Zernike Polynomials and their Spectral Representation

Transcription

1 Proceedings of the 3 International Conference on Electronics, Signal Processing and Communication Systems Zernie Polynomials and their Spectral Representation Miroslav Vlce Department of Applied Mathematics Faculty of Transportation Sciences Czech Technical University Prague, Czech Republic vlce@fd.cvut.cz Pavel Sova Department of Circuit Theory Faculty of Electrical Engineering Czech Technical University Prague, Czech Republic sova@fel.cvut.cz Abstract The Zernie polynomials Z m n (ρ, ϕ) are nown in optical physics, and they are used for the various diffractions and aberrations problems of lenses. They are defined on a circle, so that their representation decouples radial and axial coordinates. It is now that the Zernie radial polynomials R m n (ρ) are represented through Jacobi polynomials. This paper deals with Chebyshev expansions for Jacobi polynomials. We have developed the recursive evaluation for spectral coefficients used in these expansions. These consequently provide a straightforward interpretation of Fourier transform of Zernie polynomials. I. INTRODUCTION The conventional representation of Zernie radial polynomials gives unsatisfactory results for large values of degree n [3]. Some methods employ the recurrence relations [6], the other [] suggests an algorithm in the form of discrete cosine transform which overcomes other methods in terms of accuracy. Nevertheless, the final formula represents an integral whose evaluation is performed using uniform sampling of the integrand. The proposed Chebyshev expansion of Jacobi polynomials Pn α,β (x) results directly in spectral coefficients of Rn m (ρ) without need of using discrete Fourier transform. Jacobi polynomials n (x) are defined as [7] n m= ( n + α m ) ( ) n + β n m () (x ) n m (x + ) m and they satisfy the self-adjoined differential equation [ d ( x) α+ ( + x) β+ d ] dx dx P n α,β (x) +n (n + + α + β)( x) α ( + x) β Pn α,β (x) =. Later, we will use Jacobi polynomials for radial part Rn m (ρ) of the Zernie polynomials { Zn m Rn m (ρ) cos mϕ (ρ, ϕ) = Rn m (3) (ρ) sin mϕ, () Based on () we can derived for y(x) Pn α,β (x) a standard form of the Jacobi differential equation [7] as ( x )y (x) [(α + β + )x + α β]y (x) +n(n + + α + β)y(x) =. (4) In order to represent Jacobi polynomials through the Chebyshev expansions Pn α,β (x) = a (α,β) (l)t l (x). (5) b (α,β) (l)u l (x), (6) where T l (x), and U l (x) are Chebyshev polynomials of the first, and second ind, respectively, we have developed recursive evaluation of the spectral coefficients a (α,β) (l) and b (α,β) (l). A. Recursive algorithm-i Inserting in (4) y(x) = a (α,β) (l)t l (x), (7) after considerable algebra we obtain the three-point recursive formulae which are concisely summarized in Table I. The algorithm for the coefficients a (α,β) (l) was used for evaluating Jacobi polynomials (5). Two numerical results for Jacobi polynomials P (.5,.75) 8 (x) and P (7,) (x) are summarized in Table II. Note, the second polynomial is closely related to Zernie radial polynomial R 7 39(ρ), that we introduce later. B. Recursive algorithm-ii Inserting in (4) y(x) = b (α,β) (l)u l (x), (8) we need to perform quite involved algebra to obtain the fivepoint recursive formulae - Table III. The algorithm for the

