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

Abstract

Zernike polynomials have been widely used to describe the aberrations in wave-front 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 important to develop a technique for converting the Zernike coefficients obtained from one aperture size to another size. In this paper, by investigating the properties of Zernike polynomials, we propose a general method for establishing the relationship between two sets of Zernike coefficients computed with different aperture sizes.

1. Introduction

In the past decades, interest in wave-front sensing of the human eye has increased rapidly in the field of ophthalmic optics. Several techniques have been developed for measuring the aberrations of the eye.^{1, 2} In 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 coefficients are computed, an aperture radius describing the circular area in which the Zernike polynomials are defined must be specified. Such a specification is usually affected 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 can not be directly compared. Unfortunately, this type of comparison is exactly what needs to be done in repeatability 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 investigating the properties of Zernike polynomials, we present a general method for establishing the relationship 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 Schwiegerling, and moreover, our method can be easily implemented.

2. Background

Zernike polynomials have been successfully used in many scientific research fields such as image analysis,¹⁰ pattern recognition,¹¹ astronomical telescope.¹² Some efficient algorithms for fast computation of Zernike moments defined by Eq. (7) below have also been reported.^13–15 Recently, Zernike polynomials have been applied to describe the aberrations in the human eye.¹ There are several different representations of Zernike polynomials in the literature. We adopt standard OSA notation. The Zernike polynomial of order n with index m describing the azimuthal frequency of the azimuthal component is defined as

$$Z_n^m(\rho ,\theta )=\{\begin{array}{c}{N}_n^m{R}_n^m(\rho )cos(m\theta )\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m\ge 0\\ -N_n^m{R}_n^m(\rho )sin(m\theta )\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m<0\end{array},\mid m\mid \le n,n-\mid m\mid \text{even}$$ (1)

where the radial polynomial $R_n^m(\rho )$ is given by

$$R_n^m(\rho )=\underset{s=0}{\overset{(n-\mid m\mid )/2}{\Sigma }}\frac{{(-1)}^s(n-s)!}{s![(n+\mid m\mid )/2-s]![(n-\mid m\mid )/2-s]!}{\rho }^{n-2s}$$ (2)

and $N_n^m$ is the normalization factor given by

$$N_n^m=\sqrt{\frac{2(n+1)}{1+{\delta }_{m,0}}}$$ (3)

Here δ_{m, 0} is the Kronecker symbol.

Eqs. (2) and (3) show that both the radial polynomial $R_n^m(\rho )$ and the normalization factor $N_n^m$ are symmetric about m, i.e., $R_n^m(\rho )=R_n^{-m}(\rho ),N_n^m=N_n^{-m}$, for m ≥ 0. Thus, for the study of these polynomials, we can only consider the case where m ≥ 0. Let n = m + 2k with k ≥ 0, Eq. (2) can be rewritten as

$$\begin{array}{ll}{R}_{m+2k}^m(\rho )& =\underset{s=0}{\overset{k}{\Sigma }}\frac{{(-1)}^s(m+2k-s)!}{s!(k-s)!(m+k-s)!}{\rho }^{m+2k-2s}\\ & =\underset{s=k}{\overset{0}{\Sigma }}\frac{{(-1)}^{k-s}(m+k+s)!}{s!(k-s)!(m+s)!}{\rho }^{m+2s}(\text{making\hspace{0.17em}the\hspace{0.17em}change\hspace{0.17em}of\hspace{0.17em}variable}{s}^{\prime }=k-s)\\ & =\underset{s=0}{\overset{k}{\Sigma }}{c}_{k,s}^m{\rho }^{m+2s}\end{array}$$ (4)

where

$$c_{k,s}^m={(-1)}^{k-s}\frac{(m+k+s)!}{s!(k-s)!(m+s)!}$$ (5)

