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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 35 (16 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.
H.: 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, ..."
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Cited by 33 (11 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, 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 (i) 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; (ii) a runtime method for adjusting data values to reflect contour tree simplifications; (iii) an efficient way of mapping a spatial location into the contour tree to determine the applicable transfer function; and (iv) 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 I.
Sizebased Transfer Functions: A New Volume Exploration Technique
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 2008
"... The visualization of complex 3D images remains a challenge, a fact that is magnified by the difficulty to classify or segment volume data. In this paper, we introduce sizebased transfer functions, which map the local scale of features to color and opacity. Features in a data set with similar or i ..."
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Cited by 31 (3 self)
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The visualization of complex 3D images remains a challenge, a fact that is magnified by the difficulty to classify or segment volume data. In this paper, we introduce sizebased transfer functions, which map the local scale of features to color and opacity. Features in a data set with similar or identical scalar values can be classified based on their relative size. We achieve this with the use of scale fields, which are 3D fields that represent the relative size of the local feature at each voxel. We present a mechanism for obtaining these scale fields at interactive rates, through a continuous scalespace analysis and a set of detection filters. Through a number of examples, we show that sizebased transfer functions can improve classification and enhance volume rendering techniques, such as maximum intensity projection. The ability to classify objects based on local size at interactive rates proves to be a powerful method for complex data exploration.
A FeatureDriven Approach to Locating Optimal Viewpoints for Volume Visualization
 In IEEE Visualization
, 2005
"... Figure 1: Locating optimal viewpoints by individually estimating the visibility quality of each feature subvolume. The value under each image represents its corresponding estimate normalized to [0.0, 1.0]. Optimal viewpoint selection is an important task because it considerably influences the amount ..."
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Cited by 30 (1 self)
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Figure 1: Locating optimal viewpoints by individually estimating the visibility quality of each feature subvolume. The value under each image represents its corresponding estimate normalized to [0.0, 1.0]. Optimal viewpoint selection is an important task because it considerably influences the amount of information contained in the 2D projected images of 3D objects, and thus dominates their first impressions from a psychological point of view. Although several methods have been proposed that calculate the optimal positions of viewpoints especially for 3D surface meshes, none has been done for solid objects such as volumes. This paper presents a new method of locating such optimal viewpoints when visualizing volumes using direct volume rendering. The major idea behind our method is to decompose an entire volume into a set of feature components, and then find a globally optimal viewpoint by finding a compromise between locally optimal viewpoints for the components. As the feature components, the method employs interval volumes and their combinations that characterize the topological transitions of isosurfaces according to the scalar field. Furthermore, opacity transfer functions are also utilized to assign different weights to the decomposed components so that users can emphasize features of specific interest in the volumes. Several examples of volume datasets together with their optimal positions of viewpoints are exhibited in order to demonstrate that the method can effectively guide naive users to find optimal projections of volumes.
Efficient Computation of MorseSmale Complexes for ThreeDimensional Scalar Functions
, 2007
"... The 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, ..."
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Cited by 23 (11 self)
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The 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.
Topological Volume Skeletonization Using Adaptive Tetrahedralization
 in Proc. Geometric Modeling and Processing
, 2004
"... Topological volume skeletons represent levelset graphs of 3D scalar fields, and have recently become crucial to visualizing the global isosurface transitions in the volume. However, it is still a timeconsuming task to extract them especially when input volumes are largescale data and/or prone to ..."
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Cited by 23 (4 self)
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Topological volume skeletons represent levelset graphs of 3D scalar fields, and have recently become crucial to visualizing the global isosurface transitions in the volume. However, it is still a timeconsuming task to extract them especially when input volumes are largescale data and/or prone to smallamplitude noise. This paper presents an efficient method for accelerating the computation of such skeletons using adaptive tetrahedralization. The present tetrahedralization is a topdown approach to linear interpolation of the scalar fields in that it selects tetrahedra to be subdivided adaptively using several criteria. As the criteria, the method employs a topological criterion as well as a geometric one in order to pursue all the topological isosurface transitions that may contribute to the global skeleton of the volume. The tetrahedralization also allows us to avoid unnecessary tracking of minor degenerate features that hide the global skeleton. Experimental results are included to demonstrate that the present method smoothes out the original scalar fields effectively without missing any significant topological features. 1
A topological approach to simplification of threedimensional scalar functions
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS (SPECIAL ISSUE IEEE VISUALIZATION
, 2006
"... This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale complex, which provides a succinct representation of a function’s associated gradient flow field, is used to identify topological features and their significance ..."
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Cited by 23 (11 self)
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This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale 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 MorseSmale 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 and removes small features and noise.
Topological landscapes: A terrain metaphor for scientific data
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex informat ..."
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Cited by 21 (8 self)
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Abstract—Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex information easily accessible by using commonly known concepts. In this paper we propose a new metaphor, called “Topological Landscapes, ” which facilitates understanding the topological structure of scalar functions. The basic idea is to construct a terrain with the same topology as a given dataset and to display the terrain as an easily understood representation of the actual input data. In this projection from an ndimensional scalar function to a twodimensional (2D) model we preserve function values of critical points, the persistence (function span) of topological features, and one possible additional metric property (in our examples volume). By displaying this topologically equivalent landscape together with the original data we harness the natural human proficiency in understanding terrain topography and make complex topological information easily accessible.
Simple and Optimal OutputSensitive Construction of Contour Trees Using Monotone Paths
, 2004
"... Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole da ..."
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Cited by 19 (1 self)
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Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole dataset, but sorts only a subset of socalled componentcritical points. They form only a small fraction of the vertices in the dataset, for typical data that arise in practice. The algorithm is simple, achieves the optimal outputsensitive bound in running time, and works in any dimension. Our experiments show that the algorithm compares favorably with the previous best algorithm.
Topologically Clean Distance Fields
"... 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 c ..."
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Cited by 18 (12 self)
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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.