Impact of adversarial examples on deep learning models for biomedical image segmentation

Deep learning models, which are increasingly being used in the field of medical image analysis, come with a major security risk, namely, their vulnerability to adversarial examples. Adversarial examples are carefully crafted samples that force machine learning models to make mistakes during testing time. These malicious samples have been shown to be highly effective in misguiding classification tasks. However, research on the influence of adversarial examples on segmentation is significantly lacking. Given that a large portion of medical imaging problems are effectively segmentation problems, we analyze the impact of adversarial examples on deep learning-based image segmentation models. Specifically, we expose the vulnerability of these models to adversarial examples by proposing the Adaptive Segmentation Mask Attack (ASMA). This novel algorithm makes it possible to craft targeted adversarial examples that come with (1) high intersection-over-union rates between the target adversarial mask and the prediction and (2) with perturbation that is, for the most part, invisible to the bare eye. We lay out experimental and visual evidence by showing results obtained for the ISIC skin lesion segmentation challenge and the problem of glaucoma optic disc segmentation. An implementation of this algorithm and additional examples can be found at https://github.com/utkuozbulak/adaptive-segmentation-mask-attack

Discrete harmonic analysis associated with ultraspherical expansions

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calder\'on-Zygmund theorem in our discrete setting

Conical square functions associated with Bessel, Laguerre and Schr\"odinger operators in UMD Banach spaces

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions

UMD Banach spaces and square functions associated with heat semigroups for Schr\"odinger and Laguerre operators

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schr\"odinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L^p-boundedness properties for the square functions to our Banach valued setting by using \gamma-radonifying operators. We also prove that these L^p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times (0,\infty )$, where $\mathcal{L}_\lambda=\partial _t^2+\partial _x^2-\frac{\lambda (\lambda -1)}{x^2}$, with $\lambda >1$. In this paper we characterize the solutions of $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times(0,\infty )$ representable by Bessel-Poisson integrals of BMO-functions as those ones satisfying certain Carleson properties

Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap \mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_\theta$ with symbol $\theta(\cdot/|\cdot|)$, where $\theta \in C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^d$.Comment: 24 page

BMO functions and Balayage of Carleson measures in the Bessel setting

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup associated with the Bessel operator $B_\lambda =-x^{-\lambda }Dx^{2\lambda }Dx^{-\lambda}$, $\lambda >0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson