Results 1  10
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44
Discrete Multiscale Vector Field Decomposition
, 2003
"... While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finitedifference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vecto ..."
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Cited by 68 (8 self)
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While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finitedifference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vector fields on arbitrary tetrahedral grids. We introduce a variational, multiscale decomposition of vector fields into three intuitive components: a divergencefree part, a curlfree part, and a harmonic part. We show how our discrete approach matches its wellknown smooth analog, called the HelmotzHodge decomposition, and that the resulting computational tools have very intuitive geometric interpretation. We demonstrate the versatility of these tools in a series of applications, ranging from data visualization to fluid and deformable object simulation.
QuadCover – Surface Parameterization using Branched Coverings.
 COMPUT. GRAPH. FORUM
, 2007
"... We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface in ..."
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Cited by 62 (6 self)
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We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.
Vector Field Design on Surfaces
 ACM Transactions on Graphics
, 2006
"... Figure 1: This figure shows various vector fields created on surfaces using our vector field design system. The vector field shown at the right was used to guide texture synthesis shown in Figure 12 (right). Vector field design on surfaces is necessary for many graphics applications: examplebased t ..."
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Cited by 44 (14 self)
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Figure 1: This figure shows various vector fields created on surfaces using our vector field design system. The vector field shown at the right was used to guide texture synthesis shown in Figure 12 (right). Vector field design on surfaces is necessary for many graphics applications: examplebased texture synthesis, nonphotorealistic rendering, and fluid simulation. A vector field design system should allow a user to create a large variety of complex vector fields with relatively little effort. In this paper, we present a vector field design system for surfaces that allows the user to control the number of singularities in the vector field and their placement. Our system combines basis vector fields to make an initial vector field that meets the user’s specifications. The initial vector field often contains unwanted singularities. Such singularities cannot always be eliminated, due to the PoincaréHopf index theorem. To reduce the effect caused by these singularities, our system allows a user to move a singularity to a more favorable location or to cancel a pair of singularities. These operations provide topological guarantees for the vector field in that they only affect the userspecified singularities. Other editing operations are also provided so that the user may change the topological and geometric characteristics of the vector field. We demonstrate our vector field design system for several applications: examplebased texture synthesis, painterly rendering of images, and pencil sketch illustrations of smooth surfaces.
Design of tangent vector fields
 ACM Trans. Graph
, 2007
"... Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and nonphotorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of ..."
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Cited by 41 (4 self)
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Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and nonphotorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of userprovided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1forms), we obtain an intrinsic, coordinatefree formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient prefactorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 33 (2 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
NSymmetry Direction Field Design
, 2008
"... Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and ..."
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Cited by 31 (0 self)
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Many algorithms in computer graphics and geometry processing use two orthogonal smooth direction fields (unit tangent vector fields) defined over a surface. For instance, these direction fields are used in texture synthesis, in geometry processing or in nonphotorealistic rendering to distribute and orient elements on the surface. Such direction fields can be designed in fundamentally different ways, according to the symmetry requested: inverting a direction or swapping two directions may be allowed or not. Despite the advances realized in the last few years in the domain of geometry processing, a unified formalism is still lacking for the mathematical object that characterizes these generalized direction fields. As a consequence, existing direction field design algorithms are limited to use nonoptimum local relaxation procedures. In this paper, we formalize Nsymmetry direction fields, a generalization of classical direction fields. We give a new definition of their singularities to explain how they relate with the topology of the surface. Namely, we provide an accessible demonstration of the PoincaréHopf theorem in the case of Nsymmetry direction fields on 2manifolds. Based on this theorem, we explain how to control the topology of Nsymmetry direction fields on meshes. We demonstrate the validity and robustness of this formalism by deriving a highly efficient algorithm to design a smooth eld interpolating user de ned singularities and directions.
Vector field editing and periodic orbit extraction using morse decomposition
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2007
"... Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this pa ..."
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Cited by 25 (12 self)
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Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamlinebased technique that allows vector field topology to be easily identified.
Discrete OneForms on Meshes and Applications to 3D Mesh Parameterization
 Journal of CAGD
, 2006
"... We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing mesh ..."
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Cited by 25 (1 self)
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We describe how some simple properties of discrete oneforms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be nonconvex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being nonconvex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.
Efficient Morse Decompositions of Vector Fields
"... Abstract — Existing topologybased vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field an ..."
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Cited by 18 (7 self)
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Abstract — Existing topologybased vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCG’s, while fast, are overly conservative and usually results in MCG’s that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of τmaps, which typically provides finer MCG’s than existing techniques. Furthermore, the choice of τ provides a natural tradeoff between the fineness of the MCG’s and the computational costs. We provide efficient implementations of Morse decomposition based on τmaps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation.Furthermore, we propose the use of spatial τmaps in addition to the original temporal τmaps. These techniques provide additional tradeoffs between the quality of the MCG’s and the speed of computation. We demonstrate the utility of our technique with various examples in plane and on surfaces including engine simulation datasets. Index Terms — Vector field topology, Morse decomposition, τmaps, Morse connection graph, flow combinatorialization.
Oneforms on meshes and applications to 3D mesh parameterization
, 2004
"... We develop a theory of oneforms on meshes. The theory culminates in a discrete analog of the PoincareHopf index theorem for meshes. We apply this theorem to obtain some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spr ..."
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Cited by 10 (4 self)
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We develop a theory of oneforms on meshes. The theory culminates in a discrete analog of the PoincareHopf index theorem for meshes. We apply this theorem to obtain some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "springembedding " theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk by a planar tiling with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be nonconvex. We characterize when it is still possible to achieve an injective parameterization, despite these boundaries being nonconvex. The third result is an analogous Tuttelike theorem for meshes with genus 1 (topologically equivalent to the torus), paving the way for a general method to locally parameterize such meshes to the plane in a naturally seamless manner. The last result generalizes recent work of Gu and Yau. Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.