Definitions

There are even and odd Zernike polynomials. The even Zernike polynomials are defined as

:Z^{m}_n(\rho,\varphi) = R^m_n(\rho),\cos(m,\varphi) !

(even function over the azimuthal angle \varphi), and the odd Zernike polynomials are defined as

:Z^{-m}_n(\rho,\varphi) = R^m_n(\rho),\sin(m,\varphi), !

(odd function over the azimuthal angle \varphi) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), \varphi is the azimuthal angle, ρ is the radial distance 0\le\rho\le 1, and R^m_n are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1 in the unit disk, i.e. |Z^{m}_n(\rho,\varphi)| \le 1 if \rho \le 1. The radial polynomials R^m_n are defined as

:R^m_n(\rho) = \sum_{k=0}^{\tfrac{n-m}{2}} \frac{(-1)^k,(n-k)!}{k!\left (\tfrac{n+m}{2}-k \right )! \left (\tfrac{n-m}{2}-k \right)!} ;\rho^{n-2k}

for even n − m, while it is 0 for odd n − m. A special value is

:R_n^m(1)=1.

Other representations

Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:

:R_n^m(\rho)=\sum_{k=0}^{\tfrac{n-m}{2}}(-1)^k \binom{n-k}{k} \binom{n-2k}{\tfrac{n-m}{2}-k} \rho^{n-2k}.

A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:

:\begin{align} R_n^m(\rho) &= (-1)^{(n-m)/2} \rho^mP_{(n-m)/2}^{(m,0)}(1-2\rho^2) \ &= \binom{n}{\tfrac{n+m}{2}}\rho^n \ {}2F{1}\left(-\tfrac{n+m}{2},-\tfrac{n-m}{2};-n;\rho^{-2}\right) \ &= (-1)^{\tfrac{n-m}{2}}\binom{\tfrac{n+m}{2}}{m}\rho^m \ {}2F{1}\left(1+\tfrac{n+m}{2},-\tfrac{n-m}{2};1+m;\rho^2\right) \end{align}

for n − m even.

The inverse relation expands \rho ^j for fixed m\le j into R_n^m(\rho) : \rho^j = \sum_{n\equiv m \pmod 2}^j h_{j,n,m}R_n^m(\rho) with rational coefficients h_{j,n,m}{{cite journal : h_{j,n,m}= \frac{n+1}{1+\frac{j+n}{2}} \frac{\binom{(j-m)/2}{(n-m)/2}}{\binom{(j+n)/2}{(n-m)/2}} for even j-m=0,2,4,\ldots.

The factor \rho^{n-2k} in the radial polynomial R_n^m(\rho) may be expanded in a Bernstein basis of b_{s,n/2}(\rho^2) for even n or \rho times a function of b_{s,(n-1)/2}(\rho^2) for odd n in the range \lfloor n/2\rfloor-k \le s \le \lfloor n/2\rfloor. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients: :R_n^m(\rho) = \frac{1}{\binom{\lfloor n/2\rfloor}{\lfloor m/2\rfloor}} \rho^{n\mod 2} \sum_{s=\lfloor m/2\rfloor}^{\lfloor n/2\rfloor} (-1)^{\lfloor n/2\rfloor -s} \binom{s}{\lfloor m/2\rfloor}\binom{(n+m)/2}{s+\lceil m/2\rceil} b_{s,\lfloor n/2\rfloor}(\rho^2).

Rodrigues Formula

The radial polynomials satisfy the Rodrigues' formula :R_{n}^{m}(x)=\frac{1}{\left( \frac{n-m}{2} \right) ! x^m}\left( \frac{d}{d\left( x^2 \right)} \right) ^{\frac{n-m}{2}}\left[ x^{n+m}\left( x^2-1 \right) ^{\frac{n-m}{2}} \right].

Properties

Orthogonality

The orthogonality in the radial part reads :\int_0^1\sqrt{2n+2}R_n^m(\rho),\sqrt{2n'+2}R_{n'}^{m}(\rho),\rho d\rho = \delta_{n,n'}

or

\underset{0}{\overset{1}{\mathop \int }},R_{n}^{m}(\rho )R_^{m}(\rho )\rho d\rho =\frac}{2n+2}.

