Inmathematics, theZernike polynomials are asequence ofpolynomials that areorthogonal on theunit disk. Named after optical physicistFrits Zernike, laureate of the 1953Nobel Prize in Physics and the inventor ofphase-contrast microscopy, they play important roles in various optics branches such as beamoptics and imaging.^[1][2]

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

${\displaystyle Z_n^m(\rho ,\phi )=R_n^m(\rho )\cos (m\phi )}$

(even function over the azimuthal angle${\displaystyle \phi }$), and the odd Zernike polynomials are defined as

${\displaystyle Z_n^{-m}(\rho ,\phi )=R_n^m(\rho )\sin (m\phi ),}$

(odd function over the azimuthal angle${\displaystyle \phi }$) wherem andn are nonnegativeintegers withn ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials),${\displaystyle \phi }$ is theazimuthalangle,ρ is the radial distance${\displaystyle 0\le \rho \le 1}$, and${\displaystyle R_n^m}$ 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.${\displaystyle |Z_n^m(\rho ,\phi )|\le 1}$ if${\displaystyle \rho \le 1}$. The radial polynomials${\displaystyle R_n^m}$ are defined as

${\displaystyle R_n^m(\rho )=\sum _{k=0}^{\frac{n-m}{2}}\frac{(-1{)}^k(n-k)!}{k!\left(\frac{n+m}{2}-k\right)!\left(\frac{n-m}{2}-k\right)!}{\rho }^{n-2k}}$

for evenn −m, while it is 0 for oddn −m. A special value is

${\displaystyle R_n^m(1)=1.}$

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

${\displaystyle R_n^m(\rho )=\sum _{k=0}^{\frac{n-m}{2}}(-1{)}^k(\genfrac{}{}{0ex}{}{n-k}{k})(\genfrac{}{}{0ex}{}{n-2k}{\frac{n-m}{2}-k}){\rho }^{n-2k}}$.

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

forn −m even.

The inverse relation expands${\displaystyle {\rho }^j}$ for fixed${\displaystyle m\le j}$ into${\displaystyle R_n^m(\rho )}$

${\displaystyle {\rho }^j=\sum _{n\equiv m(\mathrm{mod}2)}^j{h}_{j,n,m}{R}_n^m(\rho )}$

with rational coefficients${\displaystyle h_{j,n,m}}$^[3]

${\displaystyle h_{j,n,m}=\frac{n+1}{1+\frac{j+n}{2}}\frac{(\genfrac{}{}{0ex}{}{(j-m)/2}{(n-m)/2})}{(\genfrac{}{}{0ex}{}{(j+n)/2}{(n-m)/2})}}$

for even${\displaystyle j-m=0,2,4,\dots }$.

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

${\displaystyle R_n^m(\rho )=\frac{1}{(\genfrac{}{}{0ex}{}{\lfloor n/2\rfloor }{\lfloor m/2\rfloor })}{\rho }^{n\mathrm{mod}2}\sum _{s=\lfloor m/2\rfloor }^{\lfloor n/2\rfloor }(-1{)}^{\lfloor n/2\rfloor -s}(\genfrac{}{}{0ex}{}{s}{\lfloor m/2\rfloor })(\genfrac{}{}{0ex}{}{(n+m)/2}{s+\lceil m/2\rceil })b_{s,\lfloor n/2\rfloor }({\rho }^2).}$

The radial polynomials satisfy theRodrigues' formula

${\displaystyle R_n^m(x)=\frac{1}{\left(\frac{n-m}{2}\right)!x^m}{\left(\frac{d}{d\left(x^2\right)}\right)}^{\frac{n-m}{2}}\left[x^{n+m}{\left(x^2-1\right)}^{\frac{n-m}{2}}\right].}$

The orthogonality in the radial part reads^[4]

${\displaystyle {\int }_0^1\sqrt{2n+2}{R}_n^m(\rho )\sqrt{2n^{\prime }+2}{R}_{n^{\prime }}^m(\rho )\rho d\rho ={\delta }_{n,n^{\prime }}}$

or

${\displaystyle \underset{0}{\stackrel{1}{\int }}{R}_n^m(\rho )R_{n^{\prime }}^m(\rho )\rho d\rho =\frac{{\delta }_{n,n^{\prime }}}{2n+2}.}$

Orthogonality in the angular part is represented by theelementary

${\displaystyle {\int }_0^{2\pi }\cos (m\phi )\cos (m^{\prime }\phi )d\phi ={ϵ}_m\pi {\delta }_{m,m^{\prime }},}$

${\displaystyle {\int }_0^{2\pi }\sin (m\phi )\sin (m^{\prime }\phi )d\phi =\pi {\delta }_{m,m^{\prime }};m\ne 0,}$