Since the Zernike polynomials are orthogonal over the unit circle, the polar coordinates (r, θ) must be scaled to the normalized polar coordinates (ρ, θ) by setting ρ = r/r_max, where r_max denotes the maximum radial extent of the wave-front error surface. The wave-front error, W(r, θ) can thus be represented by a finite set of the Zernike polynomials as

$$W(r,\theta )=\underset{n=0}{\overset{N}{\Sigma }}\underset{m}{\Sigma }{a}_{n,m}{Z}_n^m(r/r_{max},\theta )$$ (6)

where N denotes the maximum order used in the representation, and a_{n, m} are the Zernike coefficients given by

$$a_{n,m}={\int }_0^{r_{max}}{\int }_0^{2\pi }{Z}_n^m(r/r_{max},\theta )W(r,\theta )\mathit{rdrd}\theta$$ (7)

The above equation shows clearly that the coefficients a_{n, m} depend on the choice of r_max. This dependence makes it difficult to compare two wave-front error measures obtained under different aperture sizes. To surmount this difficulty, it is necessary to develop a method that is capable to compute the Zernike coefficients for a given aperture size r₂ based on the expansion coefficients for a different aperture size r₁. Without loss of generality, we assume that r₁ takes value 1, and the problem can be formulated as follows.

Assume that the wave-front error can be expressed as

$$W(r,\theta )=\underset{n=0}{\overset{N}{\Sigma }}\underset{m}{\Sigma }{a}_{n,m}{Z}_n^m(r,\theta )$$ (8)

where the coefficients a_{n, m} are known. The same wave-front error must be represented as

$$W(r,\theta )=\underset{n=0}{\overset{N}{\Sigma }}\underset{m}{\Sigma }{b}_{n,m}{Z}_n^m(\lambda r,\theta )$$ (9)

where λ is a parameter taking positive value. We need to find the coefficient conversion relationships between two sets of coefficients {b_{n, m}}and {a_{n, m}}.

3. Methods and Results

In this section, we propose a general method that allows a new set of Zernike coefficients {b_{n, m}} corresponding to an arbitrary aperture size to be found from an original set of coefficients {a_{n, m}}. As indicated by Schwiegerling,⁸ the new coefficients b_{n, m} depend only on the coefficients a_{n, m} that have the same azimuthal frequency m. Thus, we consider a subset of terms in Eq. (8) all of which have the same azimuthal frequency m

$$W_m(r,\theta )=\{\begin{array}{l}\left(\underset{k=0}{\overset{K}{\Sigma }}{a}_{m+2k,m}{N}_{m+2k}^m{R}_{m+2k}^m(r)\right)cos(m\theta ),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m\ge 0\\ -\left(\underset{k=0}{\overset{K}{\Sigma }}{a}_{-m+2k,m}{N}_{-m+2k}^{-m}{R}_{-m+2k}^{-m}(r)\right)sin(m\theta ),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m<0\end{array}$$ (10)

where K is given by

$$K=\{\begin{array}{l}\left(N-\mid m\mid \right)/2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{if\hspace{0.17em}}N\hspace{0.17em}\text{and}\hspace{0.17em}m\hspace{0.17em}\text{have\hspace{0.17em}the\hspace{0.17em}same\hspace{0.17em}parity}\\ \left(N-1-\mid m\mid \right)/2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{otherwise}\end{array}$$ (11)

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

$$W_m(r,\theta )=\{\begin{array}{l}\left(\underset{k=0}{\overset{K}{\Sigma }}{b}_{m+2k,m}{N}_{m+2k}^m{R}_{m+2k}^m(\lambda r)\right)cos(m\theta ),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m\ge 0\\ -\left(\underset{k=0}{\overset{K}{\Sigma }}{b}_{-m+2k,m}{N}_{-m+2k}^{-m}{R}_{-m+2k}^{-m}(\lambda r)\right)sin(m\theta ),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}m<0\end{array}$$ (12)

By equating (10) and (12), the sine and cosine dependence immediately cancels, and this leads to the following relation

$$\underset{k=0}{\overset{K}{\Sigma }}{b}_{m+2k,m}{N}_{m+2k}^m{R}_{m+2k}^m(\lambda r)=\underset{k=0}{\overset{K}{\Sigma }}{a}_{m+2k,m}{N}_{m+2k}^m{R}_{m+2k}^m(r)$$ (13)

