Morphological Adjunctions, Moore Family and Morphological Transforms

Mathematical Morphology arose in 1964 by the work of George Matheron and Jean Serra, who developed its main concepts and tools. It uses concepts from algebra and geometry. (Set theory, complete lattice theory, convexity etc,). The notion of adjunction is very fundamental in Mathematical Morphology. Morphological systems is a broad class of nonlinear signal operators that have found many applications in image analysis. Morphological Transforms are a type of non linear signal transform for morphological systems. The Moore family stands for the family of closed objects. When the ETI and DTI systems are related via an adjunction, then there is also a close relationship between their impulse responses. Namely ,let ε be an ETI system, and let∆ be its adjoint dilation. It is easy to show that ∆ is a DTI system[11], and therefore ( ) f f g ∆ = ⊕ ,where g is the lower impulse response. In this paper, we will try to present the inter-relationships between Moore family, adjunctions and Morphological transforms.