Mathematical Morphology and Distance Maps

Abstract

Mathematical morphology defines algebraic operations on geometric sets (shapes). The basic operations that correspond to addition and subtraction are the dilation δ and the erosion ε operations that are denoted as ⊕ and ⊖, respectively. For our purposes a shape is defined as the interior of a closed curve. Exploring the history of mathematical morphology that started with the work of G. Matheron and J. Serra, and the various applications especially in image synthesis and processing, is beyond the scope of this chapter. The interested reader could start with a simple introduction to this field in Dougherty’s book [60]. Here we concentrate on some of the basic operations. Motivated by [21, 177], and the related slope transform of P. Maragos [148], and Dorst and Boomgaard [59], we explore the link between curve evolution and continuous-scale morphology. We also refer to [77, 76], for efficient algorithms of the basic morphological operations with a rectangular structuring element.