Image signal ($u:\Omega \to \mathbb{R}$)

A grey-value image signal is a scalar function on an image signal domain or an array of pixel values. Continuous image processing models use spaces such as $L^2$, Sobolev spaces or the BV space, while discrete models use finite-dimensional arrays. The chosen representation distinguishes smooth transitions, noise and image edges.

Image noise is unwanted variation in an observed image signal. In an additive model $g=u+\eta$, Gaussian noise motivates a squared data-fidelity penalty, while other statistics require different penalties. Image smoothing suppresses rapid fluctuations, but a filter must distinguish noise from genuine image edges.