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Understanding the Structure of the Turbulent Mixing Layer in Hydrodynamic Instabilities
 IEEE Transactions on Visualization and Computer Graphics
"... When a heavy fluid is placed above a light fluid, tiny vertical perturbations in the interface create a characteristic structure of rising bubbles and falling spikes known as RayleighTaylor instability. RayleighTaylor instabilities have received much attention over the past halfcentury because of ..."
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Cited by 47 (22 self)
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When a heavy fluid is placed above a light fluid, tiny vertical perturbations in the interface create a characteristic structure of rising bubbles and falling spikes known as RayleighTaylor instability. RayleighTaylor instabilities have received much attention over the past halfcentury because of their importance in understanding many natural and manmade phenomena, ranging from the rate of formation of heavy elements in supernovae to the design of capsules for Inertial Confinement Fusion. We present a new approach to analyze RayleighTaylor instabilities in which we extract a hierarchical segmentation of the mixing envelope surface to identify bubbles and analyze analogous segmentations of fields on the original interface plane. We compute meaningful statistical information that reveals the evolution of topological features and corroborates the observations made by scientists. We also use geometric tracking to follow the evolution of single bubbles and highlight merge/split events leading to the formation of the large and complex structures characteristic of the later stages. In particular we (i) Provide a formal definition of a bubble; (ii) Segment the envelope surface to identify bubbles; (iii) Provide a multiscale analysis technique to produce statistical measures of bubble growth; (iv) Correlate bubble measurements with analysis of fields on the interface plane; (v) Track the evolution of individual bubbles over time. Our approach is based on the rigorous mathematical foundations of Morse theory and can be applied to a more general class of applications. Index Terms—topology, multiresolution, Morse theory
Topologybased Simplification for Feature Extraction from 3D Scalar Fields
"... This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the Morse ..."
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Cited by 44 (21 self)
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This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the MorseSmale complex by repeated application of two atomic operations that removes pairs of critical points. The simplification procedure leaves important critical points untouched, and is therefore useful for extracting features. We present a visualization of the simplified topology.
B.: Topologycontrolled volume rendering
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topologybased methods can be used to identify interesting features in volumetric data sets, to find seed sets for accelerated isosurface extraction, or ..."
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Cited by 44 (12 self)
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Abstract—Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topologybased methods can be used to identify interesting features in volumetric data sets, to find seed sets for accelerated isosurface extraction, or to treat individual connected components as distinct entities for isosurfacing or interval volume rendering. We describe a framework for direct volume rendering based on segmenting a volume into regions of equivalent contour topology and applying separate transfer functions to each region. Each region corresponds to a branch of a hierarchical contour tree decomposition, and a separate transfer function can be defined for it. The novel contributions of our work are 1) a volume rendering framework and interface where a unique transfer function can be assigned to each subvolume corresponding to a branch of the contour tree, 2) a runtime method for adjusting data values to reflect contour tree simplifications, 3) an efficient way of mapping a spatial location into the contour tree to determine the applicable transfer function, and 4) an algorithm for hardwareaccelerated direct volume rendering that visualizes the contour treebased segmentation at interactive frame rates using graphics processing units (GPUs) that support loops and conditional branches in fragment programs. Index Terms—Direct volume rendering, transfer function design, topology, contour tree, simplification. Ç 1
TopologyBased Flow Visualization, The State of the Art. In
 Topologybased Methods in Visualization,
, 2007
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A Practical Approach to MorseSmale Complex Computation: Scalability and Generality
"... Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for co ..."
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Cited by 41 (9 self)
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Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closurefinite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divideandconquer strategy allows for memoryefficient computation of the MS complex and simplification onthefly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementationspecific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on offtheshelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory. Index Terms—Topologybased analysis, MorseSmale complex, large scale data. 1
Spectral quadrangulation with orientation and alignment control
 IN ACM SIGGRAPH ASIA
, 2008
"... This paper presents a new quadrangulation algorithm, extending the spectral surface quadrangulation approach where the coarse quadrangular structure is derived from the MorseSmale complex of an eigenfunction of the Laplacian operator on the input mesh. In contrast to the original scheme, we provi ..."
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Cited by 34 (9 self)
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This paper presents a new quadrangulation algorithm, extending the spectral surface quadrangulation approach where the coarse quadrangular structure is derived from the MorseSmale complex of an eigenfunction of the Laplacian operator on the input mesh. In contrast to the original scheme, we provide flexible explicit controls of the shape, size, orientation and feature alignment of the quadrangular faces. We achieve this by proper selection of the optimal eigenvalue (shape), by adaption of the area term in the Laplacian operator (size), and by adding special constraints to the Laplace eigenproblem (orientation and alignment). By solving a generalized eigenproblem we can generate a scalar field on the mesh whose MorseSmale complex is of high quality and satisfies all the user requirements. The final quadrilateral mesh is generated from the MorseSmale complex by computing a globally smooth parametrization. Here we additionally introduce edge constraints to preserve user specified feature lines accurately.
Efficient computation of MorseSmale complexes for threedimensional scalar functions
 IEEE Trans. Vis. Comput. Graph
"... AbstractThe MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through ..."
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Cited by 32 (14 self)
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AbstractThe MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.
Analysis of scalar fields over point cloud data
 In Proceedings of the ACM/SIAM Symposium on Discrete Algorithms
, 2009
"... 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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Cited by 31 (9 self)
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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. appor t de r ech er ch e
Visual exploration of high dimensional scalar functions
 IEEE TRANS. VISUALIZATION AND COMPUTER GRAPHICS
, 2010
"... An important goal of scientific data analysis is to understand the behavior of a system or process based on a sample of the system. In many instances it is possible to observe both input parameters and system outputs, and characterize the system as a highdimensional function. Such data sets arise, ..."
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Cited by 30 (15 self)
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An important goal of scientific data analysis is to understand the behavior of a system or process based on a sample of the system. In many instances it is possible to observe both input parameters and system outputs, and characterize the system as a highdimensional function. Such data sets arise, for instance, in large numerical simulations, as energy landscapes in optimization problems, or in the analysis of image data relating to biological or medical parameters. This paper proposes an approach to analyze and visualizing such data sets. The proposed method combines topological and geometric techniques to provide interactive visualizations of discretely sampled highdimensional scalar fields. The method relies on a segmentation of the parameter space using an approximate MorseSmale complex on the cloud of point samples. For each crystal of the MorseSmale complex, a regression of the system parameters with respect to the output yields a curve in the parameter space. The result is a simplified geometric representation of the MorseSmale complex in the high dimensional input domain. Finally, the geometric representation is embedded in 2D, using dimension reduction, to provide a visualization platform. The geometric properties of the regression curves enable the visualization of additional information about each crystal such as local and global shape, width, length, and sampling densities. The method is illustrated on several synthetic examples of two dimensional functions. Two use cases, using data sets from the UCI machine learning repository, demonstrate the utility of the proposed approach on real data. Finally, in collaboration with domain experts the proposed