Search results for "Hausdorff"

Transformations by diagonal matrices in a normed space

1962

ON TOPOLOGICAL SPACES WITH A UNIQUE QUASI-PROXIMITY

1994

Abstract Trying to solve the question of whether every T 1 topological space with a unique compatible quasi-proximity should be hereditarily compact, we show that it is true for product spaces as well as for locally hereditarily Lindelof spaces.

An optimal extension of Marstrand?s plane-packing theorem

2003

We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.

Density theorems for Hausdorff and packing measures

1995

Local dimensions of measures on infinitely generated self-affine sets

2014

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space. We also give an estimate, that holds for all translation vectors, with only assuming the affine maps to be contractive.

Conservative swept volume boundary approximation

2010

We present a novel technique for approximating the boundary of a swept volume. The generator given by an input triangle mesh is rendered under all rigid transformations of a discrete trajectory. We use a special shader program that creates offset geometry of each triangle on the fly, thus guaranteeing a conservative rasterization and correct depth values. Utilizing rasterization mechanisms and the depth buffer we then get a conservative voxelization of the swept volume (SV) and can extract a triangle mesh from its surface. This mesh is simplified maintaining conservativeness as well as an error bound measured in terms of the one-sided Hausdorff distance. For this we introduce a new techniqu…

Hand Held 3D Scanning for Cultural Heritage: Experimenting Low Cost Structure Sensor Scan

2017

In the last years 3D scanning has become an important resource in many fields, in particular it has played a key role in study and preservation of Cultural Heritage. Moreover today, thanks to the miniaturization of electronic components, it has been possible produce a new category of 3D scanners, also known as handheld scanners. Handheld scanners combine a relatively low cost with the advantage of the portability. The aim of this chapter is two-fold: first, a survey about the most recent 3D handheld scanners is presented. As second, a study about the possibility to employ the handheld scanners in the field of Cultural Heritage is conducted. In this investigation, a doorway of the Benedictin…

Combinatorial proofs of two theorems of Lutz and Stull

2021

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatori…

Deep Learning Based Cardiac MRI Segmentation: Do We Need Experts?

2021

Deep learning methods are the de facto solutions to a multitude of medical image analysis tasks. Cardiac MRI segmentation is one such application, which, like many others, requires a large number of annotated data so that a trained network can generalize well. Unfortunately, the process of having a large number of manually curated images by medical experts is both slow and utterly expensive. In this paper, we set out to explore whether expert knowledge is a strict requirement for the creation of annotated data sets on which machine learning can successfully be trained. To do so, we gauged the performance of three segmentation models, namely U-Net, Attention U-Net, and ENet, trained with dif…

Dimension estimates on circular (s,t)-Furstenberg sets

2023

In this paper, we show that circular $(s,t)$-Furstenberg sets in $\mathbb R^2$ have Hausdorff dimension at least $$\max\{\frac{t}3+s,(2t+1)s-t\} \text{ for all $0<s,t\le 1$}.$$ This result extends the previous dimension estimates on circular Kakeya sets by Wolff.