## B.: Topology-controlled volume rendering

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Venue: | IEEE Transactions on Visualization and Computer Graphics |

Citations: | 34 - 11 self |

### BibTeX

@ARTICLE{Weber_b.:topology-controlled,

author = {Gunther H. Weber and Ieee Computer Society and Scott E. Dillard and Hamish Carr and Valerio Pascucci and Bernd Hamann},

title = {B.: Topology-controlled volume rendering},

journal = {IEEE Transactions on Visualization and Computer Graphics},

year = {},

pages = {330--341}

}

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### Abstract

Abstract—Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topology-based 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 hardware-accelerated direct volume rendering that visualizes the contour tree-based 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

### Citations

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Citation Context ...of the core techniques used for scientific and medical visualization, assigning optical properties to points in the 3D domain of a scalar field and computing the resulting intensity at an image plane =-=[8]-=-. The color map, or transfer function that assigns the optical properties is applied uniformly in the field, and is normally based on the density value at a given point. While this approach provides a... |

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Citation Context ...the topological properties of interest are usually the number of connected components and the genus of an isosurface, i.e., the number of independent tunnels or “holes” in an isosurface. Morse theory =-=[13]-=- shows that topological changes in scalar fields defined on manifolds occur at distinct isolated points called critical points. A Reeb graph [14] expresses the evolution of individual contours as a gr... |

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Citation Context ...ces [40]. Nothing, however, depends on this choice, and the method described in [19] can be substituted instead. Once we have computed the contour tree, we simplify it to remove symbolic perturbation =-=[41]-=-, then assign each topological zone a transfer function. Consequently, every topological detail, including noise, is assigned its own transfer function. For medium to large data sets, especially when ... |

176 | MultiDimensional Transfer Functions for Interactive VolumeRendering
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Citation Context ...sting region. There has been much work on defining transfer functions which express interesting features in the data. Improved methods utilize additional derived quantities such as gradient magnitude =-=[7]-=-, but they still apply the same transfer function uniformly throughout the domain. Other methods use segmentation information to modify a transfer function [12] or to apply different rendering methods... |

171 | The contour spectrum
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Citation Context ...od [21] of level set extraction. B. Utilizing Topology in Scientific Visualization Even if the topology is known, a user interface for effective data exploration must still be developed. Bajaj et al. =-=[22]-=- introduced an interface called the contour spectrum, where properties such as isosurface area, enclosed volume, and the contour tree were plotted alongside isosurfaces to provide users with additiona... |

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Citation Context ...h as density values in a computedtomography (CT) scan, map directly to physical properties such as tissue type. Even when volume rendering is applied to non-medical data this model is often used [2], =-=[3]-=-. While this approach provides a structural overview of an entire data set, it is unable to distinguish between distinct features that share the same scalar value. If, for example, a region of interes... |

140 | Computing contour trees in all dimensions
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Citation Context ...ree [1]. Because scientific data sets often lack holes, and contour trees are easier to compute than general Reeb graphs, they have been given special attention in scientific visualization literature =-=[11, 6, 14, 3, 9]-=-. This work extends previous contour-tree-based methods for isosurface extraction. Carr and Snoeyink [2] treat the individual contours of an isosurface as separate entities. This allows for the outer ... |

126 | Contour trees and small seed sets for isosurface traversal
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Citation Context ...ree [1]. Because scientific data sets often lack holes, and contour trees are easier to compute than general Reeb graphs, they have been given special attention in scientific visualization literature =-=[11, 6, 14, 3, 9]-=-. This work extends previous contour-tree-based methods for isosurface extraction. Carr and Snoeyink [2] treat the individual contours of an isosurface as separate entities. This allows for the outer ... |

110 | Morse-smale complexes for piecewise linear 3-manifolds
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Citation Context ...e used widely for scalar topology simplification: the volume skeleton tree that corresponds to a contour tree augmented with nodes corresponding to genus changes [29], [30] and the MorseSmale complex =-=[31]-=-–[33]. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. [35]. Carr et al. [34] simplified the contour tree with two basic... |

99 |
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Citation Context ...dent tunnels or “holes” in an isosurface. Morse theory [13] shows that topological changes in scalar fields defined on manifolds occur at distinct isolated points called critical points. A Reeb graph =-=[14]-=- expresses the evolution of individual contours as a graph that is defined by these critical points and their relationships. For simply connected domains, the Reeb graph is always a tree structure, ca... |

