Synthesis and applications of lattice image operators based on fuzzy norms

In this paper we use concepts from the lattice-based theory of morphological operators and fuzzy sets to develop generalized lattice image operators that can be expressed as nonlinear convolutions that are suprema or infima of fuzzy intersection or union norms. Our emphasis (and differences with previous works) is the construction of pairs of fuzzy dilation and erosion operators that form lattice adjunctions. This guarantees that their composition will be a valid algebraic opening or closing. The power but also the difficulty in applying these fuzzy operators to image analysis is the large variety of fuzzy norms and the absence of systematic ways in selecting them. Towards this goal, we have performed extensive experiments in applying these fuzzy operators to various nonlinear filtering and image analysis tasks, attempting first to understand the effect that the type of fuzzy norm and the shape-size of structuring function have on the resulting new image operators. Further, we have developed some new fuzzy edge gradients and optimized their usage for edge detection on test problems via a parametric fuzzy norm.