View source for Zernike polynomials

{{Short description|Polynomial sequence}}
[[File:Zernike polynomials with read-blue cmap.png|360px|thumb|The first 21 Zernike polynomials, ordered vertically by radial degree and horizontally by azimuthal degree]]

In [[Mathematics|mathematics]], the '''Zernike polynomials''' are a [[Polynomial sequence|sequence]] of [[Polynomial|polynomial]]s that are [[Orthogonal polynomials|orthogonal]] on the [[Unit disk|unit disk]]. Named after optical physicist [[Biography:Frits Zernike|Frits Zernike]], laureate of the 1953 Nobel Prize in Physics and the inventor of [[Organization:Phase-contrast microscopy|phase-contrast microscopy]], they play important roles in various optics branches such as beam [[Physics:Optics|optics]] and imaging.<ref>
{{cite journal
 |last   = Zernike
 |first  = F.
 |year   = 1934
 |title  = Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode
 |journal= Physica
 |volume=1
 |number= 8
 |doi=10.1016/S0031-8914(34)80259-5
|bibcode=1934Phy.....1..689Z
 |pages  = 689–704
}}</ref><ref>
{{cite book
|author=Born, Max
|author2=Wolf, Emil
|name-list-style=amp
|title=[[Physics:Principles of Optics|Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light]]
|year=1999
|publisher=Cambridge University Press
|location=Cambridge, UK
|isbn=9780521642224
|pages=986
|edition=7th}} (see [https://books.google.com/books?id=nUHGpfNsGyUC&q=Zernike also at Google Books])</ref>
<!-- to do: [[Wavefront|wavefront]] -->

==Definitions==
There are [[Even and odd functions|even and odd]] Zernike polynomials. The even Zernike polynomials are defined as

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

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

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

(odd function over the azimuthal angle <math>\varphi</math>) where ''m'' and ''n'' are nonnegative [[Integer|integer]]s with ''n&nbsp;≥&nbsp;m ≥ 0'' (''m'' = 0 for spherical Zernike polynomials), ''<math>\varphi</math>'' is the [[Azimuth|azimuth]]al [[Angle|angle]], ''ρ'' is the radial distance <math>0\le\rho\le 1</math>, and <math>R^m_n</math> are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. <math>|Z^{m}_n(\rho,\varphi)| \le 1</math>. The radial polynomials <math>R^m_n</math> are defined as

:<math>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}</math>

for an even number of ''n'' − ''m'', while it is 0 for an odd number of ''n'' − ''m''. A special value is

:<math>R_n^m(1)=1. </math>

===Other representations===
Rewriting the ratios of factorials in the radial part as products of [[Binomial coefficient|binomials]] shows that the coefficients are integer numbers:

:<math>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}</math>.

A notation as terminating [[Hypergeometric function|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.:

:<math>\begin{align}
R_n^m(\rho) &= \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}</math>

for ''n'' − ''m'' even.

The factor <math>\rho^{n-2k}</math> in the radial polynomial <math>R_n^m(\rho)</math> may be expanded in a [[Bernstein polynomial|Bernstein basis]] of <math>b_{s,n/2}(\rho^2)</math> for even <math>n</math> or <math>\rho</math> times a function of <math>b_{s,(n-1)/2}(\rho^2)</math> for odd <math>n</math> in the range <math>\lfloor n/2\rfloor-k \le s \le \lfloor n/2\rfloor</math>. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:
:<math>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).</math>

