Zernike polynomials#Noll's sequential indices

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

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

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

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

(odd function over the azimuthal angle $\backslash varphi$) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), $\backslash varphi$ is the azimuthal angle, ρ is the radial distance $0\backslash le\backslash rho\backslash le\; 1$, 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, i.e. $|Z^\{m\}\_n(\backslash rho,\backslash varphi)|\; \backslash le\; 1$. The radial polynomials $R^m\_n$ are defined as

:$R^m\_n(\backslash rho)\; =\; \backslash sum\_\{k=0\}^\{\backslash tfrac\{n-m\}\{2\}\}\; \backslash frac\{(-1)^k\backslash ,(n-k)!\}\{k!\backslash left\; (\backslash tfrac\{n+m\}\{2\}-k\; \backslash right\; )!\; \backslash left\; (\backslash tfrac\{n-m\}\{2\}-k\; \backslash right)!\}\; \backslash ;\backslash rho^\{n-2k\}$

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

:$R\_n^m(1)=1.$

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

:$R\_n^m(\backslash rho)=\backslash sum\_\{k=0\}^\{\backslash tfrac\{n-m\}\{2\}\}(-1)^k\; \backslash binom\{n-k\}\{k\}\; \backslash binom\{n-2k\}\{\backslash tfrac\{n-m\}\{2\}-k\}\; \backslash rho^\{n-2k\}$.

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.:

:$\backslash begin\{align\}$

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 $\backslash rho\; ^j$ for fixed $m\backslash le\; j$ into $R\_n^m(\backslash rho)$

:$$

\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

|first1=R. J.

|last1=Mathar

|doi=10.2298/SAJ0979107M

|title=Zernike Basis to Cartesian Transformations

|journal=Serbian Astronomical Journal

|volume=179

|year=2009

|bibcode=2009SerAJ.179..107M

|pages=107–120

|issue=179|arxiv = 0809.2368 |s2cid=115159231

}}

:$$

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 $j-m=0,2,4,\backslash ldots$.

The factor $\backslash rho^\{n-2k\}$ in the radial polynomial $R\_n^m(\backslash rho)$ may be expanded in a Bernstein basis of $b\_\{s,n/2\}(\backslash rho^2)$ for even $n$ or $\backslash rho$ times a function of $b\_\{s,(n-1)/2\}(\backslash rho^2)$ for odd $n$ in the range $\backslash lfloor\; n/2\backslash rfloor-k\; \backslash le\; s\; \backslash le\; \backslash lfloor\; n/2\backslash rfloor$. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:

:$R\_n^m(\backslash rho)\; =\; \backslash frac\{1\}\{\backslash binom\{\backslash lfloor\; n/2\backslash rfloor\}\{\backslash lfloor\; m/2\backslash rfloor\}\}\; \backslash rho^\{n\backslash mod\; 2\}\; \backslash sum\_\{s=\backslash lfloor\; m/2\backslash rfloor\}^\{\backslash lfloor\; n/2\backslash rfloor\}\; (-1)^\{\backslash lfloor\; n/2\backslash rfloor\; -s\}\; \backslash binom\{s\}\{\backslash lfloor\; m/2\backslash rfloor\}\backslash binom\{(n+m)/2\}\{s+\backslash lceil\; m/2\backslash rceil\}\; b\_\{s,\backslash lfloor\; n/2\backslash rfloor\}(\backslash rho^2).$

Applications often involve linear algebra, where an integral over a product of Zernike polynomials and some other factor builds a 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

|first1=R. J.

|last1=Noll

|title=Zernike polynomials and atmospheric turbulence

|journal=J. Opt. Soc. Am.

|volume=66

|year=1976

|url=https://ftp.bioeng.auckland.ac.nz/pub/pub/jtur044/references/fitting/NOLL1976.pdf

|doi=10.1364/JOSA.66.000207

|page=207

|bibcode=1976JOSA...66..207N

|issue=3}} The table of this association $Z\_n^m\; \backslash rightarrow\; Z\_j$ starts as follows {{OEIS|A176988}}.

j = \frac{n(n+1)}{2}+|m|+\left\{\begin{array}{ll}

0, & m>0 \land n \equiv \{0,1\} \pmod 4;\\

0, & m<0 \land n \equiv \{2,3\} \pmod 4;\\

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 $m>0$ obtain even indices j.
• The odd Z where odd indices j.
• Within a given n, a lower $\backslash left\backslash vert\; m\; \backslash right\backslash vert$ results in a lower j.

