Nilpotent approximations of sub-Riemannian distances for fast perceptual grouping of blood vessels in 2D and 3D

\u3cp\u3eWe propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups SE(2) and SE(3). In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate SE(n), and which are obtained via the exponential and logarithmic map on SE(n). In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on SE(n). The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that (1) sub-Riemannian geometry is essential in achieving top performance and (2) grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on R \u3csup\u3en\u3c/sup\u3e and SE(n). \u3c/p\u3

Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs

\u3cp\u3eFokker-Planck PDEs (including diffusions) for stable Lévy processes (includingWiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on SE(2). Here, we extend these approaches to 3D using Fourier transform on the Lie group SE(3) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations ℝ\u3csup\u3e3\u3c/sup\u3e ⋊ S\u3csup\u3e2\u3c/sup\u3e := SE(3)/0 × SO(2)) as the quotient in SE(3). In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker-Planck PDEs of a-stable Lévy processes on ℝ\u3csup\u3e3\u3c/sup\u3e ⋊ S\u3csup\u3e2\u3c/sup\u3e. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α = 1 (the diffusion kernel) to the kernel for α = 1/2 (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.\u3c/p\u3

Crossing-preserving multi-scale vesselness

The multi–scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot properly cope with crossings or bifurcations since it only looks for elongated structures. Therefore, we disentangle crossings/bifurcations via (multiple scale) invertible orientation scores and apply vesselness filters in this domain. This new method via scale–orientation scores performs considerably better at enhancing vessels throughout crossings and bifurcations than the Frangi version. Both methods are evaluated on a public dataset. Performance is measured by comparing ground truth data to the segmentation results obtained by basic thresholding and morphological component analysis of the filtered images. Keywords: Multi-scale vesselness filters; continuous wavelet transforms; line detection; gauge frames; retinal imagin

Vessel tracking via sub-riemannian geodesics on the projective line bundle

\u3cp\u3eWe study a data-driven sub-Riemannian (SR) curve optimization model for connecting local orientations in orientation lifts of images. Our model lives on the projective line bundle R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e, with P\u3csup\u3e1\u3c/sup\u3e = S\u3csup\u3e1\u3c/sup\u3e/~ with identification of antipodal points. It extends previous cortical models for contour perception on R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e to the data-driven case. We provide a complete (mainly numerical) analysis of the dynamics of the 1st Maxwell-set with growing radii of SR-spheres, revealing the cut-locus. Furthermore, a comparison of the cusp-surface in R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e to its counterpart in R\u3csup\u3e2\u3c/sup\u3e × S\u3csup\u3e1\u3c/sup\u3e of a previous model, reveals a general and strong reduction of cusps in spatial projections of geodesics. Numerical solutions of the model are obtained by a single wavefront propagation method relying on a simple extension of existing anisotropic fast-marching or iterative morphological scale space methods. Experiments show that the projective line bundle structure greatly reduces the presence of cusps. Another advantage of including R\u3csup\u3e2\u3c/sup\u3e × P\u3csup\u3e1\u3c/sup\u3e instead of R\u3csup\u3e2\u3c/sup\u3e × S\u3csup\u3e1\u3c/sup\u3e in the wavefront propagation is reduction of computational time.\u3c/p\u3

Template matching via densities on the roto-translation group

\u3cp\u3eWe propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations. Additionally, we propose a generalized linear regression framework for the construction of suitable templates using smoothing splines. Here, it is important to recognize a curved geometry on the position-orientation domain, which we identify with the Lie group SE(2): The roto-translation group. Templates are then optimized in a B-spline basis, and smoothness is defined with respect to the curved geometry. We achieve state-of-the-art results on three different applications: Detection of the optic nerve head in the retina (99.83 percent success rate on 1,737 images), of the fovea in the retina (99.32 percent success rate on 1,616 images), and of the pupil in regular camera images (95.86 percent on 1,521 images). The high performance is due to inclusion of both intensity and orientation features with effective geometric priors in the template matching. Moreover, our method is fast due to a cross-correlation based matching approach.\u3c/p\u3

A new retinal vessel tracking method based on orientation scores

The retinal vasculature is the only part of the body's circulatory system that can be observed non-invasively. A large variety of diseases affect the vasculature, in ways that may cause geometrical and functional changes. Retinal images are therefore not only suitable for investigation of ocular diseases such as glaucoma and age-related macular degeneration (AMD), but also for systemic diseases such as diabetes, hypertension and arteriosclerosis. This paper presents a novel method for retinal vasculature extraction, using a vessel tracking method based on multi-orientation analysis. We apply multi-orientation analysis via so-called invertible orientation scores, modeling the cortical columns in the visual system of higher mammals. This allows us to successfully deal with the many complex problems inherent to vasculature tracking, such as tracking over crossings, bifurcations, parallel tracks and tracks of varying widths. The method runs fully automatically and provides a detailed model of the retinal vasculature, which is crucial as a sound basis for further quantitative analysis of the retina, especially in screening applications

The hessian of axially symmetric functions on SE(3) and application in 3D image analysis

\u3cp\u3eWe propose a method for computation of the Hessian of axially symmetric functions on the roto-translation group SE(3). Eigen decomposition of the resulting Hessian is then used for curvature estimation of tubular structures, similar to how the Hessian matrix of 2D or 3D image data can be used for orientation estimation. This paper focuses on a new implementation of a Gaussian regularized Hessian on the roto-translation group. Furthermore we show how eigenanalysis of this Hessian gives rise to exponential curve fits on data on position and orientation (e.g. orientation scores), whose spatial projections provide local fits in 3D data. We quantitatively validate our exponential curve fits by comparing the curvature of the spatially projected fitted curve to ground truth curvature of artificial 3D data. We also show first results on real MRA data.\u3c/p\u3

Design and processing of invertible orientation scores of 3D images

\u3cp\u3eThe enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores U: R \u3csup\u3e2\u3c/sup\u3e× S \u3csup\u3e1\u3c/sup\u3e→ C were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores U: R \u3csup\u3e3\u3c/sup\u3e× S \u3csup\u3e2\u3c/sup\u3e→ C. First, we construct the orientation score from a given dataset, which is achieved by an invertible coherent state type of transform. For this transformation we introduce 3D versions of the 2D cake wavelets, which are complex wavelets that can simultaneously detect oriented structures and oriented edges. Here we introduce two types of cake wavelets: the first uses a discrete Fourier transform, and the second is designed in the 3D generalized Zernike basis, allowing us to calculate analytical expressions for the spatial filters. Second, we propose a nonlinear diffusion flow on the 3D roto-translation group: crossing-preserving coherence-enhancing diffusion via orientation scores (CEDOS). Finally, we show two applications of the orientation score transformation. In the first application we apply our CEDOS algorithm to real medical image data. In the second one we develop a new tubularity measure using 3D orientation scores and apply the tubularity measure to both artificial and real medical data. \u3c/p\u3

Roto-translation covariant convolutional networks for medical image analysis

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