===Noll's sequential indices===
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 ''l'' to a single index ''j'' has been introduced by Noll.<ref name=noll1976>{{cite journal
|first1=R. J.
|last1=Noll
|title=Zernike polynomials and atmospheric turbulence
|journal=J. Opt. Soc. Am.
|volume=66
|year=1976
|url=ftp://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}}</ref> The table of this association <math>Z_n^l \rightarrow Z_j</math> starts as follows {{OEIS|A176988}}.
<math>
j = \frac{n(n+1)}{2}+|l|+\left\{\begin{array}{ll}
0, & l>0 \land n \equiv \{0,1\} \pmod 4;\\
0,  & l<0 \land n \equiv \{2,3\} \pmod 4;\\
1,  & l \ge 0 \land n \equiv \{2,3\} \pmod 4;\\
1, & l \le 0 \land n \equiv \{0,1\} \pmod 4.
\end{array}\right.
</math>
{|class="wikitable"
!n,l
{{!!}} 0,0{{!!}}1,1{{!!}} 1,−1 {{!!}} 2,0{{!!}} 2,−2 {{!!}} 2,2{{!!}}3,−1{{!!}} 3,1 {{!!}} 3,−3 {{!!}} 3,3
|-------
! j
{{!}} 1{{!!}}2{{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}}8 {{!!}} 9{{!!}} 10
|-----
!n,l
{{!!}}4,0 {{!!}}4,2 {{!!}}4,−2{{!!}}4,4{{!!}}4,−4{{!!}}5,1{{!!}}5,−1{{!!}}5,3 {{!!}}5,−3{{!!}}5,5
|-----
! j
{{!!}}11 {{!!}}12 {{!!}}13 {{!!}}14{{!!}}15{{!!}}16{{!!}} 17 {{!!}} 18 {{!!}}19 {{!!}}20
|}
The rule is the following.

* The even Zernike polynomials ''Z'' (with even azimuthal parts <math>\cos(m\varphi)</math>, where <math>m=l</math> as <math>l</math> is a positive number) obtain even indices ''j.''
* The odd ''Z'' obtains (with odd azimuthal parts <math>\sin(m\varphi)</math>, where <math>m=\left\vert l \right\vert</math> as <math>l</math> is a negative number) odd indices ''j''.
* Within a given ''n'', a lower <math>\left\vert l \right\vert</math> results in a lower&nbsp;''j''.

===OSA/ANSI standard indices===
[[Organization:The Optical Society|OSA]]
<ref name=thibos2002>{{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
|url=http://voi.opt.uh.edu/2000-JRS-standardsforrepotingtheopticalaberrationsofeyes.pdf
|pmid=12361175
}}</ref> and ANSI single-index Zernike polynomials using:

:<math>j =\frac{n(n+2)+l}{2}</math>

{|class="wikitable"
!n,l
{{!!}} 0,0{{!!}}1,-1{{!!}} 1,1 {{!!}} 2,-2{{!!}} 2,0 {{!!}} 2,2{{!!}}3,-3{{!!}} 3,-1 {{!!}} 3,1 {{!!}} 3,3
|-------
! j
{{!}} 0 {{!!}} 1 {{!!}} 2 {{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}} 8 {{!!}} 9
|-----
!n,l
{{!!}}4,-4 {{!!}}4,-2 {{!!}}4,0{{!!}}4,2{{!!}}4,4{{!!}}5,-5{{!!}}5,-3{{!!}}5,-1 {{!!}}5,1{{!!}}5,3
|-----
! j
{{!!}} 10 {{!!}} 11 {{!!}} 12 {{!!}} 13 {{!!}} 14 {{!!}} 15 {{!!}} 16 {{!!}} 17 {{!!}} 18 {{!!}} 19
|}

===Fringe/University of Arizona indices===

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., [[Engineering:Photolithography|photolithography]].<ref>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.</ref><ref>{{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}}</ref>

<math>
j = \left(1+\frac{n+|l|}{2}\right)^2-2|l|+ \frac{1-\sgn l}{2}
</math>

where <math>
\sgn l
</math> is the [[Sign function|sign or signum function]]. The first 20 fringe numbers are listed below.

{|class="wikitable"
!n,l
{{!!}} 0,0{{!!}}1,1{{!!}} 1,−1 {{!!}} 2,0{{!!}} 2,2 {{!!}} 2,-2{{!!}}3,1{{!!}} 3,-1 {{!!}} 4,0 {{!!}} 3,3
|-------
! j
{{!}} 1{{!!}}2{{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}}8 {{!!}} 9{{!!}} 10
|-----
!n,l
{{!!}}3,-3 {{!!}}4,2 {{!!}}4,−2{{!!}}5,1{{!!}}5,−1{{!!}}6,0{{!!}}4,4{{!!}}4,-4 {{!!}}5,3{{!!}}5,-3
|-----
! j
{{!!}}11 {{!!}}12 {{!!}}13 {{!!}}14{{!!}}15{{!!}}16{{!!}} 17 {{!!}} 18 {{!!}}19 {{!!}}20
|}

