Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices

Tropical Geometry and Mathematical Morphology share the same max-plus and\nmin-plus semiring arithmetic and matrix algebra. In this chapter we summarize\nsome of their main ideas and common (geometric and algebraic) structure,\ngeneralize and extend both of them using weighted lattices and a max-$\\star$\nalgebra with an arbitrary binary operation $\\star$ that distributes over max,\nand outline applications to geometry, machine learning, and optimization.\nFurther, we generalize tropical geometrical objects using weighted lattices.\nFinally, we provide the optimal solution of max-$\\star$ equations using\nmorphological adjunctions that are projections on weighted lattices, and apply\nit to optimal piecewise-linear regression for fitting max-$\\star$ tropical\ncurves and surfaces to arbitrary data that constitute polygonal or polyhedral\nshape approximations. This also includes an efficient algorithm for solving the\nconvex regression problem of data fitting with max-affine functions.\n