Soit E un ensemble arbitraire non vide. L'ensemble P(E) de tous les sous-ensembles de E, dont l'ordre est défini par la relation d'inclusion ?, est un treillis complet pour les opérations ? (union) et ? (intersection) De plus pour chaque élément X ? P(E) il existe un unique X, Définition, vol.25 ,
De même pour l'ensemble des entiers relatifs ,
cette démarche s'étend directement aux produits de treillis de type T , c'est à dire aux fonctions multivariées ,
niv ? : pour tout pixel p, si g p > f p , alors g p = f p ? (?g) p ,
fonction marqueur g détermine le degré de simplification obtenu. Ainsi, des simplifications plus ou moins fortes peuvent être obtenues en choisissant des marqueurs plus ou moins éloignés de la fonction de référence f ,
En effet, à condition que l'image d'entrée ait été transformée pour que ses minima marquent les objets pertinents de l'image et que ses lignes de crêtes correspondent aux frontières des objets de l'image, la LPE aura partitionné l'image en régions significatives ,
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