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Contributing vertices-based Minkowski difference (CVMD) of polyhedra and applications

Contributing vertices-based Minkowski difference (CVMD) of polyhedra and applications Minkowski sum and difference, well known as dilation and erosion in image analysis, constitute the kernel of mathematical morphology. Extending the interesting results of the later theory from 2D images to 3D meshes is very promising. However, this requires the development of robust algorithms for the computation of Minkowski operations, which is far from being an easy task. The contributing vertices concept has been introduced for the computation of Minkowski sum of convex polyhedra, and later extended for the computation of non-convex/convex pairs of polyhedra. For Minkowski difference, the available literature is poor. In this work, we demonstrate the duality of the contributing vertices concept w.r.t. Minkowski operations, and propose an exact and efficient Contributing Vertices-based Minkowski Difference (CVMD) algorithm for polyhedra. Our algorithm operates on convex polyhedra and on pairs of convex/non-convex polyhedra without any modification. We also show its beneficial application where the second operand is represented in 3D by an implicit surface or a point cloud, among other possible representations. The conducted benchmarks show that CVMD largely outperforms an indirect Nef polyhedrabased approach we implemented in order to validate our results. All our implementations produce exact results and supplementary materials are provided [Figure not available: see fulltext.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png 3D Research Springer Journals

Contributing vertices-based Minkowski difference (CVMD) of polyhedra and applications

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References (34)

Publisher
Springer Journals
Copyright
Copyright © 2013 by 3D Research Center, Kwangwoon University and Springer-Verlag Berlin Heidelberg
Subject
Engineering; Signal, Image and Speech Processing; Computer Imaging, Vision, Pattern Recognition and Graphics; Optics, Optoelectronics, Plasmonics and Optical Devices
eISSN
2092-6731
DOI
10.1007/3DRes.04(2013)1
Publisher site
See Article on Publisher Site

Abstract

Minkowski sum and difference, well known as dilation and erosion in image analysis, constitute the kernel of mathematical morphology. Extending the interesting results of the later theory from 2D images to 3D meshes is very promising. However, this requires the development of robust algorithms for the computation of Minkowski operations, which is far from being an easy task. The contributing vertices concept has been introduced for the computation of Minkowski sum of convex polyhedra, and later extended for the computation of non-convex/convex pairs of polyhedra. For Minkowski difference, the available literature is poor. In this work, we demonstrate the duality of the contributing vertices concept w.r.t. Minkowski operations, and propose an exact and efficient Contributing Vertices-based Minkowski Difference (CVMD) algorithm for polyhedra. Our algorithm operates on convex polyhedra and on pairs of convex/non-convex polyhedra without any modification. We also show its beneficial application where the second operand is represented in 3D by an implicit surface or a point cloud, among other possible representations. The conducted benchmarks show that CVMD largely outperforms an indirect Nef polyhedrabased approach we implemented in order to validate our results. All our implementations produce exact results and supplementary materials are provided [Figure not available: see fulltext.]

Journal

3D ResearchSpringer Journals

Published: Sep 26, 2013

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