2.BACKGROUND 1.INTRODUCTION GeneralmethodtoderivetherelationshipbetweentwosetsofZernikecoefficientscorrespondingtodifferentaperturesizes

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

to different aperture sizes

Huazhong Shu and Limin Luo

Laboratory of Image Science and Technology, Department of Computer Science and Engineering, Southeast University, 210096 Nanjing, China

Guoniu Han

L’Institut de Recherche Mathématique Avancée, Université Louis Pasteur and Centre National de la Recherche Scientifique, 7, rue René-Descartes, 67084 Strasbourg, France

Jean-Louis Coatrieux

Laboratoire Traitement du Signal et de l’Image, Université de Rennes I, Institut National de la Santê et de la Recherche Mêdicale, U642, 35042 Rennes, France

Received November 16, 2005; revised February 15, 2006; accepted March 2, 2006; posted March 10, 2006 (Doc. ID 66010) Zernike polynomials have been widely used to describe the aberrations in wavefront sensing of the eye. The Zernike coefficients are often computed under different aperture sizes. For the sake of comparison, the same aperture diameter is required. Since no standard aperture size is available for reporting the results, it is im- portant to develop a technique for converting the Zernike coefficients obtained from one aperture size to an- other size. By investigating the properties of Zernike polynomials, we propose a general method for establish- ing the relationship between two sets of Zernike coefficients computed with different aperture sizes. © 2006 Optical Society of America

OCIS codes: 330.4460, 220.1010, 000.3870.

1. INTRODUCTION

In the past decades, interest in wavefront sensing of the human eye has increased rapidly in the field of oph- thalmic optics. Several techniques have been developed for measuring aberrations of the eye.^1,2In general, these techniques typically represent the aberrations as a wave- front error map at the corneal or pupil plane. Zernike polynomials, due to their properties such as orthogonality and rotational invariance, have been extensively used for fitting corneal surfaces.^3–6 Moreover, the lower terms of the Zernike polynomial expansion can be related to known types of aberrations such as defocus, astigmatism, coma, and spherical aberration.⁷ When the Zernike coef- ficients are computed, an aperture radius describing the circular area in which the Zernike polynomials are de- fined must be specified. Such a specification is usually af- fected by the measurement conditions and by variation in natural aperture size across the human population. Since the Zernike coefficients are often obtained under different aperture sizes, the values of the expansion coefficients cannot be directly compared. Unfortunately, this type of comparison is exactly what needs to be done in repeatabil- ity and epidemiological studies. To solve this problem, a technique for converting a set of Zernike coefficients from one aperture size to another is required.

Recently, Schwiegerling⁸ proposed a method to derive the relationship between the sets of Zernike coefficients

for two different aperture sizes, but he did not provide a full demonstration for his results. Campbell⁹ developed an algorithm based on matrix representation to find a new set of Zernike coefficients from an original set when the aperture size is changed. The advantage of Campbell’s method is its easy implementation. In this paper, by in- vestigating the properties of Zernike polynomials, we present a general method for establishing the relation- ship between two sets of Zernike coefficients computed with different aperture sizes. An explicit and rigorous demonstration of the method is given in detail. It is shown that the results derived from the proposed method are much more simple than those obtained by Schwieger- ling, and moreover, our method can be easily imple- mented.

2. BACKGROUND

Zernike polynomials have been successfully used in many scientific research fields such as image analysis,¹⁰pattern recognition,¹¹ and astronomical telescopes.¹² Some effi- cient algorithms for fast computation of Zernike moments defined by Eq. (7) below have also been reported.^13–15Re- cently, Zernike polynomials have been applied to describe the aberrations in the human eye.¹There are several dif- ferent representations of Zernike polynomials in the lit- erature. We adopt the standard Optical Society of

1084-7529/06/081960-7/$15.00 © 2006 Optical Society of America

America notation. The Zernike polynomial of ordernwith index m describing the azimuthal frequency of the azi- muthal component is defined as

