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| [[Image:ZernikePolynome6.svg|360px|thumb|
| | Nice to satisfy you, my name is Refugia. Puerto Rico is exactly where he's been living for years and he will by no means move. In her professional life she is a payroll clerk but she's always wanted her personal company. To perform baseball is the hobby he will never quit performing.<br><br>Look at my page; [http://www.sangroupindia.Co.in/members/jodijervois/activity/44269/ at home std test] |
| The first 15 Zernike polynomials, ordered vertically by radial degree and horizontally by azimuthal degree.]]
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| In [[mathematics]], the '''Zernike polynomials''' are a [[polynomial sequence|sequence]] of [[polynomial]]s that are [[orthogonal polynomials|orthogonal]] on the [[unit disk]]. Named after [[Nobel Prize]] winner and optical physicist, and inventor of the [[phase contrast microscopy]], [[Frits Zernike]], they play an important role in beam [[optics]].<ref>
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| {{cite journal
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| |last = Zernike
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| |first = F.
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| |year = 1934
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| |title = Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode
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| |journal= Physica
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| |volume=1,
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| |number= 8
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| |doi=10.1016/S0031-8914(34)80259-5
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| |bibcode=1934Phy.....1..689Z
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| |pages = 689–704
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| }}</ref><ref>
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| {{cite book
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| |author=[[Max Born|Born, Max]], and [[Emil Wolf|Wolf, Emil]]
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| |title=Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light
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| |year=1999
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| |publisher=Cambridge University Press
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| |location=Cambridge, UK
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| |isbn=9780521642224
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| |pages=986
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| |url=http://books.google.nl/books/about/Principles_of_Optics.html?id=aoX0gYLuENoC
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| |edition=7th}}</ref>
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| | |
| <!-- to do: [[wavefront]] -->
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| ==Definitions==
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| There are [[even and odd functions|even and odd]] '''Zernike polynomials'''. The even ones are defined as
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| :<math>Z^{m}_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!</math>
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| and the odd ones as
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| :<math>Z^{-m}_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!</math>
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| where ''m'' and ''n'' are nonnegative [[integer]]s with ''n''≥''m'', ''φ'' is the [[azimuth]]al [[angle]], and ''ρ'' is the radial distance <math>0\le\rho\le 1</math>. 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>.
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| The radial polynomials ''R''<sup>m</sup><sub>n</sub> are defined as
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| :<math>R^m_n(\rho) = \! \sum_{k=0}^{(n-m)/2} \!\!\! \frac{(-1)^k\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!} \;\rho^{n-2\,k}</math>
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| for ''n'' − ''m'' even, and are identically 0 for ''n'' − ''m'' odd.
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| ===Other Representations===
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| Rewriting the ratios of factorials in the radial part as products of
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| [[Binomial coefficient|binomials]] shows that the coefficients are integer numbers:
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| :<math>R_n^m(\rho)=\sum_{k=0}^{(n-m)/2}(-1)^k \binom{n-k}{k} \binom{n-2k}{(n-m)/2-k} \rho^{n-2k}</math>.
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| A notation as terminating
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| [[Hypergeometric function|Gaussian Hypergeometric Functions]]
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| is useful to reveal recurrences, to demonstrate that they are special
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| cases of [[Jacobi polynomials]], to write down the differential equations, etc.:
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| :<math>R_n^m(\rho)= \binom{n}{(n+m)/2}
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| \rho^n
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| {}_2F_{1}\left(-\frac{n+m}{2},-\frac{n-m}{2};-n;\rho^{-2}\right)
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| </math>
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| :<math>= (-1)^{(n+m)/2}\binom{(n+m)/2}{(n-m)/2}\rho^m
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| {}_2F_{1}\left(1+n,1-\frac{n-m}{2};1+\frac{n+m}{2};\rho^2\right)
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| </math>
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| for ''n'' − ''m'' even.
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| ===Noll's sequential indices===
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| Applications often involve linear algebra, where integrals over products
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| of Zernike polynomials and some other factor build the matrix elements.
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| To enumerate the rows and columns of these matrices by a single index, a conventional
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| mapping of the two indices ''n'' and ''m'' to a single index ''j'' has been
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| introduced by Noll.<ref name=noll1976>{{cite journal
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| |first1=R. J.
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| |last1=Noll
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| |title=Zernike polynomials and atmospheric turbulence
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| |journal=J. Opt. Soc. Am.
