What is a digital image?

Definition

In very specific applications, the images may have a non-rectangular geometry. For example, some sensors have hexagonal pixels. We will not deal with this kind of image as it is very rare.

A digital image can be seen as a function \(f\) from \(\mathbb{N}^d\) to \(\mathbb{R}^B\): it associates at each discrete coordinate \((m,\,n,\,\dots) \in \mathbb{N}^d\) a finite set of intensities \(\{i_1,\dots,i_B\} \in \mathbb{R}^B\):

\[\begin{split} \begin{aligned} f:\qquad\; \mathbb{N}^d &\to \mathbb{R}^B \\ m,n,\dots &\mapsto f(m,n,\dots) = \{i_1,\dots,i_B\}. \end{aligned} \end{split}\]

A digital image can also be seen as an array of \(d\) dimensions whose elements contain \(B\) numbers.

• For example, a grayscale image corresponds to \(d=2\) (the image has two dimensions) and \(B=1\) (there is only one value per location \((m,n)\): the grayscale intensity).

Using the bands red, green, and blue corresponds to the RGB color space. A color space is a mean to model the colors in some standard way. Other color space for displaying on screens exist, such as HSV (hue, saturation, value or intensity).

• A common color image corresponds to \(d=2\) and \(B=3\) bands, typically red, green, and blue.

• An MRI image corresponds to \(d=3\) (the image is three-dimensional) and \(B=1\).

In the general case of a 2-dimensional image \(f(m,n)\) of size \(M \times N\), one uses the coordinate system showed Fig. 1: the pixel at coordinates \((0,0)\) is on the top left corner of the image.

Fig. 1 Coordinate system generally used in image processing.

Diversity of images

Digital images can be categorized in various ways.

Dimension number \(d\)

Common images, such as photographs, are 2D (2-dimensional) images while other images lie in more than two dimensions. A 3D image, as seen in MRI scans, is often referred to as a “3D image” or “cube”. A 1D image is essentially a signal. The elements constituting a 2D image are called pixels (“picture element”), and those constituting a 3D image are called voxels (“volume element”).

Dimension heterogeneity

In common 2D images, the two dimensions are spatial dimensions. However, the dimensions can represent another physical domain and be different. For instance, a video can be seen as a 2D+\(t\) image (two spatial dimensions, one temporal dimension); a functional MRI sequence can be seen as a 3D+\(t\) image (three spatial dimensions, one temporal dimension); and a hyperspectral image is a 2D+\(\lambda\) image (two spatial dimensions, plus a third dimension depending on the wavelength).

Element dimension \(B\)

Each element within an image can be scalar or vector. For instance, pixels in a 2D grayscale image gather only one value: the gray intensity. Pixels in photography gather three values (the intensity of red, green and blue). Images from the Pléiades constellation are RVB–IR: they gather four values (red, green, blue, and infrared).

Element intensity set

Common images have pixel intensities within the range \(\{0,1,\dots,255\}\), but binary images have values in \(\{0,1\}\). Most of the time, the intensities are assumed to be real numbers.