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Issue No.08 - August (2011 vol.33)
pp: 1646-1658
Vanessa Robins , The Australian National University, Canberra
Peter John Wood , The Australian National University, Canberra
Adrian P. Sheppard , The Australian National University, Canberra
ABSTRACT
We present an algorithm for determining the Morse complex of a two or three-dimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e., a critical point) of the grayscale image. Since more than one critical point may be associated with a single image voxel, we model digital images by cubical complexes. A new homotopic algorithm is used to construct a discrete Morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets. We make use of discrete Morse theory and simple homotopy theory to prove correctness of this algorithm. The resulting Morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology.
INDEX TERMS
Discrete Morse theory, computational topology, persistent homology, digital topology.
CITATION
Vanessa Robins, Peter John Wood, Adrian P. Sheppard, "Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.33, no. 8, pp. 1646-1658, August 2011, doi:10.1109/TPAMI.2011.95
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