Results 1  10
of
19
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. ..."
Abstract

Cited by 10 (6 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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.
Simplification of Morse Decompositions using Morse Set Mergers
"... Abstract. A common problem of vector field topology algorithms is the large number of the resulting topological features. This paper describes a method to simplify Morse decompositions by iteratively merging pairs of Morse sets that are adjacent in the Morse Connection Graph (MCG). When Morse sets A ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. A common problem of vector field topology algorithms is the large number of the resulting topological features. This paper describes a method to simplify Morse decompositions by iteratively merging pairs of Morse sets that are adjacent in the Morse Connection Graph (MCG). When Morse sets A and B are merged, they are replaced by a single Morse set, that can be thought of as the union of A, B and all trajectories connecting A and B. Pairs of Morse sets to be merged can be picked based on a variety of criteria. For example, one can allow only pairs whose merger results in a topologically simple Morse set to be selected, and give preference to mergers leading to small Morse sets. 1
2D Vector Fields using Edge Maps
, 2010
"... Vector fields, represented as vector values sampled on the vertices of a triangulation, are commonly used to model physical phenomena. To analyze and understand vector fields, practitioners use derived properties such as the paths of massless particles advected by the flow, called streamlines. Howev ..."
Abstract
 Add to MetaCart
Vector fields, represented as vector values sampled on the vertices of a triangulation, are commonly used to model physical phenomena. To analyze and understand vector fields, practitioners use derived properties such as the paths of massless particles advected by the flow, called streamlines. However, computing streamlines requires numerical methods not guaranteed to preserve fundamental invariants such as the fact that streamlines cannot cross. The resulting inconsistencies can cause errors in the analysis, e.g. invalid topological skeletons, and thus lead to misinterpretations of the data. We propose an alternate representation for triangulated vector fields that exchanges vector values with an encoding of the transversal flow behavior of each triangle. We call this representation edge maps. This work focuses on the mathematical properties of edge maps; a companion paper discusses some of their applications [1]. Edge maps allow for a multiresolution approximation of flow by merging adjacent streamlines into an interval based mapping. Consistency is enforced at any resolution if the merged sets maintain an orderpreserving property. At the coarsest resolution, we define a notion of equivalency between edge maps, and show that there exist 23 equivalence classes describing all possible behaviors of piecewise linear