( max , min )-convolution and Mathematical Morphology

• Main Author: Angulo, Jesus
• Other Authors: Centre de Morphologie Mathématique (CMM), MINES ParisTech - École nationale supérieure des mines de Paris, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2015
• Subjects: Minkowski addition; adjunction; HamiltonJacobi PDE; fuzzy morphology; viscous morphology; [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]; Iceland
• Online Access: https://hal.archives-ouvertes.fr/hal-01257501; https://hal.archives-ouvertes.fr/hal-01257501/document; https://hal.archives-ouvertes.fr/hal-01257501/file/MaxMinConvolution_ISMM15_final.pdf; https://doi.org/10.1007/978-3-319-18720-4_41

International audience A formal denition of morphological operators in (max, min)-algebra is introduced and their relevant properties from an algebraic viewpoint are stated. Some previous works in mathematical morphology have already encountered this type of operators but a systematic study of them has not yet been undertaken in the morphological literature. It is shown in particular that one of their fundamental property is the equivalence with level set processing using Minkowski addition and subtraction. Theory of viscosity solutions of the Hamilton-Jacobi equation with Hamiltonians containing u and Du is summarized, in particular, the corresponding Hopf-Lax-Oleinik formulas as (max, min)-operators. Links between (max, min)-convolutions and some previous approaches of unconventional morphology, in particular fuzzy morphology and viscous morphology, are reviewed.