geometric properties of the skin serve as the foundation of our proofs and insights for the algorithms. The proofs can be extended to the meshing of other low degree surfaces, such as NURBS. The surfaces created by the software bring stability in finite element methods and visualization of molecular structures to scientists.

In this paper, we present a novel method for efficient 3D model comparison. The method is designed to match highly deformed models through capturing two types of information. First, we propose a feature point extraction algorithm, which is based on “Level Set Diagram”, to reliably capture the topological points of a general 3D model. These topological points represent the skeletal structure of the model. Second, we also capture both spatial and curvature information, which describes the global surface of a 3D model. This is different from traditional topological 3D matching methods that use only low-dimension local features. Our method can accurately distinguish different types of 3D models even if they have similar topology. By applying the bipartite graph matching technique, our method can achieve a high precision of 0.54 even at a recall rate of 1.0 as demonstrated in our experimental results. 1.

This paper presents our recent results in the field of surface representation based on topological coding. In particular, we investigate a possible way to adapt to discrete surface models some theoretical concepts as Morse theory and Reeb graphs which bases on differential topology. Starting from a triangulated surface, our aim is to code the relationship among critical points of the height function associated to the mesh. We named Extended Reeb Graph (ERG) the graph representation which can handle also degenerate critical points. The ERG gives an effective representation of the surface shape available for classification, simplification and restoring purposes.

In this paper a graph-based method is presented which not only characterizes topological classification of the tessellated surfaces but also simultaneously generates the substantial circles or generators on the surface. Canonical polygons cannot always be mapped back to the original surface in terms of the edges of the given triangles. Hence, instead of applying canonical transformation to the initial "word", an associated graph is constructed using the unique vertices in the word. The graph is then decomposed into its constituent loops and paths. Based on the type of edges present, the loops are classified into three types. The number of loops of each type in the graph is then used for counting the rank or genus and classification of the given surface as being open or closed, orientable or non-orientable. The image of the loops and paths on the original surface give the substantial circles and arcs on the surface respectively.

This thesis evaluates the suitability of Voronoi ball models (VBMs) as a multipurpose shape representation for applications in computer graphics, scientific visualization, and computer vision. The effectiveness of VBMs is judged with respect to six key properties, namely stability, flexibility, accuracy, complexity, efficiency, and intuitiveness. These properties have a significant impact on the range of applicability of a computational shape model. The ability of VBMs to support a number of core shape-driven operations, in particular shape extraction, simplification, matching, interpolation, manipulation, and surface reconstruction, is examined by determining the strength of the key properties in the representation. The general approach is to use VBMs in a number of representative applications, each requiring several of the shape operations being considered. These applications include image matching and interpolation, shape model extraction from image data, two and three-dimensional shape simplification, and polygonal surface reconstruc-tion. The performance of VBMs in these applications is indicative of the extent to which each key property is present. The results of the experiments are very positive. They indicate that a VBM-based shape similarity measure can be effectively applied to quantify 2D shape differences and solve the 2D/3D shape correspondence problem. The findings also show that the VBM and the medial axis can be used together to take advantage of their complementary properties; the VBM gives the medial axis greater stability, while the axis adds connectivity and topological information to the

Abstract not found

Abstract not found

The brain network is usually constructed by estimating the connectivity matrix and thresholding it at an arbitrary level. The problem with this standard method is that we do not have any generally accepted criteria for determining a proper threshold. Thus, we propose a novel multiscale framework that models all brain networks generated over every possible threshold. Our approach is based on persistent homology and its various representations such as the Rips filtration, barcodes and dendrograms. This new persistent homological framework enables us to quantify various persistent topological features at different scales in a coherent manner. The barcode is used to quantify and visualize the evolutionary changes of topological features such as the Betti numbers over different scales. By incorporating additional geometric information to the barcode, we obtain a single linkage dendrogram that shows the overall evolution of the network. The difference between the two networks is then measured by the Gromov-Hausdorff distance over the dendrograms. As an illustration, we modeled and differentiated the FDG-PET based functional brain networks of 24 attentiondeficit hyperactivity disorder children, 26 autism spectrum disorder children and 11 pediatric control subjects.