OSA

{{cite journal

|first1=L. N.

|last1=Thibos

|first2=R. A.

|last2=Applegate

|first3=J. T.

|last3=Schwiegerling

|first4=R.

|last4=Webb

|title=Standards for reporting the optical aberrations of eyes

|journal=Journal of Refractive Surgery

|volume=18

|issue=5

|year=2002

|pages=S652-60

|doi=10.3928/1081-597X-20020901-30

|pmid=12361175

}} and ANSI single-index Zernike polynomials using:

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

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.Loomis, J., "A Computer Program for Analysis of Interferometric Data," Optical Interferograms, Reduction and Interpretation, ASTM STP 666, A. H. Guenther and D. H. Liebenberg, Eds., American Society for Testing and Materials, 1978, pp. 71–86.{{cite conference|conference=Proc SPIE |volume=4771|pages=276–286|year=2002|doi=10.1117/12.482169|title=Orthogonality of Zernike polynomials | first1=V. L. |last1=Genberg |first2=G. J. |last2=Michels|first3=K. B. |last3=Doyle|book-title=Optomechanical design and Engineering 2002}}

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.

James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1).{{cite book |author=Eric P. Goodwin |author2=James C. Wyant |title=Field Guide to Interferometric Optical Testing |year=2006 |page=25 |isbn=0-8194-6510-0}} This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

They satisfy the Rodrigues' formula

:$Z\_\{n\}^\{m\}(x)=\backslash frac\{x^\{-m\}\}\{\backslash left(\; \backslash frac\{n-m\}\{2\}\; \backslash right)\; !\}\backslash left(\; \backslash frac\{d\}\{d\backslash left(\; x^2\; \backslash right)\}\; \backslash right)\; ^\{\backslash frac\{n-m\}\{2\}\}\backslash left[\; x^\{n+m\}\backslash left(\; x^2-1\; \backslash right)\; ^\{\backslash frac\{n-m\}\{2\}\}\; \backslash right]$

and can be related to the Jacobi polynomials as

:$Z\_\{n\}^\{m\}(x)=x^m\backslash frac\{P\_\{\backslash frac\{n-m\}\{2\}\}^\{(0,m)\}\backslash left(\; 2x^2-1\; \backslash right)\}\{P\_\{\backslash frac\{n-m\}\{2\}\}^\{(0,m)\}(1)\}$.

The orthogonality in the radial part reads{{cite journal|first1=V. | last1=Lakshminarayanan|first2=Andre |last2=Fleck|doi=10.1080/09500340.2011.554896|journal=J. Mod. Opt.|year=2011|bibcode=2011JMOp...58..545L|volume=58|issue=7|pages=545–561|title=Zernike polynomials: a guide| s2cid=120905947}}

:$\backslash int\_0^1\backslash sqrt\{2n+2\}R\_n^m(\backslash rho)\backslash ,\backslash sqrt\{2n\text{'}+2\}R\_\{n\text{'}\}^\{m\}(\backslash rho)\backslash ,\backslash rho\; d\backslash rho\; =\; \backslash delta\_\{n,n\text{'}\}$

or

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

Orthogonality in the angular part is represented by the elementary

:$\backslash int\_0^\{2\backslash pi\}\; \backslash cos(m\backslash varphi)\backslash cos(m\text{'}\backslash varphi)\backslash ,d\backslash varphi=\backslash epsilon\_m\backslash pi\backslash delta\_\{m,m\text{'}\},$

:$\backslash int\_0^\{2\backslash pi\}\; \backslash sin(m\backslash varphi)\backslash sin(m\text{'}\backslash varphi)\backslash ,d\backslash varphi=\backslash pi\backslash delta\_\{m,m\text{'}\};\backslash quad\; m\backslash neq\; 0,$

