CIMPA-ICTP Fellowships "Research in Pairs"

Mathematical Morphology in the view of Hamilton-Jacobi equations
and Hopf-Lax-Oleinik formulas

24 – 28 July, 2023


Jésus Angulo (MINES ParisTech)
El Hadji S. Diop (University Iba Der Thiam of Thies)

In many image and data processing, for instance, data compression, feature detection, motion analysis/detection, multiband frequency analysis, performing a multiscale analysis is very important, for either at multiple scales or resolutions. Because of its nonlinearity aspects and good shape and geometry description properties, mathematical morphology has been for over decades an efficient and powerful multiscale analysis tool. Various formulations were proposed for theoretical studies. An algebraic formulation based on complete lattice theory has been proposed, we also have a functional analysis framework yielding partial differential equations (PDEs) that are in fact a particular case of the first order Hamilton-Jacobi equations for which viscosity solutions exist and are given by Hopf-Lax-Oleinik (HLO) formulas. There are also some formulations based on graphs. Based on those formulations, many theoretical studies have been conducted, and plenty algorithms have been designed, which contribute to inspiring new breakthroughs in data analysis and processing; for instance, in machine vision, pattern recognition or artificial intelligence.

The outline of the course is as follows:

• Basics on morphological operators
• Notions of viscosity solutions & Mathematical morphology in Rn
• Robust and adaptive morphological operators
• Mathematical morphology in Riemannian manifolds & Extensions