Quaternion Zernike Spherical Polynomials
Speaker:Dr. João Pedro Morias
Postdoctoral Researcher
the University of Averio
Date & Time:16 Oct 2013 (Wednesday) 14:30
Organized by:Department of Mathematics


Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this development. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In this talk, we introduce the Zernike spherical polynomials within quaternion analysis (R(Q)ZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). R(Q)ZSPs are complete and orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second order homogeneous differential equation are also discussed. As an application, we present 3D plot simulations that demonstrate the effectiveness of our approach. R(Q)ZSPs are new in literature and have some consequences that are now under investigation.


Dr. João Morias is currently a Postdoctoral researcher at the Institute of Applied Analysis at the University of Aveiro of Portugal. Dr. Morias obtained his Dr. rer. nat. in Mathematics at Bauhaus-University Weimar in Germany. He is interested in Clifford analysis, Quaternion analysis, Initial-boundary value problems of partial differential equations, Function spaces, Quasiconformal geometry. The publications of Dr. Morias include, On Orthogonal Monogenics in Oblate Spheroidal Domains, Hadamard Three-Hyperballs Theorem and Overconvergence of Special Monogenic Simple Series , On uncertainty principle for quaternionic linear canonical transform, On convergence properties of 3D spheroidal monogenics, Generalized holomorphic Szegö kernel in 3D spheroids etc.