Generalised morphological image diffusion

Relationships between linear and morphological scale-spaces have been considered by various previous works. The aim of this paper is to study how to generalise the discrete and continuous diffusion-based approaches in order to introduce nonlinear filters whose limit effects mimic the asymmetric behaviour of morphological dilation and erosion, as well as other multiscale filters, hybrid between the standard linear and morphological filters. A methodology based on the counter-harmonic mean is adopted here. Partial differential equations are formulated and details of numerical implementation are discussed to illustrate the various studied cases: isotropic, edge-preserving and coherence-enhancing diffusion. We also found a new way to derive the classical link between Gaussian scale-space and dilation/erosion scale-spaces based on quadratic structuring functions in the discrete and continuous setting. We have included some preliminary applications of the generalised morphological diffusion to solve image processing problems such as denoising and image enhancement in the case of asymmetric bright/dark image properties.