## Topological Considerations in Isosurface Generation (1994)

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Venue: | ACM Transactions on Graphics |

Citations: | 96 - 0 self |

### BibTeX

@ARTICLE{Gelder94topologicalconsiderations,

author = {Allen Van Gelder and Jane Wilhelms},

title = {Topological Considerations in Isosurface Generation},

journal = {ACM Transactions on Graphics},

year = {1994},

volume = {13},

pages = {337--375}

}

### Years of Citing Articles

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### Abstract

A popular technique for rendition of isosurfaces in sampled data is to consider cells with sample points as corners and approximate the isosurface in each cell by one or more polygons whose vertices are obtained by interpolation of the sample data. That is, each polygon vertex is a point on a cell edge, between two adjacent sample points, where the function is estimated to equal the desired threshold value. The two sample points have values on opposite sides of the threshold, and the interpolated point is called an intersection point. When one cell face has an intersection point ineach of its four edges, then the correct connection among intersection points becomes ambiguous. An incorrect connection can lead to erroneous topology in the rendered surface, and possible discontinuities. We show that disambiguation methods, to be at all accurate, need to consider sample values in the neighborhood outside the cell. This paper studies the problems of disambiguation, reports on some solutions, and presents some statistics on the occurrence of such ambiguities. A natural way to incorporate neighborhood information is through the use of calculated gradients at cell corners. They provide insight into the behavior of a function in well-understood ways. We introduce two gradient-consistency heuristics that use calculated gradients at the corners of ambiguous faces, as well as the function values at those corners, to disambiguate at a reasonable computational cost. These methods give the correct topology on several examples that caused problems for other methods we examined.

### Citations

2077 | Marching cubes: A high resolution 3D surface construction algorithm
- Lorensen, Cline
- 1987
(Show Context)
Citation Context ...er case, in which the positive and negative values are reversed. Treating inverses as the same case, there are 14 cases, as shown in Figure 1, which we refer to as the major cases. Lorensen and Cline =-=[LC87]-=- give 15 cases because they do not consider the reflection of case 11 to be the same case. Our case numbering follows theirs with the exception of this omitted case. It is easy to see that cases 1, 2,... |

995 | Computer Graphics : Principles and Practice - Foley - 1990 |

724 | Display of surfaces from volume data - Levoy - 1988 |

374 | Introduction to implicit surfaces - Bloomenthal, Bajaj, et al. - 1997 |

284 | Data structure for soft objects
- Wyvill, McPheeters, et al.
- 1986
(Show Context)
Citation Context ... then connected the contour lines into three-dimensional isosurfaces [FKU77, CS78, WH79, CIBL83]. More recent approaches create the isosurface by examining the three dimensions at once. Wyvill et al. =-=[WMW86]-=- developed a method that represents the isosurface as a polygon mesh. They compute samples in a three-dimensional rectangular lattice and analyze cells in this lattice individually. Although they were... |

277 | Octrees for faster isosurface generation - Wilhelms, Gelder - 1990 |

161 | Optimal surface reconstruction from planar contours - Fuchs, Kedmen, et al. - 1977 |

136 |
The asymptotic decider: resolving the ambiguity in marching cubes
- Nielson, Hamann
- 1991
(Show Context)
Citation Context ...on of this work at a workshop [WVG90], other researchers have developed additional methods to resolve ambiguities using an assumption of trilinearity in each cell [Nat91], or bilinearity in each face =-=[NH91]-=-. 3.1 Nature of the Data Isosurfaces may be used to represent scalar data values that are distributed in a three-dimensional coordinate system, such as density values generated by a CT-scan or tempera... |

116 | V-BUFFER: Visible Volume Rendering - Upson, Keeler - 1988 |

95 | Guaranteed ray intersections with implicit surfaces - Kalra, Barr - 1989 |

65 | Three-Dimensional Display of Human Organs from Computed Tomograms - Herman - 1979 |

60 | A Class of Local Interpolating Splines - Catmull, Rom - 1974 |

58 |
The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm
- Artzy, Frieder, et al.
- 1981
(Show Context)
Citation Context ...xperience is reported in Section 5.2. Simple Boolean This category includes the simplest policies, such as \always connect the positive diagonal," which is implicit in early boundary tracking methods =-=[AFH81]-=-. The marching cubes algorithm [LC87, CLL + 88, GN89], uses the signs of the eight cell vertices (relative to the threshold) to index a table of 15 cases containing a polygonization of the isosurface.... |

51 | Conversion of complex contour line definitions into polygonal element mosaics - Christiansen, Sederberg - 1978 |

