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The analysis of brain networks is central to neurobiological research. In this context the following tasks often arise: (1) understand the cellular composition of a reconstructed neural tissue volume to determine the nodes of the brain network; (2) quantify connectivity features statistically; and (3) compare these to predictions of mathematical models. We present a framework for interactive, visually supported accomplishment of these tasks. Its central component, the stratification matrix viewer, allows users to visualize the distribution of cellular and/or connectional properties of neurons at different levels of aggregation. We demonstrate its use in four case studies analyzing neural network data from the rat barrel cortex and human temporal cortex.
In many applications, geodesic hierarchical models are adequate for the study of temporal observations.
We employ such a model derived for manifold-valued data to Kendall's shape space.
In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics
and employ the approach for longitudinal analysis of 2D rat skulls shapes as well as 3D shapes derived from an imaging study on osteoarthritis. Particularly, we perform hypothesis test and estimate the mean trends.
Data sets sampled in Lie groups are widespread, and as with multivariate data, it is important for many applications to assess the differences between the sets in terms of their distributions. Indices for this task are usually derived by considering the Lie group as a Riemannian manifold. Then, however, compatibility with the group operation is guaranteed only if a bi-invariant metric exists, which is not the case for most non-compact and non-commutative groups. We show here that if one considers an affine connection structure instead, one obtains bi-invariant generalizations of well-known dissimilarity measures: a Hotelling $T^2$ statistic, Bhattacharyya distance and Hellinger distance. Each of the dissimilarity measures matches its multivariate counterpart for Euclidean data and is translation-invariant, so that biases, e.g., through an arbitrary choice of reference, are avoided. We further derive non-parametric two-sample tests that are bi-invariant and consistent. We demonstrate the potential of these dissimilarity measures by performing group tests on data of knee configurations and epidemiological shape data. Significant differences are revealed in both cases.
In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall's shape space. In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics and employ the approach for longitudinal analysis of 2D rat skulls shapes as well as 3D shapes derived from an imaging study on osteoarthritis. Particularly, we perform hypothesis test and estimate the mean trends.
Thin, curved structures occur in many volumetric datasets. Their analysis using classical volume rendering is difficult because parts of such structures can bend away or hide behind occluding elements. This problem cannot be fully compensated by effective navigation alone, because structure-adapted navigation in the volume is cumbersome and only parts of the structure are visible in each view.
We solve this problem by rendering a spatially transformed view into the volume so that an unobscured visualization of the entire curved structure is obtained. As a result, simple and intuitive navigation becomes possible. The domain of the spatial transform is defined by a triangle mesh that is topologically equivalent to an open disc and that approximates the structure of interest. The rendering is based on ray-casting in which the rays traverse the original curved sub-volume. In order to carve out volumes of varying thickness, the lengths of the rays as well as the position of the mesh vertices can be easily modified in a view-controlled manner by interactive painting. We describe a prototypical implementation and demonstrate the interactive visual inspection of complex structures from digital humanities, biology, medicine, and materials science. Displaying the structure as a whole enables simple inspection of interesting substructures in their original spatial context.
Overall, we show that transformed views utilizing ray-casting-based volume rendering supported by guiding surface meshes and supplemented by local, interactive modifications of ray lengths and vertex positions, represent a simple but versatile approach to effectively visualize thin, curved structures in volumetric data.
We propose generalizations of the T²-statistics of Hotelling and the Bhattacharayya distance for data taking values in Lie groups.
A key feature of the derived measures is that they are compatible with the group structure even for manifolds that do not admit any bi-invariant metric.
This property, e.g., assures analysis that does not depend on the reference shape, thus, preventing bias due to arbitrary choices thereof.
Furthermore, the generalizations agree with the common definitions for the special case of flat vector spaces guaranteeing consistency.
Employing a permutation test setup, we further obtain nonparametric, two-sample testing procedures that themselves are bi-invariant and consistent.
We validate our method in group tests revealing significant differences in hippocampal shape between individuals with mild cognitive impairment and normal controls.
Intrinsic and parametric regression models are of high interest for the statistical analysis of manifold-valued data such as images and shapes. The standard linear ansatz has been generalized to geodesic regression on manifolds making it possible to analyze dependencies of random variables that spread along generalized straight lines. Nevertheless, in some scenarios, the evolution of the data cannot be modeled adequately by a geodesic.
We present a framework for nonlinear regression on manifolds by considering Riemannian splines, whose segments are Bézier curves, as trajectories.
Unlike variational formulations that require time-discretization, we take a constructive approach that provides efficient and exact evaluation by virtue of the generalized de Casteljau algorithm.
We validate our method in experiments on the reconstruction of periodic motion of the mitral valve as well as the analysis of femoral shape changes during the course of osteoarthritis, endorsing Bézier spline regression as an effective and flexible tool for manifold-valued regression.
We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions,
we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common,
nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example
application we have chosen 3D shapes of knee bones, reconstructed from image
data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.
The historical importance of ancient manuscripts is unique since they provide information about the heritage of ancient cultures. Often texts are hidden in rolled or folded documents. Due to recent impro- vements in sensitivity and resolution, spectacular disclosures of rolled hidden texts were possible by X-ray tomography. However, revealing text on folded manuscripts is even more challenging. Manual unfolding is often too risky in view of the fragile condition of fragments, as it can lead to the total loss of the document. X-ray tomography allows for virtual unfolding and enables non-destructive access to hid- den texts. We have recently demonstrated the procedure and tested unfolding algorithms on a mockup sample. Here, we present results on unfolding ancient papyrus packages from the papyrus collection of the Musée du Louvre, among them objects folded along approximately orthogonal folding lines. In one of the packages, the first identification of a word was achieved, the Coptic word for “Lord”.
We present visual methods for the analysis and comparison of the results of curved fibre reconstruction algorithms, i.e., of algorithms extracting characteristics of curved fibres from X-ray computed tomography scans. In this work, we extend previous methods for the analysis and comparison of results of different fibre reconstruction algorithms or parametrisations to the analysis of curved fibres. We propose fibre dissimilarity measures for such curved fibres and apply these to compare multiple results to a specified reference. We further propose visualisation methods to analyse differences between multiple results quantitatively and qualitatively. In two case studies, we show that the presented methods provide valuable insights for advancing and parametrising fibre reconstruction algorithms, and support in improving their results in characterising curved fibres.