General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes

Transcription

1 960 J. Opt. Soc. A. A/ Vol. 23, No. 8/ August 2006 Shu et al. Geneal ethod to deve the elatonshp between two sets of Zene coeffcents coespondng to dffeent apetue szes Huazhong Shu and Ln Luo Laboatoy of Iage Scence and Technology, Depatent of Copute Scence and Engneeng, Southeast Unvesty, Nanjng, Chna Guonu Han L Insttut de Recheche Mathéatque Avancée, Unvesté Lous Pasteu and Cente Natonal de la Recheche Scentfque, 7, ue René-Descates, Stasboug, Fance Jean-Lous Coateux Laboatoe Tateent du Sgnal et de l Iage, Unvesté de Rennes I, Insttut Natonal de la Santê et de la Recheche Mêdcale, U642, Rennes, Fance Receved Novebe 6, 2005; evsed Febuay 5, 2006; accepted Mach 2, 2006; posted Mach 0, 2006 (Doc. ID 6600) Zene polynoals have been wdely used to descbe the abeatons n wavefont sensng of the eye. The Zene coeffcents ae often coputed unde dffeent apetue szes. Fo the sae of copason, the sae apetue daete s equed. Snce no standad apetue sze s avalable fo epotng the esults, t s potant to develop a technque fo convetng the Zene coeffcents obtaned fo one apetue sze to anothe sze. By nvestgatng the popetes of Zene polynoals, we popose a geneal ethod fo establshng the elatonshp between two sets of Zene coeffcents coputed wth dffeent apetue szes Optcal Socety of Aeca OCIS codes: , , INTRODUCTION In the past decades, nteest n wavefont sensng of the huan eye has nceased apdly n the feld of ophthalc optcs. Seveal technques have been developed fo easung abeatons of the eye.,2 In geneal, these technques typcally epesent the abeatons as a wavefont eo ap at the coneal o pupl plane. Zene polynoals, due to the popetes such as othogonalty and otatonal nvaance, have been extensvely used fo fttng coneal sufaces. 3 6 Moeove, the lowe tes of the Zene polynoal expanson can be elated to nown types of abeatons such as defocus, astgats, coa, and sphecal abeaton. 7 When the Zene coeffcents ae coputed, an apetue adus descbng the ccula aea n whch the Zene polynoals ae defned ust be specfed. Such a specfcaton s usually affected by the easueent condtons and by vaaton n natual apetue sze acoss the huan populaton. Snce the Zene coeffcents ae often obtaned unde dffeent apetue szes, the values of the expanson coeffcents cannot be dectly copaed. Unfotunately, ths type of copason s exactly what needs to be done n epeatablty and epdeologcal studes. To solve ths poble, a technque fo convetng a set of Zene coeffcents fo one apetue sze to anothe s equed. Recently, Schwegelng 8 poposed a ethod to deve the elatonshp between the sets of Zene coeffcents fo two dffeent apetue szes, but he dd not povde a full deonstaton fo hs esults. Capbell 9 developed an algoth based on atx epesentaton to fnd a new set of Zene coeffcents fo an ognal set when the apetue sze s changed. The advantage of Capbell s ethod s ts easy pleentaton. In ths pape, by nvestgatng the popetes of Zene polynoals, we pesent a geneal ethod fo establshng the elatonshp between two sets of Zene coeffcents coputed wth dffeent apetue szes. An explct and goous deonstaton of the ethod s gven n detal. It s shown that the esults deved fo the poposed ethod ae uch oe sple than those obtaned by Schwegelng, and oeove, ou ethod can be easly pleented. 2. BACGROUND Zene polynoals have been successfully used n any scentfc eseach felds such as age analyss, 0 patten ecognton, and astonocal telescopes. 2 Soe effcent algoths fo fast coputaton of Zene oents defned by Eq. (7) below have also been epoted. 3 5 Recently, Zene polynoals have been appled to descbe the abeatons n the huan eye. Thee ae seveal dffeent epesentatons of Zene polynoals n the lteatue. We adopt the standad Optcal Socety of /06/ /$ Optcal Socety of Aeca

