arXiv subjects

Balázs Maga

Publications and source records attributed to Balázs Maga.

29 records · Page 2Linked to original sources

arXiv2020

Attention U-Net Based Adversarial Architectures for Chest X-ray Lung Segmentation

Chest X-ray is the most common test among medical imaging modalities. It is applied for detection and differentiation of, among others, lung cancer, tuberculosis, and pneumonia, the last with importance due to the COVID-19 disease. Integrating computer-aided detection methods into the radiologist diagnostic pipeline, greatly reduces the doctors' workload, increasing reliability and quantitative analysis. Here we present a novel deep learning approach for lung segmentation, a basic, but arduous task in the diagnostic pipeline. Our method uses state-of-the-art fully convolutional neural networks in conjunction with an adversarial critic model. It generalized well to CXR images of unseen datasets with different patient profiles, achieving a final DSC of 97.5% on the JSRT dataset.

eess.IV↗

arXiv2020

Characterization of lip sets

We denote the local ``little" Lipschitz constant of a function $f: {\mathbb R}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a continuous function $f$ such that $ {\mathrm{lip}}f=\mathbf{1} _E$? In an earlier paper we introduced the concept of strongly one-sided dense sets. Our main result characterizes $ {\mathrm{lip}}1$ sets as countable unions of closed sets which are strongly one-sided dense. We also show that a stronger statement is not true i.e. there are strongly one-sided dense $F _σ$ sets which are not $ {\mathrm{lip}}1$.

math.CA↗

arXiv2019

Generic Birkhoff Spectra

Suppose that $Ω= \{0, 1\}^ {\mathbb {N}}$ and $ σ$ is the one-sided shift. The Birkhoff spectrum $ \displaystyle S_{f}( α)=\dim_{H}\Big \{ ω\in Ω:\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N f(σ^n ω) = α\Big \},$ where $\dim_{H}$ is the Hausdorff dimension. It is well-known that the support of $S_{f}( α)$ is a bounded and closed interval $L_f = [α_{f, \min}^*, α_{f, \max}^*]$ and $S_{f}( α)$ on $L_{f}$ is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical $f\in C( Ω)$ in the sense of Baire category. For a dense set in $C( Ω)$ the spectrum is not continuous on $ {\mathbb {R}}$, though for the generic $f\in C( Ω)$ the spectrum is continuous on $ {\mathbb {R}}$, but has infinite one-sided derivatives at the endpoints of $L_{f}$. We give an example of a function which has continuous $S_{f}$ on $ {\mathbb {R}}$, but with finite one-sided derivatives at the endpoints of $L_{f}$. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions $f$ and $g$ are close in $C( Ω)$ then $S_{f}$ and $S_{g}$ are close on $L_{f}$ apart from neighborhoods of the endpoints.

math.DS↗

arXiv2018

Baire categorical aspects of first passage percolation II

In this paper we continue our earlier work about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence $\frac{B(t_n)}{t_n}$ for some $(t_n)\to\infty$. Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation.

math.GN↗

arXiv2018

Type $1$ and $2$ sets for series of translates of functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $1$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= {\mathbb R}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= {\mathbb R}$ modulo sets of measure zero. If $ Λ$ is not type $1$ we say that $ Λ$ is type 2. The exact characterization of type $1$ and type $2$ sets is not known. In this paper we continue our study of the properties of type $1$ and $2$ sets. We discuss sub and supersets of type $1$ and $2$ sets and we give a complete and simple characterization of a subclass of dyadic type $1$ sets. We discuss the existence of type $1$ sets containing infinitely many algebraically independent elements. Finally, we consider unions and Minkowski sums of type $1$ and $2$ sets.

math.CA↗

arXiv2018

Random constructions for translates of non-negative functions

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is type $2$ if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable "witness function" $f: {\mathbb R}\to [0,+ {\infty})$ such that both the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}$ and its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}$ are of positive Lebesgue measure. If $ Λ$ is not type $2$ we say that $ Λ$ is type $1$. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set $ Λ$ if we randomly delete its elements. Motivated by results concerning weighted sums $\sum c_n f(nx)$ and the Khinchin conjecture, we also discuss some results about weighted sums $\sum_{n=1}^{\infty}c_n f(x+λ_n)$.

math.CA↗

arXiv2018

On series of translates of positive functions III

Suppose $Λ$ is a discrete infinite set of nonnegative real numbers. We say that $ Λ$ is of type 1 if the series $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: { {\mathbb R}}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= { {\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= { {\mathbb R}}$ modulo sets of measure zero. If $ Λ$ is not of type 1 we say that $ Λ$ is of type 2. In this paper we show that there is a universal $ Λ$ with gaps monotone decreasingly converging to zero such that for any open subset $G \subset { {\mathbb R}}$ one can find a characteristic function $f_{G}$ such that $G \subset D(f_G, Λ)$ and $C(f_G, Λ)= { {\mathbb R}} {\setminus} G$ modulo sets of measure zero. We also consider the question whether $C(f, Λ)$ can contain non-degenerate intervals for continuous functions when $D(f, Λ)$ is of positive measure. The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.

math.CA↗

arXiv2017

Baire categorical aspects of first passage percolation

In the previous decades, the theory of first passage percolation became a highly important area of probability theory. In this work, we will observe what can be said about the corresponding structure if we forget about the probability measure defined on the product space of edges and simply consider topology in the terms of residuality. We focus on interesting questions arising in the probabilistic setup that make sense in this setting, too. We will see that certain classical almost sure events, as the existence of geodesics have residual counterparts, while the notion of the limit shape or time constants gets as chaotic as possible.

math.GN↗

arXiv2017

Random power series near the endpoint of the convergence interval

In this paper, we are going to consider power series $$ \sum_{n=1}^{\infty} a_nx^n, $$ where the coefficients $a_n$ are chosen independently at random from a finite set with uniform distribution. We prove that if the expected value of the coefficients is positive (resp. negative), then $$ \lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty\qquad (\text{resp. }\lim_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty) $$ with probability $1$. Also, if the expected value of the coefficients is $0$, then $$ \limsup_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=\infty,\qquad \liminf_{x\to 1-}\sum_{n=1}^{\infty} a_nx^n=-\infty $$ with probability $1$. We investigate the analogous question in terms of Baire categories.

math.CA↗

arXiv2017

Accumulation Points of Graphs of Baire-1 and Baire-2 Functions

During the last few decades E. S. Thomas, S. J. Agronsky, J. G. Ceder, and T. L. Pearson gave an equivalent definition of the real Baire class 1 functions by characterizing their graph. In this paper, using their results, we consider the following problem: let $T$ be a given subset of $[0,1]\times\mathbb{R}$. When can we find a function $f:[0,1]\rightarrow\mathbb{R}$ such that the accumulation points of its graph are exactly the points of $T$? We show that if such a function exists, we can choose it to be a Baire-2 function. We characterize the accumulation sets of bounded and not necessarily bounded functions separately. We also examine the similar question in the case of Baire-1 functions.

math.CA↗

arXiv2016

Characterizations and Properties of Graphs of Baire Functions

Let $X$ be a paracompact topological space and $Y$ be a Banach space. In this paper, we will characterize the Baire-1 functions $f:X\rightarrow{Y}$ by their graph: namely, we will show that $f$ is a Baire-1 function if and only if its graph $gr(f)$ is the intersection of a sequence $(G_n)_{n=1}^{\infty}$ of open sets in $X\times{Y}$ such that for all $x\in{X}$ and $n\in\mathbb{N}$ the vertical section of $G_n$ is a convex set, whose diameter tends to $0$ as $n\rightarrow\infty$. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

math.CA↗