:$\backslash int\_0^\{2\backslash pi\}\; \backslash cos(m\backslash varphi)\backslash sin(m\text{'}\backslash varphi)\backslash ,d\backslash varphi=0,$

where $\backslash epsilon\_m$ (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 if $m=0$ and 1 if $m\backslash neq\; 0$. 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,

:$\backslash int\; Z\_n^l(\backslash rho,\backslash varphi)Z\_\{n\text{'}\}^\{l\text{'}\}(\backslash rho,\backslash varphi)\; \backslash ,\; d^2r\; =\backslash frac\{\backslash epsilon\_l\backslash pi\}\{2n+2\}\backslash delta\_\{n,n\text{'}\}\backslash delta\_\{l,l\text{'}\},$

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

Any sufficiently smooth real-valued phase field over the unit disk $G(\backslash rho,\backslash varphi)$ 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

:$G(\backslash rho,\backslash varphi)\; =\; \backslash sum\_\{m,n\}\backslash left[\; a\_\{m,n\}\; Z^\{m\}\_n(\backslash rho,\backslash varphi)\; +\; b\_\{m,n\}\; Z^\{-m\}\_n(\backslash rho,\backslash varphi)\; \backslash right],$

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

:$\backslash langle\; F,\; G\; \backslash rangle\; :=\; \backslash int\; F(\backslash rho,\backslash varphi)G(\backslash rho,\backslash varphi)\backslash rho\; d\backslash rho\; d\backslash varphi.$

The Zernike coefficients can then be expressed as follows:

:$\backslash begin\{align\}$

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.

The reflections of trigonometric functions result that the parity with respect to reflection along the x axis is

:$Z\_n^\{l\}(\backslash rho,\backslash varphi)=Z\_n^\{l\}(\backslash rho,-\backslash varphi)$ for l ≥ 0,

:$Z\_n^\{l\}(\backslash rho,\backslash varphi)=-Z\_n^\{l\}(\backslash rho,-\backslash varphi)$ for l < 0.

The π shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates is

:$Z\_n^l(\backslash rho,\backslash varphi)\; =\; (-1)^l\; Z\_n^l(\backslash rho,\backslash varphi+\backslash pi),$

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 $Z\_n^l(\backslash rho,\backslash varphi)\; =\; Z\_n^l(\backslash rho,\backslash varphi+\backslash pi).$
• Angularly odd Zernike polynomials: Zernike polynomials with odd l so that $Z\_n^l(\backslash rho,\backslash varphi)\; =\; -\; Z\_n^l(\backslash rho,\backslash varphi+\backslash pi).$

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

:$R\_n^m(\backslash rho)=(-1)^n\; R\_n^m(-\backslash rho)=(-1)^m\; R\_n^m(-\backslash rho).$

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

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

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

The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation{{cite journal | year = 2017 | title = Quantum superintegrable Zernike system | journal = J. Math. Phys. | volume = 58 | issue = 7| doi = 10.1063/1.4990794 | last1 = Pogosyan | first1 = George S. | last2 = Salto-Alegre | first2 = Cristina | last3 = Wolf | first3 = Kurt Bernardo | last4 = Yakhno | first4 = Alexander | bibcode = 2017JMP....58g2101P | arxiv = 1702.08570 }}

:$\backslash begin\{align\}$

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 $\backslash nabla\; \backslash cdot\; (1-\backslash rho^2\; )\; \backslash nabla\; Z\_n^m\; =\; \backslash left(\; m^2\; -\; n(n+2)\; \backslash right)\; Z\_n^m$ (relating to rough surface BRDFs{{Cite web|url=https://github.com/pec27/urdf | title = A Unitary BRDF for Surfaces with Gaussian Deviations}}), which differs from the above by a factor $\backslash partial\_\{\backslash varphi\; \backslash varphi\}$.