2 Proceedings of the 3 International Conference on Electronics, Signal Processing and Communication Systems TABLE II THE COEFFICIENTS a (.5,.75) (l) AND a (7,) (l) OF JACOBI POLYNOMIALS. TABLE IV THE COEFFICIENTS b (.5,.75) (l) AND b (7,) (l) OF JACOBI POLYNOMIALS. l a (.5,.75) (l) a (7,) (l) e e e e e e e e e e e e6 l b (.5,.75) (l) b (7,) (l) e e e e e e e e e e e e x x m Fig.. Jacobi polynomial P (.5,.75) 8 (x) and its coefficients a (.5,.75) (l) from equation (5) m Fig.. Jacobi polynomial P (.5,.75) 8 (x) and its coefficients b (.5,.75) (l) from equation (6). coefficients b (α,β) (l) was used for evaluating Jacobi polynomials (6). Two numerical results for Jacobi polynomials P (.5,.75) 8 (x) and P (7,) (x) are summarized in Table IV. II. ZERNIKE AND JACOBI POLYNOMIALS The radial function Rn m (ρ) (3) of a Zernie polynomial is related to Jacobi polynomials Pn α,β (x) [5] as R m n (ρ) = () ρ m P m, ( ρ ), (9) where = n m must be an integer. Now, we use the recursive algorithm for evaluating the spectral coefficients a (α,β) (l) ( ρ ) = a (α,β) (l)t l ( ρ ), () and then for radial function we have Rn m (ρ) = () () l a (m,) (l) ρ m T l (ρ). () We can also use the spectral coefficients b (α,β) (l) to represent ( ρ ) = b (α,β) (l)u l ( ρ ). () Chebyshev polynomials of the second ind satisfy the identity U l ( ρ ) = () l U l ( ρ ) = () l U l+(ρ), (3) ρ and then for radial function we have alternatively Rn m (ρ) = () () l b (m,) (l) ρ m U l+ (ρ). (4)

3 Proceedings of the 3 International Conference on Electronics, Signal Processing and Communication Systems Using in () identity ρ m T l (ρ) = m m µ= ( m µ ) T l+m µ (ρ), (5) we can evaluate spectral representation c (m,) of the radial polynomial Rn m (ρ) = () () l c (m,) (l)t l+m (ρ). (6) Similarly, using identity m ρ m U l+ (ρ) = (m) µ= ( m µ ) U l+m µ (ρ), (7) in (4), we obtain spectral representation d (m,) of the radial polynomial Rn m (ρ) = () () l d (m,) (l)u l+m (ρ). (8) As the first Jacobi polynomials are P (m,) ( ρ ) =, (9) P (m,) ( ρ ) = (m + ) (m + )ρ, () it is not difficult to mae the crosschec of radial polynomial representations (9) either with (6), or (8). For illustration, we present an explicit method for finding spectral coefficients c (3,) (l) for Zernie radial polynomial R+3 3 (ρ). Actually, R 3 +3(ρ) = () then we substitute for () l a (3,) (l) ρ 3 T l (ρ), () ρ 3 T l (ρ) = 8 [T l 3(ρ) + 3T l (ρ) + 3T l+ (ρ) +T l+3 (ρ)]. () If we collect the coefficients a (3,) (m) that correspond to the same degree l, we obtain a simple algorithm for spectral coefficients c (3,) (l) for l = to + c (3,) (l) = [ a (3,) (l ) 3a (3,) (l) + 3a (3,) (l + ) 8 ] a (3,) (l + ). III. CONCLUSION (3) We have derived a robust evaluation of Zernie radial polynomials which derives its numerical stability from evaluation of spectral coefficients a (α,β) (l), b (α,β) (l). Instead of using discrete Fourier transform we have employed the Chebyshev expansion for a solution of the differential equation for Jacobi polynomials. Suggested method does not require any transformation (DCT or DFT) to get a spectral coefficients and therefore it is computationally efficient and numerically ρ Fig. 3. Jacobi polynomial P (3,) 4 ( ρ ) according to eq. () is closely related to Zernie radial polynomial R 3 (ρ). Corresponding spectral coefficients are () l a (3,) (l) = [ ,.375, , 5.65, ]. stable. We have presented numerical calculation of Jacobi polynomials P (.5,.75) 8 (x), P (7,) (x), and P (3,) 4 ( ρ ). Using these algorithms for Jacobi polynomials we could easily evaluate corresponding Zernie radial polynomials - Figure 4 and 5. Our algorithms provide Zernie radial polynomials of a considerable high degree n. For computation of Zernie radial polynomials Janssen and Dirsen [] used a discrete Fourier (cosine) transform of Chebyshev polynomial of the second ind R m n (ρ) = π π U n (ρ cos θ) cos mθ dθ. (4) Equations () and (4) suggest that the spectral coefficients a (α,β) (l), b (α,β) (l) can be an alternative representation of the above integral (4). ACKNOWLEDGMENT The authors wish to than the Grant Agency of the Czech Republic for supporting this research through project GAP//795: Novel selective transforms for nonstationary signal processing. REFERENCES [] T. A. Brummelaar, The Contribution of High Order Zernie Modes to Wavefront Tilt, Optics Communications, 5(995), pp [] A. J. E. M. Janssen, P. Dirsen, Computing Zernie polynomials of arbitrary degree using the discrete Fourier transform, Journal of the European Optical Society - Rapid publications, Vol. (7), 5 (995), pp [3] B. Lewis, J. H. Burge, Fitting High-Order Zernie Polynomials to Finite Data Interferometry XVI: Techniques and Analysis, Proc. of SPIE, vol. 8493, [4] R. W. Gray et al., An analytic expression for the field dependence of Zernie polynomials in rotationally symmetric optical systems, Optics Express, Vol., No. 5, Jul, pp [5] A. Wunsche, Generalized Zernie or disc polynomials, Journal of Computational and Applied Mathematics, 74 (5), pp [6] B. H. Shaibaei, R. Pramesran, Recursive fromula to compute Zernie radial polynomials, Optics Letters, Vol. 38, No. 4, Jul 3, pp [7] M. Abramowitz and I. Stegun I., Handboo of Mathematical Functions, Dover Publications, New Yor Inc., 97. 3