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| |volume=66
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| |year=1976
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| |url=ftp://ftp.bioeng.auckland.ac.nz/pub/pub/jtur044/references/fitting/NOLL1976.pdf
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| |doi=10.1364/JOSA.66.000207
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| |page=207
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| |bibcode=1976JOSA...66..207N
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| |issue=3}}</ref> The table of this association <math>Z_n^m \rightarrow Z_j</math> starts as follows {{OEIS|A176988}}
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| {|class="wikitable"
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| !n,m
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| {{!!}} 0,0{{!!}}1,1{{!!}} 1,-1 {{!!}} 2,0{{!!}} 2,-2 {{!!}} 2,2{{!!}}3,-1{{!!}} 3,1 {{!!}} 3,-3 {{!!}} 3,3
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| |-------
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| ! j
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| {{!}} 1{{!!}}2{{!!}} 3 {{!!}} 4 {{!!}} 5 {{!!}} 6 {{!!}} 7 {{!!}}8 {{!!}} 9{{!!}} 10
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| |-----
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| !n,m
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| {{!!}}4,0 {{!!}}4,2 {{!!}}4,-2{{!!}}4,4{{!!}}4,-4{{!!}}5,1{{!!}}5,-1{{!!}}5,3 {{!!}}5,-3{{!!}}5,5
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| |-----
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| ! j
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| {{!!}}11 {{!!}}12 {{!!}}13 {{!!}}14{{!!}}15{{!!}}16{{!!}} 17 {{!!}} 18 {{!!}}19 {{!!}}20
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| |}
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| The rule is that the even ''Z'' (with even azimuthal part m, <math>\cos(m\varphi)</math>)
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| obtain even indices ''j'', the odd ''Z'' odd indices ''j''. Within a given ''n'',
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| lower values of ''m'' obtain lower ''j''.
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| ==Properties==
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| ===Orthogonality===
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| The orthogonality in the radial part reads
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| :<math>\int_0^1 \rho \sqrt{2n+2}R_n^m(\rho)\,\sqrt{2n'+2}R_{n'}^{m}(\rho)d\rho = \delta_{n,n'}</math>.
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| Orthogonality in the angular part is represented by
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| :<math>\int_0^{2\pi} \cos(m\varphi)\cos(m'\varphi)d\varphi=\epsilon_m\pi\delta_{|m|,|m'|}</math>,
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| :<math>\int_0^{2\pi} \sin(m\varphi)\sin(m'\varphi)d\varphi=(-1)^{m+m'}\pi\delta_{|m|,|m'|};\quad m\neq 0</math>,
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| :<math>\int_0^{2\pi} \cos(m\varphi)\sin(m'\varphi)d\varphi=0</math>,
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| where <math>\epsilon_m</math> (sometimes called the [[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>.
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| The product of the angular and radial parts establishes the orthogonality of the Zernike functions
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| with respect to both indices if integrated over the unit disk,
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| :<math>\int Z_n^m(\rho,\varphi)Z_{n'}^{m'}(\rho,\varphi)d^2r
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| =\frac{\epsilon_m\pi}{2n+2}\delta_{n,n'}\delta_{m,m'}</math>,
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| where <math>d^2r=\rho\,d\rho\,d\varphi</math> is the [[Jacobian matrix and determinant|Jacobian]] of the
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| circular coordinate system, and where <math>n-m</math> and <math>n'-m'</math>
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| are both even.
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| A special value is
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| :<math>R_n^m(1)=1</math>.
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| ===Zernike Transform===
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| Any sufficiently smooth real-valued phase field over the unit disk <math>G(\rho,\varphi)</math> can be represented in terms
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| of its Zernike coefficients (odd and even), just as much as periodic functions find an orthogonal representation with the [[Fourier series]]. We have
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| :<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>
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| where the coefficients can be calculated using [[inner product]]s. On the space of <math>L^2</math> functions on the unit disk we have an inner product defined by
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| :<math>\langle F, G \rangle := \int F(\rho,\varphi)G(\rho,\varphi)\rho d\rho d\varphi.</math>
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| The Zernike coefficients can then be expressed as follows:
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| :<math>a_{m,n} = \frac{2n+2}{\epsilon_m\pi}\langle G(\rho,\varphi),Z^{m}_n(\rho,\varphi) \rangle,\quad b_{m,n} = \frac{2n+2}{\epsilon_m\pi}\langle G(\rho,\varphi),Z^{-m}_n(\rho,\varphi) \rangle.</math>
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| Alternatively, we can use the known values of phase function ''G'' on the circular grid to form a system of equations.