Orthogonality in the angular part is represented by the elementary

:\int_0^{2\pi} \cos(m\varphi)\cos(m'\varphi),d\varphi=\epsilon_m\pi\delta_{m,m'}, :\int_0^{2\pi} \sin(m\varphi)\sin(m'\varphi),d\varphi=\pi\delta_{m,m'};\quad m\neq 0, :\int_0^{2\pi} \cos(m\varphi)\sin(m'\varphi),d\varphi=0,

where \epsilon_m (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 if m=0 and 1 if m\neq 0. The product of the angular and radial parts establishes the orthogonality of the Zernike functions with respect to both indices if integrated over the unit disk,

:\int Z_n^l(\rho,\varphi)Z_{n'}^{l'}(\rho,\varphi) , d^2r =\frac{\epsilon_l\pi}{2n+2}\delta_{n,n'}\delta_{l,l'},

where d^2r=\rho,d\rho,d\varphi is the Jacobian of the circular coordinate system, and where n-l and n'-l' are both even.

Zernike transform

Any sufficiently smooth real-valued phase field over the unit disk G(\rho,\varphi) can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with the Fourier series. We have :G(\rho,\varphi) = \sum_{m,n}\left[ a_{m,n} Z^{m}n(\rho,\varphi) + b{m,n} Z^{-m}_n(\rho,\varphi) \right], where the coefficients can be calculated using inner products. On the space of L^2 functions on the unit disk, there is an inner product defined by :\langle F, G \rangle := \int F(\rho,\varphi)G(\rho,\varphi)\rho d\rho d\varphi. The Zernike coefficients can then be expressed as follows:

:\begin{align} a_{m,n} &= \frac{2n+2}{\epsilon_m\pi} \left \langle G(\rho,\varphi),Z^{m}n(\rho,\varphi) \right \rangle, \ b{m,n} &= \frac{2n+2}{\epsilon_m\pi} \left \langle G(\rho,\varphi),Z^{-m}_n(\rho,\varphi) \right \rangle. \end{align}

Alternatively, one can use the known values of phase function G on the circular grid to form a system of equations. The phase function is retrieved by the unknown-coefficient weighted product with (known values) of Zernike polynomial across the unit grid. Hence, coefficients can also be found by solving a linear system, for instance by matrix inversion. Fast algorithms to calculate the forward and inverse Zernike transform use symmetry properties of trigonometric functions, separability of radial and azimuthal parts of Zernike polynomials, and their rotational symmetries.

Symmetries

The reflections of trigonometric functions result that the parity with respect to reflection along the x axis is :Z_n^{l}(\rho,\varphi)=Z_n^{l}(\rho,-\varphi) for l ≥ 0, :Z_n^{l}(\rho,\varphi)=-Z_n^{l}(\rho,-\varphi) for l The π shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates is :Z_n^l(\rho,\varphi) = (-1)^l Z_n^l(\rho,\varphi+\pi), where (-1)^l could as well be written (-1)^n because n-l as even numbers are only cases to get non-vanishing Zernike polynomials. (If n is even then l is also even. If n is odd, then l is also odd.) This property is sometimes used to categorize Zernike polynomials into even and odd polynomials in terms of their angular dependence. (it is also possible to add another category with l = 0 since it has a special property of no angular dependence.)

• Angularly even Zernike polynomials: Zernike polynomials with even l so that Z_n^l(\rho,\varphi) = Z_n^l(\rho,\varphi+\pi).
• Angularly odd Zernike polynomials: Zernike polynomials with odd l so that Z_n^l(\rho,\varphi) = - Z_n^l(\rho,\varphi+\pi). (This nomenclature is not used in practice because non-vanishing polynomials have even l only combined with even n and odd l combined with odd n, so angularly even polynomials are also radially even polynomials and angularly odd polynomials are also radially odd polynomials such that the attribute angularly is superflous.)

