Background

Abstract

© The Author(s) 2014. In this chapter, we introduce the mathematical structures used to represent scalar fields and their morphology in the smooth and in the discrete cases. We provide the basic mathematical concepts, such as manifold, cell complex, regular grid, and simplicial complex (Sect. 1.1). We introduce discrete models for scalar fields defined at finite sets of points, such as regular models and simplicial models (Sect. 1.2). We present the basic notions of Morse theory, which provides a description of the morphology of functions in the smooth case (Sect. 1.3), and the watershed transform in the smooth case, which is an alternative framework to Morse theory (Sect. 1.4). We discuss the two existing approaches for extending the results of Morse theory to the discrete case: Banchoff’s piecewise-linear Morse theory (Sect. 1.5) and Forman’s discrete Morse theory (Sect. 1.6).