Z_n^m共␳,␪兲=

再

^{N−nmNRnmnmR共␳兲nm共␳兲cossin共m共m␪兲␪兲 forformmⱖ⬍00}

冎

^,

兩m兩ⱕn,n−兩m兩even, 共1兲 where the radial polynomialR_n^m共␳兲is given by

R_n^m共␳兲=^{共n−兩m兩兲/2}

兺

_s=0 s!关共n+兩m兩兲^共/2 −^{− 1兲}s^s共兴ⁿ!关共⁻n^s兲−^!兩m兩兲/2 −s兴!␳^n−2s, 共2兲 andN_n^mis the normalization factor given by

N_n^m=

冑

^{21 +共n}␦^{+ 1}m,0^兲

. 共3兲

Here␦m,0is the Kronecker symbol.

Equations (2) and (3) show that both the radial polyno- mialR_n^m共␳兲and the normalization factorN_n^mare symmet- ric about m, i.e., R_n^m共␳兲=R_n^−m共␳兲, N_n^m=N_n^−m, for mⱖ0.

Thus, for the study of these polynomials, we can only con- sider the case wheremⱖ0. Letn=m+ 2k withkⱖ0; Eq.

(2) can be rewritten as

R_m+2k^m 共␳兲=

兺

_s=0^k s!^{共− 1}共k−^兲ss^共兲^m!^{+ 2k}共m+⁻k^s−^兲!s兲!␳^m+2k−2s

=

兺

_s=k^{0 共}s^{− 1}!共k^兲k−s−s^共m兲!⁺共m^k+⁺s^s兲^兲!^!␳^m+2s

共making the change of variables =k−s兲

=

兺

_s=0^{k ck,sm␳m+2s, 共4兲}

where

C_k,s^m =共− 1兲^k−s 共m+k+s兲!

s!共k−s兲!共m+s兲!. 共5兲 Since the Zernike polynomials are orthogonal over the unit circle, the polar coordinates共r,␪兲 must be scaled to the normalized polar coordinates 共␳,␪兲 by setting ␳

=r/rmax, wherermax denotes the maximum radial extent

of the wavefront error surface. The wavefront error, W共r,␪兲, can thus be represented by a finite set of the Zernike polynomials as

W共r,␪兲=

兺

_n=0^N

兺

_m ^{an,mZnm共r/rmax,␪兲, 共6兲}

whereN denotes the maximum order used in the repre- sentation, andan,mare the Zernike coefficients given by

an,m=

冕

0 r_max

冕

0 2␲

Z_n^m共r/rmax,␪兲W共r,␪兲rdrd␪. 共7兲

Equation (7) shows clearly that the coefficientsa_n,mde- pend on the choice ofrmax. This dependence makes it dif- ficult to compare two wavefront error measures obtained under different aperture sizes. To surmount this difficulty, it is necessary to develop a method that is capable of com- puting the Zernike coefficients for a given aperture sizer2

based on the expansion coefficients for a different aper- ture sizer1. Without loss of generality, we assume thatr1

takes a value of 1, and the problem can be formulated as follows.

Assume that the wavefront error can be expressed as

W共r,␪兲=_n=0

兺

^N

兺

_m ^{an,mZnm共r,␪兲, 共8兲}

where the coefficients a_n,m are known. The same wave- front error must be represented as

W共r,␪兲=

兺

_n=0^N

兺

_m ^{bn,mZnm共␭r,␪兲, 共9兲}

where␭is a parameter taking a positive value. We need to find the coefficient conversion relationships between two sets of coefficients兵b_n,m其and兵a_n,m其.

