Voronoi Diagrams in CGAL, the Computational Geometry Algorithms Library

This work describes the implementation of computational geometry algorithms developed within the Insight Toolkit (ITK): Distance Transform (DT), Voronoi diagrams, k Nearest Neighbor (kNN) transform, and finally a K Nearest Neighbor classifier for multichannel data, that is used for supervised segmentation. We have tested this algorithm for 2D and 3D medical datasets, and the results are excellent in terms of accuracy and performance. One of the strongest points of the algorithms described here is that they can be used for many other applications, because they are based on the ordered propagation paradigm. This idea consists in actually not raster scan the image but rather in start from the image objects and propagate them until the image is totally filled. This has been demonstrated to be a good approach in many algorithms as for example, computation of Distance Transforms, Voronoi Diagrams, Fast Marching, skeletons computation, etc. We show here that these algorithms have low computational complexity and it provides excellent results for clinical applications as the segmentation of brain MRI.

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GEODESIC-PRESERVING POLYGON SIMPLIFICATION

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195914600097 ◽

2014 ◽

Vol 24 (04) ◽

pp. 307-323 ◽

Cited By ~ 3

Author(s):

OSWIN AICHHOLZER ◽

THOMAS HACKL ◽

MATIAS KORMAN ◽

ALEXANDER PILZ ◽

BIRGIT VOGTENHUBER

Keyword(s):

Data Structure ◽

Computational Geometry ◽

Linear Time ◽

Shortest Paths ◽

Voronoi Diagrams ◽

Time Dependent ◽

Post Processing ◽

Running Time ◽

Intrinsic Complexity ◽

Processing Steps

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon [Formula: see text] by a polygon [Formula: see text] such that (1) [Formula: see text] contains [Formula: see text], (2) [Formula: see text] has its reflex vertices at the same positions as [Formula: see text], and (3) the number of vertices of [Formula: see text] is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both [Formula: see text] and [Formula: see text], our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of [Formula: see text]. We describe several of these applications (including linear-time post-processing steps that might be necessary).

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