Erik J. Bekkers

This paper presents a robust and fully automatic filter-based approach for retinal vessel segmentation. We propose new filters based on 3D rotating frames in so-called orientation scores, which are functions on the Lie-group domain of positions and orientations [Formula: see text]. By means of a wavelet-type transform, a 2D image is lifted to a 3D orientation score, where elongated structures are disentangled into their corresponding orientation planes. In the lifted domain [Formula: see text], vessels are enhanced by means of multi-scale second-order Gaussian derivatives perpendicular to the line structures. More precisely, we use a left-invariant rotating derivative (LID) frame, and a locally adaptive derivative (LAD) frame. The LAD is adaptive to the local line structures and is found by eigensystem analysis of the left-invariant Hessian matrix (computed with the LID). After multi-scale filtering via the LID or LAD in the orientation score domain, the results are projected back to the 2D image plane giving us the enhanced vessels. Then a binary segmentation is obtained through thresholding. The proposed methods are validated on six retinal image datasets with different image types, on which competitive segmentation performances are achieved. In particular, the proposed algorithm of applying the LAD filter on orientation scores (LAD-OS) outperforms most of the state-of-the-art methods. The LAD-OS is capable of dealing with typically difficult cases like crossings, central arterial reflex, closely parallel and tiny vessels. The high computational speed of the proposed methods allows processing of large datasets in a screening setting.

We propose a framework for rotation and translation covariant deep learning using SE(2) group convolutions. The group product of the special Euclidean motion group SE(2) describes how a concatenation of two roto-translations results in a net roto-translation. We encode this geometric structure into convolutional neural networks (CNNs) via SE(2) group convolutional layers, which fit into the standard 2D CNN framework, and which allow to generically deal with rotated input samples without the need for data augmentation. We introduce three layers: a lifting layer which lifts a 2D (vector valued) image to an SE(2)-image, i.e., 3D (vector valued) data whose domain is SE(2); a group convolution layer from and to an SE(2)-image; and a projection layer from an SE(2)-image to a 2D image. The lifting and group convolution layers are SE(2) covariant (the output roto-translates with the input). The final projection layer, a maximum intensity projection over rotations, makes the full CNN rotation invariant. We show with three different problems in histopathology, retinal imaging, and electron microscopy that with the proposed group CNNs, state-of-the-art performance can be achieved, without the need for data augmentation by rotation and with increased performance compared to standard CNNs that do rely on augmentation.

The retinal fractal dimension (FD) is a measure of vasculature branching pattern complexity. FD has been considered as a potential biomarker for the detection of several diseases like diabetes and hypertension. However, conflicting findings were found in the reported literature regarding the association between this biomarker and diseases. In this paper, we examine the stability of the FD measurement with respect to (1) different vessel annotations obtained from human observers, (2) automatic segmentation methods, (3) various regions of interest, (4) accuracy of vessel segmentation methods, and (5) different imaging modalities. Our results demonstrate that the relative errors for the measurement of FD are significant and FD varies considerably according to the image quality, modality, and the technique used for measuring it. Automated and semiautomated methods for the measurement of FD are not stable enough, which makes FD a deceptive biomarker in quantitative clinical applications.

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton--Jacobi--Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry.