Numerical approaches for linear left-invariant diffusions on SE(2), their comparison to exact solutions, and their applications in retinal imaging

Left-invariant PDE-evolutions on the roto-translation group SE(2) (and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, is missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based finite element type of methods (equivalent to SE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions crucially in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining the left-invariant PDE-evolutions with invertible orientation scores

Cardiac motion estimation using covariant derivatives and Helmholtz decomposition

The investigation and quantification of cardiac movement is important for assessment of cardiac abnormalities and treatment effectiveness. Therefore we consider new aperture problem-free methods to track cardiac motion from 2-dimensional MR tagged images and corresponding sine-phase images. Tracking is achieved by following the movement of scale-space maxima, yielding a sparse set of linear features of the unknown optic flow vector field. Interpolation/reconstruction of the velocity field is then carried out by minimizing an energy functional which is a Sobolev-norm expressed in covariant derivatives (rather than standard derivatives). These covariant derivatives are used to express prior knowledge about the velocity field in the variational framework employed. They are defined on a fiber bundle where sections coincide with vector fields. Furthermore, the optic flow vector field is decomposed in a divergence free and a rotation free part, using our multi-scale Helmholtz decomposition algorithm that combines diffusion and Helmholtz decomposition in a single non-singular analytic kernel operator. Finally, we combine this multi-scale Helmholtz decomposition with vector field reconstruction (based on covariant derivatives) in a single algorithm and present some experiments of cardiac motion estimation. Further experiments on phantom data with ground truth show that both the inclusion of covariant derivatives and the inclusion of the multi-scale Helmholtz decomposition improves the optic flow reconstruction

Curvature Based Biomarkers for Diabetic Retinopathy via Exponential Curve Fits in SE(2)

We propose a robust and fully automatic method for the analysis of vessel tortuosity. Our method does not rely on pre-segmentation of vessels, but instead acts directly on retinal image data. The method is based on theory of best-fit exponential curves in the roto-translation group SE(2). We lift 2D images to 3D functions called orientation scores by including an orientation dimension in the domain. In the extended domain of positions and orientations (identified with SE(2)) we study exponential curves, whose spatial projections have constant curvature. By locally fitting such curves to data in orientation scores, via our new iterative stabilizing refinement method, we are able to assign to each location a curvature and confidence value. These values are then used to define global tortuosity measures. The method is validated on synthetic and retinal images. We show that the tortuosity measures can serve as effective biomarkers for diabetes and different stages of diabetic retinopathy