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15
Persistent brain network homology from the perspective of dendrogram
 IEEE T. Med. Imaging, in press
, 2012
"... Abstract—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 determiningaproper threshold. Thus, we propose a novel multiscale framew ..."
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Cited by 7 (3 self)
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Abstract—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 determiningaproper 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. Thedifferencebetweenthetwonetworks is then measured by the Gromov–Hausdorff distance over the dendrograms. As an illustration, we modeled and differentiated the FDGPET based functional brain networks of 24 attentiondeficit hyperactivity disorder children, 26 autism spectrum disorder children, and 11 pediatric control subjects. Index Terms—Barcode, functional brain network, Gromov–Hausdorff distance, persistent homology, rips complex, rips filtration, single linkage dendrogram. I.
Topological and Geometrical Reconstruction of Complex Objects on Irregular Isothetic Grids
 In Proc. of Int. Conf. of DGCI, LNCS 4245
, 2006
"... Abstract. In this paper, we address the problem of vectorization of binary images on irregular isothetic grids. The representation of graphical elements by lines is common in document analysis, where images are digitized on (sometimes verylarge scale) regular grids. Regardless of final application, ..."
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Cited by 4 (3 self)
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Abstract. In this paper, we address the problem of vectorization of binary images on irregular isothetic grids. The representation of graphical elements by lines is common in document analysis, where images are digitized on (sometimes verylarge scale) regular grids. Regardless of final application, we propose to first describe the topology of an irregular twodimensional object with its associated Reeb graph, and we recode it with simple irregular discrete arcs. The second phase of our algorithm consists of a polygonal reconstruction of this object, with discrete lines through the elementary arcs computed in the previous stage. We also illustrate the robustness of our method, and discuss applications and improvements. 1
ALGORITHMS FOR SMOOTH AND DEFORMABLE SURFACES IN 3D
, 2002
"... 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 underlying structure of Delaunay triangulations and alpha shapes. The surface serves as a deformable manifold in various disc ..."
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Cited by 4 (2 self)
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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 underlying 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.
Deformable Geometry Model Matching Using Bipartite Graph
 Proc. of Computer Graphics International (CGI
, 2004
"... 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 topolo ..."
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Cited by 3 (1 self)
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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 lowdimension 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.
Topological Techniques for Shape Understanding
 IN CENTRAL EUROPEAN SEMINAR ON COMPUTER GRAPHICS, CESCG
, 2001
"... 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 ..."
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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.
Graph Based Topological Analysis of Tessellated Surfaces
"... In this paper a graphbased 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 ..."
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In this paper a graphbased 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 &quot;word&quot;, 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 nonorientable. The image of the loops and paths on the original surface give the substantial circles and arcs on the surface respectively.
Matching and Retrieval of 3D Deformable Models (三維立體可變形模型的比對與搜尋)
, 2004
"... Due to the popularity of computer graphics, 3D games and CG movies, a large number of 3D models have been created. Many of them are freely available on the Internet and can be found in many personal homepages, repositories and even libraries. Though the number of 3D models keeps on increasing, 3D mo ..."
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Due to the popularity of computer graphics, 3D games and CG movies, a large number of 3D models have been created. Many of them are freely available on the Internet and can be found in many personal homepages, repositories and even libraries. Though the number of 3D models keeps on increasing, 3D models are time and effort consuming to build. The group of deformable models (or so called animating models or articulated models) is especially a good example. These models are usually high quality meshes that require weeks of work to construct. As such, it would be advantageous to build models by reusing or adapting from existing ones. To facilitate reuse and sharing, research on 3D model matching and retrieval becomes essential. Existing contentbased 3D model matching methods can be categorized into four major approaches: geometrybased, transformbased, imagebased and topologybased. Among these four approaches, only topologybased matching methods can handle highly deformable models, i.e., models representing the same