===Wyant indices===

[[Biography:James C. Wyant|James C. Wyant]] uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1).<ref>{{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}}</ref> This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

==Properties==

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

or

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

Orthogonality in the angular part is represented by the [[List of integrals of trigonometric functions|elementary]]

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

where <math>\epsilon_m</math> (sometimes called the [[Biography:Carl Neumann|Neumann factor]] because it frequently appears in conjunction with Bessel functions) is defined as ''2'' if <math>m=0</math> and ''1'' if <math>m\neq 0</math>. 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,

:<math>\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'},</math>

where <math>d^2r=\rho\,d\rho\,d\varphi</math> is the [[Jacobian matrix and determinant|Jacobian]] of the circular coordinate system, and where <math>n-l</math> and <math>n'-l'</math> are both even.

===Zernike transform===
Any sufficiently smooth real-valued phase field over the unit disk <math>G(\rho,\varphi)</math> 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
:<math>G(\rho,\varphi) = \sum_{m,n}\left[ a_{m,n} Z^{m}_n(\rho,\varphi) + b_{m,n} Z^{-m}_n(\rho,\varphi) \right],</math>
where the coefficients can be calculated using inner products.  On the space of <math>L^2</math> functions on the unit disk, there is an inner product defined by
:<math>\langle F, G \rangle := \int F(\rho,\varphi)G(\rho,\varphi)\rho d\rho d\varphi.</math>
The Zernike coefficients can then be expressed as follows:

:<math>\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}</math>

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 [[List of trigonometric identities|reflections of trigonometric functions]] result that the parity with respect to reflection along the ''x'' axis is
:<math>Z_n^{l}(\rho,\varphi)=Z_n^{l}(\rho,-\varphi)</math> for ''l'' ≥ 0,
:<math>Z_n^{l}(\rho,\varphi)=-Z_n^{l}(\rho,-\varphi)</math> for ''l'' < 0.
The [[List of trigonometric identities|''π'' shifts of trigonometric functions]] result that the parity with respect to point reflection at the center of coordinates is
:<math>Z_n^l(\rho,\varphi) = (-1)^l Z_n^l(\rho,\varphi+\pi),</math>
where <math>(-1)^l</math> could as well be written <math>(-1)^n</math> because <math>n-l</math> 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 <math>Z_n^l(\rho,\varphi) =  Z_n^l(\rho,\varphi+\pi).</math>
* Angularly odd Zernike polynomials: Zernike polynomials with odd ''l'' so that <math>Z_n^l(\rho,\varphi) = - Z_n^l(\rho,\varphi+\pi).</math>

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

:<math>R_n^m(\rho)=(-1)^n R_n^m(-\rho)=(-1)^m R_n^m(-\rho).</math>

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

The [[List of trigonometric identities|periodicity of the trigonometric functions]] results in invariance if rotated by multiples of <math>2\pi/l</math> radian around the center:

:<math>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.</math>

===Recurrence relations===

The Zernike polynomials satisfy the following recurrence relation which depends neither on the degree nor on the azimuthal order of the radial polynomials:<ref name="barmak">{{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 }}</ref>

:<math>\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}</math>

From the definition of <math>R_n^m</math> it can be seen that <math>R_m^m(\rho) = \rho^m</math> and <math>R_{m+2}^m(\rho) = ((m+2)\rho^{2} - (m+1))\rho^m</math>. The following three-term recurrence relation<ref name="kintner1976">
{{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
}}</ref> then allows to calculate all other <math>R_n^m(\rho)</math>:

:<math>
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{ .}
</math>

The above relation is especially useful since the derivative of <math>R_n^m</math> can be calculated from two radial Zernike polynomials of adjacent degree:<ref name=kintner1976 />

:<math>
\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{ .}
</math>

==Examples==

=== Radial polynomials ===
The first few radial polynomials are:

