Topic
Rectifiable set
About: Rectifiable set is a research topic. Over the lifetime, 47 publications have been published within this topic receiving 890 citations.
Papers
TL;DR: In this article, the authors give a necessary and sufficient condition for a given set K to lie in a rectifiable curve, which is the image of a finite interval under a Lipschitz mapping.
Abstract: Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image of a finite interval under a Lipschitz mapping. Recall that for a connected set F c C, F is a rectifiable curve (not necessarily simple) if and only if l(F) < ~ , where l(-) denotes one dimensional Hausdorff measure. This classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge (but not necessarily each vertex!) at most twice. If K is a finite set then we are essentially reduced to the classical Traveling Salesman Problem (TSP): Compute the length of the shortest Hami l ton ian cycle which hits all points of K. This is the same, up to a constant multiple, as asking for the inf imum of l(F) where F is a curve, K c F. (Such a F is called a spanning tree in TSP theory.) For infinite sets K, we cannot hope in general to have K be a subset of a Jordan curve. What we should therefore look at is connected sets which conta in K. Let Fmi n be the shortest (minimal) spanning tree. Then we cannot possibly solve our problem for sets K of infinite cardinality if we cannot find F, I(F) < C O/(Fmin) , for any finite set K. (Here and throughout the paper C, Co, C1, c o , etc. denote various universal constants.) While there are several algorithms for computing l(Fml.), these algorithms work for finite graphs satisfying the triangle inequality, and do not use the Euclidean properties of K. (See [13] for an excellent discussion of some of these algorithms.) Therefore these methods cannot solve our problem for general infinite K. We present a method which is a minor modification of a well-known algorithm ("Farthest Insert ion" see [13]) which yields a F with I(F) < C O l(Fmi,). The Farthest Insert ion algorithm has been extensively studied with large numerical calculations on computers, and is experimentally good in the sense that the F produced satisfy I(F) < C O l(F,,,i,) for all examples which have
387 citations
TL;DR: This paper explains how varifolds can encode numerically nonoriented objects both from the discrete and the continuous polynomials of any nonoriented rectifiable set.
Abstract: In this paper, we address the problem of orientation that naturally arises when representing shapes like curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient to model a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kind of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by F. Almgren, allow the representation of any non-oriented manifold (more generally any non-oriented rectifiable set). In particular, we explain how varifolds can encode numerically non-oriented objects both from the discrete and continuous point of view. We show various ways to build a Hilbert space structure on the set of varifolds based on the theory of reproducing kernels. We show that, unlike the currents' setting, these metrics are consistent with shape volume (theorem 4.1) and we derive a formula for the variation of metric with respect to the shape (theorem 4.2). Finally, we propose a generalization to non-oriented shapes of registration algorithms in the context of Large Deformations Metric Mapping (LDDMM), which we detail with a few examples in the last part of the paper.
147 citations
TL;DR: In this article, the problem of orientation of shapes is addressed in the context of varifolds, which can encode numerically nonoriented objects both from the discrete and the continuous polynomial.
Abstract: In this paper, we address the problem of orientation that naturally arises when representing shapes such as curves or surfaces as currents. In the field of computational anatomy, the framework of currents has indeed proved very efficient in modeling a wide variety of shapes. However, in such approaches, orientation of shapes is a fundamental issue that can lead to several drawbacks in treating certain kinds of datasets. More specifically, problems occur with structures like acute pikes because of canceling effects of currents or with data that consists in many disconnected pieces like fiber bundles for which currents require a consistent orientation of all pieces. As a promising alternative to currents, varifolds, introduced in the context of geometric measure theory by Almgren, allow the representation of any nonoriented manifold (more generally any nonoriented rectifiable set). In particular, we explain how varifolds can encode numerically nonoriented objects both from the discrete and the continuous po...
133 citations
TL;DR: In this article, it was shown that if E is a subset of Euclidean n-space and if the m-dimensional Hausdorff density of E exists and equals one Hm almost everywhere in E, then E is countably (Hm, m) rectifiable.
Abstract: The purpose of this paper is to prove the following theorem: If E is a subset of Euclidean n-space and if the m-dimensional Hausdorff density of E exists and equals one Hm almost everywhere in E, then E is countably (Hm, m) rectifiable. Here Hm is the m-dimensional Hausdorff measure. The proof is a generalization of the proof given by J. M. Marstrand in the special case n = 3, m = 2. 1. Notation and terminology. Throughout the whole paper m and n will be fixed integers such that 1 S m 0, E(6)= {xEERn:d(x,E)SI}. Theboundaryof E isdenotedby ME. For aERn and O
56 citations
TL;DR: In this article, a generalized version of the classical F. and M. Riesz theorem was shown to be equivalent to quantitative mutual absolute continuity of harmonic measure, and surface measure.
Abstract: Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a uniformly rectifiable set of dimension $n$. Then bounded harmonic functions in $\Omega:= \mathbb{R}^{n+1}\setminus E$ satisfy Carleson measure estimates, and are "$\varepsilon$-approximable". Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure, and surface measure.
37 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 5 |
| 2018 | 4 |
| 2017 | 3 |
| 2016 | 1 |
| 2015 | 4 |