## Arbitrary topology shape reconstruction from planar cross sections (1996)

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Venue: | Graphical Models and Image Processing |

Citations: | 77 - 12 self |

### BibTeX

@INPROCEEDINGS{Bajaj96arbitrarytopology,

author = {Chandrajit L. Bajaj and Edward J. Coyle and Kwun-nan Lin},

title = {Arbitrary topology shape reconstruction from planar cross sections},

booktitle = {Graphical Models and Image Processing},

year = {1996},

pages = {524--543}

}

### Years of Citing Articles

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### Abstract

In computed tomography, magnetic resonance imaging and ultrasound imaging, reconstruction of the 3D object from the 2D scalar-valued slices obtained by the imaging system is di cult because of the large spacings between the 2D slices. The aliasing that results from this undersampling in the direction orthogonal to the slices leads to two problems known as the correspondence problem and the tiling problem. A third problem, known as the branching problem, arises because of the structure of the objects being imaged in these applications. Existing reconstruction algorithms typically address only one or two of these problems. In this paper, we approach all three of these problems simultaneously. This is accomplished by imposing a set of three constraints on the reconstructed surface and then deriving precise correspondence and tiling rules from these constraints. The constraints ensure that the regions tiled by these rules obey physical constructs and have a natural appearance. Regions which cannot be tiled by these rules without breaking one or more constraints are tiled with their medial axis (edge Voronoi diagram). Our implementation of the above approach generates triangles of 3D isosurfaces from input which is either a set of contour data or a volume of image slices. Results obtained with synthetic and actual medical data are presented. There are still speci c cases in which our new approach can generate distorted results, but these cases are much less likely to occur than those which cause distortions in other tiling approaches. 2 1

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Citation Context ...nt surface reconstructions. Algorithms [4, 6, 7, 10, 11, 14, 15, 25, 28] which attempt to tile all contour vertices to the adjacent slice produce an unlikely topology, as shown in Fig. 3b. Boissonnat =-=[2]-=- and Barequet et al. [1] produce horizontal triangles which lie on the slice, thus avoiding tilings like those in Fig. 3b. This generates a result better than Fig. 3b without adding any intermediate v... |

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Citation Context ... certain metrics such as surface area and enclosed volume. The other is the topological correctness of the tiling. The problem of mating points between contours into triangles is formalized by Keppel =-=[15]-=- into a graph search problem. Fuchs et al. [10] provide an e cient algorithm based on an Euler tour of a toroidal graph to obtain an optimal solution. Their algorithm has a time complexityofO(n 2 log ... |

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Citation Context ...f the traced untiled polygons. If the projection of an untiled region is convex, it is triangulated with its center of gravity. Otherwise, we triangulate using its medial axis. We use Lee's algorithm =-=[16]-=- to nd the EV D (edge Voronoi diagram or medial axis). The EV D is approximated by a smaller number of line segments so fewer triangles are required to cover the untiled region. The Z values of the EV... |

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Citation Context ...the desired surface. He reduces the problem of tetrahedralization of the object delimited by the two slices into building tetrahedra from two 2D Delaunay triangulations of two adjacent slices. Geiger =-=[12]-=- improves Boissonnat's approach so it can handle complicated branching and dissimilar contours. He projects the external Voronoi skeleton (EV S) from one slice to the adjacent slice and adds the proje... |

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Citation Context ...ns. Algorithms [4, 6, 7, 10, 11, 14, 15, 25, 28] which attempt to tile all contour vertices to the adjacent slice produce an unlikely topology, as shown in Fig. 3b. Boissonnat [2] and Barequet et al. =-=[1]-=- produce horizontal triangles which lie on the slice, thus avoiding tilings like those in Fig. 3b. This generates a result better than Fig. 3b without adding any intermediate vertices. 2.3 The Branchi... |

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Citation Context ...resulting from the same cross sections as in (a). If the distance between slices is large, a priori knowledge or global information is required to determine the correct correspondence. Bresler et al. =-=[3]-=- use domain knowledge to constrain the problem. Meyers et al. [20] and Soroka [26] approximate the contours by ellipses and then assemble them into cylinders to determine the correspondence. Wang et a... |

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Citation Context ...m surface area optimization approach produces fewer abnormalities. When two corresponding contours are very di erent, it is di cult to obtain a topologically correct and natural tiling. Gitlin et al. =-=[13]-=- show one example in which two polygons cannot be tiled to form a polyhedron. Their example is a pair of extremely di erent contours. Even in a moderately dissimilar contour pair in which a polyhedron... |

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Citation Context ...e domain knowledge to constrain the problem. Meyers et al. [20] and Soroka [26] approximate the contours by ellipses and then assemble them into cylinders to determine the correspondence. Wang et al. =-=[28]-=- check the overlapping area as the criterion for the correspondence. 2.2 The Tiling Problem Tiling means using slice chords to triangulate the strip lying between contours of two adjacent slices into ... |