3. METHODS AND RESULTS

In this section, we propose a general method that allows a new set of Zernike coefficients兵b_n,m其 corresponding to an arbitrary aperture size to be found from an original set of coefficients 兵a_n,m其. As indicated by Schwiegerling,⁸ the new coefficientsb_n,mdepend only on the coefficientsa_n,m that have the same azimuthal frequencym. Thus, we con- sider a subset of terms in Eq. (8), all of which have the same azimuthal frequencym:

W_m共r,␪兲=

冦 ^冉

⁻

^兺

^k=0

^冉

^K

^兺

^{k=0Kam+2k,ma−m+2k,mNm+2kmN−m+2k−mRm+2kmR−m+2k−m共r兲}

^冊

^{cos共r兲共}

^冊

^{msin␪兲共,m␪兲, forformmⱖ⬍00}

冣

^{, 共10兲}

whereKis given by

K=

再

^{共N−兩m兩兲/2, ifNandm}have the same parity

共N− 1 −兩m兩兲/2, otherwise

冎

^{. 共11兲}

Similarly, the subset of terms in Eq. (9) with the same azimuthal frequencymcan be expressed as

W_m共r,␪兲=

冦 ^冉

⁻

^兺

^k=0

^冉

^K

^兺

^{k=0Kbm+2k,mb−m+2k,mNm+2kmN−m+2k−mRm+2kmR−m+2k−m共␭r兲}

^冊

^{共␭cosr兲共}

^冊

^{msin␪兲共,m␪兲, forformmⱖ⬍00.}

冣

^共12兲

By equating Eqs. (10) and (12), the sine and cosine depen- dence immediately cancels, and this leads to the following relation:

兺

k=0 K

b_m+2k,mN_m+2k^m R_m+2k^m 共␭r兲=

兺

_k=0^{K am+2k,mNm+2km Rm+2km 共r兲.}

共13兲 Note that we have taken into account only the case ofm ⱖ0; the case wherem⬍0 can be treated in a similar man- ner. Let

R¯

m+2k

m 共r兲=N_m+2k^m R_m+2k^m 共r兲. 共14兲 Equation (13) can be rewritten as

兺

k=0 K

b_m+2k,mR¯

m+2k

m 共␭r兲=

兺

_k=0^{K am+2k,mR¯m+2km 共r兲. 共15兲}

To solve Eq. (15), we will use the following basic re- sults.

Lemma 1. Let a functionf共r兲be expressed as

f共r兲=

兺

_n=0^{K anPn共r兲=}

兺

_n=0^{K bnPn共␭r兲, 共16兲}

whereP_n共r兲is a polynomial of orderngiven by

P_n共r兲=_k=0

兺

^{n cn,krk, cn,n⫽0; 共17兲}

then we have

b_i= 1

␭ⁱ

冋

^ai+n=i+1

^兺

^K

^{冉 兺}

^{k=in cn,k␭k−idk,i}

^冊

^an

册

^{, i}= 0,1,2, . . . ,K, 共18兲 from whichCK=共cn,k兲, with 0ⱕkⱕnⱕK, is a 共K+ 1兲⫻共K + 1兲lower triangular matrix, andD_K=共d_n,k兲is the inverse matrix ofCK.

The proof of Lemma 1 is deferred to Appendix A.

We are interested in a special case of Lemma 1 for which each polynomial order n can be expressed as n

=m+qk where m and q are given positive integers, k

= 0 , 1 , . . . ,K. The corresponding result is described in the following corollary.

Corollary. Given the positive integer numbers m, q, andK, letP_n^m共r兲be a set of polynomials defined as

P_n^m共r兲=P_m+qk^m 共r兲=

兺

_s=0^{k ck,smrm+qs, k}= 0,1,2, . . . ,K.

