By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard "coherence enhancing diffusion" equations on images as they can enhance images with crossing contours. Here we employ differential geometry on SE(2) to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions.
Duits, R., & Franken, E. M. (2010). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part II: Non-linear left-invariant diffusions on invertible orientation scores. Quarterly of Applied Mathematics, 68(2), 293-331. https://doi.org/10.1090/S0033-569X-10-01173-3