In this paper we establish some new fractional differential properties for a class of fractional circle polynomials. We apply the Zernike polynomials and  a new class of fractional circle polynomials  in modeling ophthalmic surfaces in visual optics and we compare the obtained results. The total RMS error is presented when addressing capability of these functions in fitting with surfaces, and it is showed  that the new fractional circle polynomials can be used as an alternative to the Zernike Polynomials to represent the complete anterior corneal surface.publishe

Born M

M.M. Rodrigues

N. Vieira

Samko SG

Journal of Physics Conference Series

Rodrigues, Maria Manuela Fernandes

Vieira, Nelson Felipe

Repositório Institucional da Universidade de Aveiro

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSThe orthogonality of the fractional circle polynomials and its applicationin modeling of ophthalmic surfacesTo cite this article: M.M. Rodrigues and N. Vieira 2019 J. Phys.: Conf. Ser. 1194 012094 View the article online for updates and enhancements.This content was downloaded from IP address 193.137.168.208 on 06/05/2019 at 11:41Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing LtdGroup32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/0120941The orthogonality of the fractional circle polynomialsand its application in modeling of ophthalmicsurfacesM.M. Rodrigues‡ and N. Vieira‡‡ CIDMA - Center for Research and Development in Mathematics and Applications,Department of Mathematics, University of Aveiro, Campus Universita´rio de Santiago,3810-193 Aveiro, PortugalE-mail: nloureirovieira@gmail.comAbstract. In this paper we establish some new fractional differential properties for a class offractional circle polynomials. We apply the Zernike polynomials and a new class of fractionalcircle polynomials in modeling ophthalmic surfaces in visual optics and we compare the obtainedresults. The total RMS error is presented when addressing capability of these functions in fittingwith surfaces, and it is showed that the new fractional circle polynomials can be used as analternative to the Zernike Polynomials to represent the complete anterior corneal surface.1. IntroductionThe ocular aberrations are commonly described in terms of series of Zernike polynomials thatoffer distinct advances due to their normalization on a circular pupil, however, in some caseswith slow convergence they may not be the most appropriate choice. This fact was analyzed in[1], where the authors showed a better fitting accuracy of circular Bessel Functions for abruptvariations of post-surgical corneal surfaces. The Bessel functions were also used in [2]. Here theauthors studied the of a Bessel beacon generated with a spatial light modulator as a fixationtarget for ophthalmic adaptive optics systems, rather than a conventional point-spread-function.In [2] it was showed an evidence of an increased immunity for defocus fluctuations and wasexamined the power spectral density variations of the individual Zernike terms.In this work, we consider a class of fractional circle polynomials which has orthogonalsets on a circular pupil, and we establish some new fractional differential properties for thisclass of fractional circle polynomials. We aim to investigate the applicability of this class ofspecial functions in modelling ophthalmic surfaces. Moreover, we compare our results with thecorrespondent ones obtained when Zernike polynomials are used.The structure of the paper reads as follows: in the Preliminaries section, we recall some basicfacts about fractional calculus and the g-Jacobi functions. In the 3rd section, we introduce anew class of fractional circle polynomials and we study some properties of these polynomials,namely, orthogonality relations and fractional differential properties. In the last section, weapply the introduced class of polynomials to the modulations of ophthalmic surfaces, and wecompare the fitting properties with those presented by the Zernike polynomials.Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/01209422. Fractional calculus and g-Jacobi functionsWe start by recalling the definition of fractional derivatives and fractional integrals in the senseof Riemann-Liouville (for more details see [3]).Definition 2.1 For t > 0Jνf(t) :=1Γ(ν)∫ t0(t− τ)ν−1 f(τ) dτis called the Riemann-Liouville fractional integral of the function f(t) of order ν with Re(ν) > 0.Definition 2.2 If t > 0 and m ∈ N0 such that m− 1 ≤ ν < m, then the fractional derivative off(t) of order ν is defined asDνf(t) = Dm[Jm−νf(t)], (1)(if it exists) where m− ν > 0.We now turn to the definition and basic properties of the g-Jacobi functions introducedand studied in [4]. These functions play a fundamental role in defining the fractional Zernikepolynomials, and correspond to a generalization for the fractional case of classical Jacobipolynomials.Definition 2.3 We define the (generalized or) g-Jacobi functions by the formulaP (α,β)ν (t) = (−2)−ν Γ(ν + 1)−1 (1− t)−αDν[(1− t)ν+α (1 + t)ν+β], ν > 0, (2)where α > −1, β > −1 and Dν is the Riemann-Liouville fractional differential operator (1).In the following results, we recall some properties of the g-Jacobi functions (see [4]), which areanalogous to the corresponding properties of the classical Jacobi polynomials. In the first tworesults, we provide the corresponding explicit formulas for these functions.Theorem 2.4 For the g-Jacobi functions holds the explicit formulaP (α,β)ν (t) = 2−ν∞∑k=0(ν + αν − k) (ν + βk)(t− 1)k (t+ 1)ν−k, (3)where (αβ)=Γ(1 + α)Γ(1 + β) Γ(1 + α− β)(4)is the binomial coefficient with real arguments.Theorem 2.5 The g-Jacobian functions can be represented asP (α,β)ν (t) =(ν + αν)2F1(−ν, ν + α+ β; α+ 1;1− t2)(5)=1Γ(1 + ν)∞∑k=0(νk)Γ(1 + ν + α+ β + k) Γ(1 + α+ ν)Γ(1 + ν + α+ β) Γ(1 + α+ k)(t− 12)k, (6)where 2F1(a, b; c; t) is the Gauss hypergeometric function.Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/01209433. Fractional circle polynomialsThe aim of this section is to introduce a new class of fractional circle polynomials and to studysome of their main properties. Taking into account the definition of g-Jacobi functions presentedpreviously and the ideas presented in [5], we introduce the definition for a new class of fractionalcircle polynomials.Definition 3.1 We define the fractional circle polynomials byW−mν (ρ, θ) = Rmν (ρ) sin(mθ), when [ν −m] is odd , (7)Wmν (ρ, θ) = Rmν (ρ) cos(mθ), when [ν −m] is even , (8)where m ∈ N0, ν ∈ R+, ν > m, 0 ≤ ρ ≤ 1 is the radial distance, 0 ≤ θ ≤ 2pi is the azimuthalangle, and [β] represents the integer part of β. The fractional radial function Rmν (ρ) will bedefined asRmν (ρ) = (−1)[ ν−m2] ρm P(m, 12)ν−m2(1− 2ρ2), (9)where P(α,β)ν (x) are the g-Jacobi functions defined in (2).In [6] it was proved the following orthogonality relation for the g-Jacobi polynomials Pα,βν (x):Lemma 3.2 (cf. [6]) The following relation holds∫ 1−1(1− x)α (1 + x)β P(α,β)n+θ (x)P(α,β)m+θ (x) dx = ψ (n, θ, α, β) δn,m,where δn,m denotes the Kronecker number andψ(n, θ, α, β) =(−1)θ2α+β+1Γ(α+ n+ θ + 1)Γ(β + n+ θ + 1)Γ(1− θ)Γ(1 + θ)(2n+ α+ β + θ + 1)Γ(n + α+ β + θ + 1)Γ(n + 1 + θ),for 0 ≤ θ < 1, α > −1 and β > −1.It is also possible to obtain an orthogonality relation for the fractional circle polynomialsWmν (ρ, θ), because their radial part is written in terms of the orthogonal class of g-Jacobipolynomial Pα,βν (x). The figure below show the pyramid with the first 15 modes of Wmn+ 12.Now, we establish a fractional differential property satisfied by the fractional circlepolynomials Wmn+ 12. In order to do that, we initially proved a fractional differential property forthe radial part of Wmn+ 12, i.e., for Rmν (ρ).Theorem 3.3 The fractional radial function Rmν (ρ) satisfies the following fractional partialdifferential equationDαρRmν (ρ) +ρα+ 1Dα+1ρ Rmν (ρ) =Γ(1 +m)ρm−α(α+ 1)Γ(m− α)2F1(m− ν2,m+ ν + 12; m+ 1; ρ2),where Dαρ is defined via (1).Proof: Taking into account (9) and (5) we haveDαρRmν (ρ) = (−1)[ ν−m2]Dαρ[ρm P(m, 12)ν−m2(1− 2ρ2)]= (−1)[ν−m2](ν+m2ν−m2)Dαρ[ρm 2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)].Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/0120944Figure 1. Pyramid for W .Then, we derive the explicit form for the fractional partial derivative that appears in the righthand side of the last expression. For that, we apply the Leibnitz product rule presented in [7],and after straightforward calculation we obtainDαρ[ρm 2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=∞∑j=0(αj)Dα−jρ ρm Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]= Γ(1 +m) ρm−α∞∑j=0(j − α− 1j)(−ρ)jΓ(1 +m− α+ j)Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=Γ(1 +m)Γ(1 +m− α)ρm−α∞∑j=0Γ(j − α) Γ(1 +m− α) (−ρ)jΓ(−α) Γ(1 +m− α+ j) j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=Γ(1 +m) (m− α)Γ(1 +m− α) (−α− 1)ρm−α×∞∑j=0Γ(j − α) Γ(m− α) (−ρ)jΓ(−α− 1) Γ(1 +m− α+ j) j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=Γ(1 +m) (m− α)(m− α)Γ(m− α) (−α− 1)ρm−α×∞∑j=0Γ(j − α) Γ(m− α) (−ρ)jΓ(−α− 1) Γ(1 +m− α+ j) j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/0120945=Γ(1 +m)Γ(m− α) (−α− 1)ρm−α×∞∑j=0Γ(j − α) Γ(m− α) (−ρ)jΓ(−α− 1) Γ(1 +m− α+ j) j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]. (10)Analogously, we deduceDα+1ρ[ρm 2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=Γ(1 +m)Γ(m− α)ρm−α−1∞∑j=0Γ(j − α− 1) Γ(m− α) (−ρ)jΓ(−α− 1) Γ(m− α+ j) j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)].(11)So,Dαρ[ρm 2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]−ρ−α− 1Dα+1ρ[ρm 2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]=Γ(1 +m)ρm−α(−α− 1)Γ(m− α)×∞∑j=0(Γ(j − α)Γ(1 +m− α+ j)−Γ(j − α− 1)Γ(m− α+ j))Γ(m− α) (−ρ)jΓ(−α− 1)j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)].(12)The series present in (12) can be looked as a convergent Mengoli’s series, and we have∞∑j=0(Γ(j − α)Γ(1 +m− α+ j)−Γ(j − α− 1)Γ(m− α+ j))Γ(m− α) (−ρ)jΓ(−α− 1)j!Djρ[2F1(m− ν2,m+ ν + 12; m+ 1; ρ2)]= −2F1(m− ν2,m+ ν + 12; m+ 1; ρ2).The previous theorem immediately implies the following corollary.Corollary 3.4 The fractional circle polynomials W±mν (ρ, θ) satisfy the following fractionalpartial differential equationDαρW±mν (ρ, θ) +ρα+ 1Dα+1ρ W±mν (ρ, θ) =Γ(1 +m)ρm−α(α+ 1)Γ(m− α)2F1(m− ν2,m+ ν + 12; m+ 1; ρ2).4. Application in modeling of ophthalmic surfacesIn this section, we compare the behaviour of introduced fractional circle polynomials with theZernike circle polynomials when both are applied in the modulation of ophthalmic surfaces. Theevaluation of the RMS error in fitting these polynomials with the specific surfaces is chosen as aneffective indicator for the comparison of these polynomials. In each case, we use 15 polynomialsof the orthogonal set. The surface of interest can be modeled by:C(r, φ) =P∑p=1apψp(r, φ) + εp(r, φ), (13)Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/0120946where the index p is a polynomial ordering-number, ψp(r, φ), with p = 1, . . . , P , is the pthpolynomial, ap, with p = 1, . . . , P , is the coefficient associated with ψp(r, φ), P is the order, ris the normalised distance from the origin, φ is the angle and εp(r, φ) represents the modelingerror. Throughout this analysis, we chose the polar coordinate system for convenience. Using aset of such discrete orthogonal polynomials, we can form a linear modelC = ψa+ ε, (14)where C is a D−element vector column of surface evaluated at discrete points (rd, φd), withd = 1, . . . ,D, ψ is a (D × P ) matrix of discrete orthogonal polynomials ψp(rd, φd), a is aP−element vector column of coefficients, and ε represents a D-element vector column of themeasurement and modeling error. For this model, the coefficient vector a can be estimatedusing the least squares method, that isâ = (ψTψ)−1ψTC, (15)where T denotes the transposition since the inverse exists. The RMS error can be found byRMSerror =√∑Pp=1(εp(r, φ))2P. (16)For Zernike’s polynomials, the most widely used representation of its radial function is (see [5]):R|m|n (r) =(n−|m|)/2∑s=0(−1)s(n− s)!s!