1,708 research outputs found
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 of 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 blows
up as approaches 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–SlobodeckiÄ 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–SlobodeckiÄ 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–SlobodeckiÄ 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 , where is the
Newtonian repulsive potential in the whole space with the
nonlinear convex mobility , and . 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 . For linear mobility it was shown that there is a
special solution in the form of a disk vortex of constant intensity in space
supported in a ball that spreads in time like , thus
showing a discontinuous leading front or shock. Our present results are in
sharp contrast with the case of concave mobilities of the form
, with 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
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