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25
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 10 (5 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.
Simple Quad Domains for Field Aligned Mesh Parametrization
"... Figure 1: (Left) An input mesh of quads induces a cross field with an entangled graph of separatrices defining almost eight thousand domains; (center) the graph is disentangled with small distortion from the input field to obtain just twenty parametrization domains; (right) parametrization is smooth ..."
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Cited by 7 (4 self)
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Figure 1: (Left) An input mesh of quads induces a cross field with an entangled graph of separatrices defining almost eight thousand domains; (center) the graph is disentangled with small distortion from the input field to obtain just twenty parametrization domains; (right) parametrization is smoothed to make it conformal; an example of remeshing from the parametrization. We present a method for the global parametrization of meshes that preserves alignment to a cross field in input while obtaining a parametric domain made of few coarse axisaligned rectangular patches, which form an abstract base complex without Tjunctions. The method is based on the topological simplification of the cross field in input, followed by global smoothing.
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 6 (5 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.
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 6 (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.
Edge maps: Representing flow with bounded error
 In Proceedings of IEEE Pacific Visualization Symposium 2011
, 2011
"... Figure 1: Edge maps enable new views of vector field stability, illustrated with a vector field on this wavy surface. Top row (middle right): A visualization of some colored regions where flow shares the same source (green spheres) and sink (red spheres) is augmented to show how these regions overla ..."
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Cited by 6 (1 self)
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Figure 1: Edge maps enable new views of vector field stability, illustrated with a vector field on this wavy surface. Top row (middle right): A visualization of some colored regions where flow shares the same source (green spheres) and sink (red spheres) is augmented to show how these regions overlap when error is introduced. Bottom row (middle right): Streamwaves (colored green to red as they grow) show the advection of a single particle. In the presence of error, waves can widen and narrow, and bifurcate or merge. Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Many analysis techniques rely on computing streamlines, a task often hampered by numerical instabilities. Approaches that ignore the resulting errors can lead to inconsistencies that may produce unreliable visualizations and ultimately prevent indepth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with linear maps defined on its boundary. This representation, called edge maps, is equivalent to computing all possible streamlines at a user
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 5 (4 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
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
Morse connection graphs for piecewise constant vector fields on surfaces. Computer Aided Geometric Design
"... We describe an algorithm for constructing Morse Connection Graphs (MCGs) of Piecewise Constant (PC) vector fields on surfaces. The main novel aspect of our algorithm is its way of dealing with false positives that could arise when computing Morse sets from an inexact graph representation. First, our ..."
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Cited by 4 (2 self)
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We describe an algorithm for constructing Morse Connection Graphs (MCGs) of Piecewise Constant (PC) vector fields on surfaces. The main novel aspect of our algorithm is its way of dealing with false positives that could arise when computing Morse sets from an inexact graph representation. First, our MCG does not contain trivial Morse sets that may not contain any vector field features, or contain features that cancel each other. Second, we provide a simple criterion that can be used to rigorously verify MCG edges, i.e. to determine if a respective connecting chain of trajectories indeed exists. We also introduce an adaptive refinement scheme for the transition graph that aims to minimize the number of MCG arcs that the algorithm is not able to positively verify.
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.
Hierarchy of Stable Morse Decompositions
"... We introduce an algorithm for construction of the Morse hierarchy, i.e. a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse ..."
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Cited by 2 (1 self)
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We introduce an algorithm for construction of the Morse hierarchy, i.e. a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of userprescribed stability. More stable Morse decompositions are coarser, i.e. they consist of larger Morse sets. In this work, we develop an algorithm for tracking the growth of Morse sets and topological events (mergers) that they undergo as their stability is gradually increased. The resulting Morse hierarchy can be explored interactively. We provide examples demonstrating that it can provide a useful coarse overview of the vector field topology.