Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

Abstract

Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherenceenhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherenceenhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore,

Design and Processing of Invertible Orientation Scores of 3D Images

“…Methods that deal with crossings by applying coherenceenhancing diffusion via 2D orientation scores have been developed for 2D data [37,64]. Here, we propose an extension Fig.…”

Retinal Vascular Tortuosity in Hospitalized Patients with Type 2 Diabetes and Diabetic Retinopathy in China

“…This brain-inspired method is based on the theory of best-fit exponential curves in the roto-translation group SE (2) developed by Duits, Franken and Janssen [15]- [17], and, importantly, does not rely on explicit segmentation of the blood vessels. 2D images are lifted to 3D functions called "orientation scores" by adding an orientation dimension to the domain [18].…”

Left Invariant Evolution Equations on Gabor Transforms

“…. , 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.…”

“…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…”