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Greedy routing with guaranteed delivery using ricci flows
 In Proc. of the 8th International Symposium on Information Processing in Sensor Networks (IPSN’09
, 2009
"... Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In parti ..."
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Cited by 39 (17 self)
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Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with nontriangular faces as holes, by either using the nodes ’ location or using a landmarkbased scheme without node location. The conformal map is computed with Ricci flow such that all the nontriangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossipstyle computation. The method applies to both unit disk graph models and quasiunit disk graph models. Simulation results are presented for these scenarios.
Almost isometric mesh parameterization through abstract domains
 621–635, July/August 2010. [Online]. Available: http://vcg.isti.cnr.it/Publications/ 2010/PTC10
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Resilient Routing for Sensor Networks Using Hyperbolic Embedding of Universal Covering Space
"... Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. ..."
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Cited by 18 (8 self)
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Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms. I.
Intrinsic Geometric Scale Space by Shape Diffusion
"... Abstract—This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the a ..."
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Cited by 12 (3 self)
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Abstract—This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the axiomatic causality property. Within the theoretical framework, we further present a featurebased shape representation derived from IGSS processing, which is shown to be theoretically plausible and practically effective. By integrating the concept of scaledependent saliency into the shape description, this representation is not only highly descriptive of the local structures, but also exhibits several desired characteristics of global shape representations, such as being compact, robust to noise and computationally efficient. We demonstrate the capabilities of our approach through salient geometric feature detection and highly discriminative matching of 3D scans. Index Terms—Scale space, feature extraction, geometric flow, Riemannian manifolds. 1
Texture Mapping via Optimal Mass Transport
, 2010
"... In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the “earthmover’s metric”). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation co ..."
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Cited by 12 (0 self)
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In this paper, we present a novel method for texture mapping of closed surfaces. Our method is based on the technique of optimal mass transport (also known as the “earthmover’s metric”). This is a classical problem that concerns determining the optimal way, in the sense of minimal transportation cost, of moving a pile of soil from one site to another. In our context, the resulting mapping is area preserving and minimizes angle distortion in the optimal mass sense. Indeed, we first begin with an anglepreserving mapping (which may greatly distort area) and then correct it using the mass transport procedure derived via a certain gradient flow. In order to obtain fast convergence to the optimal mapping, we incorporate a multiresolution scheme into our flow. We also use ideas from discrete exterior calculus in our computations.
Global parametrization by incremental flattening
 ACM Transactions on Graphics (TOG
, 2012
"... Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial scree ..."
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Cited by 11 (2 self)
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Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specifi c permission and/or a fee. Permissions may be
Packing circles and spheres on surfaces
 TO APPEAR IN THE ACM SIGGRAPH CONFERENCE PROCEEDINGS
, 2009
"... Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate ci ..."
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Cited by 11 (6 self)
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Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsionfree support structure, hybrid trihex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry.
X.: Directproduct volumetric parameterization of handlebodies via harmonic fields
 In Proc. Shape Modeling Intl. (2010
"... Abstract—Volumetric parameterization plays an important role for geometric modeling. Due to the complicated topological nature of volumes, it is much more challenging than the surface case. This work focuses on the parameterization of volumes with a boundary surface embedded in 3D space. The intuit ..."
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Cited by 10 (0 self)
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Abstract—Volumetric parameterization plays an important role for geometric modeling. Due to the complicated topological nature of volumes, it is much more challenging than the surface case. This work focuses on the parameterization of volumes with a boundary surface embedded in 3D space. The intuition is to decompose the volume as the direct product of a two dimensional surface and a one dimensional curve. We first partition the boundary surface into ceiling, floor and walls. Then we compute the harmonic field in the volume with a Dirichlet boundary condition. By tracing the integral curve along the gradient of the harmonic function, we can parameterize the volume to the parametric domain. The method is guaranteed to produce bijection for handlebodies with complex topology, including topological balls as a degenerate case. Furthermore, the parameterization is regular everywhere. We apply the proposed parameterization method to construct hexahedral mesh. KeywordsSolid modeling, volume parameterization, hexahedral mesh, polycube, harmonic fields, handlebody, directproduct. I.
Scalable and Fully Distributed Localization With Mere Connectivity
"... Abstract—This work proposes a novel connectivitybased localization algorithm, well suitable for largescale sensor networks with complex shapes and nonuniform nodal distribution. In contrast to current stateofart connectivitybased localization methods, the proposed algorithm is fully distribute ..."
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Cited by 9 (1 self)
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Abstract—This work proposes a novel connectivitybased localization algorithm, well suitable for largescale sensor networks with complex shapes and nonuniform nodal distribution. In contrast to current stateofart connectivitybased localization methods, the proposed algorithm is fully distributed, where each node only needs the information of its neighbors, without cumbersome partitioning and merging process. The algorithm is highly scalable, with limited error propagation and linear computation and communication cost with respect to the size of the network. Moreover, the algorithm is theoretically guaranteed and numerically stable. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing superior performance of the proposed algorithm. I.
Computing Teichmüller Shape Space
 SUBMITTED TO IEEE TVCG
"... Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmüller theory. Two surfaces are conformal equivalent, if there exists a bijective anglepreserving map between them. ..."
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Cited by 8 (3 self)
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Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmüller theory. Two surfaces are conformal equivalent, if there exists a bijective anglepreserving map between them. The Teichmüller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to Identity. A curve in the Teichmüller space represents a deformation process from one class to the other. In this work, we apply Teichmüller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic, invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmüller coordinates is practical, stable and efficient. The algorithms for the Teichmüller coordinates of surfaces with positive or zero Euler numbers have been studied before. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with −1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmüller space.