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24
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.
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 11 (3 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
"... 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 ..."
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Cited by 9 (8 self)
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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.
Parallel Computation of 2D MorseSmale Complexes
"... Abstract—The MorseSmale complex is a useful topological data structure for the analysis and visualization of scalar data. This paper describes an algorithm that processes all mesh elements of the domain in parallel to compute the MorseSmale complex of large twodimensional data sets at interactive ..."
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Cited by 9 (3 self)
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Abstract—The MorseSmale complex is a useful topological data structure for the analysis and visualization of scalar data. This paper describes an algorithm that processes all mesh elements of the domain in parallel to compute the MorseSmale complex of large twodimensional data sets at interactive speeds. We employ a reformulation of the MorseSmale complex using Forman’s Discrete Morse Theory and achieve scalability by computing the discrete gradient using local accesses only. We also introduce a novel approach to merge gradient paths that ensures accurate geometry of the computed complex. We demonstrate that our algorithm performs well on both multicore environments and on massively parallel architectures such as the GPU. Index Terms—Topologybased methods, discrete Morse theory, large datasets, gradient pairs, multicore, 2D scalar functions.
Flow visualization with quantified spatial and temporal errors using edge maps
, 2011
"... 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 suc ..."
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Cited by 9 (2 self)
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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.
Visualizing Robustness of Critical Points for 2D TimeVarying Vector Fields
"... Figure 1: Robustness assignment for critical point trajectories for a 2D timevarying vector field from a combustion simulation. The trajectories are mapped to colors based on their (dynamic) robustness values. The zoomedin versions show robustness pairings among the trajectories: same color segmen ..."
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Cited by 4 (3 self)
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Figure 1: Robustness assignment for critical point trajectories for a 2D timevarying vector field from a combustion simulation. The trajectories are mapped to colors based on their (dynamic) robustness values. The zoomedin versions show robustness pairings among the trajectories: same color segments are paired to each other. (a) and (c) involve fold and bluesky bifurcations. (b) and (d) are part of a long trajectory with high robustness values but different partners. Analyzing critical points and their temporal evolutions plays a crucial role in understanding the behavior of vector fields. A key challenge is to quantify the stability of critical points: more stable points may represent more important phenomena or vice versa. The topological notion of robustness is a tool which allows us to quantify rigorously the stability of each critical point. Intuitively, the robustness of a critical point is the minimum amount of perturbation necessary to cancel it within a local neighborhood, measured under an appropriate metric. In this paper, we introduce a new analysis and visualization framework which enables interactive exploration of robustness of critical points for both stationary and timevarying 2D vector fields. This framework allows the endusers, for the first time, to investigate how the stability of a critical point evolves over time. We show that this depends heavily on the global properties of the vector field and that structural changes can correspond to interesting behavior. We demonstrate the practicality of our theories and techniques on several datasets involving combustion and oceanic eddy simulations and obtain some key insights regarding their stable and unstable features.
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.
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 ..."
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Cited by 3 (1 self)
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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
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 3 (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.
B.: Interpreting feature tracking through the lens of robustness. TopoInVis (accepted
, 2013
"... Abstract A key challenge in the study of a timevarying vector fields is to resolve the correspondences between features in successive time steps and to analyze the dynamic behaviors of such features, socalled feature tracking. Commonly tracked features, such as volumes, areas, contours, boundaries ..."
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Cited by 2 (2 self)
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Abstract A key challenge in the study of a timevarying vector fields is to resolve the correspondences between features in successive time steps and to analyze the dynamic behaviors of such features, socalled feature tracking. Commonly tracked features, such as volumes, areas, contours, boundaries, vortices, shock waves and critical points, represent interesting properties or structures of the data. Recently, the topological notion of robustness, a relative of persistent homology, has been introduced to quantify the stability of critical points. Intuitively, the robustness of a critical point is the minimum amount of perturbation necessary to cancel it. In this chapter, we offer a fresh interpretation of the notion of feature tracking, in particular, critical point tracking, through the lens of robustness. We infer correspondences between critical points based on their closeness in stability, measured by robustness, instead of just distance proximities within the domain. We prove formally that robustness helps us understand the sampling conditions under which we can resolve the correspondence problem based on region overlap techniques, and the uniqueness and uncertainty associated with such techniques. These conditions also give a theoretical basis for visualizing the piecewise linear realizations of critical point trajectories over time. 1