2 Shu et al. Vol. 23, No. 8/ August 2006/J. Opt. Soc. A. A 96 Aeca notaton. The Zene polynoal of ode n wth ndex descbng the azuthal fequency of the azuthal coponent s defned as Z n n, =N R n cos fo 0 N n R n sn fo 0, n, n even, the adal polynoal R n s gven by n /2 R n = s n s! s! n + /2 s! n /2 s! n 2s, and N n s the noalzaton facto gven by 2n + N n =. 3 +,0 Hee,0 s the onece sybol. Equatons (2) and (3) show that both the adal polynoal R n and the noalzaton facto N n ae syetc about,.e., R n =R n, N n =N n, fo 0. Thus, fo the study of these polynoals, we can only consde the case 0. Let n=+2 wth 0; Eq. (2) can be ewtten as R +2 = s +2 s! s! s! + s! +2 2s = 0 s + + s! s= s! s! + s! +2s ang the change of vaables = s = c,s +2s, + + s! C,s = s s! s! + s!. 5 Snce the Zene polynoals ae othogonal ove the unt ccle, the pola coodnates, ust be scaled to the noalzed pola coodnates, by settng =/ ax, ax denotes the axu adal extent 2 4 of the wavefont eo suface. The wavefont eo, W,, can thus be epesented by a fnte set of the Zene polynoals as N W, = a n, Z n / ax,, 6 N denotes the axu ode used n the epesentaton, and a n, ae the Zene coeffcents gven by a n, =0 ax 0 2 Z n / ax,w,dd. Equaton (7) shows clealy that the coeffcents a n, depend on the choce of ax. Ths dependence aes t dffcult to copae two wavefont eo easues obtaned unde dffeent apetue szes. To suount ths dffculty, t s necessay to develop a ethod that s capable of coputng the Zene coeffcents fo a gven apetue sze 2 based on the expanson coeffcents fo a dffeent apetue sze. Wthout loss of genealty, we assue that taes a value of, and the poble can be foulated as follows. Assue that the wavefont eo can be expessed as N W, = a n, Z n,, 8 the coeffcents a n, ae nown. The sae wavefont eo ust be epesented as N W, = b n, Z n,, 9 s a paaete tang a postve value. We need to fnd the coeffcent conveson elatonshps between two sets of coeffcents b n, and a n,. 3. METHODS AND RESULTS In ths secton, we popose a geneal ethod that allows a new set of Zene coeffcents b n, coespondng to an abtay apetue sze to be found fo an ognal set of coeffcents a n,. As ndcated by Schwegelng, 8 the new coeffcents b n, depend only on the coeffcents a n, that have the sae azuthal fequency. Thus, we consde a subset of tes n Eq. (8), all of whch have the sae azuthal fequency : 7 W, = =0 a +2, N +2 =0 R +2 a +2, N +2 cos, fo 0 R +2 sn, fo 0, 0 s gven by N /2, f N and have the sae paty =N /2, othewse. Slaly, the subset of tes n Eq. (9) wth the sae azuthal fequency can be expessed as

3 962 J. Opt. Soc. A. A/ Vol. 23, No. 8/ August 2006 Shu et al. W, = =0 b +2, N +2 =0 R +2 b +2, N +2 cos, fo 0 R +2 sn, fo 0. 2 By equatng Eqs. (0) and (2), the sne and cosne dependence edately cancels, and ths leads to the followng elaton: f = =0 a +q, P +q = =0 b +q, P +q ; 20 =0 b +2, N +2 R +2 = =0 a +2, N +2 R Note that we have taen nto account only the case of 0; the case 0 can be teated n a sla anne. Let R +2 = N +2 Equaton (3) can be ewtten as =0 b +2, R +2 = =0 R +2. a +2, R To solve Eq. (5), we wll use the followng basc esults. Lea. Let a functon f be expessed as f = a n P n = b n P n, P n s a polynoal of ode n gven by then we have b = a + n 6 P n = =0 c n,, c n,n 0; 7 n=+ n c n, d, = a n, = 0,,2,...,, 8 fo whch C =c n,, wth 0n, sa+ + lowe tangula atx, and D =d n, s the nvese atx of C. The poof of Lea s defeed to Appendx A. We ae nteested n a specal case of Lea fo whch each polynoal ode n can be expessed as n =+q and q ae gven postve nteges, =0,,...,. The coespondng esult s descbed n the followng coollay. Coollay. Gven the postve ntege nubes, q, and, let P n be a set of polynoals defned as P n = P +q = c,s +qs, = 0,,2,...,. then we have +q b +q = a +q + =+ c,j d j, j= +q j qa, = 0,,2,...,, 2 fo whch D =d,j s the nvese atx of C =c,j ; both atces ae a ++ lowe tangle atx. Both Lea and Coollay ae vald fo any type of polynoals. To apply the, an essental step conssts of fndng the nvese atx D o D when the ognal atx C o C s nown. Fo the pupose of ths pape, we ae patculaly nteested n the use of Zene polynoals. Fo the adal polynoals R +2 defned by Eq. (4), we have the followng poposton. Poposton. Fo the lowe tangula atx C whose eleents c,s ae defned by Eq. (5), the eleents of the nvese atx D ae gven as follows: +2s +! +! d,s = s! + + s +!. 22 The poof of Poposton s defeed to Appendx A. Fo the noalzed adal polynoals R +2 defned by Eq. (4), t can be ewtten as R +2 c,s = = N +2 = R , ,0 c,s R +2 = c,s +2s, s! = s +,0 s! s! + s!. 24 Let f be a functon that can be epesented as 9 Snce the noalzaton facto N +2 depends only on and, by usng Poposton, we can easly deve the followng esult wthout poof.

