Hamilton-Jacobi Equations and Mathematical Morphology in Pseudo-Riemannian Manifolds

Partial differential equations (PDEs) are very suitable in nonlinear data modeling and analysis. Hamilton-Jacobi (HJ) equations constitute a particular family in the PDEs area with many connections in various science fields, and admit under some assumptions viscosity solutions known as Hopf-Lax-Oleinik (HLO) formulas. Mathematical morphology (MM) is an efficient nonlinear image and data analysis method, which can be formulated as first order HJ PDEs. In this work, we propose to formulate HJ PDEs equations in compact Pseudo-Riemannian manifolds and prove their viscosity solutions. We also prove the viscosity solutions for a particular Hamiltonian, which makes the link to MM, constituting a new extension of classical morphological operators in compact pseudo-Riemannian manifolds. Obtained experimental results on real images show interesting capabilities of the proposed approach in multiscale analysis and image filtering.