## Delaunay Reconstruction from Multiaxial Planar Cross-Sections (1997)

Citations: | 7 - 0 self |

### BibTeX

@TECHREPORT{Dance97delaunayreconstruction,

author = {C. R. Dance and R. W. Prager},

title = {Delaunay Reconstruction from Multiaxial Planar Cross-Sections},

institution = {},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

this report is as follows:

### Citations

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Citation Context ... total can be obtained in O(n) time 3 . This is interesting since it implies that the reconstruction problem is easier than triangulating a set of n points in a plane, for which O(n log n) is optimal =-=[48]-=-. In a weaker sense it is also easier than triangulating a simple n-gon since for this problem only formidably complex algorithms O(n) are known [41]. 2 Strictly, it is required to find a simple close... |

484 |
M.: Zippered polygon meshes from range images
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Citation Context ...ith those proposed in this report may be obtained from [17, 64]. In some situations, such as laser range imaging, the points or contours are occluded, so multiple views of the object must be combined =-=[61]-=-. The problem of reconstruction in stereo vision is harder still since only poor knowledge of point depth is available [21, 20]. A more constrained problem which is intimately related to the approach ... |

463 |
Algorithms in C
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Citation Context ...ly considers the case of arrangements with no bounding region. For discussion of the linear algebra involved in finding intersections and interior points of lines and planes the reader is referred to =-=[41, 13, 52]-=-. Recall from Section 5.2 that a sub-arrangement may be optimally constructed incrementally. At each stage, the set of faces in an existing arrangement cut by the new plane (the zone of the plane) is ... |

136 | A physically based approach to 2-D shape blending
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Citation Context ...n. Finally, intermediate sections of reconstructions from parallel sections are interpolations of the shapes of the contours in those sections. Thus they may be used for morphing in computer graphics =-=[51]-=- and for the averaging of random shapes. Similarly, techniques from those fields may come to be useful in reconstruction. 3 The Problem 3.1 General Considerations original object plane of section sect... |

89 |
Surface coding based on Morse theory
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Citation Context ...e centroids and difference between their principal axis lengths. Methods guaranteeing topological consistency have been developed which build trees describing the inside-outside relations of contours =-=[54, 55]-=- and combine them with minimum spanning tree heuristics [26]. 4.2 Volume Reconstruction from Parallel Sections As described in Section 3 volume reconstruction methods form a mesh over the region of in... |

71 |
Efficient ThreeDimensional Delaunay Triangulation With Automatic Point Creation and Imposed Boundary Constraints
- Weatherill, Hassan
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(Show Context)
Citation Context ..., 20]. A more constrained problem which is intimately related to the approach to reconstruction taken here is that of meshing a specified volume. This is usually performed for finite element analysis =-=[65]-=-, computer graphics or computer aided design. Finally, intermediate sections of reconstructions from parallel sections are interpolations of the shapes of the contours in those sections. Thus they may... |

57 |
Constructing a Reeb graph automatically from cross sections
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Citation Context ...e centroids and difference between their principal axis lengths. Methods guaranteeing topological consistency have been developed which build trees describing the inside-outside relations of contours =-=[54, 55]-=- and combine them with minimum spanning tree heuristics [26]. 4.2 Volume Reconstruction from Parallel Sections As described in Section 3 volume reconstruction methods form a mesh over the region of in... |

29 |
Closed Object Boundaries from Scattered Points
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Citation Context ...erform global interpolation. Fitting of splines requires a set of location parameters to be determined. This often requires an existing underlying surface which is normally generated by triangulation =-=[34, 64]-=- or an expensive optimisation procedure. It is difficult to guarantee that splines are smooth in 3D space (visual continuity) and that they do not selfintersect, although solutions to both these probl... |

19 |
Reconstructing 3D objects from cross-sections, in Computation of Curves and Surfaces
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Citation Context ...an [11, 53]. Volume based approaches [5] and reconstruction from non-parallel planes [46] have only been studied in the last decade. Reviews of the surface from cross-sections problem may be found in =-=[50, 34, 37]-=-. 4.1 Surface Reconstruction from Parallel Sections Optimal Tiling. Tiling has been addressed by constrained optimisation of various costs. The earliest solution maximised the volume of the reconstruc... |