The Zernike polynomials satisfy the following recurrence relation which depends neither on the degree nor on the azimuthal order of the radial polynomials:{{cite journal | year = 2013 | title = Recursive formula to compute Zernike radial polynomials | journal = Opt. Lett. | volume = 38 | issue = 14| pages = 2487–2489 | doi = 10.1364/OL.38.002487 | last1 = Honarvar Shakibaei | first1 = Barmak| pmid = 23939089 | bibcode = 2013OptL...38.2487H }}

:$\backslash begin\{align\}$

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 $R\_m^m(\backslash rho)\; =\; \backslash rho^m$ and $R\_\{m+2\}^m(\backslash rho)\; =\; ((m+2)\backslash rho^\{2\}\; -\; (m+1))\backslash rho^m$. The following three-term recurrence relation

{{cite journal

|first1=E. C.

|last1=Kintner

|doi=10.1080/713819334

|title=On the mathematical properties of the Zernike Polynomials

|journal=Opt. Acta

|volume=23

|year=1976

|pages=679–680

|bibcode=1976AcOpt..23..679K

|issue=8

}} then allows to calculate all other $R\_n^m(\backslash rho)$:

:$$

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 above relation is especially 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).

The first few radial polynomials are:

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

:$R^1\_1(\backslash rho)\; =\; \backslash rho\; \backslash ,$

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

:$R^2\_2(\backslash rho)\; =\; \backslash rho^2\; \backslash ,$

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

:$R^3\_3(\backslash rho)\; =\; \backslash rho^3\; \backslash ,$

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

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

:$R^4\_4(\backslash rho)\; =\; \backslash rho^4\; \backslash ,$

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

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

:$R^5\_5(\backslash rho)\; =\; \backslash rho^5\; \backslash ,$

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

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

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

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

The first few Zernike modes, at various indices, are shown below. They are normalized such that: $\backslash int\_0^\{2\backslash pi\}\; \backslash int\_0^1\; Z^2\backslash cdot\backslash rho\backslash ,d\backslash rho\backslash ,d\backslash phi\; =\; \backslash pi$, which is equivalent to $\backslash operatorname\{Var\}(Z)\_\backslash text\{unit\; circle\}\; =\; 1$.

{{Further|Optical_aberration#Zernike_model_of_aberrations}}

File:ZernikeAiryImage.jpg

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

|bibcode=2013JOSAA..30..726T|journal=J. Opt. Soc. Am. A|volume=30|issue=4|pages=726–32|arxiv=1302.7106|s2cid=23491106}}

{{cite journal|first1=A. J. E. M.|last1=Janssen|title=New analytic results for the Zernike Circle Polynomials from a basic result in the Nijboer-Zernike diffraction theory|doi=10.2971/jeos.2011.11028|year=2011|journal=Journal of the European Optical Society: Rapid Publications| volume=6|pages=11028|bibcode=2011JEOS....6E1028J|doi-access=free}}

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 $\backslash rho\backslash approx\; 1$, which often leads attempts to define other orthogonal functions over the circular disk.{{cite journal|first1=Richard|last1=Barakat|title=Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials|doi=10.1364/JOSA.70.000739|year=1980|journal=J. Opt. Soc. Am.|volume=70|issue=6|pages=739–742|bibcode=1980JOSA...70..739B}}{{cite arXiv|first1=A. J. E. M.|last1=Janssen|title=A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory|eprint=1110.2369|year=2011|class=math-ph}}{{cite arXiv|first1=R. J.|last1=Mathar|title=Orthogonal basis function over the unit circle with the minimax property|eprint=1802.09518|year=2018|class=math.NA}}

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.{{Cite journal|last1=Akondi|first1=Vyas|last2=Dubra|first2=Alfredo|date=2020-06-22|title=Average gradient of Zernike polynomials over polygons|journal=Optics Express|language=EN|volume=28|issue=13|pages=18876–18886|doi=10.1364/OE.393223|pmid=32672177|pmc=7340383|bibcode=2020OExpr..2818876A|issn=1094-4087|doi-access=free}}

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

| first = A.

