# Images and image streams

An image \(a\) is a function \(a : \mathbf{X} \rightarrow \mathbb{F}\) from some point set \(\mathbf{X}\) to some value set \(\mathbb{F}\)

**Each element \((x, a(x)) \in a)\) is called a *pixel** where \(x\) is its location and \(a(x)\) its value.

The spatial doman \(\mathbf{X}\) and the set of values of the image are denoted \(\texttt{domain}(a)\) and \(\texttt{range}(a) \equiv \{a(x) : x \in \mathbf{X}\}\). We say \(a\) is an $\mathbb{F}$-valued image on \(\mathbf{X}\). The set of all images from \(\mathbf{X}\) to \(\mathbb{F}\) is denoted as \(\mathbb{F}^{\mathbf{X}}\).

## Georeferencing

Conceptually georeferencing can be denoted as a function of the form \(g : \mathbf{X} \rightarrow \mathbb{R}^2\), where \(\mathbf{X} \subseteq \mathbb{R}^2\).

## Value sets

A value set is an algebraic system, that is, a set together with a finite number of operations, e.g. \(\mathbb{Z}, \mathbb{R}\). Since remote sensors are able to measure radiance at multiple spectral bands, vectors of the types are also common value sets, e.g. \(\mathbb{Z}^b\) where \(b\) is the number of bands.

## Image streams

Let \(\mathbf{U}\) and \(\mathbb{F}\) be a point set and a value set. An image stream over \(\mathbf{U}\) and \(\mathbb{F}\) are unbounded, timestamp-ordered sequence of $\mathbb{F}$-valued images coming from a sensor or operator, \(\alpha \equiv \langle a_1, a_2, \dots, a_t, \dots \rangle\), where \(\texttt{domain}(a_t) \subseteq \mathbf{U}\) for each image \(a_t\) in \(\alpha\).

\(\mathbf{U}\) is referred to as the **field of view** of the stream, also denoted as \(\mathbf{FV}(\alpha)\).

By definition \(a_t |_\mathbf{U} = a_t\).

The notation \(\alpha \equiv \langle \dots, a \rangle\) is used to explicitly denote the current image \(a\) in a stream \(\alpha\).