(n−|m|2 − s)!(n+|m|2 − s)!rn−2s, (17)where n ≥ m so the parity of a polynomial is the same as the corresponding n. The Zernikepolynomials are given in the normalized form by:Zmn (r, φ) = Rmn (r) cos(mφ), for even mZ−mn (r, φ) = Rmn (r) sin(mφ), for odd m. (18)In the case of the fractional circle polynomials on a circular disk, they are given byWmn+ 12(r, φ) = r|m|P(α,β)n+ 12(2r2 − 1) exp(miφ). (19)In order to evaluate the RMS error in fitting these polynomials with the specific surfaces, weselected 15 modes of each orthogonal set.• Arrangement of type 1: In the case of the Zernike circle polynomials, from the standardpyramid we consider the first 15 modes to our analysis (m = −4,−3, . . . , 3, 4 and n =0, 1, 2, 3, 4). For the introduced fractional circle polynomials we consider β = 1/2, α = 1,m = 1 and n = 1, . . . , 15.• Arrangement of type 2: In the case of the Zernike circle polynomials, from the standardpyramid we consider the first 15 modes. Concerning the fractional circle polynomials, weconsider the first 15 modes of the pyramid presented in Figure 1.In our analysis, we have considered an interesting case that is the model of the total anterior eyesurface (Fgure 2), which includes the anterior surface of the cornea, limbus and sclera. This wasdone by etching together two spherical surfaces of different radius. To produce this model weemployed typical parameters for anterior corneal radius, visible iris and diameter of the eye. Asshown in table below, the smallest error is given by the fractional circle polynomials in the caseof arrangement of type 1. On the other hand, for arrangement of type 2 the Zernike polynomialspresent the smallest fitting error.Group32IOP Conf. Series: Journal of Physics: Conf. Series 1194 (2019) 012094IOP Publishingdoi:10.1088/1742-6596/1194/1/0120947Figure 2. Total eye model with typical parameters.Arrangement of type 1 Arrangement of type 2Zernike Polynomials 9.49µm 9.49µmFractional Circle Polynomials 2.9µm 142µmBased on these examples, it is clear that in the case of the fractional circle polynomials themodels with more irregularities lead to an increase of the RMS error.Funding InformationThe work of M.M. Rodrigues and N. Vieira was supported by Portuguese funds throughthe CIDMA - Center for Research and Development in Mathematics and Applications, andthe Portuguese Foundation for Science and Technology (“FCT–Fundac¸a˜o para a Cieˆncia e aTecnologia”), within project UID/MAT/ 0416/2013.N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref:IF/00271/2014).References[1] Trevino JP, Gomez-Correa JE, Iskander DR and Chavez-Cerda S 2013 Zernike vs.Bessel circular functions invisual optics Ophthalmic and Physiological Optics 33 394-402[2] Lambert AJ, Daly EM and Dainty CJ 2013 Improved fixation quality provided by a Bessel beacon in anadaptive optics system Ophthalmic and Physiological Optics 33 403-11[3] Samko SG, Kilbas AA and Marichev OI 1993 Fractional integrals and derivatives: theory and applications(New York: Gordon and Breach)[4] Mirevski SP, Boyadijiev L and Scherer R 2007 On the Riemann-Liouville fractional calculus, g-Jabobi functionsand F-Gauss functions Appl. Math. Comput. 187(1) 315-25.[5] Born M and Wolf E 1959 Principles of Optics (London-New York-Paris-Los Angeles: Pergamon Press)[6] Ford NJ, Moayyed H and Rodrigues MM 2018 Orthogonality for a class of generalised Jacobi polynomialPα,βν (x) Fractional Differ. Calc. 8(1) 95-110[7] Herrmann R 2011 Fractional calculus: an introduction for physicists (Hackensack, NJ: World Scientific)

The orthogonality of the fractional circle polynomials and its application in modeling of ophthalmic surfaces

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The orthogonality of the fractional circle polynomials and its application in modeling of ophthalmic surfaces

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