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

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

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

TLR1/TLR2 Heterodimers Play an Important Role in the Recognition of Borrelia Spirochetes

After infection with Borrelia species, the risk for developing Lyme disease varies significantly between individuals. Recognition of Borrelia by the immune system is mediated by pattern recognition receptors (PRRs), such as TLRs. While TLR2 is the main recognition receptor for Borrelia spp., little is known about the role of TLR1 and TLR6, which both can form functionally active heterodimers with TLR2. Here we investigated the recognition of Borrelia by both murine and human TLR1 and TLR6. Peritoneal macrophages from TLR1- and TLR6- gene deficient mice were isolated and exposed to Borrelia. Human PBMCs were stimulated with Borrelia with or without specific TLR1 and TLR6 blocking using specific antibodies. Finally, the functional consequences of TLR polymorphisms on Borrelia-induced cytokine production were assessed. Splenocytes isolated from both TLR1−/− and TLR6−/− mice displayed a distorted Th1/Th2 cytokine balance after stimulation with B.burgdorferi, while no differences in pro-inflammatory cytokine production were observed. In contrast, blockade of TLR1 with specific neutralizing antibodies led to decreased cytokine production by human PBMCs after exposure to B.burgdorferi. Blockade of human TLR6 did not lead to suppression of cytokine production. When PBMCs from healthy individuals bearing polymorphisms in TLR1 were exposed to B.burgdorferi, a remarkably decreased in vitro cytokine production was observed in comparison to wild-type controls. TLR6 polymorphisms lead to a minor modified cytokine production. This study indicates a dominant role for TLR1/TLR2 heterodimers in the induction of the early inflammatory response by Borrelia spirochetes in humans