Anisotropic Diffusion in Riemannian Colour Geometry

Abstract Anisotropic diffusion has long been an important tool in image processing. More recently, it has also found its way to colour imaging. Until now, mainly Euclidean colour spaces have been considered in this context, but recent years have seen a renewed interest in and importance of non-Euclidean colour geometry. The main contribution of this paper is the derivation of the equations for anisotropic diffusion in Riemannian colour geometry. It is demonstrated that it contains several well-known solutions such as Perona–Malik diffusion and Tschumperlé–Deriche diffusion as special cases. Furthermore, it is shown how it is non-trivially connected to Sochen’s general framework for low-level vision. The main significance of the method is that it decouples the coordinates used for solving the diffusion equation from the ones that define the metric of the colour manifold, and thus directs the magnitude and direction of the diffusion through the diffusion tensor. It also enables the use of non-Euclidean colour manifolds and metrics for applications such as denoising, inpainting, and demosaicing, based on anisotropic diffusion.