Note that we have taken into account only the case of m ≥ 0; the case where m < 0 can be treated in a similar manner. Let

$${\overline{R}}_{m+2k}^m(r)=N_{m+2k}^m{R}_{m+2k}^m(r)$$ (14)

Eq. (13) can be rewritten as

$$\underset{k=0}{\overset{K}{\Sigma }}{b}_{m+2k,m}{\overline{R}}_{m+2k}^m(\lambda r)=\underset{k=0}{\overset{K}{\Sigma }}{a}_{m+2k,m}{\overline{R}}_{m+2k}^m(r)$$ (15)

In order to solve Eq. (15), we will use the following basic results.

Lemma 1

Let a function f(r) be expressed as

$$f(r)=\underset{n=0}{\overset{K}{\Sigma }}{a}_n{P}_n(r)=\underset{n=0}{\overset{K}{\Sigma }}{b}_n{P}_n(\lambda r)$$ (16)

where P_n(r) is a polynomial of order n given by

$$P_n(r)=\underset{k=0}{\overset{n}{\Sigma }}{c}_{n,k}{r}^k,\hspace{0.17em}\text{with\hspace{0.17em}}{c}_{n,n}\ne 0,$$ (17)

then we have

$$b_i=\frac{1}{{\lambda }^i}\left[a_i+\underset{n=i+1}{\overset{K}{\Sigma }}\left(\underset{k=i}{\overset{n}{\Sigma }}\frac{c_{n,k}{d}_{k,i}}{{\lambda }^{k-i}}\right)a_n\right],i=0,1,2,\hspace{0.17em}\dots ,K$$ (18)

from which C_K = (c_n, _k), with 0 ≤ K ≤ n ≤ K, is a (K + 1) × (K + 1) lower triangular matrix, and D_K = (d_n, _k) is the inverse matrix of C_K.

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, and K. let $P_n^m(r)$ be a set of polynomials defined as

$$P_n^m(r)=P_{m+qk}^m(r)=\underset{s=0}{\overset{k}{\Sigma }}{c}_{k,s}^m{r}^{m+qs},k=0,1,2,\dots ,K$$ (19)

Let f(r) be a function that can be represented as

$$f(r)=\underset{k=0}{\overset{K}{\Sigma }}{a}_{m+qk,m}{P}_{m+qk}^m(r)=\underset{k=0}{\overset{K}{\Sigma }}{b}_{m+qk,m}{P}_{m+qk}^m(\lambda r)$$ (20)

Then we have

$$b_{m+qk}=\frac{1}{{\lambda }^{m+qk}}\left[a_{m+qk}+\underset{i=k+1}{\overset{K}{\Sigma }}\left(\underset{j=k}{\overset{i}{\Sigma }}\frac{c_{i,j}^m{d}_{j,k}^m}{{\lambda }^{(j-k)q}}\right)a_{m+qi}\right],k=0,1,2,\dots ,K$$ (21)

from which $D_K^m=\left(d_{i,j}^m\right)$ is the inverse matrix of $C_K^m=\left(c_{i,j}^m\right)$, both matrices are (K + 1) × (K + 1) lower triangle matrix.

Both Lemma 1 and Corollary are valid for any type of polynomials. In order to apply them, an essential step consists of finding the inverse matrix D_K or $D_K^m$ when the original matrix C_K or $C_K^m$ is known. For the purpose of the paper, we are particularly interested in the use of Zernike polynomials. For the radial polynomials $R_{m+2k}^m(r)$ defined by Eq. (4), we have the following proposition.

Proposition 1

For the lower triangular matrix $C_K^m$ whose elements $c_{k,s}^m$ are defined by Eq. (5), the elements of the inverse matrix $D_K^m$ are given as follows

$$d_{k,s}^m=\frac{(m+2s+1)k!(m+k)!}{(k-s)!(m+k+s+1)!}$$ (22)

The proof of Proposition 1 is deferred to Appendix A.

