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Rotational Symmetry Field Design on Surfaces
"... tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). ..."
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Cited by 63 (8 self)
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tensor smoothed as a 4RoSy field, (c) topological editing operations applied to (b), and (d) more global smoothing performed on (b). Notice that treating the curvature tensor as a 4RoSy field (b) leads to fewer unnatural singularities and therefore less visual artifacts than as a 2RoSy field (a). In addition, both topological editing (c) and global smoothing (d) can be used to remove more singularities from (b). However, topological editing (c) provides local control while excessive global smoothing (d) can cause hatch directions to deviate from their natural orientations (neck and chest). Designing rotational symmetries on surfaces is a necessary task for a wide variety of graphics applications, such as surface parameterization and remeshing, painterly rendering and penandink sketching, and texture synthesis. In these applications, the topology of a rotational symmetry field such as singularities and separatrices can have a direct impact on the quality of the results. In this paper, we present a design system that provides control over the topology of rotational symmetry fields on surfaces. As the foundation of our system, we provide comprehensive analysis for rotational symmetry fields on surfaces and present efficient algorithms to identify singularities and separatrices. We also describe design operations that allow a rotational symmetry field to be created and modified in an intuitive fashion by using the idea of basis fields and relaxation. In particular, we provide control over the topology of a rotational symmetry field by allowing the user to remove singularities from the field or to move them to more desirable locations.
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 35 (19 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.
Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces
, 2011
"... Numerical simulations and experimental observations are inherently imprecise. Therefore, most vector fields of interest in scientific visualization are known only up to an error. In such cases, some topological features, especially those not stable enough, may be artifacts of the imprecision of the ..."
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Cited by 21 (6 self)
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Numerical simulations and experimental observations are inherently imprecise. Therefore, most vector fields of interest in scientific visualization are known only up to an error. In such cases, some topological features, especially those not stable enough, may be artifacts of the imprecision of the input. This paper introduces a technique to compute topological features of userprescribed stability with respect to perturbation of the input vector field. In order to make our approach simple and efficient, we develop our algorithms for the case of piecewise constant (PC) vector fields. Our approach is based on a supertransition graph, a common graph representation of all PC vector fields whose vector value in a mesh triangle is contained in a convex set of vectors associated with that triangle. The graph is used to compute a Morse decomposition that is coarse enough to be correct for all vector fields satisfying the constraint. Apart from computingstableMorsedecompositions, ourtechniquecanalsobeused to estimate the stability of Morse sets with respect to perturbation of the vector field or to compute topological features of continuous vector fields using the PC framework.
Asymmetric Tensor Analysis for Flow Visualization
 IEEE Trans. on Visualization and Computer Graphics
"... Abstract—The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectorybased vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient te ..."
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Cited by 20 (13 self)
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Abstract—The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectorybased vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field that can represent either a 2D compressible flow or the projection of a 3D compressible or incompressible flow onto a 2D manifold. To illustrate the structures in asymmetric tensor fields, we introduce the notions of eigenvalue manifold and eigenvector manifold. These concepts afford a number of theoretical results that clarify the connections between symmetric and antisymmetric components in tensor fields. In addition, these manifolds naturally lead to partitions of tensor fields, which we use to design effective visualization strategies. Moreover, we extend eigenvectors continuously into the complex domains which we refer to as pseudoeigenvectors. We make use of evenly spaced tensor lines following pseudoeigenvectors to illustrate the local linearization of tensors everywhere inside complex domains simultaneously. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparameterization with physical meaning. This allows us to relate our tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide a physical interpretation of our tensordriven vector field analysis in the context of fluid mechanics. To demonstrate the utility of our approach, we have applied our visualization techniques and interpretation to the study of the Sullivan Vortex as well as computational fluid dynamics simulation data. Index Terms—Tensor field visualization, flow analysis, asymmetric tensors, flow segmentation, tensor field topology, surfaces. Ç 1
Combinatorial 2d vector field topology extraction and simplification
"... Summary: This paper investigates a combinatorial approach to vector field topology. The theoretical basis is given by Robin Forman’s work on a combinatorial Morse theory for dynamical systems defined on general simplicial complexes. We formulate Forman’s theory in a graph theoretic setting and provi ..."
