Computational geometry: an introduction
Computational geometry: an introduction
Computer Vision, Graphics, and Image Processing
The quickhull algorithm for convex hulls
ACM Transactions on Mathematical Software (TOMS)
Computational geometry in C (2nd ed.)
Computational geometry in C (2nd ed.)
Polynomial/rational approximation of Minkowski sum boundary curves
Graphical Models and Image Processing
Using generic programming for designing a data structure for polyhedral surfaces
Computational Geometry: Theory and Applications - Special issue on applications and challenges
Fast penetration depth estimation using rasterization hardware and hierarchical refinement
Proceedings of the nineteenth annual symposium on Computational geometry
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Accurate Minkowski sum approximation of polyhedral models
Graphical Models - Special issue on PG2004
Spatial Planning: A Configuration Space Approach
IEEE Transactions on Computers
Contributing vertices-based Minkowski sum computation of convex polyhedra
Computer-Aided Design
Algorithmica - Special Issue: European Symposium on Algorithms 2007, Guest Editors: Larse Arge and Emo Welzl
Determining the directional contact range of two convex polyhedra
Computer-Aided Design
Discrete critical values: a general framework for silhouettes computation
SGP '09 Proceedings of the Symposium on Geometry Processing
A New Algorithm for the Computation of the Minkowski Difference of Convex Polyhedra
SMI '10 Proceedings of the 2010 Shape Modeling International Conference
Contributing vertices-based Minkowski sum of a nonconvex--convex pair of polyhedra
ACM Transactions on Graphics (TOG)
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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.]