Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Introduction to Solid Modeling
Introduction to Solid Modeling
Structural operators for modeling 3-manifolds
SMA '97 Proceedings of the fourth ACM symposium on Solid modeling and applications
Proceedings of the sixth ACM symposium on Solid modeling and applications
Introduction to Combinatorial Pyramids
Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000]
Computer Graphics and Geometric Modelling: Mathematics
Computer Graphics and Geometric Modelling: Mathematics
Euler operators for stratified objects with incomplete boundaries
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
GWB: A Solid Modeler with Euler Operators
IEEE Computer Graphics and Applications
Computing homology group generators of images using irregular graph pyramids
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Insertion and expansion operations for n-dimensional generalized maps
DGCI'08 Proceedings of the 14th IAPR international conference on Discrete geometry for computer imagery
Pyramids of n-dimensional generalized maps
GbRPR'05 Proceedings of the 5th IAPR international conference on Graph-Based Representations in Pattern Recognition
Removal operations in nd generalized maps for efficient homology computation
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
Topological operators on cell complexes in arbitrary dimensions
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
Multi-resolution cell complexes based on homology-preserving euler operators
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
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We propose a set of atomic modeling operators for simplifying and refining cell complexes in arbitrary dimensions. Such operators either preserve the homology of the cell complex, or they modify it in a controlled way. We show that such operators form a minimally complete basis for updating cell complexes, and we compare them with various operators previously proposed in the literature. Based on the new operators, we define a hierarchical model for cell complexes, that we call a Hierarchical Cell Complex (HCC), and we discuss its properties. An HCC implicitly encodes a virtually continuous set of complexes obtained from the original complex through the application of our operators. Then, we describe the implementation of a version of the HCC based on the subset of the proposed modeling operators which preserve homology. We apply the homology-preserving HCC to enhance the efficiency in extracting homology generators at different resolutions. To this aim, we propose an algorithm which computes homology generators on the coarsest representation of the original complex, and uses the hierarchical model to propagate them to complexes at any intermediate resolution, and we prove its correctness. Finally, we present experimental results showing the efficiency and effectiveness of the proposed approach.