The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Differential Geometry for Model Independent Analysis of Images and Other Non-Euclidean Data: Recent Developments

This article provides an exposition of recent methodologies for nonparametric analysis of digital observations on images and other non-Euclidean objects. Fr\'echet means of distributions on metric spaces, such as manifolds and stratified spaces, have played an important role in this endeavor. Apart from theoretical issues of uniqueness of the Fr\'echet minimizer and the asymptotic distribution of the sample Fr\'echet mean under uniqueness, applications to image analysis are highlighted. In addition, nonparametric Bayes theory is brought to bear on the problems of density estimation and classification on manifolds

Random change on a Lie group and mean glaucomatous projective shape change detection from stereo pair images

In this article we develop a nonparametric methodology for estimating the mean change for matched samples on a Lie group. We then notice that for k>=5, a manifold of projective shapes of k-ads in 3D has the structure of a 3k-15 dimensional Lie group that is equivariantly embedded in a Euclidean space, therefore testing for mean change amounts to a one sample test for extrinsic means on this Lie group. The Lie group technique leads to a large sample and a nonparametric bootstrap test for one population extrinsic mean on a projective shape space, as recently developed by Patrangenaru, Liu and Sughatadasa. On the other hand, in the absence of occlusions, the 3D projective shape of a spatial k-ad can be recovered from a stereo pair of images, thus allowing one to test for mean glaucomatous 3D projective shape change detection from standard stereo pair eye images.Lie group 3D projective shape Extrinsic means One sample test statistic on manifolds Nonparametric bootstrap Stereo pairs 3D scene reconstruction from a pair of noncalibrated camera views Medical imaging of the eye

Command shaping for nonlinear crane dynamics,”

Abstract: Motion-induced vibration can be greatly reduced by properly shaping the reference command. Input shaping is one type of reference shaping method that is based largely on linear superposition. In this paper we document the impact of nonlinear crane dynamics on the effectiveness of input shaping. As typical bridge cranes are driven using Cartesian motions, they behave nearly linearly for low-and moderate-velocity motions. On the other hand, the natural rotational motions of tower cranes make them more nonlinear. The nonlinear equations of motion for both bridge and tower cranes are presented and experimentally verified using two portable cranes. The effectiveness of input shaping on the near-linear bridge crane is explained. Then, a command-shaping algorithm is developed to improve vibration reduction during the more nonlinear slewing motions of the tower crane. Experimental results demonstrate the effectiveness of the proposed approach over a wide range of operating conditions