4 Shu et al. Vol. 23, No. 8/ August 2006/J. Opt. Soc. A. A 963 Poposton 2. Fo the lowe tangula atx C whose eleents c,s ae defned by Eq. (24), the eleents of the nvese atx D ae gven as follows: d,s = +,0 2 +2s + d,s = +,0 +2s +! +! 2 s! + + s +!. 25 We ae now eady to establsh the elatonshp between the two sets of Zene coeffcents b,,b +2,, b +4,,...,b +2, and a,,a +2,,a +4,,..., a +2, that appea n Eq. (3). Applyng the Coollay to the noalzed adal polynoals R +2 wth q=2 and usng Eqs. (24) and (25), we have Theoe. Theoe. Fo gven nteges and, and eal postve nube, let b,,b +2,,b +4,,...,b +2, and a,,a +2,,a +4,,...,a +2, be two sets of Zene coeffcents coespondng to the apetue szes and, espectvely; we then have b +2, = +2a +2, + =+ =+ j= c,j d j, +2, 2j a = +2a +2, + C,,a +2,, =0,,...,, 26 C,, = j= j 2j + + j! j! j! + j + +! fo = +, +2,...,0. 27 The elatonshp establshed n Theoe s explct, and the coeffcent b +2, depends only on the set of coeffcents a +2,,a +2+,,...,a +2, ; thus, t s oe sple than that gven by Schwegelng. 8 Note also that even though the above esults wee deonstated fo the case 0, they ean vald fo 0 due to the syety popety of the adal polynoals R n about. Table shows the conveson elatonshp between the coeffcents b n, and a n, fo Zene polynoal expansons up to 45 tes (up to ode 8). The esults ae the sae as those gven by Schwegelng 8 except fo b,. As coectly ndcated by Schwegelng, 8 an nteestng featue can be obseved fo Table : Fo a gven adal polynoal ode n, the conveson fo the ognal to the new coeffcents has the sae fo egadless of the azuthal fequency. Ths can be deonstated as follows. Theoe 2. Let C,, defned by Eq. (27) be the coeffcent of a +2, n the expanson of b +2, gven by Eq. (26), and let C+2l, l, l be the coeffcent of a +2,+2l n the expanson of b +2,+2l l s an ntege nube less than o equal to ; then we have C,, = C +2l, l, l. 28 Poof. Fo Eq. (27), we have l C +2l, l, l= j= t l j + l + + j! 2j +l l j! j + l! + l + j + +! = j= l j + + j! 2j j! j! + j + +!. 29 Table. Coeffcent Conveson Relatonshps fo Zene Polynoal Expansons up to Ode 8 n New Expanson Coeffcents b n, 0 0 b 0, =a 0, 3 2 a 2, a , +3 0, b, = a, 2 2 a 2 3, a , a 4, a , a 5, ,0,2 b 2, = 2 a 2, 5 2 a 4, a 6, a , 3 3,,,3 b 3, = a 3 3, 2 6 a 2 5, a 2 4 7, 4 4, 2,0,2,4 b 4, = 4 a 4, a 6, a 8, 5 5, 3,,,3,5 b 5, = 5 a 5, a 7, 6 6, 4, 2,0,2,4,6 b 6, = a 6 6, 3 7 a 2 8, 7 7, 5, 3,,,3,5,7 b 7, = a 7 7, 8 8, 6, 4, 2,0,2,4,6,8 b 8, = a 8 8,