The radial polynomials are also either even or odd, depending on the order n or the azimuthal index m:

:R_n^m(\rho)=(-1)^n R_n^m(-\rho)=(-1)^m R_n^m(-\rho).

These equalities are easily seen since R_n^m(\rho) with an odd (even) m contains only odd (even) powers to ρ (see examples of R_n^m(\rho) below).

The periodicity of the trigonometric functions results in invariance if rotated by multiples of 2\pi/l radian around the center:

:Z_n^l \left (\rho, \varphi+ \tfrac{2\pi k}{l} \right )=Z_n^l(\rho,\varphi),\qquad k= 0, \pm 1,\pm 2,\cdots.

As eigenfunctions of a differential operator

The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation :\begin{align} L\left[f\right] = \nabla^2 f - ({\bf r}\cdot \nabla)^2 f - 2{\bf r}\cdot \nabla f \end{align} self-adjoint over the unit disk, with negative eigenvalues L[Z_n^m] = -n(n+2)Z_n^m. Other self-adjoint differential operators can be constructed for which the Zernike polynomials form a spectrum, for example \nabla \cdot (1-\rho^2 ) \nabla Z_n^m = \left( m^2 - n(n+2) \right) Z_n^m (relating to rough surface BRDFs), which differs from the above by a factor \partial_{\varphi \varphi}.

Recurrence relations

The Zernike polynomials satisfy the following recurrence relation:

:\begin{align} R_n^m(\rho)+R_{n-2}^m(\rho)=\rho\left[R_{n-1}^{\left|m-1\right|}(\rho)+R_{n-1}^{m+1}(\rho)\right] \text{ .} \end{align}

From the definition of R_n^m it can be seen that R_m^m(\rho) = \rho^m and R_{m+2}^m(\rho) = ((m+2)\rho^{2} - (m+1))\rho^m. The following three-term recurrence relation

then allows to calculate all other R_n^m(\rho):

: R_n^m(\rho) = \frac{2(n-1)(2n(n-2)\rho^2-m^2-n(n-2))R_{n-2}^m(\rho) - n(n+m-2)(n-m-2)R_{n-4}^m(\rho)}{(n+m)(n-m)(n-2)} \text{ .}

The main use of these recurrences is to avoid cancellation of digits that occurs for large n in the accumulation of the oscillatory binomial terms in the power series notation .

The above relation is also useful since the derivative of R_n^m can be calculated from two radial Zernike polynomials of adjacent degree:

: \frac{\operatorname{d}}{\operatorname{d}! \rho} R_n^m(\rho) = \frac{(2 n m (\rho^2 - 1) + (n-m)(m + n(2\rho^2 - 1))) R_n^m(\rho) - (n+m)(n-m) R_{n-2}^m(\rho)}{2 n \rho (\rho^2 - 1)} \text{ .}

The differential equation of the Gaussian Hypergeometric Function is equivalent to : \rho^2(\rho^2-1) \frac{d^2}{d\rho^2} R_n^m(\rho) = [n(n+2)\rho^2-m^2]R_n^m(\rho)+\rho(1-3\rho^2)\frac{d}{d\rho} R_n^m(\rho).

Nomenclature

Noll's sequential indices

Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds matrix elements. To enumerate the rows and columns of these matrices by a single index, a conventional mapping of the two indices n and m to a single index j has been introduced by Noll.{{cite journal

(In this section the m is the signed upper index of Z which may be positive or negative or zero.) The table of this association Z_n^m \rightarrow Z_j starts as follows .

j = \frac{n(n+1)}{2}+|m|+\left{\begin{array}{ll} 0, & m0 \land n \equiv {0,1} \pmod 4;\ 0, & m 1, & m \ge 0 \land n \equiv {2,3} \pmod 4;\ 1, & m \le 0 \land n \equiv {0,1} \pmod 4. \end{array}\right.

The rule is the following.

• The even Zernike polynomials Z with m0 obtain even indices j.
• The odd Z where m obtain odd indices j.
• Within a given n, a lower \left\vert m \right\vert results in a lower j.

