By Loïc Mazo (auth.), Massimo Ferri, Patrizio Frosini, Claudia Landi, Andrea Cerri, Barbara Di Fabio (eds.)
This booklet constitutes the lawsuits of the 4th foreign Workshop on Computational Topology in photo Context, CTIC 2012, held in Bertinoro, Italy, in may well 2012. The sixteen papers awarded during this quantity have been rigorously reviewed and chosen for inclusion during this publication. They specialise in the topology and computation in snapshot context. The workshop is dedicated to computational tools utilizing topology for the research and comparability of pictures. The concerned study fields contain computational topology and geometry, discrete topology and geometry, geometrical modeling, algebraic topology for photograph functions, and the other box concerning a geometric-topological method of picture processing.
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Additional resources for Computational Topology in Image Context: 4th International Workshop, CTIC 2012, Bertinoro, Italy, May 28-30, 2012. Proceedings
For that, it removes degree two cells and dangling cells. Then we can compute homology on the reduced n-Gmap and project the generator on the original object. Some results show the interest of the simpliﬁcation step, both in memory space and in computation time. Some questions are still open. The ﬁrst question is about the conditions on removed cells. Is it possible to remove some other type of cells while preserving the homology? The answer is no in 2D and 3D, but still open in higher dimension.
03s To compute the homology generators, we iterate through all the cells of the nGmap and we compute incidence matrices (which describes the boundary of the cells) using the incidence number deﬁnition. Then we reduce incidence matrices into their Smith-Agoston normal form for computing homology generators . Compared to the classical Smith normal form, the speciﬁcity of the Agoston reduced normal form is that for a given dimension d, the basis of the boundaries Bp is a subset of the basis of cycles Zp , thus the quotient group Hp = Zp /Bp can directly be obtained by simply removing from Zp the boundaries of inﬁnite order.
Vuc¸ini and Kropatsch  proposed to reduce the necessity for visual inspection by using topological information derived from Homology analysis. A schematic view of this reconstruction pipeline is displayed in Fig. 1. Offline Inspect Uniform Data Non-uniform Point Set Reconstruct with Resolution Nx RMSE Low? No Yes Artefacts? Visually Inspect Uniform Data No Visualize/Use Uniform Data Yes Increase Resolution Nx Fig. 1. Schematic view of a pipeline for the reconstruction of non-uniform point sets to uniform representations when the target resolution is unknown.