共19兲 Letf共r兲be a function that can be represented as

f共r兲=

兺

_k=0^{K am+qk,mPm+qkm 共r兲=}

兺

_k=0^{K bm+qk,mPm+qkm 共␭r兲; 共20兲}

then we have

b_m+qk= 1

␭^m+qk

冋

^am+qk+i=k+1

^兺

^K

^{冉 兺}

^{j=ki c␭i,jm共j−k兲qdj,km}

^冊

^am+qi

册

^,

k= 0,1,2, . . . ,K, 共21兲

from which D_K^m=共d_i,j^m兲 is the inverse matrix ofC_K^m=共c_i,j^m兲; both matrices are a共K+ 1兲⫻共K+ 1兲lower triangle matrix.

Both Lemma 1 and Corollary are valid for any type of polynomials. To apply them, an essential step consists of finding the inverse matrix D_K or D_K^m when the original matrixC_KorC_K^mis known. For the purpose of this paper, we are particularly interested in the use of Zernike poly- nomials. For the radial polynomials R_m+2k^m 共r兲 defined by Eq. (4), we have the following proposition.

Proposition 1. For the lower triangular matrix C_K^m whose elementsc_k,s^m are defined by Eq. (5), the elements of the inverse matrixD_K^mare given as follows:

d_k,s^m =共m+ 2s+ 1兲k!共m+k兲!

共k−s兲!共m+k+s+ 1兲!. 共22兲 The proof of Proposition 1 is deferred to Appendix A.

For the normalized radial polynomialsR¯

m+2k

m 共r兲defined by Eq. (14), it can be rewritten as

R¯

m+2k

m 共r兲=N_m+2k^m R_m+2k^m 共r兲

=

冑

^{2共m1 ++ 2k}␦m,0^{+ 1兲}

R_m+2k^m 共r兲=

兺

_s=0^{k ¯ck,smrm+2s,}

共23兲

where

c

¯_k,s^m =

冑

^{2共m1 ++ 2k}␦m,0^{+ 1兲}

c_k,s^m

=共− 1兲^k−s

冑

^{2共m1 ++ 2k}␦m,0^{+ 1兲}

共m+k+s兲! s!共k−s兲!共m+s兲!.

共24兲

Since the normalization factorN_m+2k^m depends only onm andk, by using Proposition 1, we can easily derive the fol- lowing result without proof.

Proposition 2. For the lower triangular matrix C¯

K m

whose elements¯c_k,s^m are defined by Eq. (24), the elements of the inverse matrixD¯

K

mare given as follows:

d¯

k,s

m =

冑

²共m^{1 +}+ 2s^␦m,0+ 1兲d_k,s^m

=

冑

^{共1 +␦m,0兲共m2 + 2s+ 1兲}共k−s兲^k!^!共m^共m+⁺k^k+^兲!s+ 1兲!. 共25兲 We are now ready to establish the relationship between the two sets of Zernike coefficients 兵b_m,m,b_m+2,m, b_m+4,m, . . . ,b_m+2K,m其 and 兵a_m,m,a_m+2,m,a_m+4,m, . . . , am+2K,m其 that appear in Eq. (13). Applying the Corollary to the normalized radial polynomialsR¯

m+2k

m 共r兲 with q= 2 and using Eqs. (24) and (25), we have Theorem 1.

Theorem 1.For given integersmandK, and real posi- tive number␭, let兵b_m,m,b_m+2,m,b_m+4,m, . . . ,b_m+2K,m其 and 兵am,m,am+2,m,am+4,m, . . . ,am+2K,m其 be two sets of Zernike coefficients corresponding to the aperture sizes 1 and ␭, respectively; we then have

b_m+2k,m= 1

␭^m+2k

冋

^{am+2k,m+i=k+1}

^兺

^K

^兺

^j=ki

^冉

^{¯c␭i,jm2共j−k兲d¯j,km}

^冊

^am+2i,m

册

= 1

␭^m+2k

冋

^{am+2k,m+i=k+1}

^兺

^{K C共m,k,i兲am+2i,m}

册

^,

k= 0,1, . . . ,K, 共26兲 where

C共m,k,i兲=

冑

共m+ 2i+ 1兲共m+ 2k+ 1兲

⫻

兺

_j=k^{i 共}_␭^{− 12共j−k兲兲i−j}_共i−j兲!共j−^共mk兲⁺!ⁱ共m^+j+^兲!j+k+ 1兲!

