Quaternion Zernike spherical polynomials

Morais, J.; Cação, I.

doi:10.1090/S0025-5718-2014-02888-3

Quaternion Zernike spherical polynomials
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by J. Morais and I. Cação PDF

Math. Comp. 84 (2015), 1317-1337 Request permission

Abstract:

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 the present paper we introduce the Zernike spherical polynomials within quaternionic analysis ((R)QZSPs), 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 $\mathbb {R}^3$ and $\mathbb {R}^4$). (R)QZSPs are 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 provide the reader with plot simulations that demonstrate the effectiveness of our approach.

(R)QZSPs are new in literature and have some consequences that are now under investigation.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
A. B. Bhatia and E. Wolf, On the circle polynomials of Zernike and related orthogonal sets, Proc. Cambridge Philos. Soc. 50 (1954), 40–48. MR 58021
S. Bock and K. Gürlebeck, On a generalized Appell system and monogenic power series, Math. Methods Appl. Sci. 33 (2010), no. 4, 394–411. MR 2641616, DOI 10.1002/mma.1213
I. Cação, K. Gürlebeck, and H. Malonek, Special monogenic polynomials and $L_2$-approximation, Adv. Appl. Clifford Algebras 11 (2001), no. S2, 47–60. MR 2075341, DOI 10.1007/BF03219121
I. Cação, Constructive Approximation by Monogenic polynomials. Ph.D. thesis, Universidade de Aveiro, 2004.
I. Cação, K. Gürlebeck and S. Bock, Complete orthonormal systems of spherical monogenics - A constructive approach. L. H. Son et al. (ed.), Methods of Complex and Clifford Analysis, SAS International Publications, Delhi, 241-260 (2005).
I. Cação, K. Gürlebeck, and S. Bock, On derivatives of spherical monogenics, Complex Var. Elliptic Equ. 51 (2006), no. 8-11, 847–869. MR 2234103, DOI 10.1080/17476930600689084
I. Cação and H. Malonek, Remarks on some properties of monogenic polynomials, in: T.E. Simos, G.Psihoyios, Ch. Tsitouras, Special Volume of Wiley-VCH, ICNAAM, 596-599 (2006).
Isabel Cação, Complete orthonormal sets of polynomial solutions of the Riesz and Moisil-Teodorescu systems in $\Bbb R^3$, Numer. Algorithms 55 (2010), no. 2-3, 191–203. MR 2720627, DOI 10.1007/s11075-010-9411-z
N. Canterakis, 3D Zernike Moments and Zernike Affine Invariants For 3d Image Analysis and Recognition. In 11th Scandinavian Conf. on Image Analysis, 1999.
R. Delanghe, On homogeneous polynomial solutions of the Riesz system and their harmonic potentials, Complex Var. Elliptic Equ. 52 (2007), no. 10-11, 1047–1061. MR 2374971, DOI 10.1080/17476930701466630
Richard Delanghe, On homogeneous polynomial solutions of the Moisil-Théodoresco system in $\Bbb R^3$, Comput. Methods Funct. Theory 9 (2009), no. 1, 199–212. MR 2478271, DOI 10.1007/BF03321722
Klaus Gürlebeck and Wolfgang Sprössig, Quaternionic analysis and elliptic boundary value problems, Mathematical Research, vol. 56, Akademie-Verlag, Berlin, 1989. MR 1056478
K. Gürlebeck and W. Sprössig, Quaternionic Calculus for Engineers and Physicists. John Wiley and Sons, Chichester, 1997.
K. Gürlebeck and Helmuth R. Malonek, A hypercomplex derivative of monogenic functions in $\textbf {R}^{n+1}$ and its applications, Complex Variables Theory Appl. 39 (1999), no. 3, 199–228. MR 1717571, DOI 10.1080/17476939908815192
K. Gürlebeck and J. Morais, On orthonormal polynomial solutions of the Riesz system in $\mathbb {R}^3$. Recent Advances in Computational and Applied Mathematics, Springer Netherlands, 143-158 (2011).
