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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 30 (5 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 20 (8 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.
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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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 10 (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
Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces,” Computer Graphics Forum, to appear
, 2011
"... Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. ..."
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Cited by 10 (6 self)
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Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three wellknown simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster. Index Terms—Morse decomposition, vector field topology. Ç 1
Morse set classification and hierarchical refinement using conley index
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Morse decomposition provides a numerically stable topological representation of vector fields that is crucial for their rigorous interpretation. However, Morse decomposition is not unique, and its granularity directly impacts its computational cost. In this paper, we propose an automatic re ..."
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Cited by 6 (5 self)
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Abstract—Morse decomposition provides a numerically stable topological representation of vector fields that is crucial for their rigorous interpretation. However, Morse decomposition is not unique, and its granularity directly impacts its computational cost. In this paper, we propose an automatic refinement scheme to construct the Morse Connection Graph (MCG) of a given vector field in a hierarchical fashion. Our framework allows a Morse set to be refined through a local update of the flow combinatorialization graph, as well as the connection regions between Morse sets. The computation is fast because the most expensive computation is concentrated on a small portion of the domain. Furthermore, the present work allows the generation of a topologically consistent hierarchy of MCGs, which cannot be obtained using a global method. The classification of the extracted Morse sets is a crucial step for the construction of the MCG, for which the Poincaré index is inadequate. We make use of an upper bound for the Conley index, provided by the Betti numbers of an index pair for a translation along the flow, to classify the Morse sets. This upper bound is sufficiently accurate for Morse set classification and provides supportive information for the automatic refinement process. An improved visualization technique for MCG is developed to incorporate the Conley indices. Finally, we apply the proposed techniques to a number of synthetic and realworld simulation data to demonstrate their utility. Index Terms—Morse decomposition, vector field topology, upper bound of Conley index, topology refinement, hierarchical refinement. 1
TADD: A Computational Framework for Data Analysis using Discrete Morse Theory
"... Abstract. This paper presents a computational framework that allows for a robust extraction of the extremal structure of scalar and vector fields on 2D manifolds embedded in 3D. This structure consists of critical points, separatrices, and periodic orbits. The framework is based on Forman’s discrete ..."
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Cited by 5 (5 self)
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Abstract. This paper presents a computational framework that allows for a robust extraction of the extremal structure of scalar and vector fields on 2D manifolds embedded in 3D. This structure consists of critical points, separatrices, and periodic orbits. The framework is based on Forman’s discrete Morse theory, which guarantees the topological consistency of the computed extremal structure. Using a graph theoretical formulation of this theory, we present an algorithmic pipeline that computes a hierarchy of extremal structures. This hierarchy is defined by an importance measure and enables the user to select an appropriate level of detail.
Stable Feature Flow Fields
"... Abstract—Feature Flow Fields are a wellaccepted approach for extracting and tracking features. In particular, they are often used to track critical points in timedependent vector fields and to extract and track vortex core lines. The general idea is to extract the feature or its temporal evolution ..."
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Cited by 4 (1 self)
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Abstract—Feature Flow Fields are a wellaccepted approach for extracting and tracking features. In particular, they are often used to track critical points in timedependent vector fields and to extract and track vortex core lines. The general idea is to extract the feature or its temporal evolution using a stream line integration in a derived vector field – the socalled Feature Flow Field (FFF). Hence, the desired feature line is a stream line of the FFF. As we will carefully analyze in this paper, the stream lines around this feature line may diverge from it. This creates an unstable situation: if the integration moves slightly off the feature line due to numerical errors, then it will be captured by the diverging neighborhood and carried away from the real feature line. The goal of this paper is to define a new FFF with the guarantee that the neighborhood of a feature line has always converging behavior. This way, we have an automatic correction of numerical errors: if the integration moves slightly off the feature line, it automatically moves back to it during the ongoing integration. This yields results which are an order of magnitude more accurate than the results from previous schemes. We present new stable FFF formulations for the main applications of tracking critical points and solving the Parallel Vectors operator. We apply our method to a number of data sets. 1
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 3 (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.