:<math> R^0_0(\rho) = 1 \,</math>

:<math> R^1_1(\rho) = \rho \,</math>

:<math> R^0_2(\rho) = 2\rho^2 - 1 \,</math>

:<math> R^2_2(\rho) = \rho^2 \,</math>

:<math> R^1_3(\rho) = 3\rho^3 - 2\rho \, </math>

:<math> R^3_3(\rho) = \rho^3 \,</math>

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

:<math> R^2_4(\rho) = 4\rho^4 - 3\rho^2 \,</math>

:<math> R^4_4(\rho) = \rho^4 \,</math>

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

:<math> R^3_5(\rho) = 5\rho^5 - 4\rho^3 \,</math>

:<math> R^5_5(\rho) = \rho^5 \,</math>

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

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

:<math> R^4_6(\rho) = 6\rho^6 - 5\rho^4 \,</math>

:<math> R^6_6(\rho) = \rho^6. \,</math>

=== Zernike polynomials ===
The first few Zernike modes, at various indices, are shown below. They are normalized such that: <math>\int_0^{2\pi} \int_0^1 Z^2\cdot\rho\,d\rho\,d\phi = \pi</math>, which is equivalent to <math> \operatorname{Var}(Z)_\text{unit circle} = 1 </math>.
{| class="wikitable sortable"
|-
! <math>Z_n^l</math>|| [[Zernike polynomials#OSA/ANSI standard indices|OSA/ANSI]]<br/>index<br/>(<math>j</math>) || [[Zernike polynomials#Noll's sequential indices|Noll]]<br/>index<br/> (<math>j</math>) || [[Zernike polynomials#Wyant indices|Wyant]]<br/>index<br/> (<math>j</math>) || [[Zernike polynomials#Fringe/University of Arizona indices|Fringe/UA]]<br/>index<br/>(<math>j</math>) !! Radial<br/>degree<br/>(<math>n</math>) !! Azimuthal<br/>degree<br/>(<math>l</math>) !! <math>Z_j</math> !! Classical name
|-
| <math>Z_0^0</math> || {{0}}0 || {{0}}1 || {{0}}0 || {{0}}1 || 0 || {{0}}0 || <math>1</math> || [[Physics:Piston (optics)|Piston]] (see, [[Wigner semicircle distribution]])
|-
| <math>Z_1^{-1}</math> || {{0}}1 || {{0}}3 || {{0}}2 || {{0}}3 || 1 || −1 || <math>2 \rho \sin \phi</math> || [[Physics:Tilt (optics)|Tilt]] (Y-Tilt, vertical tilt)
|-
| <math>Z_1^1</math> || {{0}}2 || {{0}}2 || {{0}}1 || {{0}}2 || 1 || +1 || <math>2 \rho \cos \phi</math> || [[Physics:Tilt (optics)|Tilt]] (X-Tilt, horizontal tilt)
|-
| <math>Z_2^{-2}</math> || {{0}}3 || {{0}}5 || {{0}}5 || {{0}}6 || 2 || −2 || <math>\sqrt{6} \rho^2 \sin 2 \phi</math> || [[Physics:Astigmatism (optical systems)|Oblique astigmatism]]
|-
| <math>Z_2^0</math> || {{0}}4 || {{0}}4 || {{0}}3 || {{0}}4 || 2 || {{0}}0 || <math>\sqrt{3} (2 \rho^2 - 1)</math> || [[Physics:Defocus aberration|Defocus]] (longitudinal position)
|-
| <math>Z_2^2</math> || {{0}}5 || {{0}}6 || {{0}}4 || {{0}}5 || 2 || +2 || <math>\sqrt{6} \rho^2 \cos 2 \phi</math> || Vertical astigmatism
|-
| <math>Z_3^{-3}</math> || {{0}}6 || {{0}}9 || 10 || 11 || 3 || −3 || <math>\sqrt{8} \rho^3 \sin 3 \phi</math> || Vertical trefoil
|-
| <math>Z_3^{-1}</math> || {{0}}7 || {{0}}7 || {{0}}7 ||  {{0}}8 || 3 || −1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \sin \phi</math> || [[Physics:Coma (optics)|Vertical coma]]
|-
| <math>Z_3^1</math> || {{0}}8 || {{0}}8 || {{0}}6 ||  {{0}}7 || 3 || +1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \cos \phi</math> || Horizontal coma
|-
| <math>Z_3^3</math> || {{0}}9 || 10 || {{0}}9 || 10 || 3 || +3 || <math>\sqrt{8} \rho^3 \cos 3 \phi</math> || Oblique trefoil
|-
| <math>Z_4^{-4}</math> || 10 || 15 || 17 || 18 ||  4 || −4 || <math>\sqrt{10} \rho^4 \sin 4 \phi</math> || Oblique quadrafoil
|-
| <math>Z_4^{-2}</math> || 11 || 13 || 12 || 13 ||  4 || −2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \sin 2 \phi</math> || Oblique secondary astigmatism
|-
| <math>Z_4^0</math> || 12 || 11 || {{0}}8 || {{0}}9 || 4 || {{0}}0 || <math>\sqrt{5} (6 \rho^4 - 6 \rho^2 +1)</math> || [[Physics:Spherical aberration|Primary spherical]]
|-
| <math>Z_4^2</math> || 13 || 12 || 11 || 12 || 4 || +2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \cos 2 \phi</math> || Vertical secondary astigmatism
|-
| <math>Z_4^4</math> || 14 || 14 || 16 || 17 || 4 || +4 || <math>\sqrt{10} \rho^4 \cos 4 \phi</math> || Vertical quadrafoil

