1.4. Image models

To answer the question of what an image mathematically speaking is, we differentiate between two settings for image models, namely continuous and discrete models. While describing images in a continuous setting is mathematically more accurate, their respective analysis is often more challenging, which is one reason to use discretised images instead. Additionally, performing image processing on high-resolution images leads to a high computational complexity. For this reason, one may sacrifice a bit of accuracy in favour of faster computational methods.

Definition 1.1 (Banach space)

A Banach space \(( X,\lVert . \rVert )\) is defined as a vector space \(X\) paired with a norm \(\lVert . \rVert\) in which all Cauchy sequences converge. Hence, a Banach space is a complete, normed vector space.

Definition 1.2 (Continuous image)

Let \(\Omega \subset \mathbb{R}^n\) be an open and bounded set. We model an image \(f:\Omega \xrightarrow{ } \mathbb{R}^m\) as a piece-wise continuous function \(f \in X\) in a Banach space \(X\).

We will later discuss required properties of continuous images, which will further characterise the structure of the Banach space \(X\).

Definition 1.3 (Discrete grid)

We define a subset \(\Omega_h \subset \Omega \subset \mathbb{R}^n\) as a discrete, \(n\)-dimensional grid with indices \(\{i_1,...,i_n\}\subset \{0,...,N_1-1\} \times \ldots \times \{0,...,N_n-1\}\) as

\[\Omega_h=\{x_{i_1,...,i_n}\in \Omega \mid x_{i_1,...,i_n} = (a_1 + i_1h, ... , a_n + i_nh) \},\]

for which \(h>0\) is a fixed grid size for all dimensions and \(a = \left(a_1,...,a_{n} \right)\) is a base corner of the \(n\)-dimensional grid.

Definition 1.4 (Discrete image)

We define a discrete image \(f_h\) as a map on a discrete grid to a finite-dimensional Euclidean vector space, i.e.

\[f_h:\Omega_h \xrightarrow{ } \mathbb{R}^m.\]

Remark 1.1

The discrete image can be interpreted as a sampling of an underlying continuous function \(f\). With increasing resolution, i.e., for grid size \(h\xrightarrow{ } 0\) we demand \(f_h \xrightarrow{ } f\). One could say, with infinitesimal resolution the discrete image converges to a continuous one.

Recalling the different types of images, one might ask how the target space can be characterised, in which the range of an image is embedded. In general, we model a continuous image \(f\) as a map

\[f: \Omega \subset \mathbb{R}^n \xrightarrow{ } \mathbb{R}^m\]

So the dimension \(m \in \N\) of the target space is directly linked with the type of images regarded.

A monochrome image contains only different shades of a single color, so the target space only requires one dimension, i.e., \(m = 1\). RGB images consist of three color channels (red, green, and blue) and hence the required dimension is \(m=3\), while for hyperspectral imaging one typically has thousands of channels representing different wavelengths, so we can assume \(m >\!\!> 3\) for this case.