Deformation of surfaces in 2D persistent homology

In the context of 2D persistent homology a new metric has been recently introduced, the coherent matching distance. In order to study this metric, the filtering function is required to present particular “regularity” properties, based on a geometrical construction of the real plane, called extended Pareto grid. This dissertation shows a new result for modifying the extended Pareto grid associated to a filtering function defined on a smooth closed surface, with values in the real plane. In future, the technical result presented here could be used to prove the genericity of the regularity conditions assumed for the filtering function.