Tracking features in image sequences using discrete Morse functions

The goal of this contribution is to present an application of discrete Morse theory to tracking features in image sequences. The proposed algorithm can be used for tracking moving figures in a filmed scene, for tracking moving particles, as well as for detecting canals in a CT scan of the head, or similar features in other types of data. The underlying idea which is used is the parametric discrete Morse theory presented in [13], where an algorithm for constructing the bifurcation diagram of a discrete family of discrete Morse functions was given. The original algorithm is improved here for the specific purpose of tracking features in images and other types of data, in order to produce more realistic results and eliminate irregularities which appear as a result of noise and excess details in the data

Geometric constructions on spheres and planes in Rn

Using Lie geometry and the Lie product in Rn+3, we give an algebraic description of geometric objects constructed from spheres and planes of dimension n−k, k ≥ 1 in Rn. We define algebraic invariants, which characterize geometric properties of these objects, and their position in Rn

Ascending and descending regions of a discrete Morse function

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.Comment: 23 pages, 12 figure

Coincidence points of maps on Zpα_p{\alpha}-spaces

Sia X uno spazio con una azione libera del gruppo ciclico Z$_{p^{\alpha}}$ ed f : X $\rightarrow$ M una mappa continua. Lo scopo di questo articolo è stimare per mezzo dell'indice Z$_{p^{\alpha}}$ $$A_{f}=\left\{ x\epsilon X\mid f\left(gx\right)=f\left(x\right)\: for\: all\: g\:\epsilon Z_{p^{\alpha}}\right\}$$ quando l'indice dello spazio X è noto ed M verifica opportune proprietà.Let X be a space with a free action of the cyclic group Z$_{p^{\alpha}}$ and f : X $\rightarrow$ M a continuous map. The purpose of this paper is to estimate by means of the Z$_{p^{\alpha}}$- index the size of the set $$A_{f}=\left\{ x\epsilon X\mid f\left(gx\right)=f\left(x\right)\: for\: all\: g\:\epsilon Z_{p^{\alpha}}\right\}$$ when the index of the space X is known, and the space M satisfies certain conditions

PERFECT DISCRETE MORSE FUNCTIONS ON CONNECTED SUMS

We study perfect discrete Morse functions on closed, connected, oriented n-dimensional manifolds. We show how to compose such functions on connected sums of manifolds of arbitrary dimensions and how to decompose them on connected sums of closed oriented surfaces