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Citation Context ..., where n is the total number of vertices on the contours bounding the triangles. Sloan et al. [25] locate bottlenecks and improve the speed of Fuchs' algorithm by a constant factor. Shinagawa et al. =-=[24]-=- generalize the discrete toroidal graph into a continuous one. Homotopy is used for reconstructing smooth surfaces from the toroidal graph. Homotopy is similar to metamorphosis (morphing) in which one... |

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Citation Context ...ained, the goal is to enable a human to easily visualize, in 3D, this large collection of data. Many algorithms have been developed for this purpose, but they can all be classi ed into two categories =-=[8]-=-: volume rendering methods and surface reconstruction methods. This paper concentrates on surface reconstruction methods, all of which proceed by extracting the isosurfaces corresponding to a speci ed... |

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Citation Context ...the shaded result are presented in (h) and (i). In the case of one untiled region's projection enclosing the projection of another untiled region, one can use the EV D algorithm of Srinivasan et. al. =-=[27]-=- to calculate the medial axis of nested polygons. Because our algorithm tries to do as much tiling as possible during the last tiling pass, the number of untiled regions left is near minimum. The case... |

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Citation Context ...her a hole or the beginning/end of a vertical feature. The possibility of branching signi cantly complicates the task of tiling. It creates the problem of branching surface reconstruction. Lin et al. =-=[17]-=- model branching regions by interpolating many intermediate contours. This method generates a smooth surface at the cost of a large number of triangles. Other branch processing approaches can be class... |

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Citation Context ...ur of a toroidal graph to obtain an optimal solution. Their algorithm has a time complexityofO(n 2 log n), where n is the total number of vertices on the contours bounding the triangles. Sloan et al. =-=[25]-=- locate bottlenecks and improve the speed of Fuchs' algorithm by a constant factor. Shinagawa et al. [24] generalize the discrete toroidal graph into a continuous one. Homotopy is used for reconstruct... |

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Citation Context ...is large, a priori knowledge or global information is required to determine the correct correspondence. Bresler et al. [3] use domain knowledge to constrain the problem. Meyers et al. [20] and Soroka =-=[26]-=- approximate the contours by ellipses and then assemble them into cylinders to determine the correspondence. Wang et al. [28] check the overlapping area as the criterion for the correspondence. 2.2 Th... |

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Citation Context ... C1 L C2 C1 C2 L C3 C3 L (a) (b) (c) (d) (e) Figure 4: Di erent reconstructions for branching contours: (a) branching contours on adjacent slices; (b)-(e) di erent surface reconstructions. Algorithms =-=[4, 6, 7, 10, 11, 14, 15, 25, 28]-=- which attempt to tile all contour vertices to the adjacent slice produce an unlikely topology, as shown in Fig. 3b. Boissonnat [2] and Barequet et al. [1] produce horizontal triangles which lie on th... |

8 |
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Citation Context ...ur onto its convex hull to look up the tiling pair from the convex hull tiling. This method avoids some abnormalities produced by Christiansen's algorithm. After comparing di erent algorithms, Meyers =-=[19]-=- points out that the minimum surface area optimization approach produces fewer abnormalities. When two corresponding contours are very di erent, it is di cult to obtain a topologically correct and nat... |

7 |
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Citation Context ... is used for reconstructing smooth surfaces from the toroidal graph. Homotopy is similar to metamorphosis (morphing) in which one contour is gradually changed into another contour. Kehtarnavaz et al. =-=[14]-=- represent the search problem as a Levenshtein graph and use dynamic programming to nd its minimum cost path. Wang et al. [28] present a method which rst assigns an initial merit to each triangle. It ... |

5 |
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(Show Context)
Citation Context ...ghts, and it nishes by utilizing the A search algorithm to nd a triangulation with minimum weight. Some fast heuristic methods have also been developed for tiling. The strategy of Christiansen et al. =-=[4]-=- is based on selection of shortest slice chords. Ganapathy et al. [11] use the concept of least tension as a heuristic guideline to tiling. These heuristic methods [4, 11] usually work quickly and wor... |

2 |
On the scaling heuristic for reconstruction from slices
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(Show Context)
Citation Context ...2b and 22c could be correctly generated by translating and scaling the contours until Theorem 6 generates the actual correspondence, and then translating and scaling back the tiling results. O'Rourke =-=[21]-=- points out a potential aw in scaling and suggests using a uniform scaling (with the same amount in both x and y). In this latter case, though, a signi cant amount of side information must be present ... |

1 |
TnT — A Statistical Partof-Speech Tagger
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(Show Context)
Citation Context ...al triangles. In terms of topological correctness, the best branching handling is (b) because it corresponds to the expected physical object better than the others do. Christiansen et al. [4], Shantz =-=[23]-=-, and Shinagawa et al. [24] use the method in Fig. 4d. They dip down the middle of the bridge to model the saddle point of the branching region. This approach works well only in simple branching cases... |