91 |
Surface coding based on morse theory
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Citation Context ...ions. Future work will also be directed at interface design. We plan to consider different visual representations of the contour tree, such as representations that show nesting properties of contours =-=[42]-=-, [43] and also to incorporate genus changes [19] in the user interface. While adding these saddles to the augmented contour tree does not change the segmentation as no branching occurs, they can stil... |

75 |
Automatic isosurface propagation using an extrema graph and sorted boundary cell lists
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Citation Context ...ree [1]. Because scientific data sets often lack holes, and contour trees are easier to compute than general Reeb graphs, they have been given special attention in scientific visualization literature =-=[11, 6, 14, 3, 9]-=-. This work extends previous contour-tree-based methods for isosurface extraction. Carr and Snoeyink [2] treat the individual contours of an isosurface as separate entities. This allows for the outer ... |

75 | H.: High-quality two-level volume rendering of segmented data sets on consumer graphics hardware
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Citation Context ...iate fully between separate features. Other methods use segmentation information to apply different transfer functions to classified regions [5], [6] or even apply different rendering modalities [7], =-=[8]-=-. These techniques require segmentation information for each volume and rely either on manual segmentation or domain-specific segmentation algorithms. The underlying weakness of all of these methods i... |

71 | On marching cubes
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Citation Context ...hardware. Our framework supports interactive volumetric visualization of noisy or topologically complex data. A. Contour Trees II. RELATED WORK In addition to its importance for isosurface extraction =-=[12]-=-, topology has become valuable for more general exploration of scalar fields. Topology, up to now, has been used mainly in the context of isosurface extraction, where the topological properties of int... |

57 | A Topological Hierarchy for Functions on Triangulated Surfaces
- Bremer, Edelsbrunner, et al.
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(Show Context)
Citation Context ...widely for scalar topology simplification: the volume skeleton tree that corresponds to a contour tree augmented with nodes corresponding to genus changes [29], [30] and the Morse-Smale complex [31], =-=[32]-=-, [33]. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. [35]. Carr et al. [34] simplified the contour tree with two basi... |

49 |
Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographical Elevation Data
- Takahashi, Ikeda, et al.
- 1995
(Show Context)
Citation Context |

48 | High quality rendering of attributed volume data
- Tiede, Schiemann, et al.
- 1998
(Show Context)
Citation Context ...ed quantities such as gradient magnitude [7], but they still apply the same transfer function uniformly throughout the domain. Other methods use segmentation information to modify a transfer function =-=[12]-=- or to apply different rendering methods to different regions [5]. Recently, statistical learning-based methods have been introduced for interactive volume data segmentation, see [13], for example. Th... |

47 | An intelligent system approach to higherdimensional classification of volume data
- Tzeng
- 2005
(Show Context)
Citation Context ...ransfer function [12] or to apply different rendering methods to different regions [5]. Recently, statistical learning-based methods have been introduced for interactive volume data segmentation, see =-=[13]-=-, for example. The idea underlying such methods is to have a user specify interactively what regions in a data set constitute a “feature.” By pointing out such regions, it is then possible to characte... |

45 |
de Panne, “Simplifying flexible isosurfaces using local geometric measures
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(Show Context)
Citation Context ...e simplified tree, and are then propagated to the multitude of edges which were removed. The simplified tree is stored as a hierarchical structure, so that fine details can be revealed on demand. See =-=[4]-=- for details on simplification of contour trees. We have implemented our rendering algorithm on graphics hardware, running at interactive frame rates. This allows for transfer functions to be modified... |

40 |
Automating transfer function design for comprehensible volume rendering based on 3d field topology analysis (case study
- Fujishiro, Azuma, et al.
- 1999
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Citation Context ...ields based on detecting critical points and critical regions. Cox et al. [25] defined a digital Morse theory and also used critical points and regions to explore medical data. Fujishiro et al. [26], =-=[27]-=- used the contour tree to detect significant isovalues automatically for transfer function design. Carr and Snoeyink [20] extended the idea of seeded isosurface extraction based on the contour tree [1... |

