OPEN ACCESS
The morphological filters require to enable idempotence and growing properties before to be used. If the proof of these properties is natural on uni-dimensional values, it is not the same for vectors. This work covers the validation of idempotence property for opening and closing operators. In the framework of a distance based ordering, we show why this property is not enabled, then we propose a modified writing form solving this problem and a generic framework to write vectorial ordering functions.
Extended Abstract
Mathematical morphology allows to develop all the possible image processing tools, just using basic operations like erosion, dilation and complementary transforms. The ordering is at the core of the mathematical morphology to construct the lattices required to develop the mathematical construction. However,ordering colour or multivariate data is not straight forward, especially when the physical or perceptual constraints of the data, typically the colour, must be respected.
In Ledoux, Richard, Capelle-Laizé (2012), an adapted construction for colour ordering respecting the human visual systems properties was proposed and compared with other approaches. This new ordering scheme respect all the required properties to produce a total order and to develop high level processing computed in a vector way, as texture features and Hit-or-Miss transformsLedoux etal.(2013).This work address the following level. To construct advance morphological filters and processing, a new set of morphological properties must be verified, including the idempotency and the growing properties. If the proof of these properties is natural on uni-dimensional values, it is not the same for vectors.
In the framework of a distance based ordering, we show that the construction proposed by Ledoux, Richard, Capelle-Laizé (2012) don’t allow to enable the idempotency property. Due to the dual construction used to extract the maximum and the minimum operation, the idempotency is not reachable without modifications on the expression. This modification must link in the n-dimensional space the maximum as being the minimum for the dual operator and inversely. Thanks to this result we expresses the generic framework to develop a new ordering scheme based on perceptual distance functions, respecting the fundamentals of the ordering, duality and idempotency.
RÉSUMÉ
Le filtrage morphologique avancé nécessite que certaines propriétés soient validées avant d’être utilisées : idempotence et croissance. Si la preuve de ces propriétés est directe sur des objets unidimensionnels, iln’en est pas de même sur des objets multivalués. Ce travail porte sur la validation de la propriété d’idempotence des opérateurs d’ouverture et de fermeture couleur. Pour l’ordonnancement basé distance, nous montrons pourquoi cette propriété n’est pas vérifiée, nous proposons ensuite une modification résolvant ce problème puis un cadre générique d’écriture des fonctions d’ordonnancement vectorielles.
mathematical morphology, idempotence, duality, ordering, distance.
MOTS-CLÉS
morphologie mathématique, idempotence, dualité, ordonnancement, distance.
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