For the normalized radial polynomials ${\overline{R}}_{m+2k}^m(r)$ defined by Eq. (14), it can be rewritten as

$${\overline{R}}_{m+2k}^m(r)=N_{m+2k}^m{R}_{m+2k}^m(r)=\sqrt{\frac{2(m+2k+1)}{1+{\delta }_{m,0}}}{R}_{m+2k}^m(r)=\underset{s=0}{\overset{k}{\Sigma }}{\overline{c}}_{k,s}^m{r}^{m+2s}$$ (23)

where

$${\overline{c}}_{k,s}^m=\sqrt{\frac{2(m+2k+1)}{1+{\delta }_{m,0}}}{c}_{k,s}^m={(-1)}^{k-s}\sqrt{\frac{2(m+2k+1)}{1+{\delta }_{m,0}}}\frac{(m+k+s)!}{s!(k-s)!(m+s)!}$$ (24)

Since the normalization factor $N_{m+2k}^m$ depends only on m and k, by using the Proposition 1, we can easily derive the following result without proof.

Proposition 2

For the lower triangular matrix ${\overline{C}}_K^m$ whose elements ${\overline{c}}_{k,s}^m$ are defined by Eq. (24), the elements of the inverse matrix ${\overline{D}}_K^m$ are given as follows

$${\overline{d}}_{k,s}^m=\sqrt{\frac{1+{\delta }_{m,0}}{2(m+2s+1)}}{d}_{k,s}^m=\sqrt{\frac{(1+{\delta }_{m,0})(m+2s+1)}{2}}\frac{k!(m+k)!}{(k-s)!(m+k+s+1)!}$$ (25)

We are now ready to establish the relationship between the two set of Zernike coefficients {b_{m, m}, b _{m+2, m}, b _{m+4, m}, … b_{m+2K, m}} and {a_{m, m}, a _{m+2, m}, a _{m+4, m}, … a_{m+2K, m}} appeared in Eq. (13). Applying Corollary to the normalized radial polynomials ${\overline{R}}_{m+2k}^m(r)$ with q = 2 and using Eqs. (24) and (25), we have

Theorem 1

For given integers m and K, and real positive number λ, let {b_{m, m}, b _{m+2, m}, b _{m+4, m}, … b_{m+2K, m}} and {a_{m, m}, a_{m+2, m}, a _{m+4, m}, … a_{m+2K, m}} be two sets of Zernike coefficients corresponding to the aperture sizes 1 and λ, respectively, we have

$$\begin{array}{ll}{b}_{m+2k,m}& =\frac{1}{{\lambda }^{m+2k}}\left[a_{m+2k,m}+\underset{i=k+1}{\overset{K}{\Sigma }}\left(\underset{j=k}{\overset{i}{\Sigma }}\frac{{\overline{c}}_{i,j}^m{\overline{d}}_{j,k}^m}{{\lambda }^{2(j-k)}}\right)a_{m+2i,m}\right]\\ & =\frac{1}{{\lambda }^{m+2k}}\left[a_{m+2k,m}+\underset{i=k+1}{\overset{K}{\Sigma }}C(m,k,i)a_{m+2i,m}\right]\end{array},k=0,1,\dots ,K,$$ (26)

where

$$\begin{array}{r}C(m,k,i)=\sqrt{(m+2i+1)(m+2k+1)}\underset{j=k}{\overset{i}{\Sigma }}\frac{{(-1)}^{i-j}}{{\lambda }^{2(j-k)}}\frac{(m+i+j)!}{(i-j)!(j-k)!(m+j+k+1)!}\\ \text{for\hspace{0.17em}}i=k+1,k+2,\dots ,K,\end{array}$$ (27)

The relationship established in Theorem 1 is explicit, and the coefficient b_{m+2K, m} depends only on the set of coefficients {a_{m+2k, m}, a _{m+2(k + 1), m}, … a_{m+2K, m}} thus, it is more simple than that given by Schwiegerling.⁸ Note also that even though the above results were demonstrated for the case m ≥ 0, they remain valid for m < 0 due to the symmetry property of the radial polynomials $R_n^m(r)$ about m.

Table 1 shows the conversion relationship between the coefficients b_{n, m} and a_{n, m} for Zernike polynomial expansions up to 45 terms (up to order 8). The results are the same as those given by Schwiegerling⁸ except for b_{1, m}.