A. I. Kamzolov, The best approximation of classes of functions $W^{\alpha }_{p}\,(S^{n})$ by polynomials in spherical harmonics, Mat. Zametki 32 (1982), no. 3, 285–293, 425 (Russian). MR 677597
K. Kou, J. Morais, and Y. Zhang, Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis, Math. Methods Appl. Sci. 36 (2013), no. 9, 1028–1041. MR 3066725, DOI 10.1002/mma.2657
Vladislav V. Kravchenko and Michael V. Shapiro, Integral representations for spatial models of mathematical physics, Pitman Research Notes in Mathematics Series, vol. 351, Longman, Harlow, 1996. MR 1429392
Vladislav V. Kravchenko, Applied quaternionic analysis, Research and Exposition in Mathematics, vol. 28, Heldermann Verlag, Lemgo, 2003. MR 1952055
R. J. Mathar, Zernike basis to cartesian transformations. Serbian Astronomical Journal 179, 107–120 (2009).
J. Morais, Approximation by homogeneous polynomial solutions of the Riesz system in $\mathbb {R}^3$. Ph.D. thesis, Bauhaus-Universität Weimar, 2009.
J. Morais and H. T. Le, Orthogonal Appell systems of monogenic functions in the cylinder, Math. Methods Appl. Sci. 34 (2011), no. 12, 1472–1486. MR 2849009, DOI 10.1002/mma.1457
J. Morais, H. T. Le, and W. Sprößig, On some constructive aspects of monogenic function theory in $\Bbb R^4$, Math. Methods Appl. Sci. 34 (2011), no. 14, 1694–1706. MR 2833822, DOI 10.1002/mma.1474
J. Morais, A complete orthogonal system of spheroidal monogenics, JNAIAM. J. Numer. Anal. Ind. Appl. Math. 6 (2011), no. 3-4, 105–119 (2012). MR 2950034
J. Morais and K. Gürlebeck, Real-part estimates for solutions of the Riesz system in $\Bbb R^3$, Complex Var. Elliptic Equ. 57 (2012), no. 5, 505–522. MR 2903479, DOI 10.1080/17476933.2010.504838
J. Morais, An orthogonal system of monogenic polynomials over prolate spheroids in $\Bbb R^3$, Math. Comput. Modelling 57 (2013), no. 3-4, 425–434. MR 3011170, DOI 10.1016/j.mcm.2012.06.020
J. Morais, K. I. Kou, and W. Sprößig, Generalized holomorphic Szegö kernel in 3D spheroids, Comput. Math. Appl. 65 (2013), no. 4, 576–588. MR 3011442, DOI 10.1016/j.camwa.2012.10.011
J. Morais and K. I. Kou. Constructing prolate spheroidal quaternion wave signals on the sphere, Submitted for publication.
B. R. A. Nijboer, The diffraction theory of aberrations. Ph.D. dissertation, University of Groningen, Groningen, The Netherlands, 1942.
G. Sansone, Orthogonal functions, Pure and Applied Mathematics, Vol. IX, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959. Revised English ed; Translated from the Italian by A. H. Diamond; with a foreword by E. Hille. MR 0103368
M. Shapiro and N. L. Vasilevski, Quaternionic $\psi$-hyperholomorphic functions, singular operators and boundary value problems I. Complex Variables, Theory Appl., 1995.
M. Shapiro and N. L. Vasilevski, Quaternionic $\psi$-hyperholomorphic functions, singular operators and boundary value problems II. Complex Variables, Theory Appl., 1995.
A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224. MR 516081, DOI 10.1017/S0305004100055638
W. J. Tango, The circle polynomials of Zernike and their application in optics, Appl. Phys. 13, 327-332 (1977).
F. Zernike, Diffraction theory of the knife-edge test and its improved version, the phase-contrast method, Physica (Amsterdam) 1, 689-704 (1934).
F. Zernike, H. C. Brinkman, Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome. Proc. Royal Acad. Amsterdam 38, 161-170 (1935).