Effective persistent homology of digital images
Ana Secundina Méndez Romero2014
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Abstract
In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been implemented as extensions of the Kenzo system and have shown a good performance when applied on some actual images extracted from a public dataset.
Figures (3)
![The maps d', f, g and h are then given by the formulas: In this way, the maps d’, f, g and h which provide the reduction p can be explicitly constructed. where the incidence number e(a,7T) is the coefficient of o in the differential dr. Let us remark that e(o,V(c)) = +1, but the other incidence numbers are arbitrary. Thanks to Theorem 2 and the explicit isomorphism H,,(C) ~ H,(C°) de- duced from the reduction p: C=C, one can compute the homology groups of the big (possibly infinite) chain complex C’ by means of the homology groups of C° (whenever C° is of finite type). In [17], this result is used to produce, for example, the effective homology of twisted cartesian products and Eilenberg- MacLane spaces.](https://figures.academia-assets.com/53154117/figure_001.jpg)
![The DVF has 4 components, and the 4 x 4 top left-hand submatrix is trian- gular unimodular. The reduction produces the matrix [—1] which of course can be reduced to the void matrix.](https://figures.academia-assets.com/53154117/figure_002.jpg)
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