Solving Problems in Mathematical Morphology through Reductions to the U-Curve Problem

Abstract

The U-curve problem is an optimization problem that consists in, given a finite set S, a Boolean lattice $(\mathcal{P}(S), \subseteq)$ and a chain $\mathcal{L}$, minimize a function $c:\mathcal{P}(S) \rightarrow \mathcal{L}$ that satisfies an extension of Matheron’s increasing-decreasing decomposition (i.e., a function that is decomposable in U-shaped curves). This problem may be used to model problems in the domain of Mathematical Morphology, for instance, morphological operator design and some types of combinatorial optimization problems. Recently, we introduced the U-Curve-Search (UCS) algorithm, which is a solver to the U-curve problem. In this paper, we recall the principles of the UCS algorithm, present a constrained version of Serra’s formulation of the Tailor problem, prove that this problem is a U-curve problem, apply the UCS algorithm to solve it and compare the performance of UCS with another optimization algorithm. Besides, we present applications of UCS in the context of W-operator design.