Basic Sets in the Digital Plane

Abstract

A set K in the plane ℝ² is basic if each continuous function $f \: K \to \mathbb R$ can be expressed as a sum f(x,y) = g(x) + h(y) with $g, h \: \mathbb R \to \mathbb R$ continuous functions. Analogously we define a digital set K _k in the digital plane to be basic if for each digital function $f: {K_k} \to {\mathbb R}$ there exist digital functions on the digital unit interval such that f(x,y) = g(x) + h(y) for each pixel (x,y) ∈ K _k . Basic subsets of the plane were characterized by Sternfeld and Skopenkov. In this paper we prove a digital analogy of this result. Moreover we explore the properties of digital basic sets, and their possible use in image analysis.