Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well-defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations $U:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R}$ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group $SE(d)$, $d=2,3$. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in $SE(d)$. These curve fits minimize first or second order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on $SE(d)$. We include these gauge frames in differential invariants and crossing preserving PDE-flows acting on extended data representation $U$ and we show their advantage compared to the standard left-invariant frame on $SE(d)$. Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores

Vesselness via multiple scale orientation scores

The multi-scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot cope with crossings or bifurcations, since it only looks for elongated structures. Therefore, we disentangle crossing structures in the image via (multiple scale) invertible orientation scores. The described vesselness filter 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

Infrastructure for Retinal Image Analysis

This paper introduces a retinal image analysis infrastructure for the automatic assessment of biomarkers related to early signs of diabetes, hypertension and other systemic diseases. The developed application provides several tools, namely normalization, vessel enhancement and segmentation, optic disc and fovea detection, junction detection, bifurcation/crossing discrimination, artery/vein classification and red lesion detection. The pipeline of these methods allows the assessment of important biomarkers characterizing dynamic properties of retinal vessels, such as tortuosity, width, fractal dimension and bifurcation geometry features

Left-invariant evolutions of wavelet transforms on the Similitude Group

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

Segmentation of perivascular spaces in 7 T MR image using auto-context model with orientation-normalized features

Quantitative study of perivascular spaces (PVSs) in brain magnetic resonance (MR) images is important for understanding the brain lymphatic system and its relationship with neurological diseases. One of major challenges is the accurate extraction of PVSs that have very thin tubular structures with various directions in three-dimensional (3D) MR images. In this paper, we propose a learning-based PVS segmentation method to address this challenge. Specifically, we first determine a region of interest (ROI) by using the anatomical brain structure and the vesselness information derived from eigenvalues of image derivatives. Then, in the ROI, we extract a number of randomized Haar features which are normalized with respect to the principal directions of the underlying image derivatives. The classifier is trained by the random forest model that can effectively learn both discriminative features and classifier parameters to maximize the information gain. Finally, a sequential learning strategy is used to further enforce various contextual patterns around the thin tubular structures into the classifier. For evaluation, we apply our proposed method to the 7T brain MR images scanned from 17 healthy subjects aged from 25 to 37. The performance is measured by voxel-wise segmentation accuracy, cluster- wise classification accuracy, and similarity of geometric properties, such as volume, length, and diameter distributions between the predicted and the true PVSs. Moreover, the accuracies are also evaluated on the simulation images with motion artifacts and lacunes to demonstrate the potential of our method in segmenting PVSs from elderly and patient populations. The experimental results show that our proposed method outperforms all existing PVS segmentation methods

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We 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) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

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