# 9

# The Spatial Domain

Before processing and analyzing gray-tone images, it should be necessary to mathematically structure the spatial domain, that is to say, roughly speaking, the space of pixels or shortly the pixel space. The purpose of this chapter is to present the very first mathematical fundamentals associated with the spatial domain.

# 9.1. Paradigms

In a particular physical setting, the spatial support of gray-tone images designates the loci where the gray-tone images are considered.

In the continuous case, these spatial locations are simply the points in a subset of the *n*-dimensional Euclidean space (in practice, *n* = 1, 2 or 3), while in the discrete case, these spatial locations are represented by means of a grid representation.

# 9.2. Mathematical structures

The mathematical discipline of reference is the theory of sets, for short *Set Theory* [RUB 67, DEV 93, BOU 04c].

The second discipline is *topology* [KEL 75, JÄN 84].

*Discrete Topology* that focuses on discrete spaces is particularly useful for digital imaging. The terms *Digital Topology* and *Digital Geometry* are used even now [KLE 04b]. *Algebraic Topology* [ROT 88], which covers the cellular spaces [KLE 04b], is less easy to address, but it is well suited when the pixels are regarded as cells, and not as points.

# 9.3. Main approaches for IPA

## 9.3.1. *Pixels*

The *spatial support*, denoted by *D*, is composed of spatial locations on which gray-tone images are defined. These *spatial locations* are called pixels (i.e. the contraction ...