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| The phase function is retrieved by the unknown-coefficient weighted product with (known values) of Zernike polynomial across
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| the unit grid. Hence coefficients can also be found by solving 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.
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| ===Symmetries===
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| The parity with respect to reflection along the ''x'' axis is
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| :<math>Z_n^{m}(\rho,\varphi)=(-1)^m Z_n^{m}(\rho,-\varphi)</math>.
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| The parity with respect to point reflection at the center of coordinates is
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| :<math>Z_n^m(\rho,\varphi) = (-1)^m Z_n^m(\rho,\varphi+\pi)</math>,
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| where <math>(-1)^m</math> could as well be written <math>(-1)^n</math>
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| because <math>n-m</math> is even for the relevant, non-vanishing values.
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| The radial polynomials are also either even or odd, depending on order ''n'' or ''m'':
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| :<math>R_n^m(\rho)=(-1)^n R_n^m(-\rho)=(-1)^m R_n^m(-\rho)</math>.
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| The periodicity of the trigonometric functions implies invariance if rotated by multiples of <math>2\pi/m</math> radian
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| around the center:
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| :<math>Z_n^m(\rho,\varphi+2\pi k/m)=Z_n^m(\rho,\varphi),\quad k= 0, \pm 1,\pm 2,\ldots</math>.
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| ==Examples==
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| === Radial polynomials ===
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| The first few radial polynomials are:
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| :<math> R^0_0(\rho) = 1 \,</math>
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| :<math> R^1_1(\rho) = \rho \,</math>
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| :<math> R^0_2(\rho) = 2\rho^2 - 1 \,</math>
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| :<math> R^2_2(\rho) = \rho^2 \,</math>
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| :<math> R^1_3(\rho) = 3\rho^3 - 2\rho \, </math>
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| :<math> R^3_3(\rho) = \rho^3 \,</math>
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| :<math> R^0_4(\rho) = 6\rho^4 - 6\rho^2 + 1 \,</math>
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| :<math> R^2_4(\rho) = 4\rho^4 - 3\rho^2 \,</math>
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| :<math> R^4_4(\rho) = \rho^4 \,</math>
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| :<math> R^1_5(\rho) = 10\rho^5 - 12\rho^3 + 3\rho \,</math>
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| :<math> R^3_5(\rho) = 5\rho^5 - 4\rho^3 \,</math>
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| :<math> R^5_5(\rho) = \rho^5 \,</math>
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| :<math> R^0_6(\rho) = 20\rho^6 - 30\rho^4 + 12\rho^2 - 1 \,</math>
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| :<math> R^2_6(\rho) = 15\rho^6 - 20\rho^4 + 6\rho^2 \,</math>
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| :<math> R^4_6(\rho) = 6\rho^6 - 5\rho^4 \,</math>
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| :<math> R^6_6(\rho) = \rho^6. \,</math>
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| === Zernike polynomials ===
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| The first few Zernike modes, ordered by Noll index <math>j</math> are<ref name=noll1976 />
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| {| class="wikitable"
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| |-
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| ! Noll index (<math>j</math>) !! Radial degree (<math>n</math>) !! Azimuthal degree (<math>m</math>) !! <math>Z_j</math> !! Classical name
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| |-
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| | 1 || 0 || 0 || <math>1</math> || [[Piston (optics)|Piston]]
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| |-
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| | 2 || 1 || 1 || <math>2 \rho \cos \theta</math> || [[Tilt (optics)|Tip]] (lateral position) (X-Tilt)
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| |-
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| | 3 || 1 || -1 || <math>2 \rho \sin \theta</math> || [[Tilt (optics)|Tilt]] (lateral position) (Y-Tilt)
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| |-
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| | 4 || 2 || 0 || <math>\sqrt{3} (2 \rho^2 - 1)</math> || [[Defocus aberration|Defocus]] (longitudinal position)
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| |-
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| | 5 || 2 || -2 || <math>\sqrt{6} \rho^2 \sin 2 \theta</math> || [[Astigmatism]]
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| |-
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| | 6 || 2 || 2 || <math>\sqrt{6} \rho^2 \cos 2 \theta</math> || Astigmatism
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| |-
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| | 7 || 3 || -1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \sin \theta</math> || [[Coma (optics)|Coma]]
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| |-
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| | 8 || 3 || 1 || <math>\sqrt{8} (3 \rho^3 - 2\rho) \cos \theta</math> || Coma
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| |-
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| | 9 || 3 || -3 || <math>\sqrt{8} \rho^3 \sin 3 \theta</math> || Trefoil
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| |-
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| | 10 || 3 || 3 || <math>\sqrt{8} \rho^3 \cos 3 \theta</math> || Trefoil
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| |-
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| | 11 || 4 || 0 || <math>\sqrt{5} (6 \rho^4 - 6 \rho^2 +1)</math> || Third order [[Spherical aberration|spherical]]
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| |-
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| | 12 || 4 || 2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \cos 2 \theta</math> || —