45 | E.C.: Marching cubes: a high resolution 3D surface construction algorithm - William, Harvey - 1987 |

43 |
Shading 3D-images from CT using gray-level gradients
- Höhne, Bernstein
- 1986
(Show Context)
Citation Context ...behavior of the sampled data between sample points, when extracting isosurfaces [CLL + 88], for direct volume rendering [Lev88, UK88], and for producing more sample points from those already provided =-=[HB86]-=-. It is attractive because it is simple to calculate. 13 The trilinear model of the cell may be used for connectivity decisions, as described by Natarajan [Nat91]. Nielson and Hamann considered biline... |

36 |
Letters: Additional reference to ”marching cubes
- Dürst
- 1988
(Show Context)
Citation Context ... based on a precomputed table of 15 topologically distinct plus-minus patterns of cell corners. In the original implementation, \marching cubes" did not recognize ambiguities. As pointed out by Durst =-=[Dur88]-=-, that method could yield a discontinuity between cells. Later researchers [Bak89, Nat91, Kal91], as well as the original authors, described modi cations that ensured continuity. Some variants are dis... |

30 |
Topological considerations in isosurface generation – extended abstract
- Wilhelms, Gelder
- 1990
(Show Context)
Citation Context ...not address ambiguities. Winget also briefly discusses isosurfaces resulting from finite element analysis 3 [Win88]. Subsequent to the presentation of a preliminary version of this work at a workshop =-=[WVG90]-=-, other researchers have developed additional methods to resolve ambiguities using an assumption of trilinearity in each cell [Nat91], or bilinearity in each face [NH91]. 3.1 Nature of the Data Isosur... |

26 | An Efficient Method of Triangulating Equivalued Surfaces by Using Tetrahedral Cells - Doi, Koide - 1991 |

23 | Representation of three-dimensional digital images - Srihari - 1981 |

20 | Building surfaces of evolution: The weaving wall - Baker - 1989 |

20 | Surface Shading in the Cuberille Environment - Chen, Herman, et al. - 1985 |

19 | 3D reconstruction of the brain from magnetic resonance images using a connectivity algorithm - Cline, Dumoulin, et al. - 1987 |

16 | An efficient 3-d visualization technique for finite element models and other coarse volumes. Computer Graphics
- Gallagher, Nagtegaal
- 1989
(Show Context)
Citation Context ...n cells. Later researchers [Bak89, Nat91, Kal91], as well as the original authors, described modifications that ensured continuity. Some variants are discussed in Section 4.1. Gallagher and Nagtegaal =-=[GN89]-=- generalized the above approaches to irregular lattices that frequently occur in finite element analysis, and considered higher-degree surfaces instead of triangulation. They also do not address ambig... |

11 | Polygonizaion of Implicite Surface,” Computer-Aided Geometric Design - Bloomenthal - 1988 |

11 |
Boundary Detection in Multidimensions
- Udupa, Srihari, et al.
- 1982
(Show Context)
Citation Context ...s that the surface changes if positive and negative are inverted; the changes may be topological as well as spatial. These changes result from the difference between 1-connectivity and 3-connectivity =-=[USH82]-=- (see Definition 3.2). See Section 5.1 for further discussion. Beveled surfaces permit patches with any orientation, so have the potential of providing a closer approximation to the underlying surface... |

9 |
Polyhedral approximation approach to molecular orbit graphics
- Koide, Doi, et al.
- 1986
(Show Context)
Citation Context ...s (the facial average method, discussed later). Koide et al. reported a more complex method based on decomposition of the cell into tetrahedra, and polygonizing the isosurface within each tetrahedron =-=[KDK86]-=-. Ambiguities are implicitly resolved by the decomposition. Lorensen et al. reported a simpler approach [LC87, CLL + 88], where the data samples were essentially the only information available about t... |

8 | A Three-Dimensional Display System for Diagonostic Imaging Applications - COOK, DWYER, et al. - 1983 |

7 |
Segmentation and surface-based modeling of objects in three-dimensional biomedical images
- Kalvin
- 1991
(Show Context)
Citation Context ...nto two triangles by another interior chord. Kalvin points out that it is important that the final surfaces not intersect themselves or each other, so that they describe physically realizable objects =-=[Kal91]. We shall-=- call this the non-intersection property . He proves that the "cuberille method" of disambiguation (always connect positive diagonals in ambiguous faces) has this property. There are certain... |

7 | Two Algorithms for the Reconstruction of Surfaces from Tomograms - CLINE, LORENSEN, et al. - 1988 |

