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 is a visual representation of an array of number that represents a physical phenomenon. It 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 where each element gathers \(B\) numbers. Some examples are now given.

• A grayscale image corresponds to \(d=2\) (the image has two dimensions) and \(B=1\): each element \((m,n)\) corresponds to only one number coding the grayscale intensity.

• A common color image corresponds to \(d=2\) and \(B=3\) bands,

• the three bands code typically the amount of 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. 15: the pixel at coordinates \((0,0)\) is on the top left corner of the image.

Fig. 15 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 (\(B=1\)) or vector (\(B>1\)). 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.