An Integrated Approach to Seismic Event Location: 1. Evaluating How Method of Location Affects the Volume of Groups of Hypocenters

When seismic events occur in spatially compact clusters, the volume and geometric characteristics of these clusters often provides information about the relative effectiveness of different location methods, or about physical processes occurring within the hypocentral region. This report defines and explains how to determine the convex polyhedron of minimum volume (CPMV) surrounding a set of points. We evaluate both single-event and joint hypocenter determination (JHD) relocations for three rather different clusters of seismic events: 1) nuclear explosions from Mururoa relocated using P and PKP phases reported by the ISC, 2) intermediate depth earthquakes near Bucaramanga, Colombia, relocated using P and PKP phases reported by the ISC, and 3) shallow earthquakes near Vanuatu (formerly, the New Hebrides), relocated using P and S phases from a local station network. This analysis demonstrates that different location methods markedly affect the volume of the CPMV, however, volumes for JHD relations are �not always smaller than volumes for single-event relocations.Phillips Laboratory, Directorate of Geophysics, Air Force Material Command, Hanscom Air Force Base, MassachusettsInstitute for Geophysic

Template-Cut: A Pattern-Based Segmentation Paradigm

We present a scale-invariant, template-based segmentation paradigm that sets up a graph and performs a graph cut to separate an object from the background. Typically graph-based schemes distribute the nodes of the graph uniformly and equidistantly on the image, and use a regularizer to bias the cut towards a particular shape. The strategy of uniform and equidistant nodes does not allow the cut to prefer more complex structures, especially when areas of the object are indistinguishable from the background. We propose a solution by introducing the concept of a "template shape" of the target object in which the nodes are sampled non-uniformly and non-equidistantly on the image. We evaluate it on 2D-images where the object's textures and backgrounds are similar, and large areas of the object have the same gray level appearance as the background. We also evaluate it in 3D on 60 brain tumor datasets for neurosurgical planning purposes.Comment: 8 pages, 6 figures, 3 tables, 6 equations, 51 reference

The dynamical environment of asteroid 21 Lutetia according to different internal models

One of the most accurate models currently used to represent the gravity field of irregular bodies is the polyhedral approach. In this model, the mass of the body is assumed to be homogeneous, which may not be true for a real object. The main goal of the present paper is to study the dynamical effects induced by three different internal structures (uniform, three- and four-layers) of asteroid (21) Lutetia, an object that recent results from space probe suggest being at least partially differentiated. The Mascon gravity approach used in the present work, consists of dividing each tetrahedron into eight parts to calculate the gravitational field around the asteroid. The zero-velocity curves show that the greatest displacement of the equilibrium points occurs in the position of the E4 point for the four-layers structure and the smallest one occurs in the position of the E3 point for the three-layers structure. Moreover, stability against impact shows that the planar limit gets slightly closer to the body with the four-layered structure. We then investigated the stability of orbital motion in the equatorial plane of (21) Lutetia and propose numerical stability criteria to map the region of stable motions. Layered structures could stabilize orbits that were unstable in the homogeneous model.Comment: 10 pages, 7 figures, and 4 Tables. Accepted for publication in MNRA

Extracting curve-skeletons from digital shapes using occluding contours

Curve-skeletons are compact and semantically relevant shape descriptors, able to summarize both topology and pose of a wide range of digital objects. Most of the state-of-the-art algorithms for their computation rely on the type of geometric primitives used and sampling frequency. In this paper we introduce a formally sound and intuitive definition of curve-skeleton, then we propose a novel method for skeleton extraction that rely on the visual appearance of the shapes. To achieve this result we inspect the properties of occluding contours, showing how information about the symmetry axes of a 3D shape can be inferred by a small set of its planar projections. The proposed method is fast, insensitive to noise, capable of working with different shape representations, resolution insensitive and easy to implement

Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations. We propose and study a novel framework to study sets of phylogenetic trees based on tropical geometry. In particular, we focus on characterizing our framework for statistical analyses of evolutionary biological processes represented by phylogenetic trees. Our setting exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics, as well as increased computational efficiency over the current state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl

Polytopes, Feasible Regions and Occlusions in the n-view Reconstruction Problem

This paper assesses the question, given a arbitrary point in P3, can it be reconstructed by a given camera orbit? We show that a solution to this problem can be found by intersecting the frustrums of the cameras in the sequence creating a polyhedron that bounds the area in P3 observed by all cameras. For a projective set of cameras this can be considered as an expansion of the chetral inequalities. We also show an exception to this basic principle is encounted when the point in P3 is occluded. Thus giving a weak condition for occlusion of an arbitrary point in P3