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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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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.
Distributed Merge Trees
"... Improved simulations and sensors are producing datasets whose increasing complexity exhausts our ability to visualize and comprehend them directly. To cope with this problem, we can detect and extract significant features in the data and use them as the basis for subsequent analysis. Topological met ..."
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Improved simulations and sensors are producing datasets whose increasing complexity exhausts our ability to visualize and comprehend them directly. To cope with this problem, we can detect and extract significant features in the data and use them as the basis for subsequent analysis. Topological methods are valuable in this context because they provide robust and general feature definitions. As the growth of serial computational power has stalled, data analysis is becoming increasingly dependent on massively parallel machines. To satisfy the computational demand created by complex datasets, algorithms need to effectively utilize these computer architectures. The main strength of topological methods, their emphasis on global information, turns into an obstacle during parallelization. We present two approaches to alleviate this problem. We develop a distributed representation of the merge tree that avoids computing the global tree on a single processor and lets us parallelize subsequent queries. To account for the increasing number of cores per processor, we develop a new data structure that lets us take advantage of multiple sharedmemory cores to parallelize the work on a single node. Finally, we present experiments that illustrate the strengths of our approach as well as help identify future challenges.
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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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
A point calculus for interlevel set homology
 Pattern Rec. Lett
"... The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of inte ..."
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The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a realvalued function. Our account of the subject is expository, devoid of proofs, and written for nonexperts in algebraic topology.
Gipsalab
, 2013
"... We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NPcomplete even if K is embedded in R 3. As a consequence, we show that it is NPhard to simplif ..."
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We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NPcomplete even if K is embedded in R 3. As a consequence, we show that it is NPhard to simplify level and sublevel sets of scalar functions on S 3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NPhard. 1.
Homological Reconstruction and Simplification in R3
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.