Left-invariant parabolic evolutions on $SE(2)$ and contour enhancement via invertible orientation scores Part II: Nonlinear left-invariant diffusions on invertible orientation scores

Abstract

Abstract. 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.

Left Invariant Evolution Equations on Gabor Transforms

“…We adapt the group theoretical approach developed for the Euclidean motion groups in the recent works [9,18,12,13,15,11], thus illustrating the scope of the methods devised for general Lie groups in [10] in signal and image processing. Reassignment will be seen to be a special case of left-invariant convection.…”

“…. , a 2d ) T and/or conductivity matrix D. We will use ideas similar to our previous work on adaptive diffusions on invertible orientation scores [17], [12], [11], [13] (where we employed evolution equations for the 2D-Euclidean motion group). We use the absolute value to adapt the diffusion and convection to avoid oscillations.…”

“…So far our motivation for (13) has been group theoretical. There is one issue we did not address yet, namely the omission of ∂ s = A 2d+1 in (13).…”

“…There is one issue we did not address yet, namely the omission of ∂ s = A 2d+1 in (13). Here we first motivate this omission and then consider the differential geometrical consequence that (adaptive) convection and diffusion takes place along so-called horizontal curves.…”

“…Akin to our framework of linear evolutions on orientation scores, cf. [13,17], this means that we enforce horizontal diffusion and convection, i.e. transport and diffusion only takes place along so-called horizontal curves in H r which are curves t → (p(t), q(t), s(t)) ∈ H r , with s(t) ∈ (0, 1), along which connection between (p(0), q(0), s(0)) and (p(t), q(t), s(t)) and the actual horizontal…”