# 5

# Key Concepts and Notions for IPA

The purpose of this chapter is to introduce the key concepts and notions of image processing and analysis, which will be used in this book.

# 5.1. Dimensionality

The concept of **dimension** empirically refers to the ‘size’. The main dimensions of a simple geometric object are its length, width, depth or thickness, or its diameter. Indeed, this term represents several concepts of which the important ones for Mathematical Imaging will be recalled in this section.

## 5.1.1. *Dimension in Physics*

A **physical quantity** is a physical property that can be quantified by measurement or calculation. Its possible values are expressed using a number, usually accompanied by a **unit of measurement**. The dimension of a physical quantity is thus expressed by a unit, which is a combination of the base units of the **International System of Units** (SI, from the French *Le Système International d’Unités*) [TAY 06a]. Unitless numbers are used for a dimensionless quantity.

The dimension of a physical object or a physical space is the number of variables that are used to describe that object or space, and thus to define a state, an event, etc.

## 5.1.2. *Dimension in Mathematics*

The concept of **vector space** is important in Image Processing and Analysis because it is very often the basic mathematical framework that will be used to represent intensity images.

First, the vector dimension will designate the domain of definition of the images (i.e. dimension 1, 2 or 3). The **Euclidean dimension ...**