5 964 J. Opt. Soc. A. A/ Vol. 23, No. 8/ August 2006 Shu et al. Copang Eqs. (27) and (29), we obtan the esult of the theoe. Anothe nteestng featue was also obseved that s suazed n the followng theoe. Theoe 3. Fo a fxed value of N, let N=+2= +2; fo Theoe, we have b +2 l, = N 2la +2 l, + = l+ C, l l,a +2, = N 2la +2 l, + C, =0 b +2 l, = N 2la +2 l, + C, l,a +2, = l+ l = N 2la +2 l, + C, =0 l, + l +a N+2 2l+2,, l =0,,...,. Then C, l, + l + = C, l, + l +. 3 l, + l +a N+2 2l+2,, l =0,,...,, fo =0,,...,l,l = 0,,...,n, Poof. Fo Eq. (27), we have C, l, + l + = l l + + l+ j= l ++ l j + + l + j +! 2j +l + l + j! j + l! + j + l +! = N +2 2l +3N 2l + j=0 + + j 2j N + 2l + j +! j! + j! N + j l +!. 33 Slaly, C, l, + l + = l l + + l+ ++ l j + + l + j +! j= l 2j +l + l + j! j + l! + j + l +! = N +2 2l +3N 2l + j=0 + + j 2j N + 2l + j +! j! + j! N + j l +!. 34 Table 2. Coeffcent Conveson Relatonshps fo Dffeent Values of and Whee N=+2=7 New Expanson Coeffcents b n, 5 b 7,5 = a 7,5 7 b 5,5 = a 5 5,5 4 3 a 2 7,5 3 2 b 7,3 = a 7 7,3 b 5,3 = a 5 5,3 4 3 a 2 7,3 b 3,3 = a 3 3,3 2 6 a 2 5, a 2 4 7,3 b 7, = a 7 7, b 5, = a 5 5, 4 3 a 2 7, 3 b 3, = a 3 3, 2 6 a 2 5, a 2 4 7, b, = a, 2 2 a 2 3, a 2 4 5, a , Copason of Eqs. (33) and (34) shows that Eq. (32) s vald. Table 2 shows the case of N=+2=7 fo dffeent values of and. 4. CONCLUSION We have developed a ethod that s sutable to detene a new set of Zene coeffcents fo an ognal set when the apetue sze s changed. An explct and goous deonstaton of the poposed appoach was gven, and soe useful featues have been obseved and poved. The new algoth allows a fa copason of abeatons, descbed n tes of Zene expanson coeffcents that wee coputed wth dffeent apetue szes. The poposed ethod s sple, and can be easly pleented.

6 Shu et al. Vol. 23, No. 8/ August 2006/J. Opt. Soc. A. A 965 Note that the foulas deved n ths pape ae atheatcally coect fo all values of = / 2, and 2 epesent the ognal and new apetue szes. But fo applcaton puposes, t s stll ecoended to ae 2 less than. In the case 2 s geate than, the wavefont eo data ust be extapolated outsde the egon of the ognal ft. It s woth entonng that such a pocess could poduce eoneous esults snce the Zene polynoals ae no longe othogonal n ths egon and they have hgh-fequency vaatons n the pephees. 8 APPENDIX A Poof of Lea. Equaton (6) can be expessed n atx fo as Usng Eq. (7), we have P f = a 0,a,a 2,...,a P0 P 2 P C P0 P P 2 P = 2 2 P = b 0,b,b 2,...,b P0 P 2 P. P0 P P 2 C P =, 2 = C..., dag,,2, 2 A A2. Substtuton of Eqs. (A2) and (A3) nto Eq. (A) yelds a 0,a,a 2,...,a C 2 A3 b 0,b,b 2,...,b = a 0,a,a 2,...,a C dag,, 2,..., C = a 0,a,a 2,...,a C dag,, 2,..., D. A5 Equaton (8) can be easly obtaned by expandng Eq. (A5). Poof of Poposton. To pove the poposton, we need to deonstate the followng elaton: c,s d s,l =,l, 0 l. A6 Fo =l, by usng Eqs. (5) and (22), we have c, d, +2! =! +! Fo l, we have Let +2 +! +! +2 +! c,s d s,l = s +2l s! s l! s! + s + l +! F,,l,s = = +2l + F,,l,s, s + + s! s l! s! + s + l +!. =. A7 A8 A9 G,,l,s s+ + + s! + ss l = s l! + s! + l + s! l + + l +. A0 It can then be easly vefed that Thus F,,l,s = G,,l,s + G,,l,s. F,,l,s = G,,l,s + G,,l,s A = G,,l, + G,,l,l =0. A2 Thus = b 0,b,b 2,...,b C dag,, 2,...,. 2 A4 We deduce fo Eq. (A8) that c,s d s,l = 0 fo l. A3 The poof s now coplete. Note that the poof of Poposton was nsped by a technque poposed by Petovse et al. 6