|}

==Applications==
[[File:ZernikeAiryImage.jpg|360px|thumb|Result of the first 21 Zernike polynomials (as above) introduced as aberrations on a flat-top beam. The beam is imaged by a lens, effecting a Fourier transform, whose intensity is represented in this picture]]

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.<ref>{{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}}</ref><ref>
{{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}}
</ref> 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 <math>\rho\approx 1</math>, which often leads attempts to define other orthogonal functions over the circular disk.<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref>

In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses. In wavefront slope sensors like the [[Engineering:Shack–Hartmann wavefront sensor|Shack-Hartmann]], Zernike coefficients of the wavefront can be obtained by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures.<ref>{{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}}</ref> 
In [[Medicine:Optometry|optometry]] and [[Medicine:Ophthalmology|ophthalmology]], Zernike polynomials are used to describe [[Physics:Optical aberration|wavefront aberrations]] of the [[Medicine:Cornea|cornea]] or lens from an ideal spherical shape, which result in refraction errors. They are also commonly used in [[Astronomy:Adaptive optics|adaptive optics]], where they can be used to characterize atmospheric distortion. Obvious applications for this are IR or visual astronomy and [[Astronomy:Satellite imagery|satellite imagery]].

Another application of the Zernike polynomials is found in the Extended Nijboer–Zernike theory of [[Physics:Diffraction|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 (geometry)|scaling]] and the [[Translation (geometry)|translation]] of the object in a [[Region of interest|region of interest]] (ROI), their [[Magnitude (mathematics)|magnitudes]] are independent of the rotation angle of the object.<ref>{{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 = [[Place:Isfahan|Isfahan]], [[Religion:Iran|Iran]]
| pages = 1–4
| doi = 10.1109/ICBME.2010.5704941
}}</ref> Thus, they can be utilized to extract [[Feature (computer vision)|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 [[Medicine:Breast cancer|breast masses]]<ref>
{{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=[[Medicine:Computers in Biology and Medicine|Computers in Biology and Medicine]]
 |volume=41 |issue=8
 |pages=726–735
 |doi=10.1016/j.compbiomed.2011.06.009
|pmid=21722886
 }}</ref> or the surface of vibrating disks.<ref>{{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}}</ref> Zernike Moments also have been used to quantify shape of osteosarcoma cancer cell lines in single cell level.<ref>{{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}}</ref> 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.<ref>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.</ref>

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

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

(Note that a factor <math>\sqrt{2n+D}</math> is absorbed in the definition of ''R'' here, whereas in <math>D=2</math> 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<ref>{{cite journal
|first1=R. J.
|last1=Mathar
|doi=10.2298/SAJ0979107M
|title=Zernike Basis to Cartesian Transformations
|journal=[[Astronomy:Serbian Astronomical Journal|Serbian Astronomical Journal]]
|volume=179
|year=2009
|bibcode=2009SerAJ.179..107M
|pages=107–120
|issue=179|arxiv = 0809.2368 |s2cid=115159231
}}</ref>

:<math>\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}</math>

for even <math>n-l\ge 0</math>, else identical to zero.

==See also==
* [[Jacobi polynomials]]
* Nijboer–Zernike theory
* [[Pseudo-Zernike polynomials]]

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==External links==
* [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)]
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[[Category:Orthogonal polynomials]]

{{Sourceattribution|Zernike polynomials}}