fori=k+ 1,k+ 2, . . . ,0. 共27兲 The relationship established in Theorem 1 is explicit, and the coefficientb_m+2k,mdepends only on the set of co- efficients 兵am+2k,m,am+2共k+1兲,m, . . . ,am+2K,m其; thus, it is more simple than that given by Schwiegerling.⁸Note also that even though the above results were demonstrated for the case mⱖ0, they remain valid for m⬍0 due to the symmetry property of the radial polynomialsR_n^m共r兲about m.

Table 1 shows the conversion relationship between the coefficientsb_n,m and a_n,mfor Zernike polynomial expan- sions up to 45 terms (up to order 8). The results are the same as those given by Schwiegerling⁸except forb_1,m.

As correctly indicated by Schwiegerling,⁸an interesting feature can be observed from Table 1: For a given radial polynomial order n, the conversion from the original to the new coefficients has the same form regardless of the azimuthal frequencym. This can be demonstrated as fol- lows.

Theorem 2.LetC共m,k,i兲defined by Eq. (27) be the co- efficient of a_m+2i,m in the expansion ofb_m+2k,m given by Eq. (26), and let C共m+ 2l,k−l,i−l兲 be the coefficient of am+2i,m+2lin the expansion ofbm+2k,m+2lwherel is an in- teger number less than or equal tok; then we have

C共m,k,i兲=C共m+ 2l,k−l,i−l兲. 共28兲 Proof.From Eq. (27), we have

C共m+ 2l,k−l,i−l兲=

冑

共m+ 2i+ 1兲共m+ 2k+ 1兲_j=k−t

兺

^{i−l 共}_␭^{− 12共j−k+l兲兲i−l−j}_{共i−l−j兲!共j−}^共m_k⁺₊^l_l兲⁺_!ⁱ_共m^+j₊^兲!_{l+j+k+ 1兲!}

=

冑

共m+ 2i+ 1兲共m+ 2k+ 1兲_j=k−l

兺

^{i 共}_␭^{− 12共j−k兲兲i−j}_{共i−j兲!共j−}^共m_k兲⁺_!ⁱ_共m^+j₊^兲!_{j+k+ 1兲!}^{. 共29兲}

Table 1. Coefficient Conversion Relationships for Zernike Polynomial Expansions up to Order 8

