In some sense, the world is composed of shapes and words, of continuous things and discrete things. The recognition and study of continuous objects in the form of shapes occupies a significant part of the effort of unraveling many geometric questions
. Shapes can be rep- resented with great generality by objects called currents. While the enormous variety and representational power of currents is useful for representing a huge variety of phenomena, it also leads to the problem that knowing something is a respectable current tells you little about how nice or regular it is. In these brief notes I give an intuitive explanation of a result that says that an important class of minimal shape decompositions will be nice if the input shape (current) is nice. These notes are an exposition of the paper by Ibrahim, Krishnamoorthy and Vixie which can be found on the arXiv:1411.0882 and any reference to these notes, should include a reference to that paper as well.
Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space o
f differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $mathbb{R}^2$.
We describe and provide code and examples for a polygonal edge matching method.
We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) equiv int | abla u | dx + lambda int |u - f| dx $; $u,f:Bbb{R}^n to Bbb{R}$. If we restrict to $u = chi_{Sigma}$ and $f = chi_{Omega}$, $Sigma, Omega in Bbb{R
}^n$, the $L^1$TV functional reduces to $E(Sigma) = Per(Sigma) + lambda |Sigmavartriangle Omega |$. We show that there is a minimizer $Sigma$ such that its boundary $partialSigma$ lies between the union of all balls of radius $frac{n}{lambda}$ contained in $Omega$ and the corresponding union of $frac{n}{lambda}$-balls in $Omega^c$. We also show that if a ball of radius $frac{n}{lambda} + epsilon$ is almost contained in $Omega$, a slightly smaller concentric ball can be added to $Sigma$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results.