7 966 J. Opt. Soc. A. A/ Vol. 23, No. 8/ August 2006 Shu et al. ACNOWLEDGMENTS Ths eseach s suppoted by the Natonal Basc Reseach Poga of Chna unde gant 2003CB7602, the Natonal Natual Scence Foundaton of Chna unde gant , and the Poga fo New Centuy Excellent Talents n Unvesty unde gant NCET We than the efeees fo the helpful coents and suggestons. The coespondng autho s H. Shu. E-al addesses: H. Shu, shu.lst@seu.edu.cn; L.Luo, luo.lst@seu.edu.cn; G. Han, guonu@ath.u.stasbg.f; J.-L. Coateux, jeanlous.coateux@unv-ennes.f. REFERENCES. J. Lang, W. G, S. Goelz, and J. F. Blle, Objectve easueent of the wave abeatons of the huan eye usng a Hatann Shac wave-font senso, J. Opt. Soc. A. A, (994). 2. J. C. He, S. Macos, R. H. Webb, and S. A. Buns, Measueent of the wave-font abeatons of the eye by a fast psychophyscal pocedue, J. Opt. Soc. A. A 5, (998). 3. J. P. Caoll, A ethod to descbe coneal topogaphy, Opto. Vson Sc. 7, (994). 4. J. Schwegelng, J. E. Gevenap, and J.. Mlle, Repesentaton of vdeoeatoscopc heght data wth Zene polynoals, J. Opt. Soc. A. A 2, (995). 5. D. R. Isande, M. J. Collns, and B. Davs, Optal odelng of coneal sufaces wth Zene polynoals, IEEE Tans. Boed. Eng. 48, (200). 6. V. A. Sca, J. Coppens, T. P. van den Beg, and R. L. van de Hejde, Coneal suface econstucton algoth that uses Zene polynoal epesentaton, J. Opt. Soc. A. A 2, (2004). 7. D. R. Isande, M. R. Moelande, M. J. Collns, and B. Davs, Modelng of coneal sufaces wth adal polynoals, IEEE Tans. Boed. Eng. 49, (2002). 8. J. Schwegelng, Scalng Zene expanson coeffcents to dffeent pupl szes, J. Opt. Soc. A. A 9, (2002). 9. C. E. Capbell, Matx ethod to fnd a new set of Zene coeffcents fo an ognal set when the apetue adus s changed, J. Opt. Soc. A. A 20, (2003). 0. M. R. Teague, Iage analyss va the geneal theoy of oents, J. Opt. Soc. A. 70, (980).. R. R. Baley and M. Snath, Othogonal oent featues fo use wth paaetc and non-paaetc classfes, IEEE Tans. Patten Anal. Mach. Intell. 8, (996). 2. J. Y. Wang and D. E. Slva, Wave-font ntepetaton wth Zene polynoals, Appl. Opt. 9, (980). 3. S. O. Belas, M. Ahad, and M. Shdha, Effcent algoth fo fast coputaton of Zene oents, J. Fanln Inst. 333, (996). 4. J. Gu, H. Z. Shu, C. Tououln, and L. M. Luo, A novel algoth fo fast coputaton of Zene oents, Patten Recogn. 35, (2002). 5. C. W. Chong, P. Raveendan, and R. Muundan, A copaatve analyss of algoths fo fast coputaton of Zene oents, Patten Recogn. 36, (2003). 6. M. Petovse, H. S. Wlf, and D. Zelbege, A=B (A Petes, 996) (avalable on lne at the Unvesty of Pennsylvana).