38 | Topological volume skeletonization and its application to transfer function design
- Takahashi, Takeshima, et al.
(Show Context)
Citation Context ...ur tree, two topological structures are used widely for scalar topology simplification: the volume skeleton tree that corresponds to a contour tree augmented with nodes corresponding to genus changes =-=[29]-=-, [30] and the MorseSmale complex [31]–[33]. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. [35]. Carr et al. [34] simp... |

34 | Topology-based simplification for feature extraction from 3d scalar fields
- Gyulassy, Natarajan, et al.
- 2005
(Show Context)
Citation Context ...d widely for scalar topology simplification: the volume skeleton tree that corresponds to a contour tree augmented with nodes corresponding to genus changes [29], [30] and the MorseSmale complex [31]–=-=[33]-=-. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. [35]. Carr et al. [34] simplified the contour tree with two basic oper... |

33 |
Path seeds and Flexible Isosurfaces: Using Topology for Exploratory Visualization
- Carr, Snoeyink
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(Show Context)
Citation Context ...hs, they have been given special attention in scientific visualization literature [11, 6, 14, 3, 9]. This work extends previous contour-tree-based methods for isosurface extraction. Carr and Snoeyink =-=[2]-=- treat the individual contours of an isosurface as separate entities. This allows for the outer component of a surface to be removed so that inner details are revealed. 2. METHOD We generalize the met... |

29 | Volume data mining using 3D field topology analysis
- Fujishiro, Azuma, et al.
- 2000
(Show Context)
Citation Context ...alar fields based on detecting critical points and critical regions. Cox et al. [25] defined a digital Morse theory and also used critical points and regions to explore medical data. Fujishiro et al. =-=[26]-=-, [27] used the contour tree to detect significant isovalues automatically for transfer function design. Carr and Snoeyink [20] extended the idea of seeded isosurface extraction based on the contour t... |

28 | Two-level volume rendering - fusing MIP and DVR
- Hauser, Mroz, et al.
- 2000
(Show Context)
Citation Context ... the same transfer function uniformly throughout the domain. Other methods use segmentation information to modify a transfer function [12] or to apply different rendering methods to different regions =-=[5]-=-. Recently, statistical learning-based methods have been introduced for interactive volume data segmentation, see [13], for example. The idea underlying such methods is to have a user specify interact... |

24 | Topological volume skeletonization using adaptive tetrahedralization
- Takahashi, Takeshima, et al.
- 2004
(Show Context)
Citation Context ...e, two topological structures are used widely for scalar topology simplification: the volume skeleton tree that corresponds to a contour tree augmented with nodes corresponding to genus changes [29], =-=[30]-=- and the MorseSmale complex [31]–[33]. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. [35]. Carr et al. [34] simplified... |

23 |
Hybrid techniques for real-time radar simulation
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Citation Context ... components split or merge, such as when two peaks meet at a saddle. If the domain of the scalar field contains holes, the Reeb graph may contain cycles. Otherwise, the graph is called a contour tree =-=[1]-=-. Because scientific data sets often lack holes, and contour trees are easier to compute than general Reeb graphs, they have been given special attention in scientific visualization literature [11, 6,... |

22 | Parallel computation of the topology of level sets
- Pascucci, Cole-McLaughlin
(Show Context)
Citation Context |

22 |
A Feature-Based Approach for Smooth Surfaces
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- 1997
(Show Context)
Citation Context ...Future work will also be directed at interface design. We plan to consider different visual representations of the contour tree, such as representations that show nesting properties of contours [42], =-=[43]-=- and also to incorporate genus changes [19] in the user interface. While adding these saddles to the augmented contour tree does not change the segmentation as no branching occurs, they can still prov... |

18 | Exploring scalar fields using critical isovalues
- Weber, Scheuermann, et al.
- 2002
(Show Context)
Citation Context ...ectrum, where properties such as isosurface area, enclosed volume, and the contour tree were plotted alongside isosurfaces to provide users with additional cues to interesting isovalues. Weber et al. =-=[23]-=-, [24] devised tools for exploring scalar fields based on detecting critical points and critical regions. Cox et al. [25] defined a digital Morse theory and also used critical points and regions to ex... |

