Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
An introduction to solid modeling
An introduction to solid modeling
Hierarchical morse complexes for piecewise linear 2-manifolds
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Morse-smale complexes for piecewise linear 3-manifolds
Proceedings of the nineteenth annual symposium on Computational geometry
Topological analysis and characterization of discrete scalar fields
Proceedings of the 11th international conference on Theoretical foundations of computer vision
Morse-Smale decompositions for modeling terrain knowledge
COSIT'05 Proceedings of the 2005 international conference on Spatial Information Theory
A topological hierarchy for functions on triangulated surfaces
IEEE Transactions on Visualization and Computer Graphics
Morphology analysis of 3D scalar fields based on morse theory and discrete distortion
Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
Simplifying morphological representations of 2D and 3D scalar fields
Proceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
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Morse theory studies the relationship between the topology of a manifold M and the critical points of a scalar function f defined on M. The Morse-Smale complex associated with f induces a subdivision of M into regions of uniform gradient flow, and represents the topology of M in a compact way. Function f can be simplified by cancelling its critical points in pairs, thus simplifying the topological representation of M, provided by the Morse-Smale complex. Here, we investigate the effect of the cancellation of critical points of f in Morse-Smale complexes in two and three dimensions by showing how the change of connectivity of a Morse-Smale complex induced by a cancellation can be interpreted and understood in a more intuitive and straightforward way as a change of connectivity in the corresponding ascending and descending Morse complexes. We consider a discrete counterpart of the Morse-Smale complex, called a quasi-Morse complex, and we present a compact graph-based representation of such complex and of its associated discrete Morse complexes, showing also how such representation is affected by a cancellation.