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

A Region-based Randers Geodesic Approach for Image Segmentation

The minimal path model based on the Eikonal partial differential equation has served as a fundamental tool for the applications of image segmentation and boundary detection in the passed two decades. However, the existing approaches commonly only exploit the image edge-based features for computing minimal paths, potentially limiting their performance in complicated segmentation situations. In this paper, we introduce a new variational image segmentation model based on the minimal path framework and the eikonal PDE, where the region-based appearance term that defines then regional homogeneity features can be taken into account for estimating the associated minimal paths. This is done by constructing a Randers geodesic metric interpretation to the region-based active contour energy. As a result, the minimization of the active contour energy is transformed to finding the solution to the Randers eikonal PDE. We also suggest a practical interactive image segmentation strategy, where the target boundary can be delineated by the concatenation of the piecewise geodesic paths. We invoke the Finsler variant of the fast marching method to estimate the geodesic distance map, yielding an efficient implementation of the proposed Eikonal region-based active contour model. Experimental results on both synthetic and real images exhibit that our model indeed achieves encouraging segmentation performance

Modeling and Visualization of Multi-material Volumes

The terminology of multi-material volumes is discussed. The classification of the multi-material volumes is given from the spatial partitions, spatial domain for material distribution, types of involved scalar fields and types of models for material distribution and composition of several materials points of view. In addition to the technical challenges of multi-material volume representations, a range of key challenges are considered before such representations can be adopted as mainstream practice

Multiscale Geometric Methods for Isolating Exercise Induced Morphological Adaptations in the Proximal Femur

The importance of skeletal bone in the functioning of the human body is well-established and acknowledged. Less pervasive among the populace, is the understanding of bone as an adaptive tissue which modulates itself to achieve the most construction sufficient for the role it is habituated to. These mechanisms are more pronounced in the long load bearing bones such as the femur. The proximal femur especially, functions under significant loads and does so with high degree of articulation, making it critical to mobility. Thus, exercising to buttress health and reinforce tissue quality is just as applicable to bone as it is to muscles. However, the efficiency of the adaptive (modelling/remodelling) processes is subdued after maturity, which makes the understanding of its potential even more important. Classically, studies have translated the evaluation of strength in terms of its material and morphology. While the morphology of the femur is constrained within a particular phenotype, minor variations can have a significant bearing on its capability to withstand loads. Morphology has been studied at different scales and dimensions wherein parameters quantified as lengths, areas, volumes and curvatures in two and three dimensions contribute towards characterising strength. The challenge has been to isolate the regions that show response to habitual loads. This thesis seeks to build on the principles of computational anatomy and develop procedures to study the distribution of mechanically relevant parameters. Methods are presented that increase the spatial resolution of traditional cross-sectional studies and develop a conformal mapping procedure for proximal femur shape matching. In addition, prevalent methods in cross-sectional analyses and finite element simulations are employed to analyse the morphology of the unique dataset. The results present the spatial heterogeneity and a multi-scale understanding of the adaptive response in the proximal femur morphology to habitual exercise loading

Displays for Exploration and Comparison of Nested or Intersecting Surfaces

The surfaces of real-world objects almost never intersect, so the human visual system is ill prepared to deal with this rare case. However, the comparison of two similar models or approximations of the same surface can require simultaneous estimation of individual global shape, estimation of point or feature correspondences, and local comparisons of shape and distance between the two surfaces. A key supposition of this work is that these relationships between intersecting surfaces, especially the local relationships, are best understood when the surfaces are displayed such that they do intersect. For instance, the relationships between radiation iso-dose levels and healthy and tumorous tissue is best studied in context with all intersections clearly shown. This dissertation presents new visualization techniques for general layered surfaces, and intersecting surfaces in particular, designed for scientists with problems that require such display. The techniques are enabled by a union/intersection refactoring of intersecting surfaces that converts them into nested surfaces, which are more easily treated for visualization. The techniques are aimed at exploratory visualization, where accurate performance of a variety of tasks is desirable, not just the best technique for one particular task. User studies, utilizing tasks selected based on interviews with scientists, are used to evaluate the effectiveness of the new techniques, and to compare them to some existing, common techniques. The studies show that participants performed the user study tasks more accurately with the new techniques than with the existing techniques