16 |
Simplicial subdivisions and sampling artifacts
- Carr, Möller, et al.
- 2001
(Show Context)
Citation Context ...n two topologically similar but distinct objects. Moreover, their methods are based on the topology of tetrahedral meshes, which can introduce geometric and topological artifacts in the output images =-=[39]-=-. Furthermore, they computed opacities for the volume rendering integral by interpolating over tetrahedra, instead of looking up opacities directly in the relevant transfer function, which can lead to... |

15 | Topological Manipulation of Isosurfaces
- Carr
- 2004
(Show Context)
Citation Context ...near interpolant may be substituted instead. To simplify processing, we construct a widget in each cube in which the mesh edges are augmented by edges between body saddles, face saddles, and vertices =-=[40]-=-. Nothing, however, depends on this choice, and the method described in [19] can be substituted instead. Once we have computed the contour tree, we simplify it to remove symbolic perturbation [41], th... |

13 | Multi-resolution computation and presentation of contour trees
- Pascucci, Cole-McLaughlin, et al.
- 2004
(Show Context)
Citation Context ...nding to genus changes [29], [30] and the MorseSmale complex [31]–[33]. Our work is based on topology simplification methods introduced by Takahashi et al. [29], Carr et al. [34], and Pascucci et al. =-=[35]-=-. Carr et al. [34] simplified the contour tree with two basic operations: leaf pruning and node collapses. Leaf pruning removes a leaf and the arc incident to the leaf from the contour tree. Since con... |

12 |
Interval Volume Decomposer: A Topological Approach to Volume Traversal
- Takahashi, Fujishiro, et al.
- 2005
(Show Context)
Citation Context ...olor p. The functionality of “hiding” regions which occlude features of interest can be duplicated by using transfer functions which are 100% transparent for these occluding regions. Takahashi et al. =-=[10]-=- describe transfer functions which express topological features. Our method allows these tools to be applied pricesly on a per-contour basis, 71srather than globally across all contours. Often, data a... |

11 |
Detecting critical regions in scalar fields
- Weber, Scheuermann, et al.
(Show Context)
Citation Context ..., where properties such as isosurface area, enclosed volume, and the contour tree were plotted alongside isosurfaces to provide users with additional cues to interesting isovalues. Weber et al. [23], =-=[24]-=- devised tools for exploring scalar fields based on detecting critical points and critical regions. Cox et al. [25] defined a digital Morse theory and also used critical points and regions to explore ... |

10 | Introducing Topological Attributes for Objective-Based Visualization of Simulated Datasets,” Volume Graphics
- Takeshima, Takahashi, et al.
- 2005
(Show Context)
Citation Context ...le specialized segmentations are applicable to particular problem domains, the power of our topology-based segmentation lies in its generality. Our approach is related to the work of Takeshima et al. =-=[10]-=-, who used a topological abstraction called volume skeleton tree to define topological attributes as additional input for multi-dimensional transfer functions. Our method extends their ideas and is mo... |

9 |
Optical models for volume rendering
- Max
- 1995
(Show Context)
Citation Context ..., such as density values in a computedtomography (CT) scan, map directly to physical properties such as tissue type. Even when volume rendering is applied to non-medical data this model is often used =-=[2]-=-, [3]. While this approach provides a structural overview of an entire data set, it is unable to distinguish between distinct features that share the same scalar value. If, for example, a region of in... |

8 |
Interaction Techniques and Vessel Analysis for Preoperative Planning
- Preim, Selle, et al.
- 2000
(Show Context)
Citation Context ...e interfaces, still do not allow a viewer to differentiate fully between separate features. Other methods use segmentation information to apply different transfer functions to classified regions [5], =-=[6]-=- or even apply different rendering modalities [7], [8]. These techniques require segmentation information for each volume and rely either on manual segmentation or domain-specific segmentation algorit... |

7 |
HighQuality Two-Level Volume Rendering
- Hadwiger, Berger, et al.
(Show Context)
Citation Context ...e fully between separate features. Other methods use segmentation information to apply different transfer functions to classified regions [5], [6] or even to apply different rendering modalities [7], =-=[8]-=-. These techniques require segmentation information for each volume and rely either on manual segmentation or domain-specific segmentation algorithms. The underlying weakness of all of these methods i... |