18 |
Connect-the-dots: a new heuristic
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Citation Context ...ea. In later papers, these authors started from the Delaunay triangulation (see Section 5.4) of the points: Boissonnat [4] sculpted to minimise the change in surface area at each step, while O'Rourke =-=[43]-=- minimised the length of the Voronoi skeleton (see Section 5.4.3) of the reconstructed object. Eventually, Boissonnat applied the Delaunay triangulation to the reconstruction from parallel contours [5... |

18 |
Surface definition for branching contour defined objects
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Citation Context ...ions of optimal tiling for the case of one contour per slice. Soon after, ways of improving reconstruction speed by heuristic tiling [11] were introduced and generalisation to multiple contours began =-=[11, 53]-=-. Volume based approaches [5] and reconstruction from non-parallel planes [46] have only been studied in the last decade. Reviews of the surface from cross-sections problem may be found in [50, 34, 37... |

17 |
Toga , “Surface reconstruction by multiaxial triangulation
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- 1994
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Citation Context ...al case of arbitrarily oriented (multi-axial) planes, which has received much less attention. Indeed, it appears that only one algorithm for this case has previously been published (by Payne and Toga =-=[46]-=-). Ideas from [46] are incorporated in the procedure developed here, but the restrictive assumptions on the shape of the contours which are required by their method are removed. This is achieved by ge... |

16 |
National Library of Medicine. “The Visible Human Project
- S
- 1994
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Citation Context ...tificial data of an ellipsoid and a more complex object with holes. Then some examples with real data are demonstrated: firstly the reconstruction of a pelvis with data from the Visible Human Project =-=[62]-=- and secondly reconstruction of 3D ultrasound data of gall bladders. The latter example is also used for volume estimation. 8.1 Ellipsoid Consider the reconstruction of an ellipsoid from random planar... |

15 | 3-D shape representation by contours
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Citation Context ...f. Whether the surface or volume paradigm is chosen, there are many ways of solving each subproblem. The general principles which should guide any shape reconstruction mechanism have been examined in =-=[66]-=-. For many applications, a successful solution may only be judged by its consistency with human perception and knowledge. In other cases, performance is quantifiable, for example by comparing distance... |

14 |
Interactive Segmentation a d Boundary Surface Formation for 3-D
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Citation Context ... it is assumed that data is available as a 3D grid (or cuberille), each cell of 2. RELATED PROBLEMS 3 which is known as a voxel. The simplest such approach gives a surface as a set of faces of voxels =-=[63]-=-. Often the grid is more widely spaced in one direction than another, for example to ensure minimal radiation dosages in CT. The set of voxels lying on the boundary of the object of interest would be ... |

13 |
Surface reconstruction between simple polygons via angle criteria
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Citation Context ...has been pointed out in [66] that a reconstruction algorithm should be invariant of net rotations or scalings of its input. Three other natural criteria are that a reconstruction preserve 6 convexity =-=[67]-=-, that it be self-consistent and that it be fair. Preservation of convexity means that if the slices consist of convex regions, the resulting reconstruction should also be convex. Self-consistency mea... |

10 |
Hexahedral Mesh Generation by Medial Axis Subdivision: II. Solids with Flat and Concave
- Price, Armstrong
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Citation Context ...s at a 4 time. More sections must be treated simultaneously to ensure curvature constraints are satisfied. Finally, the use of other linear surfaces constructed from higher order polygons is possible =-=[49]-=-. Such surfaces approximate rather than interpolate the data because of their linearity requirement. 2.3 Related Topics Several important but less constrained problems are closely related to reconstru... |

10 |
Generalised cones from serial sections
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Citation Context ... [10]. 4. PREVIOUS SOLUTIONS 9 Correspondence. Decisions about which regions to connect in multiple contour cases (the correspondence problem) have only been addressed recently, with the exception of =-=[57]-=-, where a rule-based system using generalised cones was proposed. Previously it was assumed that the topology of the object is either obvious or needs to be determined by user interaction [26]. The si... |

