Search results for "Hausdorff"
showing 10 items of 162 documents
Random cutout sets with spatially inhomogeneous intensities
2015
We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Ahlfors-regular metric spaces. We obtain formulas for the Hausdorff dimension of such cutouts in self-similar and self-conformal spaces using the multifractal decomposition of the average densities for the natural measures.
Removable singularities for div v=f in weighted Lebesgue spaces
2018
International audience; Let $w\in L^1_{loc}(\R^n)$ be apositive weight. Assuming that a doubling condition and an $L^1$ Poincar\'e inequality on balls for the measure $w(x)dx$, as well as a growth condition on $w$, we prove that the compact subsets of $\R^n$ which are removable for the distributional divergence in $L^{\infty}_{1/w}$ are exactly those with vanishing weighted Hausdorff measure. We also give such a characterization for $L^p_{1/w}$, $1<p<+\infty$, in terms of capacity. This generalizes results due to Phuc and Torres, Silhavy and the first author.
On Upper Conical Density Results
2010
We report a recent development on the theory of upper conical densities. More precisely, we look at what can be said in this respect for other measures than just the Hausdorff measure. We illustrate the methods involved by proving a result for the packing measure and for a purely unrectifiable doubling measure.
A 3D Deep Neural Network for Liver Volumetry in 3T Contrast-Enhanced MRI.
2020
To create a fully automated, reliable, and fast segmentation tool for Gd-EOB-DTPA-enhanced MRI scans using deep learning. Datasets of Gd-EOB-DTPA-enhanced liver MR images of 100 patients were assembled. Ground truth segmentation of the hepatobiliary phase images was performed manually. Automatic image segmentation was achieved with a deep convolutional neural network. Our neural network achieves an intraclass correlation coefficient (ICC) of 0.987, a Sørensen-Dice coefficient of 96.7 ± 1.9 % (mean ± std), an overlap of 92 ± 3.5 %, and a Hausdorff distance of 24.9 ± 14.7 mm compared with two expert readers who corresponded to an ICC of 0.973, a Sørensen-Dice coefficient of 95.2 ± 2.8 %, and…
A hybrid framework of multiple active appearance models and global registration for 3D prostate segmentation in MRI.
2012
International audience; Real-time fusion of Magnetic Resonance (MR) and Trans Rectal Ultra Sound (TRUS) images aid in the localization of malignant tissues in TRUS guided prostate biopsy. Registration performed on segmented contours of the prostate reduces computational complexity and improves the multimodal registration accuracy. However, accurate and computationally efficient 3D segmentation of the prostate in MR images could be a challenging task due to inter-patient shape and intensity variability of the prostate gland. In this work, we propose to use multiple statistical shape and appearance models to segment the prostate in 2D and a global registration framework to impose shape restri…
The probability that $x$ and $y$ commute in a compact group
2010
We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…
3d mesh denoising using normal based myriad filter
2011
We propose a new filtering scheme for denoising of 3D objects which are represented by a triangular mesh. This scheme consists on applying myriad filter to face normals and then updating the vertices positions in order to preserve the original shape of the object. The choice of the Myriad is justified by the assumption of Cauchy distributed angles between surface normals. This filter improves the performance of a normal-based method which is adapted to the underlying mesh structure. To evaluate these methods of filtering, we use three error metrics. The first is based on the vertices, the second is based on the normals and the third is based on Hausdorff distance. Experimental results demon…
Comments on the paper "COINCIDENCE THEOREMS FOR SOME MULTIVALUED MAPPINGS" by B. E. RHOADES, S. L. SINGH AND CHITRA KULSHRESTHA
2011
The aim of this note is to point out an error in the proof of Theorem 1 in the paper entitled “Coincidence theorems for some multivalued mappings” by B. E. Rhoades, S. L. Singh and Chitra Kulshrestha [Internat. J. Math. & Math. Sci., 7 (1984), 429-434], and to indicate a way to repair it.
The “λ-medial axis”
2005
Medial axis is known to be unstable for nonsmooth objects. For an open set O, we define the weak feature size, wfs, minimum distance between Oc and the critical points of the function distance to Oc. We introduce the "lambda-medial axis" Mλ of O, a subset of the medial axis of O which captures the homotopy type of O when λ < wfs. We show that, at least for some "regular" values of λ, Mλ remains stable under Hausdorff distance perturbations of Oc.
A common extension of Arhangel'skii's Theorem and the Hajnal-Juhasz inequality
2019
AbstractWe present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s so…