3 |
Two-Level Volume Rendering—Fusing
- Hauser, Mroz, et al.
- 2000
(Show Context)
Citation Context ...ntiate fully between separate features. Other methods use segmentation information to apply different transfer functions to classified regions [5], [6] or even to apply different rendering modalities =-=[7]-=-, [8]. These techniques require segmentation information for each volume and rely either on manual segmentation or domain-specific segmentation algorithms. The underlying weakness of all of these meth... |

2 |
Topological zone segmentation of scalar volume data
- Cox, Karron, et al.
- 2003
(Show Context)
Citation Context ...s to provide users with additional cues to interesting isovalues. Weber et al. [23], [24] devised tools for exploring scalar fields based on detecting critical points and critical regions. Cox et al. =-=[25]-=- defined a digital Morse theory and also used critical points and regions to explore medical data. Fujishiro et al. [26], [27] used the contour tree to detect significant isovalues automatically for t... |

2 |
Statictically quantitative volume rendering
- Kniss, Uitert, et al.
- 2005
(Show Context)
Citation Context ...m by scalar field behavior in a local neighborhood, and to use the resulting characterization for segmentation. However, learningbased methods require manual training to define features. Kniss et al. =-=[38]-=- improved on the classical classification by observing that a unique classification of samples is not always possible. In particular, in the proximity of boundaries it can improve visualization result... |

2 |
Parallel Computation of the Topology
- Pascucci, Cole-McLaughlin
- 2003
(Show Context)
Citation Context ...es. Algorithms for computing the contour tree [15], [16], [17], [18] were first defined for tetrahedral meshes using linear interpolation and later for hexahedral meshes using trilinear interpolation =-=[19]-=-. The contour tree may be augmented by additional nodes representing changes in topological genus of contours [19], nodes representing noncritical vertices of the mesh, or nodes representing any arbit... |

1 |
Course notes for SIGGRAPH 2004 course #28: Realtime volume graphics,” available at http://www.vrvis.at/via/resources/ course-volgraphics-2004/course28.pdf or on the SIGGRAPH 2004 conference DVD
- Engel, Hadwiger, et al.
(Show Context)
Citation Context ...olume rendering while Takeshima et al.’s approach uses pre-classification. While the use of postclassification for medical data is controversial, it significantly reduces artifacts in rendered images =-=[11]-=- and is particularly suitable for simulated data. Furthermore, instead of using linear interpolation applied to a tetrahedron we consider trilinear interpolation for a hexaheadral cell, as hexahedral ... |

1 | A featuredriven approach to locationg optimal viewpoints for volume visualization - Takahashi, Fujishiro, et al. - 2005 |

1 |
Display of Surfaces fromVolumeData
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Citation Context ... inscientific visualization. It is based on defining “optical” properties for points in the three-dimensional (3D) domain of a scalar field and computing resulting pixel intensities in an image plane =-=[1]-=-. Optical properties at a given point are normally based wholly or partially on the function value (and sometimes gradient magnitude). This approach assumes that scalar values, such as density values ... |

1 |
Real-Time Volume Graphics, chapter 4
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- 2006
(Show Context)
Citation Context ...volume rendering while Takeshima et al.’s approach uses preclassification. While the use of postclassification for medical data is controversial, it significantly reduces artifacts in rendered images =-=[11]-=- and is particularly suitable for simulated data. Furthermore, instead of using linear interpolation applied to a tetrahedron we consider trilinear interpolation for a hexaheadral cell, as hexahedral ... |

1 |
Detecting Critical Regions in Scalar Fields,” Data Visualization Conf
- Weber, Scheuermann, et al.
- 2003
(Show Context)
Citation Context ..., where properties such as isosurface area, enclosed volume, and the contour tree were plotted alongside isosurfaces to provide users with additional cues to interesting isovalues. Weber et al. [23], =-=[24]-=- devised tools for exploring scalar fields based on detecting critical points and critical regions. Cox et al. [25] defined a digital Morse theory and also used critical points and regions to explore ... |

1 |
Topological Zone Segmentation of Scalar Volume
- Cox, Karron, et al.
- 2003
(Show Context)
Citation Context ...s to provide users with additional cues to interesting isovalues. Weber et al. [23], [24] devised tools for exploring scalar fields based on detecting critical points and critical regions. Cox et al. =-=[25]-=- defined a digital Morse theory and also used critical points and regions to explore medical data. Fujishiro et al. [26], [27] used the contour tree to detect significant isovalues automatically for t... |