( max , min )-convolution and Mathematical Morphology

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...

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Bibliographic Details
Main Author: Angulo, Jesus
Other Authors: Centre de Morphologie Mathématique (CMM), Mines Paris - PSL (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
Description
Summary: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.

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