OSA/ANSI standard indices

OSA

:j =\frac{n(n+2)+l}{2}

OSA/ANSI indices can be converted back to standard indices as follows:

n = \left\lfloor \frac{\sqrt{8j + 1} - 1}{2} \right\rfloor, \qquad l = 2j - n(n+2).

Fringe/University of Arizona indices

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.

j = \left(1+\frac{n+|l|}{2}\right)^2-2|l|+ \left\lfloor\frac{1-\sgn l}{2}\right\rfloor

where \sgn l is the sign or signum function. The first 20 fringe numbers are listed below.

Wyant indices

James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1). This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

Examples

Radial polynomials

The first few radial polynomials are:

: R^0_0(\rho) = 1 ,

: R^1_1(\rho) = \rho ,

: R^0_2(\rho) = 2\rho^2 - 1 ,

: R^2_2(\rho) = \rho^2 ,

: R^1_3(\rho) = 3\rho^3 - 2\rho ,

: R^3_3(\rho) = \rho^3 ,

: R^0_4(\rho) = 6\rho^4 - 6\rho^2 + 1 ,

: R^2_4(\rho) = 4\rho^4 - 3\rho^2 ,

: R^4_4(\rho) = \rho^4 ,

: R^1_5(\rho) = 10\rho^5 - 12\rho^3 + 3\rho ,

: R^3_5(\rho) = 5\rho^5 - 4\rho^3 ,

: R^5_5(\rho) = \rho^5 ,

: R^0_6(\rho) = 20\rho^6 - 30\rho^4 + 12\rho^2 - 1 ,

: R^2_6(\rho) = 15\rho^6 - 20\rho^4 + 6\rho^2 ,

: R^4_6(\rho) = 6\rho^6 - 5\rho^4 ,

: R^6_6(\rho) = \rho^6. ,

Zernike polynomials

The first few Zernike modes, at various indices, are shown below. In this table they are normalized differently than in the remaining sections: \int_0^{2\pi} \int_0^1 [Z_j(\rho,\phi)]^2\cdot\rho,d\rho,d\phi = \pi, which is equivalent to \operatorname{Var}(Z)_\text{unit circle} = 1 .

Applications

The functions are a basis defined over the circular support area, typically the pupil planes in classical optical imaging at visible and infrared wavelengths through systems of lenses and mirrors of finite diameter. Their advantages are the simple analytical properties inherited from the simplicity of the radial functions and the factorization in radial and azimuthal functions; this leads, for example, to closed-form expressions of the two-dimensional Fourier transform in terms of Bessel functions.{{cite journal|first1=E. |last1=Tatulli|doi=10.1364/JOSAA.30.000726 |pmid=23595334|year=2013|title=Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures Their disadvantage, in particular if high n are involved, is the unequal distribution of nodal lines over the unit disk, which introduces ringing effects near the perimeter \rho\approx 1, which often leads attempts to define other orthogonal functions over the circular disk.

In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses. In wavefront slope sensors like the Shack-Hartmann, Zernike coefficients of the wavefront can be obtained by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures. In optometry and ophthalmology, Zernike polynomials are used to describe wavefront aberrations of the cornea or lens from an ideal spherical shape, which result in refraction errors. They are also commonly used in adaptive optics, where they can be used to characterize atmospheric distortion. Obvious applications for this are IR or visual astronomy and satellite imagery.

Another application of the Zernike polynomials is found in the Extended Nijboer–Zernike theory of diffraction and aberrations.

Zernike polynomials are widely used as basis functions of image moments. Since Zernike polynomials are orthogonal to each other, Zernike moments can represent properties of an image with no redundancy or overlap of information between the moments. Although Zernike moments are significantly dependent on the scaling and the translation of the object in a region of interest (ROI), their magnitudes are independent of the rotation angle of the object.{{cite conference

Higher dimensions

The concept translates to higher dimensions D if multinomials x_1^ix_2^j\cdots x_D^k in Cartesian coordinates are converted to hyperspherical coordinates, \rho^s, s\le D, multiplied by a product of Jacobi polynomials of the angular variables. In D=3 dimensions, the angular variables are spherical harmonics, for example. Linear combinations of the powers \rho^s define an orthogonal basis R_n^{(l)}(\rho) satisfying

:\int_0^1 \rho^{D-1}R_n^{(l)}(\rho)R_{n'}^{(l)}(\rho)d\rho = \delta_{n,n'}.