Table 1.

Coefficient conversion relationships for Zernike polynomial expansions up to order 8

n	m	New expansion coefficients bn,m
0	0	$\begin{array}{l}{b}_{0,m}=a_{0,m}-\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{2,m}+\sqrt{5}\left(1-\frac{3}{{\lambda }^2}+\frac{2}{{\lambda }^4}\right)a_{4,m}\\ -\sqrt{7}\left(1-\frac{6}{{\lambda }^2}+\frac{10}{{\lambda }^4}-\frac{5}{{\lambda }^6}\right)a_{6,m}+3\left(1-\frac{10}{{\lambda }^2}+\frac{30}{{\lambda }^4}-\frac{35}{{\lambda }^6}+\frac{14}{{\lambda }^8}\right)a_{8,m}\end{array}$
1	−1, 1	$b_{1,m}=\frac{1}{\lambda }\left[a_{1,m}-2\sqrt{2}\left(1-\frac{1}{{\lambda }^2}\right)a_{3,m}+\sqrt{3}\left(3-\frac{8}{{\lambda }^2}+\frac{5}{{\lambda }^4}\right)a_{5,m}-4\left(2-\frac{10}{{\lambda }^2}+\frac{15}{{\lambda }^4}-\frac{7}{{\lambda }^6}\right)a_{7,m}\right]$
2	−2, 0, 2	$b_{2,m}=\frac{1}{{\lambda }^2}\left[a_{2,m}-\sqrt{15}\left(1-\frac{1}{{\lambda }^2}\right)a_{4,m}+\sqrt{21}\left(2-\frac{5}{{\lambda }^2}+\frac{3}{{\lambda }^4}\right)a_{6,m}-\sqrt{3}\left(10-\frac{45}{{\lambda }^2}+\frac{63}{{\lambda }^4}-\frac{28}{{\lambda }^6}\right)a_{8,m}\right]$
3	−3, −1, 1, 3	$b_{3,m}=\frac{1}{{\lambda }^3}\left[a_{3,m}-2\sqrt{6}\left(1-\frac{1}{{\lambda }^2}\right)a_{5,m}+2\sqrt{2}\left(5-\frac{12}{{\lambda }^2}+\frac{7}{{\lambda }^4}\right)a_{7,m}\right]$
4	−4 −2, 0, 2, 4	$b_{4,m}=\frac{1}{{\lambda }^4}\left[a_{4,m}-\sqrt{35}\left(1-\frac{1}{{\lambda }^2}\right)a_{6,m}+3\sqrt{5}\left(3-\frac{7}{{\lambda }^2}+\frac{4}{{\lambda }^4}\right)a_{8,m}\right]$
5	−5, −3, −1, 1, 3, 5	$b_{5,m}=\frac{1}{{\lambda }^5}\left[a_{5,m}-4\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{7,m}\right]$
6	−6, −4 −2, 0, 2, 4, 6	$b_{6,m}=\frac{1}{{\lambda }^6}\left[a_{6,m}-3\sqrt{7}\left(1-\frac{1}{{\lambda }^2}\right)a_{8,m}\right]$
7	−7, −5, −3, −1, 1, 3, 5, 7	$b_{7,m}=\frac{1}{{\lambda }^7}{a}_{7,m}$
8	−8 −6, −4, −2, 0, 2, 4, 6, 8	$b_{8,m}=\frac{1}{{\lambda }^8}{a}_{8,m}$

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

Theorem 2

Let C(m, k, i) defined by Eq. (27) be the coefficient of a_{m+2i, m} in the expansion of b_{m+2k, m} given by Eq. (26), and C(m+2l, k−l, i−l) be the coefficient of a_{m+2i, m+2l} in the expansion of b_{m+2k, m+2l} where l is an integer number less than or equal to k, then we have

$$C(m,k,i)=C(m+2l,k-l,i-l)$$ (28)

