Graphical Models and Image Processing
On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees
Computational Geometry: Theory and Applications
Surface Approximation and Geometric Partitions
SIAM Journal on Computing
Integral Operators for Computing Homology Generators at Any Dimension
CIARP '08 Proceedings of the 13th Iberoamerican congress on Pattern Recognition: Progress in Pattern Recognition, Image Analysis and Applications
Directly computing the generators of image homology using graph pyramids
Image and Vision Computing
Irregular Graph Pyramids and Representative Cocycles of Cohomology Generators
GbRPR '09 Proceedings of the 7th IAPR-TC-15 International Workshop on Graph-Based Representations in Pattern Recognition
Polyhedral approximation and practical convex hull algorithm for certain classes of voxel sets
Discrete Applied Mathematics
On the cohomology of 3D digital images
Discrete Applied Mathematics - Special issue: Advances in discrete geometry and topology (DGCI 2003)
Invariant representative cocycles of cohomology generators using irregular graph pyramids
Computer Vision and Image Understanding
Convex hulls in a 3-dimensional space
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
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Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P(I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H*(P(I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H*(P(I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space.