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ABSTRACT: The Reeb graph of a scalar function tracks the evolution of the topology of its level sets. This paper describes a fast algorithm to compute the Reeb graph of a piecewise linear function defined over manifolds and non-manifolds. The key idea in the proposed approach is to maximally leverage the efficient contour tree algorithm to compute the Reeb graph. The algorithm proceeds by dividing the input into a set of subvolumes that have loop-free Reeb graphs using the join tree of the scalar function and computes the Reeb graph by combining the contour trees of all the subvolumes. Since the key ingredient of this method is a series of union-find operations, the algorithm is fast in practice. Experimental results demonstrate that it outperforms current generic algorithms by a factor of up to two orders of magnitude, and has a performance on par with algorithms that are catered to restricted classes of input. The algorithm also extends to handle large data that do not fit in memory.
IEEE transactions on visualization and computer graphics. 04/2012;
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ABSTRACT: Extracting features from point-based representations of geometric surface models is becoming increasingly important for purposes
such as model classification, matching, and exploration. In an earlier paper, we proposed a multiphase segmentation process
to identify elongated features in point-sampled surface models without the explicit construction of a mesh or other surface
representation. The preliminary results demonstrated the strength and potential of the segmentation process, but the resulting
segmentations were still of low quality, and the segmentation process could be slow. In this paper, we describe several algorithmic
improvements to overcome the shortcomings of the segmentation process. To demonstrate the improved quality of the segmentation
and the superior time efficiency of the new segmentation process, we present segmentation results obtained for various point-sampled
surface models. We also discuss an application of our segmentation process to extract ridge-separated features in point-sampled
surfaces of CAD models.
KeywordsPoint sets-Sampling-Features-Geodesic distance-Normalizedcut-Topologicalmethods-Spectralanalysis-Multiphase segmentation-Hierarchical segmentation
The Visual Computer 04/2012; 26(12):1421-1433. · 0.58 Impact Factor
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ABSTRACT: The diffusion equation based modeling of near infrared light propagation in tissue is achieved by using finite element mesh for imaging real-tissue types, such as breast and brain. The finite element mesh size (number of nodes) dictates the parameter space in the optical tomographic imaging. Most commonly used finite element meshing algorithms does not provide the flexibility of distinct nodal spacing in different regions of imaging domain to take the sensitivity of the problem into con-sideration. This work aims to present a computationally efficient mesh simplification method that can be used as a preprocessing step to iterative image reconstruction, where the finite element mesh is simplified by using an edge collapsing algorithm to reduce the parameter space at regions where the sensitivity of the problem is relatively low. It is shown, using simulations and experimental phantom data for simple meshes/domains, that a significant reduction in parameter space could be achieved without compromising on the reconstructed image quality. The maximum errors observed by using the simplified meshes were less than 0.27% in the forward problem and 5% for inverse problem.
IEEE Journal of Selected Topics in Quantum Electronics 01/2012; · 3.78 Impact Factor
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ABSTRACT: The Morse-Smale 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 Morse-Smale complex of large two-dimensional data sets at interactive speeds. We employ a reformulation of the Morse-Smale 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.
IEEE transactions on visualization and computer graphics. 12/2011;
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ABSTRACT: Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. Though geometric symmetry has been well studied within areas like shape processing, identifying symmetry in scalar fields has remained largely unexplored due to the high computational cost of the associated algorithms. We propose a computationally efficient algorithm for detecting symmetric patterns in a scalar field distribution by analysing the topology of level sets of the scalar field. Our algorithm computes the contour tree of a given scalar field and identifies subtrees that are similar. We define a robust similarity measure for comparing subtrees of the contour tree and use it to group similar subtrees together. Regions of the domain corresponding to subtrees that belong to a common group are extracted and reported to be symmetric. Identifying symmetry in scalar fields finds applications in visualization, data exploration, and feature detection. We describe two applications in detail: symmetry-aware transfer function design and symmetry-aware isosurface extraction.
IEEE transactions on visualization and computer graphics. 12/2011; 17(12):2035-44.
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IEEE Trans. Vis. Comput. Graph. 01/2011; 17:709-710.
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Comput. Graph. Forum. 01/2011; 30:1101-1110.
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ABSTRACT: Interactive visualization applications benefit from simplification techniques that generate good quality coarse meshes from high resolution meshes that represent the domain. These meshes often contain interesting substructures, called embedded structures, and it is desirable to preserve the topology of the embedded structures during simplification, in addition to preserving the topology of the domain. This paper describes a proof that link conditions, proposed earlier, are sufficient to ensure that edge contractions preserve topology of the embedded structures and the domain. Excluding two specific configurations, the link conditions are also shown to be necessary for topology preservation. Repeated application of edge contraction on an extended complex produces a coarser representation of the domain and the embedded structures. An extension of the quadric error metric is used to schedule edge contractions, resulting in a good quality coarse mesh that closely approximates the input domain and the embedded structures.
IEEE transactions on visualization and computer graphics. 06/2010;
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ABSTRACT: We introduce a variation density function that profiles the relationship between multiple scalar fields over isosurfaces of a given scalar field. This profile serves as a valuable tool for multi-field data exploration because it provides the user with cues to identify interesting isovalues of scalar fields. Existing isosurface-based techniques for scalar data exploration like Reeb graphs, contour spectra, isosurface statistics, etc., study a scalar field in isolation. We argue that the identification of interesting isovalues in a multi-field data set should necessarily be based on the interaction between the different fields. We demonstrate the effectiveness of our approach by applying it to explore data from a wide variety of applications.