Proof

From Eq. (27), we have

$$\begin{array}{l}C(m+2l,k-l,i-l)\\ =\sqrt{(m+2i+1)(m+2k+1)}\underset{j=k-l}{\overset{i-l}{\Sigma }}\frac{{(-1)}^{i-l-j}}{{\lambda }^{2(j-k+l)}}\frac{(m+l+i+j)!}{(i-l-j)!(j-k+l)!(m+l+j+k+1)!}\\ =\sqrt{(m+2i+1)(m+2k+1)}\underset{j=k}{\overset{i}{\Sigma }}\frac{{(-1)}^{i-j}}{{\lambda }^{2(j-k)}}\frac{(m+i+j)!}{(i-j)!(j-k)!(m+j+k+1)!}\end{array}$$ (29)

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

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

Theorem 3

For a fixed value of N, let N = m + 2K = m′ + 2K′, from Theorem 1, we have

$$\begin{array}{l}{b}_{m+2(K-l),m}=\frac{1}{{\lambda }^{N-2l}}\left[a_{m+2(K-l),m}+\underset{i=K-l+1}{\overset{K}{\Sigma }}C(m,K-l,i)a_{m+2i,m}\right]\hfill \\ =\frac{1}{{\lambda }^{N-2l}}\left[a_{m+2(K-l),m}+\underset{i=0}{\overset{l-1}{\Sigma }}C(m,K-l,i+K-l+1)a_{N+2i-2l+2,m}\right]\hfill \end{array},\hspace{0.17em}l=0,1,\dots ,K$$ (30)

and

$$\begin{array}{l}{b}_{m^{\prime }+2(K^{\prime }-l),m}=\frac{1}{{\lambda }^{N-2l}}\left[a_{m^{\prime }+2(K^{\prime }-l),m^{\prime }}+\underset{i=K^{\prime }-l+1}{\overset{K^{\prime }}{\Sigma }}C(m^{\prime },K^{\prime }-l,i)a_{m^{\prime }+2i,m^{\prime }}\right]\hfill \\ =\frac{1}{{\lambda }^{N-2l}}\left[a_{m^{\prime }+2(K^{\prime }-l),m^{\prime }}+\underset{i=0}{\overset{l-1}{\Sigma }}C(m^{\prime },K^{\prime }-l,i+K^{\prime }-l+1)a_{N+2i-2l+2,m^{\prime }}\right]\hfill \end{array},\hspace{0.17em}l=0,1,\dots ,K^{\prime }$$ (31)

then

$$C(m,K-l,i+K-l+1)=C(m^{\prime },K^{\prime }-l,i+K^{\prime }-l+1)$$ (32)

for i = 0,1,…, l−1, l = 0, l,…, min(K, K′)

Proof

From Eq. (27), we have

$$\begin{array}{l}C(m,K-l,i+K-l+1)\\ =\sqrt{(m+2i+2K-2l+3)(m+2K-2l+1)}\\ \times \underset{j=K-l}{\overset{i+K-l+1}{\Sigma }}\frac{{(-1)}^{i+K+1-l-j}}{{\lambda }^{2(j-K+l)}}\frac{(m+i+K-l+j+1)!}{(i+K-l+1-j)!(j-K+l)!(m+j+K-l+1)!}\\ =\sqrt{(N+2i-2l+3)(N-2l+1)}\underset{j=0}{\overset{i+1}{\Sigma }}\frac{{(-1)}^{i+1-j}}{{\lambda }^{2j}}\frac{(N+i-2l+j+1)!}{j!(i+1-j)!(N+j-l+1)!}\end{array}$$ (33)

Similarly,

$$\begin{array}{l}C(m^{\prime },K^{\prime }-l,i+K^{\prime }-l+1)\\ =\sqrt{(m^{\prime }+2i+2K^{\prime }-2l+3)(m+2K^{\prime }-2l+1)}\\ \times \underset{j=K^{\prime }-l}{\overset{i+K^{\prime }-l+1}{\Sigma }}\frac{{(-1)}^{i+K^{\prime }+1-l-j}}{{\lambda }^{2(j-K^{\prime }+l)}}\frac{(m^{\prime }+i+K^{\prime }-l+j+1)!}{(i+K^{\prime }-l+1-j)!(j-K^{\prime }+l)!(m^{\prime }+j+K^{\prime }-l+1)!}\\ =\sqrt{(N+2i-2l+3)(N-2l+1)}\underset{j=0}{\overset{i+1}{\Sigma }}\frac{{(-1)}^{i+1-j}}{{\lambda }^{2j}}\frac{(N+i-2l+j+1)!}{j!(i+1-j)!(N+j-l+1)!}\end{array}$$ (34)

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

Table 2 shows the case of N = m + 2K = 7 for different values of m and K.