4 Proceedings of the 3 International Conference on Electronics, Signal Processing and Communication Systems.5 R 3 (ρ) R 3 (ρ) Fig. 4. Zernie radial polynomial R 3 (ρ) evaluated by cosine transform as in [] and corresponding evaluation using spectral representation of Jacobi polynomial P (3,) 4 ( ρ ).5 R 7 39 (ρ) R 7 39 (ρ) Fig. 5. Zernie radial polynomial R39 7 (ρ) evaluated by cosine transform as in [] and corresponding evaluation using spectral representation of Jacobi polynomial P (7,) ( ρ ). It is worth noting, that for higher degree n the evaluation of Zernie radial polynomial through spectral representation is free of disturbances for. 4

5 Proceedings of the 3 International Conference on Electronics, Signal Processing and Communication Systems TABLE I RECURSIVE EVALUATION OF SPECTRAL COEFFICIENTS a (α,β) (l) FOR A GENERAL JACOBI POLYNOMIAL a (α,β) (l)t l (x). given initial values n, α, β a (α,β) (n) = (n) Γ(n + α + β + ) Γ(n + )Γ(n + α + β + ) a (α,β) (α β)n (n ) = α + β + n a(α,β) (n) body (for = to n end) a (α,β) (n ) = a (α,β) () a (α,β) ()/ (α β) (n ) ( + ) (α + β + n ) a(α,β) (n ) (n ) (α + β + + ) + ( + ) (α + β + n ) a(α,β) (n ) TABLE III RECURSIVE EVALUATION OF SPECTRAL COEFFICIENTS b (α,β) (l) FOR A GENERAL JACOBI POLYNOMIAL b (α,β) (l)u l (x) given initial values n, α, β b (α,β) (n + ) = b (α,β) (n + ) = b (α,β) (n) = (n) Γ(n + α + β + ) Γ(n + )Γ(n + α + β + ) b (α,β) (n ) = body (for = to n 4 (α β)n α + β + n b(α,β) (n) b (α,β) (n 4) [n(n + ) (n 4)(n ) + ( + 4)(α + β )] = + b (α,β) (n 3) (n + 3)(α β) + b (α,β) (n ) [n(n + ) (n )(n ) + ( + )(α + β )] b (α,β) (n ) (n + )(α β) end) + b (α,β) (n ) [n(n + ) (n )(n + ) + (n + )(α + β )] 5