3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image $I$ is well-composed if the boundary surface of its continuous analog is a 2D manifold. Since 3D images are not often well-composed, there are several voxel-based methods ("repairing" algorithms) for turning them into well-composed ones but these methods either do not guarantee the topological equivalence between the original image and its corresponding well-composed one or involve sub-sampling the whole image. In this paper, we present a method to locally "repair" the cubical complex $Q(I)$ (embedded in $\mathbb{R}^3$) associated to $I$ to obtain a polyhedral complex $P(I)$ homotopy equivalent to $Q(I)$ such that the boundary of every connected component of $P(I)$ is a 2D manifold. The reparation is performed via a new codification system for $P(I)$ under the form of a 3D grayscale image that allows an efficient access to cells and their faces

Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical H\'enon map.Comment: laTeX, 9 figures, 32 page

Subset Warping: Rubber Sheeting with Cuts

Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself

On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists

The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current preprint is from December 2014; it has been submitted to an AIMS journa

Toric topology

We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in "Sugaku" vol. 62 (2010), 386-41