Table 2.

Coefficient conversion relationships for different values of m and K where N = m + 2K = 7

m	K	New expansion coefficients bn,m
5	1	$\begin{array}{l}{b}_{7,5}=\frac{1}{{\lambda }^7}{a}_{7,5}\\ b_{5,5}=\frac{1}{{\lambda }^5}\left[a_{5,5}-4\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{7,5}\right]\end{array}$
3	2	$\begin{array}{l}{b}_{7,3}=\frac{1}{{\lambda }^7}{a}_{7,3}\\ b_{5,3}=\frac{1}{{\lambda }^5}\left[a_{5,3}-4\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{7,3}\right]\\ b_{3,3}=\frac{1}{{\lambda }^3}\left[a_{3,3}-2\sqrt{6}\left(1-\frac{1}{{\lambda }^2}\right)a_{5,3}+2\sqrt{2}\left(5-\frac{12}{{\lambda }^2}+\frac{7}{{\lambda }^4}\right)a_{7,3}\right]\end{array}$
1	3	$\begin{array}{l}{b}_{7,1}=\frac{1}{{\lambda }^7}{a}_{7,1}\\ b_{5,1}=\frac{1}{{\lambda }^5}\left[a_{5,1}-4\sqrt{3}\left(1-\frac{1}{{\lambda }^2}\right)a_{7,1}\right]\\ b_{3,1}=\frac{1}{{\lambda }^3}\left[a_{3,1}-2\sqrt{6}\left(1-\frac{1}{{\lambda }^2}\right)a_{5,1}+2\sqrt{2}\left(5-\frac{12}{{\lambda }^2}+\frac{7}{{\lambda }^4}\right)a_{7,1}\right]\\ b_{1,1}=\frac{1}{\lambda }\left[a_{1,1}-2\sqrt{2}\left(1-\frac{1}{{\lambda }^2}\right)a_{3,1}+\sqrt{3}\left(3-\frac{8}{{\lambda }^2}+\frac{5}{{\lambda }^4}\right)a_{5,1}-4\left(2-\frac{10}{{\lambda }^2}+\frac{15}{{\lambda }^4}-\frac{7}{{\lambda }^6}\right)a_{7,1}\right]\end{array}$

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 proposed method is simple, and can be easily implemented.

Note that the formulae derived in this paper are mathematically correct for all values of λ = r₁/r₂ where r₁ and r₂ represent the original and new aperture sizes. But for application purpose, it is still recommended to make r₂ less than r₁. In the case where r₂ is greater than r₁, the wave-front error data must be extrapolated outside the region 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

Eq. (16) can be expressed in matrix form as

$$f(r)=(a_0,a_1,a_2,\dots ,a_K)\left(\begin{array}{c}{P}_0(r)\\ P_1(r)\\ P_2(r)\\ ⋮\\ P_K(r)\end{array}\right)=(b_0,b_1,b_2,\dots ,b_K)\left(\begin{array}{c}{P}_0(\lambda r)\\ P_1(\lambda r)\\ P_2(\lambda r)\\ ⋮\\ P_K(\lambda r)\end{array}\right)$$ (A1)