n m New Expansion Coefficientsb_n,m

0 0 b_0,m=a_0,m−

冑

³共^{1 −}_␭¹2兲^a2,m+

冑

⁵共^{1 −}_␭³2+²

␭⁴兲^a4,m−

冑

⁷共¹

−_␭⁶2+¹⁰_␭4−_␭⁵6兲^a6,m+ 3共^{1 −10}_␭2+³⁰_␭4−³⁵_␭6+¹⁴_␭8兲^a8,m

1 −1 , 1 b_1,m=¹_␭关^a1,m− 2

冑

²共^{1 −}_␭¹2兲^a3,m+

冑

³共^{3 −}_␭⁸2+⁵

␭⁴兲^a5,m− 4共²

−¹⁰_␭2+¹⁵_␭4−_␭⁷6兲^a7,m兴

2 −2 , 0 , 2 b_2,m=¹

␭²关^a2,m−

冑

¹⁵共^{1 −}_␭¹2兲^a4,m+

冑

²¹共^{2 −}_␭⁵2+³

␭⁴兲^a6,m

−

冑

³共¹⁰⁻⁴⁵_␭2+⁶³

␭⁴−²⁸

␭⁶兲^a8,m兴

3 −3 , −1 , 1 , 3 b_3,m=¹

␭³关^a3,m− 2

冑

⁶共^{1 −}_␭¹2兲^a5,m+ 2

冑

²共^{5 −12}_␭2+⁷

␭⁴兲^a7,m兴

4 −4 , −2 , 0 , 2 , 4 b_4,m=¹

␭⁴关^a4,m− 2

冑

³⁵共^{1 −}_␭¹2兲^a6,m+ 3

冑

⁵共^{3 −}_␭⁷2+⁴

␭⁴兲^a8,m兴

5 −5 , −3 , −1 , 1 , 3 , 5 b_5,m=¹

␭⁵关^a5,m− 4

冑

³共^{1 −}_␭¹4兲^a7,m兴

6 −6 , −4 , −2 , 0 , 2 , 4 , 6 b_6,m=¹

␭⁶关^a6,m− 3

冑

⁷共^{1 −}_␭¹2兲^a8,m兴

7 −7 , −5 , −3 , −1 , 1 , 3 , 5 , 7 b_7,m=¹

␭⁷a_7,m

8 −8 , −6 , −4 , −2 , 0 , 2 , 4 , 6 , 8 b_8,m=¹

␭⁸a_8,m

Comparing Eqs. (27) and (29), we obtain the result of the theorem.

Another interesting feature was also observed that is summarized in the following theorem.

Theorem 3.For a fixed value of N, letN=m+ 2K=m⬘ + 2K⬘; from Theorem 1, we have

b_{m+2共K−l兲,m}= 1

␭^N−2l

冋

^{am+2共K−l兲,m+i=K−l+1}

^兺

^{K C共m,K}

−l,i兲a_m+2

i,m

册

^=␭N−2l1

冋

^{am+2共K−l兲,m+}

^兺

^{i=0l−1C共m,K}

−l,i+K−l+ 1兲aN+2i−2l+2,m

册

^{, l}= 0,1, . . . ,K, 共30兲

b_{m⬘+2共K⬘−l兲,m}= 1

␭^N−2l

冋

^{am⬘+2共K⬘−l兲,m⬘}

+

兺

i=K⬘^−l+1

K⬘

C共m⬘^,K⬘^−l,i兲a_m⬘+2i,m⬘

册

= 1

␭^N−2l

冋

^{am⬘+2共K⬘−l兲,m⬘+}

^兺

^{i=0l−1C共m⬘,K⬘}

−l,i+K⬘^{−l+ 1}兲aN+2i−2l+2,m⬘

册

^,

l= 0,1, . . . ,K. 共31兲 Then

C共m,K−l,i+K−l+ 1兲=C共m⬘^,K⬘^−l,i+K⬘^{−l+ 1}兲. fori= 0,1, . . . ,l− 1,l= 0,1, . . . ,min共K,K⬘兲. 共32兲 Proof. From Eq. (27), we have

C共m,K−l,i+K−l+ 1兲=

冑

共m+ 2i+ 2K− 2l+ 3兲共m+ 2K− 2l+ 1兲

⫻^i+K−l+1_j=k−l

兺

^{共− 1}_␭^{2共j−K+l兲兲i+K+1−l−j}_{共i+K−l+ 1 −}^共m_j兲⁺_!共ⁱ_j⁺₋^K_K⁻₊^l_l⁺_兲!^j_共^{+ 1}_m+^兲!_{j+K−l+ 1兲!}

=

冑

共N+ 2i− 2l+ 3兲共N− 2l+ 1兲

兺

_j=0^{i+1共− 1}_␭^兲2ji+1−jj!共i^共+ 1 −^N+ij^{− 2l}兲!共N⁺+^{j+ 1}j−^兲l^!+ 1兲!. 共33兲 Similarly,

C共m⬘^,K⬘^−l,i+K⬘^{−l+ 1}兲=

冑

共m⬘^{+ 2i+ 2K}⬘^{− 2l+ 3}兲共m+ 2K⬘^{− 2l+ 1}兲

⫻

兺

j=k⬘^−l

i+K⬘^−l+1共− 1兲^{i+K⬘+1−l−j}

␭^{2共j−K⬘+l兲}

共m⬘^+i+K⬘^{−l+j+ 1}兲!