7 | V-bu er: Visible volume rendering - Upson, Keeler - 1988 |

6 |
Boundary and object labelling in three-dimensional images
- Udupa, Ajjanagadde
- 1990
(Show Context)
Citation Context ...r cell is the same case upside down. The picture is similar to Part (c, right) of Figure 4. Udupa and Ajjanagadde have described three simple boolean connectivity policies (their terminology differs) =-=[UA90]. Positive values are considered to repr-=-esent the "objects". (In the cited paper "R u = R 1 " corresponds to our label "UA-R1", etc.; the "UA" gives the authors' initials.) UA-R1 Always connect negati... |

6 | ISOSURF - An Algorithm for Plotting Iso-Valued Surfaces of a Function of Three Variables - WRIGHT, HUMBRECHT - 1979 |

5 |
On generating topologically correct isosurfaces from uniform samples
- Natarajan
- 1991
(Show Context)
Citation Context ...e presentation of a preliminary version of this work at a workshop [WVG90], other researchers have developed additional methods to resolve ambiguities using an assumption of trilinearity in each cell =-=[Nat91]-=-, or bilinearity in each face [NH91]. 3.1 Nature of the Data Isosurfaces may be used to represent scalar data values that are distributed in a three-dimensional coordinate system, such as density valu... |

5 | Conversion of complex contour line de nitions into polygonal element mosaics - Christiansen, Sederberg - 1978 |

4 |
Letters: Additional reference to "marching cubes
- Durst
- 1988
(Show Context)
Citation Context ... based on a precomputed table of 15 topologically distinct plus-minus patterns of cell corners. In the original implementation, "marching cubes" did not recognize ambiguities. As pointed out=-= by Durst [Dur88]-=-, that method could yield a discontinuity between cells. Later researchers [Bak89, Nat91, Kal91], as well as the original authors, described modifications that ensured continuity. Some variants are di... |

4 | Display of 3d digital images: Computational foundations and medical applications - Herman, Udupa - 1983 |

4 | Visualization in anthropology: Reconstruction of human fossils from multiple pieces - Kalvin, Dean, et al. - 1992 |

3 |
Advanced graphics hardware for finite element results display
- Winget
- 1987
(Show Context)
Citation Context ...ment analysis, and considered higher-degree surfaces instead of triangulation. They also do not address ambiguities. Winget also briefly discusses isosurfaces resulting from finite element analysis 3 =-=[Win88]-=-. Subsequent to the presentation of a preliminary version of this work at a workshop [WVG90], other researchers have developed additional methods to resolve ambiguities using an assumption of trilinea... |

2 | Display of medical objects and their interactive manipulation - Udupa - 1989 |

2 |
An e cient 3-D visualization technique for nite element models
- Gallagher, Nagtegaal
- 1989
(Show Context)
Citation Context ...en cells. Later researchers [Bak89, Nat91, Kal91], as well as the original authors, described modi cations that ensured continuity. Some variants are discussed in Section 4.1. Gallagher and Nagtegaal =-=[GN89]-=- generalized the above approaches to irregular lattices that frequently occur in nite element analysis, and considered higher-degree surfaces instead of triangulation. They also do not address ambigui... |

2 | Three-dimensional display ofhuman organs from computer tomograms - Herman, Liu - 1979 |

1 | Three-dimensional skeletonization - Lobregt, Verbeck - 1980 |

1 |
Quantitative and qualitative comparison of volumetric and surface rendering techniques
- Rusinek, Noz, et al.
- 1991
(Show Context)
Citation Context ...r on the side of disconnecting thin objects that should be connected. Rusinek et al. have reported difficulty in visualizing thin bones with Kalvin's "Alligator" method, which uses UA-R1 con=-=nectivity [RNMK91]-=-. Their experiment used a CT-scan of a dry skull. On similar data, we have also noticed that small tunnels sometimes appear in thin bones using UA-R1 connectivity, but they disappear using others. How... |

1 | The theory, design, implementation, and evaluation of a three-dimensional surface detection algorithm - ARTZV, FRtEDER, et al. - 1981 |

1 | Building surfaces of evolution: The weaving wall. ht - BLOOMENTHAL - 1989 |

1 | Conversion of complex contour line definitions into polygonal element mosaics - CHRISTENSEN, ANDSEDERBERG - 1978 |

1 | An efficient method of triangulating equi-valued surfaces by using tetrahedral cells - Dot, ANDKOIDE - 1991 |

1 | Three-dimensional display of human organs from computer tomography - LJU - 1979 |

1 | Polyhedral approximation approach ta molecular orbit graphics - KAJIORA - 1986 |