Using Eq. (17), we have

$$\left(\begin{array}{c}{P}_0(r)\\ P_1(r)\\ P_2(r)\\ ⋮\\ P_K(r)\end{array}\right)=C_K\left(\begin{array}{l}1\hfill \\ r\hfill \\ r^2\hfill \\ ⋮\hfill \\ r^K\hfill \end{array}\right)$$ (A2)

and

$$\left(\begin{array}{c}{P}_0(\lambda r)\\ P_1(\lambda r)\\ P_2(\lambda r)\\ ⋮\\ P_K(\lambda r)\end{array}\right)=C_K\left(\begin{array}{l}1\hfill \\ \lambda r\hfill \\ {\lambda }^2{r}^2\hfill \\ ⋮\hfill \\ {\lambda }^K{r}^K\hfill \end{array}\right)=C_K\mathit{diag}(1,\lambda ,{\lambda }^2,\dots ,{\lambda }^K)\left(\begin{array}{l}1\hfill \\ r\hfill \\ r^2\hfill \\ ⋮\hfill \\ r^K\hfill \end{array}\right)$$ (A3)

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

$$(a_0,a_1,a_2,\dots ,a_K)C_K\left(\begin{array}{l}1\hfill \\ r\hfill \\ r^2\hfill \\ ⋮\hfill \\ r^K\hfill \end{array}\right)=(b_0,b_1,b_2,\dots ,b_K)C_K\mathit{diag}(1,\lambda ,{\lambda }^2,\dots ,{\lambda }^K)\left(\begin{array}{l}1\hfill \\ r\hfill \\ r^2\hfill \\ ⋮\hfill \\ r^K\hfill \end{array}\right)$$ (A4)

thus

$$\begin{array}{ll}(b_0,b_1,b_2,\dots ,b_K)& =(a_0,a_1,a_2,\dots ,a_K)C_K{\left(\mathit{diag}(1,\lambda ,{\lambda }^2,\dots ,{\lambda }^K)\right)}^{-1}{C}_K^{-1}\\ & =(a_0,a_1,a_2,\dots ,a_K)C_K\mathit{diag}(1,{\lambda }^{-1},{\lambda }^{-2},\dots ,{\lambda }^{-K})D_K\end{array}$$ (A5)

Eq. (18) can be easily obtained by expanding Eq. (A5).

Proof of Proposition 1

To prove the proposition, we need to demonstrate the following relation

$$\underset{s=l}{\overset{k}{\Sigma }}{c}_{k,s}^m{d}_{s,l}^m={\delta }_{k,l},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0\le l\le k\le K$$ (A6)

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

$$c_{k,k}^m{d}_{k,k}^m=\frac{(m+2k)!}{k!(m+k)!}\times \frac{(m+2k+1)k!(m+k)!}{(m+2k+1)!}=1$$ (A7)

For l < k, we have

$$\begin{array}{ll}\underset{s=l}{\overset{k}{\Sigma }}{c}_{k,s}^m{d}_{s,l}^m& =\underset{s=l}{\overset{k}{\Sigma }}\frac{{(-1)}^{k-s}(m+2l+1)(m+k+s)!}{(s-l)!(k-s)!(m+s+l+1)!}\\ & ={(-1)}^k(m+2l+1)\underset{s=l}{\overset{k}{\Sigma }}F(m,k,l,s)\end{array}$$ (A8)

where

$$F(m,k,l,s)=\frac{{(-1)}^s(m+k+s)!}{(s-l)!(k-s)!(m+s+l+1)!}$$ (A9)

Let

$$G(m,k,l,s)=\frac{{(-1)}^{s+1}(m+k+s)!}{(s-l)!(k+1-s)!(m+l+s)!}\frac{(k+1-s)(s-l)}{(k-l)(m+k+l+1)}$$ (A10)

it can be easily verified that

$$F(m,k,l,s)=G(m,k,l,s+1)-G(m,k,l,s)$$ (A11)

thus

$$\underset{s=l}{\overset{k}{\Sigma }}F(m,k,l,s)=\underset{s=l}{\overset{k}{\Sigma }}\left[G(m,k,l,s+1)-G(m,k,l,s)\right]=G(m,k,l,k+1)-G(m,k,l,l)=0$$ (A12)

We deduce from Eq. (A8) that

$$\underset{s=l}{\overset{k}{\Sigma }}{c}_{k,s}^m{d}_{s,l}^m=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{for\hspace{0.17em}}l<k.$$ (A13)

The proof is now complete.

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