共i+K⬘^{−l+ 1 −j}兲!共j−K⬘^+l兲!共m⬘^+j+K⬘^{−l+ 1}兲!

=

冑

共N+ 2i− 2l+ 3兲共N− 2l+ 1兲

兺

_j=0^{i+1 共− 1}_␭^兲2ji+1−j_j!共i^共_{+ 1 −}^N+i_j^{− 2l}_兲!共N⁺₊^{j+ 1}_j−^兲_l^!_{+ 1兲!}^{. 共34兲}

Comparison of Eqs. (33) and (34) shows that Eq. (32) is valid.

Table 2 shows the case ofN=m+ 2K= 7 for different val- ues ofmandK.

4. CONCLUSION

We have developed a method that is suitable to determine a new set of Zernike coefficients from an original set when the aperture size is changed. An explicit and rigorous demonstration of the proposed approach was given, and some useful features have been observed and proved. The new algorithm allows a fair comparison of aberrations, described in terms of Zernike expansion coefficients that were computed with different aperture sizes. The pro- posed method is simple, and can be easily implemented.

Table 2. Coefficient Conversion Relationships for Different Values ofmandKWhereN=m+ 2K= 7 m K New Expansion Coefficientsb_n,m 5 1 b_7,5=_␭¹

7a_7,5 b_5,5=¹

␭⁵关^a5,5− 4

冑

³共^{1 −}_␭¹2兲^a7,5兴

3 2 b_7,3=¹

␭⁷a_7,3 b_5,3=¹

␭⁵关^a5,3− 4

冑

³共^{1 −}_␭¹2兲^a7,3兴

b_3,3=_␭¹3关^a3,3− 2

冑

⁶共^{1 −}_␭¹²兲^a5,3+ 2

冑

²共^{5 −12}_␭²⁺_␭⁷⁴兲^a7,3兴

b_7,1=¹

␭⁷a_7,1 b_5,1=¹

␭⁵关^a5,1− 4

冑

³共^{1 −}_␭¹²兲^a7,1兴

1 3 b_3,1=_␭¹3关^a3,1− 2

冑

⁶共^{1 −}_␭¹2兲^a5,1+ 2

冑

²共^{5 −12}_␭2+_␭⁷4兲^a7,1兴

b_1,1=¹_␭关^a1,1− 2

冑

²共^{1 −}_␭¹2兲^a3,1+

冑

³共^{3 −}_␭⁸2+⁵

␭⁴兲^a5,1− 4共²

−¹⁰

␭²+¹⁵

␭⁴−⁷

␭⁶兲^a7,1兴

Note that the formulas derived in this paper are math- ematically correct for all values of␭=r1/r2, wherer1 and r₂represent the original and new aperture sizes. But for application purposes, it is still recommended to maker2

less thanr₁. In the case wherer₂is greater thanr₁, the wavefront error data must be extrapolated outside the re- gion of the original fit. It is worth mentioning that such a process could produce erroneous results since the Zernike polynomials are no longer orthogonal in this region and they have high-frequency variations in the peripheries.⁸

APPENDIX A

Proof of Lemma 1. Equation (16) can be expressed in matrix form as

f共r兲=共a₀,a₁,a₂, . . . ,a_K兲

冤

^{PPPPK012⯗共共共共rrrr兲兲兲兲}

冥

=共b₀,b₁,b₂, . . . ,b_K兲

冤

^{PPPPK012共␭共␭共␭⯗共␭rrrr兲兲兲兲}

冥

^{. 共A1兲}

Using Eq. (17), we have

冤

^{PPPPK012⯗共共共共rrrr兲兲兲兲}

冥

^=CK

冤

^rr1r⯗K2

冥

^{, 共A2兲}

冤

^{PPPPK012共␭共␭共␭⯗共␭rrrr兲兲兲兲}

冥

^=CK

冤

^{␭␭␭K12⯗rrr2K}

冥

^{=CKdiag共1,␭,␭2, . . . ,␭K兲}

冤

^rr1⯗r2K

冥

^.