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| |-
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| | 13 || 4 || -2 || <math>\sqrt{10} (4 \rho^4 - 3\rho^2) \sin 2 \theta</math> || —
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| |-
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| | 14 || 4 || 4 || <math>\sqrt{10} \rho^4 \cos 4 \theta</math> || —
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| |-
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| | 15 || 4 || -4 || <math>\sqrt{10} \rho^4 \sin 4 \theta</math> || —
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| | |
| |}
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| ==Applications==
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| The functions are a basis defined over the circular support area, typically
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| the pupil planes in classical optical imaging at visible and infrared wavelengths
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| through systems of lenses and mirrors
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| of finite diameter. Their advantages are the simple analytical properties
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| inherited from the simplicity of the radial functions and the factorization
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| in radial and azimuthal functions; this leads, for example, to closed form expressions
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| of the two-dimensional [[Fourier transform]] in terms of Bessel Functions.
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| Their disadvantage, in particular if high ''n'' are involved, is the unequal
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| distribution of nodal lines over the unit disk, which introduces ringing effects
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| near the perimeter <math>\rho\approx 1</math>, which often leads attempts to define other
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| orthogonal functions over the circular disk.
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| In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses, in order to achieve desired system performance.
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| In [[optometry]] and [[ophthalmology]] Zernike polynomials are used to describe [[Optical aberration|aberrations]] of the [[cornea]] or [[lens (vision)|lens]] from an ideal spherical shape, which result in [[refraction error]]s. | |
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| They are commonly used in [[adaptive optics]] where they can be used to effectively cancel out [[Atmospheric turbulence|atmospheric distortion]]. Obvious applications for this are IR or visual astronomy and [[Satellite imagery]]. For example, one of the Zernike terms (for ''m'' = 0, ''n'' = 2) is called 'de-focus'. By coupling the output from this term to a control system, an automatic focus can be implemented.
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| Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of [[diffraction]] and [[Optical aberrations|aberrations]].
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| 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 an [[Region of interest|ROI]], their [[Magnitude (mathematics)|magnitudes]] are independent of the rotation angle of the object.<ref>{{cite conference
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| | first = A.
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| | last = Tahmasbi
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| | year = 2010
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| | title = An Effective Breast Mass Diagnosis System using Zernike Moments
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| | conference = 17th Iranian Conf. on Biomedical Engineering (ICBME'2010)
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| | publisher = [[IEEE]]
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| | location = [[Isfahan]], [[Iran]]
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| | pages = 1–4
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| | doi = 10.1109/ICBME.2010.5704941
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| }}</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 [[Breast cancer|breast masses]].<ref>
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| {{cite journal
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| |last=Tahmasbi |first=A.
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| |last2=Saki |first2=F.
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| |last3=Shokouhi |first3=S.B.
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| |year=2011
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| |title=Classification of Benign and Malignant Masses Based on Zernike Moments
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| |journal=[[Computers in Biology and Medicine]]
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| |volume=41 |pages=726–735
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| |doi=10.1016/j.compbiomed.2011.06.009
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| }}</ref><ref>{{cite conference
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| | first = A.
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| | last = Tahmasbi
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| | year = 2011
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| | title = A Novel Breast Mass Diagnosis System based on Zernike Moments as Shape and Density Descriptors
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| | conference = 18th Iranian Conf. on Biomedical Engineering (ICBME'2011)
| |
| | publisher = [[IEEE]]
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| | location = [[Tehran]], [[Iran]]
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| | pages = 100–104
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| | doi = 10.1109/ICBME.2011.6168532
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| }}</ref>
| |
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| ==Higher Dimensions==
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| The concept translates to higher dimensions <math>D</math> if multinomials <math>x_1^ix_2^j\cdots x_D^k</math> in Cartesian coordinates are converted to [[Hyperspherical_coordinates#Hyperspherical_coordinates|hyperspherical coordinates]],
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| <math>\rho^s</math>, <math>s\le D</math>, multiplied by a product of Jacobi Polynomials of the angular
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| variables.
