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 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 all 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, crucial 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 left-invariant PDE-evolutions with invertible orientation scores.Comment: A final and corrected version of the manuscript is Published in Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9), p.1-50, 201

Feature vector similarity based on local structure

Local feature matching is an essential component of many image retrieval algorithms. Euclidean and Mahalanobis distances are mostly used in order to compare two feature vectors. The first distance does not give satisfactory results in many cases and is inappropriate in the typical case where the components of the feature vector are incommensurable, whereas the second one requires training data. In this paper a stability based similarity measure (SBSM) is introduced for feature vectors that are composed of arbitrary algebraic combinations of image derivatives. Feature matching based on SBSM is shown to outperform algorithms based on Euclidean and Mahalanobis distances, and does not require any training

Front-End Vision: A Multiscale Geometry Engine

Abstract. The paper is a short tutorial on the multiscale differential geometric possibilities of the front-end visual receptive fields, modeled by Gaussian derivative kernels. The paper is written in, and interactive through the use of Mathematica 4, so each statement can be run and modified by the reader on images of choice. The notion of multiscale invariant feature detection is presented in detail, with examples of second, third and fourth order of differentiation

Stability Analysis of Fractal Dimension in Retinal Vasculature

Fractal dimension (FD) has been considered as a potential biomarker for retina-based disease detection. However, conflicting findings can be found in the reported literature regarding the association of the biomarker with diseases. This motivates us to examine the stability of the FD on different (1) vessel segmentations obtained from human observers, (2) automatic segmentation methods, (3) threshold values, and (4) region-of-interests. Our experiments show that the corresponding relative errors with respect to reference ones, computed per patient, are generally higher than the relative standard deviation of the reference values themselves (among all patients). The conclusion of this paper is that we cannot fully rely on the studied FD values, and thus do not recommend their use in quantitative clinical applications

Ultra-High Field MRI Post Mortem Structural Connectivity of the Human Subthalamic Nucleus, Substantia Nigra, and Globus Pallidus

Introduction: The subthalamic nucleus, substantia nigra, and globus pallidus, three nuclei of the human basal ganglia, play an important role in motor, associative, and limbic processing. The network of the basal ganglia is generally characterized by a direct, indirect, and hyperdirect pathway. This study aims to investigate the mesoscopic nature of these connections between the subthalamic nucleus, substantia nigra, and globus pallidus and their surrounding structures. Methods: A human post mortem brain specimen including the substantia nigra, subthalamic nucleus, and globus pallidus was scanned on a 7 T MRI scanner. High resolution diffusion weighted images were used to reconstruct the fibers intersecting the substantia nigra, subthalamic nucleus, and globus pallidus. The course and density of these tracks was analyzed. Results: Most of the commonly established projections of the subthalamic nucleus, substantia nigra, and globus pallidus were successfully reconstructed. However, some of the reconstructed fiber tracks such as the connections of the substantia nigra pars compacta to the other included nuclei and the connections with the anterior commissure have not been shown previously. In addition, the quantitative tractography approach showed a typical degree of connectivity previously not documented. An example is the relatively larger projections of the subthalamic nucleus to the substantia nigra pars reticulata when compared to the projections to the globus pallidus internus. Discussion: This study shows that ultra-high field post mortem tractography allows for detailed 3D reconstruction of the projections of deep brain structures in humans. Although the results should be interpreted carefully, the newly identified connections contribute to our understanding of the basal ganglia