(Note that a factor \sqrt{2n+D} is absorbed in the definition of R here, whereas in D=2 the normalization is chosen slightly differently. This is largely a matter of taste, depending on whether one wishes to maintain an integer set of coefficients or prefers tighter formulas if the orthogonalization is involved.) The explicit representation is

:\begin{align} R_n^{(l)}(\rho) &= \sqrt{2n+D}\sum_{s=0}^{\tfrac{n-l}{2}} (-1)^s {\tfrac{n-l}{2} \choose s}{n-s-1+\tfrac{D}{2} \choose \tfrac{n-l}{2}}\rho^{n-2s} \ &=(-1)^{\tfrac{n-l}{2}} \sqrt{2n+D} \sum_{s=0}^{\tfrac{n-l}{2}} (-1)^s {\tfrac{n-l}{2} \choose s} {s-1+\tfrac{n+l+D}{2} \choose \tfrac{n-l}{2}} \rho^{2s+l} \ &=(-1)^{\tfrac{n-l}{2}} \sqrt{2n+D} {\tfrac{n+l+D}{2}-1 \choose \tfrac{n-l}{2}} \rho^l \ {}_2F_1 \left ( -\tfrac{n-l}{2},\tfrac{n+l+D}{2}; l+\tfrac{D}{2}; \rho^2 \right ) \end{align}

for even n-l\ge 0, else identical to zero, with special case R_n^{(n)}(\rho) = \sqrt{2n+D}\rho^n.

Its differential equation for the Gaussian Hypergeometric Function is equivalent to

\rho^2(\rho^2-1)\frac{d^2}{d\rho^2}R_n^{(l)}(\rho)

\left[n\rho^2(n+D)-l(D-2+l)\right]R_n^{(l)}(\rho) + \rho\left[D-1-(D+1)\rho^2\right] \frac{d}{d\rho}R_n^{(l)}(\rho).

Kintner's recurrence for fixed l and variable n\pm 2 quoted for D=2 above is in the general form :\begin{align} & -(1+\frac{n-l}{2})(1-n-\frac{D}{2})\frac{n+l+D}{2} \frac{R_{n+2}^{(l)}(\rho)}{\sqrt{2(n+2)+D}} \ =& \frac{n-l}{2}(1+n+\frac{D}{2})(1-\frac{n+l+D}{2}) \frac{R_{n-2}^{(l)}(\rho)}{\sqrt{2(n-2)+D}} \ &+(n+\frac{D}{2})\left[(1+n+\frac{D}{2})(1-n-\frac{D}{2})(1-\rho^2) +\frac12(n-l)(D+n+l) +l+\frac{D}{2}-1\right] \frac{R_n^{(l)}(\rho)}{\sqrt{2n+D}} . \end{align} For D=3 this was proposed by Deng and Gwo.{{cite conference| first1=An-Wen|last1=Deng |access-date=

For fixed n and variable l\pm 2 the recurrence is :\begin{align} &(l+\frac{D}{2}-1) \left[(l+\frac{D}{2}) (l-2+\frac{D}{2})-\frac12(l^2+lD+Dn+D^2/2+n^2-2l-D)r^2\right] R_n^{(l)}(r) \ & = -(l+\frac{D}{2}) (\frac{n+l+D}{2}-1) (1+\frac{n-l}{2}) r^2R_n^{(l-2)}(r)

(\frac{n-l}{2})(\frac{n+l+D}{2})(l-2+\frac{D}{2}) r^2 R_n^{(l+2)}(r) . \end{align} The case for D=2 was published by Chong et al.

References

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