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| In <math>D=3</math> dimensions, the angular variables are [[Spherical harmonics]], for example.
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| Linear combinations of the powers <math>\rho^s</math> define an orthogonal
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| basis <math>R_n^{(l)}(\rho)</math> satisfying
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| :<math>\int_0^1 \rho^{D-1}R_n^{(l)}(\rho)R_{n'}^{(l)}(\rho)d\rho = \delta_{n,n'}</math>.
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| (Note that a factor <math>\sqrt{2n+D}</math> is absorbed in the definition of <math>R</math> here,
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| whereas in <math>D=2</math> the normalization is chosen slightly differently. This is largely
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| a matter of taste, depending on whether one wishes to maintain an integer set of coefficients
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| or prefers tighter formulas if the orthogonalization is involved.)
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| The explicit representation is
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| :<math>R_n^{(l)}(\rho) = \sqrt{2n+D}\sum_{s=0}^{(n-l)/2}
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| (-1)^s{(n-l)/2 \choose s}{n-s-1+D/2 \choose (n-l)/2}\rho^{n-2s}</math>
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| :<math>=(-1)^{(n-l)/2}\sqrt{2n+D}\sum_{s=0}^{(n-l)/2}
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| (-1)^s{(n-l)/2 \choose s}
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| {s-1+(n+l+D)/2 \choose (n-l)/2} \rho^{2s+l}</math>
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| :<math>=(-1)^{(n-l)/2}\sqrt{2n+D}{ (D+n+l)/2-1 \choose (n-l)/2}\rho^l
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| {}_2F_1( -(n-l)/2,(n+l+D)/2;l+D/2;\rho^2)</math>
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| for even <math>n-l\ge 0</math>, else identical to zero.
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| | |
| ==See also==
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| {{commons category|Zernike Polynomials}}
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| * [[Jacobi polynomials]]
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| * [[Nijboer-Zernike theory]]
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| * [[Pseudo-Zernike polynomials]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{Use dmy dates|date=July 2011}}
| |
| * {{MathWorld|title=Zernike Polynomial|urlname=ZernikePolynomial}}
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| * {{cite journal
| |
| |first1=P. G.
| |
| |last1=Callahan
| |
| |first2=M.
| |
| |last2=De Graef
| |
| |title=Precipitate shape fitting and reconstruction by means of 3D Zernike functions
| |
| |doi=10.1088/0965-0393/20/1/015003
| |
| |year=2012
| |
| |journal= Model. Simul. Mat. Sci. Engin.
| |
| |volume=20
| |
| |pages=015003
| |
| |bibcode=2012MSMSE..20a5003C
| |
| }}
| |
| * {{cite journal
| |
| |first1=C. E.
| |
| |last1= Campbell
| |
| |doi=10.1364/JOSAA.20.000209
| |
| |title=Matrix method to find a new set of Zernike coefficients form an original set when the aperture radius is changed
| |
| |journal=J. Opt. Soc. Am. A
| |
| |volume=20
| |
| |year=2003
| |
| |bibcode=2003JOSAA..20..209C
| |
| |page=209
| |
| |issue=2}}
| |
| * {{cite journal
| |
| |first1=C.
| |
| |last1=Cerjan
| |
| |title=The Zernike-Bessel representation and its application to Hankel transforms
| |
| |journal=J. Opt. Soc. Am. A
| |
| |volume=24
| |
| |year=2007
| |
| |page=1609
| |
| |doi=10.1364/JOSAA.24.001609
| |
| |bibcode=2007JOSAA..24.1609C
| |
| |issue=6}}
| |
| * {{cite journal
| |
| |first1=S. A.
| |
| |last1=Comastri
| |
| |first2=L. I.
| |
| |last2=Perez
| |
| |first3=G. D.
| |
| |last3=Perez
| |
| |first4=G.
| |
| |last4=Martin
| |
| |first5=K.
| |
| |last5=Bastida Cerjan
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| ==External links==
| |
| * [http://www.nijboerzernike.nl The Extended Nijboer-Zernike website.]
| |
| * [http://www.utdallas.edu/~a.tahmasbi/Zernike.html 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]
| |
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| [[Category:Orthogonal polynomials]]
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| [[Category:Dutch inventions]]
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