Pupil Center Detection Approaches: A comparative analysis

In the last decade, the development of technologies and tools for eye tracking has been a constantly growing area. Detecting the center of the pupil, using image processing techniques, has been an essential step in this process. A large number of techniques have been proposed for pupil center detection using both traditional image processing and machine learning-based methods. Despite the large number of methods proposed, no comparative work on their performance was found, using the same images and performance metrics. In this work, we aim at comparing four of the most frequently cited traditional methods for pupil center detection in terms of accuracy, robustness, and computational cost. These methods are based on the circular Hough transform, ellipse fitting, Daugman's integro-differential operator and radial symmetry transform. The comparative analysis was performed with 800 infrared images from the CASIA-IrisV3 and CASIA-IrisV4 databases containing various types of disturbances. The best performance was obtained by the method based on the radial symmetry transform with an accuracy and average robustness higher than 94%. The shortest processing time, obtained with the ellipse fitting method, was 0.06 s.Comment: 15 pages, 9 figures, submitted to the journal "Computaci\'on y Sistemas

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann-Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.Instituto de Matemática Interdisciplinar (IMI)Fac. de Ciencias MatemáticasFALSEUnión EuropeaMinisterio de Ciencia e InnovaciónSwiss National Science Foundationunpu

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann--Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense

Singular solutions for fractional parabolic boundary value problems

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial and boundary value problem can still be solved on the condition that the non-zero boundary data must be singular, i.e., the solution $u(t,x)$ blows up as $x$ approaches $\partial \Omega$ in a definite way. In this paper we construct a theory of existence and uniqueness of solutions of the parabolic problem with singular data taken in a very precise sense, and also admitting initial data and a forcing term. When the boundary data are zero we recover the standard fractional heat semigroup. A general class of integro-differential operators may replace the classical fractional Laplacian operators, thus enlarging the scope of the work. As further results on the spectral theory of the fractional heat semigroup, we show that a one-sided Weyl-type law holds in the general class, which was previously known for the restricted and spectral fractional Laplacians, but is new for the censored (or regional) fractional Laplacian. This yields bounds on the fractional heat kernel

Characterisation of homogeneous fractional Sobolev spaces

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces Ds,p(Rn) and their embeddings, for s∈ (0 , 1] and p≥ 1. They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For sp<n or s= p= n= 1 we show that Ds,p(Rn) is isomorphic to a suitable function space, whereas for sp≥n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminor

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

We consider density solutions for gradient flow equations of the form $u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm N(u))$, where $\mathrm N$ is the Newtonian repulsive potential in the whole space $\mathbb R^d$ with the nonlinear convex mobility $\gamma(u)=u^\alpha$, and $\alpha>1$. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case $\gamma(u)=u$. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space $u=c_1t^{-1}$ supported in a ball that spreads in time like $c_2t^{1/d}$, thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form $\gamma(u)=u^\alpha$, with $0<\alpha<1$ studied in [9]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detail analysis allowing us to show a waiting time phenomena. This is a typical behavior for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations ilustrating the proven results and showcasing some open problems