共A3兲 Substitution of Eqs. (A2) and (A3) into Eq. (A1) yields

共a₀,a₁,a₂, . . . ,a_K兲C_K

冤

^rr1⯗rK2

冥

=共b₀,b₁,b₂, . . . ,b_K兲C_Kdiag共1,␭,␭², . . . ,␭^K兲

冤

^rr1⯗rK2

冥

^{. 共A4兲}

Thus

共b₀,b₁,b₂, . . . ,b_K兲

=共a₀,a₁,a₂, . . . ,a_K兲C_K共diag共1,␭,␭², . . . ,␭^K兲兲⁻¹C_K⁻¹

=共a₀,a₁,a₂, . . . ,a_K兲C_Kdiag共1,␭⁻¹,␭⁻², . . . ,␭^−K兲D_K. 共A5兲 Equation (18) can be easily obtained by expanding Eq.

(A5).

Proof of Proposition 1. To prove the proposition, we need to demonstrate the following relation:

兺

_s=l^{k ck,smds,lm =␦k,l, 0ⱕlⱕkⱕK. 共A6兲}

Fork=l, by using Eqs. (5) and (22), we have

c_k,k^md_k,k^m = 共m+ 2k兲!

k!共m+k兲!⫻共m+ 2k+ 1兲k!共m+k兲!

共m+ 2k+ 1兲! = 1.

共A7兲 Forl⬍k, we have

兺

_s=l^{k ck,smds,lm=}

兺

_s=l^{k 共}_共^{− 1}s−^兲l^k−s兲!^共共^mk−^{+ 2l}s兲!^{+ 1}共m^兲共+^ms⁺+^kl+ 1^+s兲^兲!^!

=共− 1兲^k共m+ 2l+ 1兲

兺

_s=l^{k F共m,k,l,s兲, 共A8兲}

where

F共m,k,l,s兲= 共− 1兲^s共m+k+s兲!

共s−l兲!共k−s兲!共m+s+l+ 1兲!. 共A9兲 Let

G共m,k,l,s兲

= 共− 1兲^s+1共m+k+s兲! 共s−l兲!共k+ 1 −s兲!共m+l+s兲!

共k+ 1 −s兲共s−l兲 共k−l兲共m+k+l+ 1兲.

共A10兲 It can then be easily verified that

F共m,k,l,s兲=G共m,k,l,s+ 1兲−G共m,k,l,s兲. 共A11兲 Thus

兺

_s=l^{k F共m,k,l,s兲=}

兺

_s=l^{k 关G共m,k,l,s+ 1兲−G共m,k,l,s兲兴}

=G共m,k,l,k+ 1兲−G共m,k,l,l兲= 0.

共A12兲 We deduce from Eq. (A8) that

兺

_s=l^{k ck,smds,lm = 0 forl⬍k. 共A13兲}

The proof is now complete.

Note that the proof of Proposition 1 was inspired by a technique proposed by Petkovseket al.¹⁶

ACKNOWLEDGMENTS

This research is supported by the National Basic Re- search Program of China under grant 2003CB716102, the National Natural Science Foundation of China under grant 60272045, and the Program for New Century Excel- lent Talents in University under grant NCET-04–0477.

We thank the referees for their helpful comments and suggestions.

The corresponding author is H. Shu. E-mail addresses:

H. Shu, shu.list@seu.edu.cn; L.Luo, luo.list@seu.edu.cn;

G. Han, guoniu@math.u.strasbg.fr; J.-L. Coatrieux, jean- louis.coatrieux@univ-rennes1.fr.

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