IEEE transactions on visualization and computer graphics. 04/2010;
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ABSTRACT: The Morse-Smale 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 Morse-Smale complex of large two-dimensional data sets at interactive speeds. We employ a reformulation of the Morse-Smale 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.
01/2010;
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ABSTRACT: The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization
and geometric modeling. This paper describes a near-optimal two-step algorithm that constructs the Reeb graph of a Morse function
defined over manifolds in any dimension. The algorithm first identifies the critical points of the input manifold, and then
connects these critical points in the second step to obtain the Reeb graph. A simplification mechanism based on topological
persistence aids in the removal of noise and unimportant features. A radial layout scheme results in a feature-directed drawing
of the Reeb graph. Experimental results demonstrate the efficiency of the Reeb graph construction in practice and its applications.
11/2008: pages 556-567;
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ABSTRACT: Surface models of biomolecules have become crucially important for the study and understanding of interaction between biomolecules
and their environment. We argue for the need for a detailed understanding of biomolecular surfaces by describing several applications
in computational and structural biology. We review methods used to model, represent, characterize, and visualize biomolecular
surfaces focusing on the role that geometry and topology play in identifying features on the surface. These methods enable
the development of efficient computational and visualization tools for studying the function of biomolecules.
12/2007: pages 237-255;
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ABSTRACT: Analysis of the results obtained from material simulations is important in the physical sciences. Our research was motivated by the need to investigate the properties of a simulated porous solid as it is hit by a projectile. This paper describes two techniques for the generation of distance fields containing a minimal number of topological features, and we use them to identify features of the material. We focus on distance fields defined on a volumetric domain considering the distance to a given surface embedded within the domain. Topological features of the field are characterized by its critical points. Our first method begins with a distance field that is computed using a standard approach, and simplifies this field using ideas from Morse theory. We present a procedure for identifying and extracting a feature set through analysis of the MS complex, and apply it to find the invariants in the clean distance field. Our second method proceeds by advancing a front, beginning at the surface, and locally controlling the creation of new critical points. We demonstrate the value of topologically clean distance fields for the analysis of filament structures in porous solids. Our methods produce a curved skeleton representation of the filaments that helps material scientists to perform a detailed qualitative and quantitative analysis of pores, and hence infer important material properties. Furthermore, we provide a set of criteria for finding the "difference" between two skeletal structures, and use this to examine how the structure of the porous solid changes over several timesteps in the simulation of the particle impact.
IEEE Transactions on Visualization and Computer Graphics 12/2007; · 2.21 Impact Factor
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ABSTRACT: Applications in biomedical science and life science produce large data sets using increasingly powerful imaging devices and computer simulations. It is becoming increasingly difficult for scientists to explore and analyze these data using traditional tools. Interactive data processing and visualization tools can support scientists to overcome these limitations.
We show that new data processing tools and visualization systems can be used successfully in biomedical and life science applications. We present an adaptive high-resolution display system suitable for biomedical image data, algorithms for analyzing and visualization protein surfaces and retinal optical coherence tomography data, and visualization tools for 3D gene expression data.
We demonstrated that interactive processing and visualization methods and systems can support scientists in a variety of biomedical and life science application areas concerned with massive data analysis.
BMC Cell Biology 02/2007; 8 Suppl 1:S10. · 2.59 Impact Factor
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ABSTRACT: Abstract
Background
Applications in biomedical science and life science produce large data sets using increasingly powerful imaging devices and computer simulations. It is becoming increasingly difficult for scientists to explore and analyze these data using traditional tools. Interactive data processing and visualization tools can support scientists to overcome these limitations.
Results
We show that new data processing tools and visualization systems can be used successfully in biomedical and life science applications. We present an adaptive high-resolution display system suitable for biomedical image data, algorithms for analyzing and visualization protein surfaces and retinal optical coherence tomography data, and visualization tools for 3D gene expression data.
Conclusion
We demonstrated that interactive processing and visualization methods and systems can support scientists in a variety of biomedical and life science application areas concerned with massive data analysis.
BMC Cell Biology. 01/2007;
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IEEE Trans. Vis. Comput. Graph. 01/2007; 13:1432-1439.
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ABSTRACT: This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The Morse-Smale complex, which provides a succinct representation of a function's associated gradient flow field, is used to identify topological features and their significance. The simplification process, guided by the Morse-Smale complex, proceeds by repeatedly applying two atomic operations that each remove a pair of critical points from the complex. Efficient storage of the complex results in execution of these atomic operations at interactive rates. Visualization of the simplified complex shows that the simplification preserves significant topological features while removing small features and noise.
IEEE Transactions on Visualization and Computer Graphics 08/2006; 12(4):474-484. · 2.21 Impact Factor
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Algorithms and Computation, 17th International Symposium, ISAAC 2006, Kolkata, India, December 18-20, 2006, Proceedings; 01/2006
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ABSTRACT: In this paper, we present a topological approach for simplifying continuous functions defined on volumetric domains. We introduce two atomic operations that remove pairs of critical points of the function and design a combinatorial algorithm that simplifies the Morse-Smale complex by repeated application of these operations. The Morse-Smale complex is a topological data structure that provides a compact representation of gradient flow between critical points of a function. Critical points paired by the Morse-Smale complex identify topological features and their importance. The simplification procedure leaves important critical points untouched, and is therefore useful for extracting desirable features. We also present a visualization of the simplified topology.
Visualization, 2005. VIS 05. IEEE; 11/2005
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16th IEEE Visualization Conference (VIS 2005), 23-28 October 2005, Minneapolis, MN, USA; 01/2005
