The Ellipsoid Factor for quantification of rods, plates and intermediate forms in 3D geometries

The Ellipsoid Factor (EF) is a method for the local determination of the rod- or plate-like nature of porous or spongy continua. EF at a point within a 3D structure is defined as the difference in axis ratios of the greatest ellipsoid which fits inside the structure and which contains the point of interest, and ranges from -1 for strongly oblate (discus-shaped) ellipsoids, to +1 for strongly prolate (javelin-shaped) ellipsoids. For an ellipsoid with axes a ≤ b ≤ c, EF = a/b – b/c. Here, EF is demonstrated in a Java plugin, Ellipsoid Factor for ImageJ, distributed in the BoneJ plugin collection. Ellipsoid Factor utilises an ellipsoid optimisation algorithm which assumes that maximal ellipsoids are centred on the medial axis, then dilates, rotates and translates slightly each ellipsoid until it cannot increase in volume any further. Ellipsoid Factor successfully identifies rods, plates and intermediate structures within trabecular bone, and summarises the distribution of geometries with an overall EF mean and standard deviation, EF histogram and Flinn diagram displaying a/b versus b/c. Ellipsoid Factor is released to the community for testing, use, and improvement

The N-Vortex Problem on a Symmetric Ellipsoid: A Perturbation Approach

We consider the N-vortex problem on a ellipsoid of revolution. Applying standard techniques of classical perturbation theory we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations the equations of motion for the N-vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid's generating ellipse. First order equations are obtained explicitly. We show numerically that the truncated first order system for the three-vortices system on the symmetric ellipsoid is non-integrable.Comment: 14 pages, 1 figur

Definable ellipsoid method, sums-of-squares proofs, and the isomorphism problem

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the graph isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree. © 2018 ACM.Peer ReviewedPostprint (author's final draft

Kernel Ellipsoidal Trimming

Ellipsoid estimation is an issue of primary importance in many practical areas such as control, system identification, visual/audio tracking, experimental design, data mining, robust statistics and novelty/outlier detection. This paper presents a new method of kernel information matrix ellipsoid estimation (KIMEE) that finds an ellipsoid in a kernel defined feature space based on a centered information matrix. Although the method is very general and can be applied to many of the aforementioned problems, the main focus in this paper is the problem of novelty or outlier detection associated with fault detection. A simple iterative algorithm based on Titterington's minimum volume ellipsoid method is proposed for practical implementation. The KIMEE method demonstrates very good performance on a set of real-life and simulated datasets compared with support vector machine methods