Lattice Fuzzy Signal Operators and Generalized Image Gradients

In this paper we use concepts from the lattice-based theory of morphological operators and fuzzy sets to develop generalized lattice image operators that are nonlinear convolutions that can be expressed as supremum (resp. infimum) of fuzzy intersection (resp. union) norms. Our emphasis and differences with many previous works is the construction of pairs of fuzzy dilation (sup of fuzzy intersection) and erosion (inf of fuzzy implication) operators that form lattice adjunctions. This guarantees that their composition will be a valid algebraic opening or closing. We have experimented with applying these fuzzy operators to various nonlinear filtering and image analysis tasks, attempting to understand the effect that the type of fuzzy norm and the shape-size of structuring function have on the resulting new image operators. We also present some theoretical and experimental results on using the lattice fuzzy operators, in combination with morphological systems or by themselves, to develop some new edge detection gradients which show improved performance in noise.