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Cited by 13 (3 self)
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Summary: This paper investigates a combinatorial approach to vector field topology. The theoretical basis is given by Robin Forman’s work on a combinatorial Morse theory for dynamical systems defined on general simplicial complexes. We formulate Forman’s theory in a graph theoretic setting and provide a simple algorithm for the construction and topological simplification of combinatorial vector fields on 2D manifolds. Given a combinatorial vector field we are able to extract its topological skeleton including all periodic orbits. Due to the solid theoretical foundation we know that the resulting structure is always topologically consistent. We explore the applicability and limitations of this combinatorial approach with several examples and determine its robustness with respect to noise. 1
Evenly spaced streamlines for surfaces: An imagebased approach
 Computer Graphics Forum
"... We introduce a novel, automatic streamline seeding algorithm for vector fields defined on surfaces in 3D space. The algorithm generates evenly spaced streamlines fast, simply and efficiently for any general surfacebased vector field. It is general because it handles large, complex, unstructured, ad ..."
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Cited by 13 (0 self)
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We introduce a novel, automatic streamline seeding algorithm for vector fields defined on surfaces in 3D space. The algorithm generates evenly spaced streamlines fast, simply and efficiently for any general surfacebased vector field. It is general because it handles large, complex, unstructured, adaptive resolution grids with holes and discontinuities, does not require a parametrization, and can generate both sparse and dense representations of the flow. It is efficient because streamlines are only integrated for visible portions of the surface. It is simple because the imagebased approach removes the need to perform streamline tracing on a triangular mesh, a process which is complicated at best. And it is fast because it makes effective, balanced use of both the CPU and the GPU. The key to the algorithm’s speed, simplicity and efficiency is its imagebased seeding strategy. We demonstrate our algorithm on complex, realworld simulation data sets from computational fluid dynamics and compare it with objectspace streamline visualizations.
Analysis of Recurrent Patterns in Toroidal Magnetic Fields
, 2010
"... In the development of magnetic confinement fusion which will potentially be a future source for low cost power, physicists must be able to analyze the magnetic field that confines the burning plasma. While the magnetic field can be described as a vector field, traditional techniques for analyzing t ..."
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Cited by 12 (0 self)
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In the development of magnetic confinement fusion which will potentially be a future source for low cost power, physicists must be able to analyze the magnetic field that confines the burning plasma. While the magnetic field can be described as a vector field, traditional techniques for analyzing the field’s topology cannot be used because of its Hamiltonian nature. In this paper we describe a technique developed as a collaboration between physicists and computer scientists that determines the topology of a toroidal magnetic field using fieldlines with near minimal lengths. More specifically, we analyze the Poincaré map of the sampled fieldlines in a Poincaré section including identifying critical points and other topological features of interest to physicists. The technique has been deployed into an interactive parallel visualization tool which physicists are using to gain new insight into simulations of magnetically confined burning plasmas.
Flow visualization with quantified spatial and temporal errors using edge maps
 IEEE Transactions on Visualization and Computer Graphics
, 2012
"... Abstract—Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical error ..."
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Cited by 9 (2 self)
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Abstract—Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent indepth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures.
Zhang E.: EvenlySpaced Streamlines for Surfaces: An ImageBased Approach. Computer Graphics Forum 28
"... We introduce a novel, automatic streamline seeding algorithm for vector fields defined on surfaces in 3D space. The algorithm generates evenlyspaced streamlines fast, simply, and efficiently for any general surfacebased vector field. It is general because it handles large, complex, unstructured, a ..."
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Cited by 9 (8 self)
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We introduce a novel, automatic streamline seeding algorithm for vector fields defined on surfaces in 3D space. The algorithm generates evenlyspaced streamlines fast, simply, and efficiently for any general surfacebased vector field. It is general because it handles large, complex, unstructured, adaptive resolution grids with holes and discontinuities, does not require a parameterization, and can generate both sparse and dense representations of the flow. It is efficient because streamlines are only integrated for visible portions of the surface. It is simple because the imagebased approach removes the need to perform streamline tracing on a triangular mesh, a process which is complicated at best. And it is fast because it makes effective, balanced use of both the CPU and the GPU. The key to the algorithm’s speed, simplicity, and efficiency is its imagebased seeding strategy. We demonstrate our algorithm on complex, realworld simulation data sets from computational fluid dynamics and compare it with objectspace streamline visualizations.