Vector ellipsoidal harmonics are introduced here for the first
time and their analytic peculiarities, as well as their
limitations, are analyzed. A novelty of these vectorial base
functions is that we need to introduce two different inner
products in order to obtain orthogonality on the surface of any
ellipsoid. Furthermore, in contrast to the vector spherical
harmonics which are independent of the radial variable, the vector
ellipsoidal harmonics can not be defined uniformly over a family
of confocal ellipsoids. An expansion theorem is proved which
secures completeness of the vectorial harmonics as well as a
non-trivial algorithm that determines the coefficients of the
expansion. Then, these new functions are used to prove
that, for the realistic ellipsoidal model of the human head, there exists a
component of the neuronal current that is invisible by the
electroencephalographic measurements while it is detectable through
magnetoencephalographic measurements in the exterior of the head.
Furthermore, in contrast to the case of the sphere, where no part of the
current contributes both to the electric potential and to the magnetic
field, we prove here that, in the case of the ellipsoid, there is a part of
the current that influences the electroencephalographic as well as the
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C3.
George Dassios, Michalis N. Tsampas. Vector ellipsoidal harmonics and neuronal current decomposition in
the brain. Inverse Problems & Imaging,
Ying Lin, Qi Ye.
Support vector machine classifiers by non-Euclidean margins.
Mathematical Foundations of Computing,
Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih.
A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition.
Discrete & Continuous Dynamical Systems - B,
Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu.
Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory.
Inverse Problems & Imaging,