| last = Tahmasbi

| year = 2010

| title = An Effective Breast Mass Diagnosis System using Zernike Moments

| conference = 17th Iranian Conf. on Biomedical Engineering (ICBME'2010)

| publisher = IEEE

| location = Isfahan, Iran

| pages = 1–4

| doi = 10.1109/ICBME.2010.5704941

}} Thus, they can be utilized to extract features from images that describe the shape characteristics of an object. For instance, Zernike moments are utilized as shape descriptors to classify benign and malignant breast masses

{{cite journal

|last1=Tahmasbi |first1=A.

|last2=Saki |first2=F.

|last3=Shokouhi |first3=S.B.

|year=2011

|title=Classification of Benign and Malignant Masses Based on Zernike Moments

|journal=Computers in Biology and Medicine

|volume=41 |issue=8

|pages=726–735

|doi=10.1016/j.compbiomed.2011.06.009

|pmid=21722886

}} or the surface of vibrating disks.{{cite journal|first1=W. P. |last1=Rdzanek|doi=10.1016/j.jsv.2018.07.035|title=Sound radiation of a vibrating elastically supported circular plate embedded into a flat screen revisited using the Zernike circle polynomials|year=2018|volume=434|pages=91–125|journal=J. Sound Vib.|bibcode=2018JSV...434...92R|s2cid=125512636}} Zernike Moments also have been used to quantify shape of osteosarcoma cancer cell lines in single cell level.{{cite journal|last1=Alizadeh|first1=Elaheh|last2=Lyons|first2=Samanthe M|last3=Castle|first3=Jordan M|last4=Prasad|first4=Ashok|title=Measuring systematic changes in invasive cancer cell shape using Zernike moments|journal=Integrative Biology|date=2016|volume=8|issue=11|pages=1183–1193|doi=10.1039/C6IB00100A|pmid=27735002}} Moreover, Zernike Moments have been used for early detection of Alzheimer's disease by extracting discriminative information from the MR images of Alzheimer's disease, Mild cognitive impairment, and Healthy groups.Gorji, H. T., and J. Haddadnia. "A novel method for early diagnosis of Alzheimer’s disease based on pseudo Zernike moment from structural MRI." Neuroscience 305 (2015): 361–371.

The concept translates to higher dimensions D if multinomials $x\_1^ix\_2^j\backslash cdots\; x\_D^k$ in Cartesian coordinates are converted to hyperspherical coordinates, $\backslash rho^s,\; s\backslash le\; D$, 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 $\backslash rho^s$ define an orthogonal basis $R\_n^\{(l)\}(\backslash rho)$ satisfying

:$\backslash int\_0^1\; \backslash rho^\{D-1\}R\_n^\{(l)\}(\backslash rho)R\_\{n\text{'}\}^\{(l)\}(\backslash rho)d\backslash rho\; =\; \backslash delta\_\{n,n\text{'}\}$.

(Note that a factor $\backslash sqrt\{2n+D\}$ 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

:$\backslash begin\{align\}$

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 $n-l\backslash ge\; 0$, 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).

• Jacobi polynomials
• Nijboer–Zernike theory
• Pseudo-Zernike polynomials

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• [http://www.nijboerzernike.nl The Extended Nijboer-Zernike website]
• [http://www.mathworks.com/matlabcentral/fileexchange/38900-zernike-moments MATLAB code for fast calculation of Zernike moments]
• [https://github.com/tvwerkhoven/libtim-py/blob/master/libtim/zern.py Python/NumPy library for calculating Zernike polynomials]
• [http://www.telescope-optics.net/zernike_aberrations.htm Zernike aberrations] at [http://www.telescope-optics.net Telescope Optics]
• [http://www.wolframalpha.com/input/?i=Plot%5BZernikeR%5B7%2C3%2Cr%5D%2C%7Br%2C0%2C1%7D%5D Example: using WolframAlpha to plot Zernike Polynomials]
• [https://github.com/nschloe/orthopy#disk-s2 orthopy, a Python package computing orthogonal polynomials (including Zernike polynomials)]

Category:Orthogonal polynomials