Nonlinear operators on graphs via stacks

International audienceWe consider a framework for nonlinear operators on functions evaluated on graphs via stacks of level sets. We investigate a family of transformations on functions evaluated on graph which includes adaptive flat and non-flat erosions and dilations in the sense of mathematical morphology. Additionally, the connection to mean motion curvature on graphs is noted. Proposed operators are illustrated in the cases of functions on graphs, textured meshes and graphs of images

Topological Characterization and Advanced Noise-Filtering Techniques for Phase Unwrapping of Interferometric Data Stacks

This chapter addresses the problem of phase unwrapping interferometric data stacks, obtained by multiple SAR acquisitions over the same area on the ground, with a twofold objective. First, a rigorous gradient-based formulation for the multichannel phase unwrapping (MCh-PhU) problem is systematically established, thus capturing the intrinsic topological character of the problem. The presented mathematical formulation is consistent with the theoretical foundation of the discrete calculus. Then within the considered theoretical framework, we formally describe an innovative procedure for the noise filtering of time-redundant multichannel multilook interferograms. The strategy underlying the adopted multichannel noise filtering (MCh-NF) procedure arises from the key observation that multilook interferograms are not fully time consistent due to multilook operations independently applied on each single interferogram. Accordingly, the presented MCh-NF procedure suitably exploits the temporal mutual relationships of the interferograms. Finally, we present some experimental results on real data and show the effectiveness of our approach applied within the well-known small baseline subset (SBAS) processing chain, thus finally retrieving the relevant Earth’s surface deformation time series for geospatial phenomena analysis and understanding

New moduli spaces of pointed curves and pencils of flat connections

It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero. Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}^1$'s stabilized by marked points of two types. The union of all $\bar{L}_n$'s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union.Comment: 37 pages, AMSTex. Several typos corrected, a reference added, subsection 3.2.2 revised, subsection 3.2.4 adde

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research