Density of Range Capturing Hypergraphs

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $H = H(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as convex distance function Delaunay graphs. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $H(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all "round" sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.Comment: new version with a tight result and shorter proo

Modelling multi-scale microstructures with combined Boolean random sets: A practical contribution

Boolean random sets are versatile tools to match morphological and topological properties of real structures of materials and particulate systems. Moreover, they can be combined in any number of ways to produce an even wider range of structures that cover a range of scales of microstructures through intersection and union. Based on well-established theory of Boolean random sets, this work provides scientists and engineers with simple and readily applicable results for matching combinations of Boolean random sets to observed microstructures. Once calibrated, such models yield straightforward three-dimensional simulation of materials, a powerful aid for investigating microstructure property relationships. Application of the proposed results to a real case situation yield convincing realisations of the observed microstructure in two and three dimensions

Finsler Active Contours

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TPAMI.2007.70713In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery

On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages

We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of the test cases reveals that MMS has, in comparison to standard sampling-based algorithms, a significant advantage in scenarios containing high-dimensional narrow passages. This provokes a novel characterization of narrow passages which attempts to capture their dimensionality, an attribute that had been (to a large extent) unattended in previous definitions.Comment: 20 page

Volume-based Estimates of Left Ventricular Blood Pool Volume and Ejection Fraction from Multi-plane 2D Ultrasound Images

Accurate estimations of left ventricular (LV) blood pool volume and left ventricular ejection fraction (LVEF) are crucial for the clinical diagnosis of cardiac disease, patient management, or other therapeutic treatment decisions, especially given a patient’s LVEF often affects his or her candidacy for cardiovascular intervention. Ultrasound (US) imaging is the most common and least expensive imaging modalities used to non-invasively image the heart to estimate the LV blood pool volume and assess LVEF. Despite advances in 3D US imaging, 2D US images are routinely used by cardiologists to image the heart and their interpretation is inherently based on the 2D LV blood pool area information immediately available in the US images, rather than 3D LV blood pool volume information. This work proposes a method to reconstruct the 3D geometry of the LV blood pool from three tri-plane 2D US images to estimate the LV blood pool volume and subsequently the LVEF. This technique uses a statistical shape model (SSM) of the LV blood pool characterized by several anchor points – the mitral valve hinges, apex, and apex-to-mitral valve midpoints – identified from the three multi-plane 2D US images. Given a new patient image dataset, the diastolic and systolic LV blood pool volumes are estimated using the SSM either as a linear combination of the n-closest LV geometries according to the Mahalanobis distance or based on the n-most dominant principal components identified after projecting the new patient into the principal component space defined by the training dataset. The performance of the proposed method was assessed by comparing the estimated LV blood pool volume and LVEF to those measured using the EchoPac PC clinical software on a dataset consisting of 66 patients, and several combinations of 50-16 used for training and validation, respectively. The studies show the proposed method achieves LV volume and LVEF estimates within 5% of those computed using the clinical software. Lastly, this work proposes an approach that requires minimal user interaction to obtain accurate 3D estimates of LV blood pool volume and LVEF using multi-plane 2D US images and confirms its performance similar to the ground truth clinical measurements

A survey of some arithmetic applications of ergodic theory in negative curvature

This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in $\mathbb R$, $\mathbb C$ and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference "Chaire Jean Morlet: G\'eom\'etrie et syst\`emes dynamiques", at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey.Comment: 31 pages, 15 figure

Bio-inspired variable imaging system simplified to the essentials: Modelling accommodation and gaze movement

A combination of an aspherical hybrid diffractive-refractive lens with a flexible fluidic membrane lens allows the implementation of a light sensitive and wide-aperture optical system with variable focus. This approach is comparable to the vertebrate eye in air, in which the cornea offers a strong optical power and the flexible crystalline lens is used for accommodation. Also following the natural model of the human eye, the decay of image quality with increasing field position is compensated, in the optical system presented here, by successively addressing different tilting angles which mimics saccadic eye-movements. The optical design and the instrumental implementation are presented and discussed, and the working principle is demonstrated