9 |
Smooth surface approximation to serial cross-sections. Computer-Aided Design
- Park, Kim
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(Show Context)
Citation Context ...f parameters to 3D space (parametric surfaces) or as the zeros of a scalar function with 3D argument (implicit surfaces). B-splines are the most widely used parametric surfaces. Nonrational B-splines =-=[44]-=- and their rational counterparts [47] have recently been used for reconstruction from contours. Both of these works provide interesting references on the smooth surface from cross-sections problem. Th... |

8 | Optimal two-dimensional triangulations
- Tan
- 1993
(Show Context)
Citation Context ...ersions of the problem. In 4D, the major difficulty is the fact that a guaranteed method of making a Delaunay triangulation conform to an arbitrary polyhedral surface is presently an unsolved problem =-=[58]-=-: it is easy to make the edges of a surface conform but a surface may not be present in a triangulation even if its edges are. 6.2 Arrangements Recall from Section 5.2 that a sub-arrangement is a desc... |

6 |
Algorithm for approximate nurbs skinning
- Piegl, L
- 1997
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Citation Context ...umetry in ultrasound [27] and in the microscopy of cells [33]. ffl Industry. In computer aided design, lofting techniques specify and then generate the geometry of an object as a sequence of sections =-=[47]-=- 1 . Robotics and reverse engineering use laser range imaging which often leads to reconstruction from sections. Non-destructive evaluation sometimes employs interpolation of contours from ultrasound ... |

6 | An optimal bound for high-quality conforming triangulations. GEOMETRY: Discrete and Computational Geometry 15
- Tan
- 1996
(Show Context)
Citation Context ...may be placed anywhere to make a Delaunay triangulation conform. For a planar graph with m edges and n points \Omega\Gamma mn) points are necessary but the best algorithm to date adds O(m 2 n) points =-=[59]-=-. 5. THE IDEA 23 5.4.3 Voronoi Diagrams and Delaunay Splits It has just been indicated that new points must be added in order that the Delaunay triangulation contain the edges of a polygon to be inter... |

1 |
Evaluation of gall bladder volume and emptying with a novel 3-dimensional ultrasound system -- comparison with the sum-ofcylinders and the ellipsoid methods
- Pauletzki, Sackmann, et al.
- 1996
(Show Context)
Citation Context ...s not the case for other regularly scanned organs like the liver and thyroid. Finally, volume measurement, an interesting application for reconstruction, has been well-documented for the gall bladder =-=[45]-=-. All scan data was acquired from a single subject during a one hour session. The subject had fasted for over 6 hours prior to scanning, and continued this fast during the scanning period. This ensure... |

1 |
Pessimal guesses may be optimal
- Sloan, Painter
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Citation Context ...ds were designed for the case of a single pair of contours, it is possible to generalise them by techniques outlined below. Heuristic Tiling. Optimal tiling is slow despite attempts to make it faster =-=[22, 56]-=-. Therefore, heuristic approaches have been suggested, based on first transforming contours into more standardised shapes, then rapidly triangulating by local criteria (local advancement rules) and fi... |

1 | CTI: automatic construction of complex 3D surfaces from contours using the Delaunay triangulation
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(Show Context)
Citation Context ...53]. However, Geiger indicated how these points could be efficiently obtained from the Delaunay triangulations of individual sections. Other authors have since implemented the same ideas, for example =-=[60]-=-. 10 4.3 Surface Reconstruction from Non-Parallel Sections Several authors on reconstruction from parallel cross-sections have suggested that their methods might be extended to work in the non-paralle... |

1 |
Computing Shape: an introduction to the representation of component and assembly geometry for computer-aided engineering. Butterworths
- Woodwark
- 1986
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Citation Context ... of transverse ribs to guide the shape and then nailing longitudinal strips to these. This takes up much space and therefore used to be done in the lofts of shipyard buildings, hence the name lofting =-=[68]-=-. 2 Intersecting planes can definitively show that regions are connected: the data may provide absolute constraints on topology between planes. ffl Object Shape. Objects whose shape variation is not w... |