**Geometric discrete Morse complexes**

*Thomas Lewiner*

## Abstract:

Differential geometry provides an intuitive way of understanding smooth objects in the space. However, with the evolution of geometric modeling by computer, this tool became both necessary and difficult to transpose to the discrete setting. The power of Morse theory relies on the link it created between differential topology and geometry. Starting from a combinatorial point of view, Forman s discrete Morse theory relates rigorously discrete objects to their topology, opening Morse theory to discrete structures. This work proposes a constructive definition of geometric discrete Morse functions and their corresponding discrete Morse-Smale complexes, where the geometry is defined as a smooth function sampled on the vertices of the discrete structure. This construction required some homology computations that turned out to be a significant improvement over existing methods by itself. The resulting Morse-Smale decomposition can then be efficiently computed, and used for applications to persistence computation, Reeb graph generation, noise removal...## Downloads:

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## BibTeX:

@phdthesis{tomlew_phd_puc,author = {Thomas Lewiner},

title = {Geometric discrete Morse complexes},

year = {2005},

month = {august},

school = {Department of Mathematics, PUC-Rio},

url = {\url{http://thomas.lewiner.org/pdfs/tomlew_phd_puc.pdf}}

}