Tropical Geometry, Mathematical Morphology and Weighted Lattices

Mathematical Morphology and Tropical Geometry share the same max/min-plus scalar arithmetic and matrix algebra. In this paper we summarize their common ideas and algebraic structure, generalize and extend both of them using weighted lattices and a max-star algebra with an arbitrary binary operation star that distributes over max, and outline applications to geometry, image analysis, and optimization. Further, we outline the optimal solution of max-star equations using weighted lattice adjunctions, and apply it to optimal regression for fitting max-star tropical curves on arbitrary data.