${\displaystyle {\int }_0^{2\pi }\cos (m\phi )\sin (m^{\prime }\phi )d\phi =0,}$

where${\displaystyle {ϵ}_m}$ (sometimes called theNeumann factor because it frequently appears in conjunction with Bessel functions) is defined as2 if${\displaystyle m=0}$ and1 if${\displaystyle m\ne 0}$. 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,

${\displaystyle \int Z_n^l(\rho ,\phi )Z_{n^{\prime }}^{l^{\prime }}(\rho ,\phi )d^{2}r=\frac{{ϵ}_l\pi }{2n+2}{\delta }_{n,n^{\prime }}{\delta }_{l,l^{\prime }},}$

where${\displaystyle d^{2}r=\rho d\rho d\phi }$ is theJacobian of the circular coordinate system, and where${\displaystyle n-l}$ and${\displaystyle n^{\prime }-l^{\prime }}$ are both even.

Any sufficiently smooth real-valued phase field over the unit disk${\displaystyle G(\rho ,\phi )}$ can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with theFourier series. We have

${\displaystyle G(\rho ,\phi )=\sum _{m,n}\left[a_{m,n}{Z}_n^m(\rho ,\phi )+b_{m,n}{Z}_n^{-m}(\rho ,\phi )\right],}$

where the coefficients can be calculated usinginner products. On the space of${\displaystyle L^2}$ functions on the unit disk, there is an inner product defined by

${\displaystyle ⟨F,G⟩:=\int F(\rho ,\phi )G(\rho ,\phi )\rho d\rho d\phi .}$

The Zernike coefficients can then be expressed as follows:

Alternatively, one can use the known values of phase functionG 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 oftrigonometric functions, separability of radial and azimuthal parts of Zernike polynomials, and their rotational symmetries.

Thereflections of trigonometric functions result that the parity with respect to reflection along thex axis is

${\displaystyle Z_n^l(\rho ,\phi )=Z_n^l(\rho ,-\phi )}$ forl ≥ 0,

${\displaystyle Z_n^l(\rho ,\phi )=-Z_n^l(\rho ,-\phi )}$ forl < 0.

Theπ shifts of trigonometric functions result that the parity with respect topoint reflection at the center of coordinates is

${\displaystyle Z_n^l(\rho ,\phi )=(-1{)}^l{Z}_n^l(\rho ,\phi +\pi ),}$

where${\displaystyle (-1{)}^l}$ could as well be written${\displaystyle (-1{)}^n}$ because${\displaystyle n-l}$ as even numbers are only cases to get non-vanishing Zernike polynomials. (Ifn is even thenl is also even. Ifn is odd, thenl 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 withl = 0 since it has a special property of no angular dependence.)

• Angularly even Zernike polynomials: Zernike polynomials with evenl so that${\displaystyle Z_n^l(\rho ,\phi )=Z_n^l(\rho ,\phi +\pi ).}$
• Angularly odd Zernike polynomials: Zernike polynomials with oddl so that${\displaystyle Z_n^l(\rho ,\phi )=-Z_n^l(\rho ,\phi +\pi ).}$

(This nomenclature is not used in practice because non-vanishing polynomials have evenl only combined with evenn andoddl combined with oddn, so angularly even polynomials are also radially even polynomials and angularly odd polynomials are also radially odd polynomials such that the attributeangularly is superflous.)

The radial polynomials are also either even or odd, depending on the ordern or the azimuthal indexm:

${\displaystyle R_n^m(\rho )=(-1{)}^n{R}_n^m(-\rho )=(-1{)}^m{R}_n^m(-\rho ).}$

These equalities are easily seen since${\displaystyle R_n^m(\rho )}$ with an odd (even)m contains only odd (even) powers toρ (see examples of${\displaystyle R_n^m(\rho )}$ below).

Theperiodicity of the trigonometric functions results in invariance if rotated by multiples of${\displaystyle 2\pi /l}$ radian around the center:

${\displaystyle Z_n^l\left(\rho ,\phi +\frac{2\pi k}{l}\right)=Z_n^l(\rho ,\phi ),k=0,\pm 1,\pm 2,\cdots .}$

The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation^[5]

${\displaystyle \begin{array}{r}L\left[f\right]={\mathrm{\nabla }}^{2}f-(\mathbf{r}\cdot \mathrm{\nabla }{)}^{2}f-2\mathbf{r}\cdot \mathrm{\nabla }f\end{array}}$

self-adjoint over the unit disk, with negative eigenvalues${\displaystyle L[Z_n^m]=-n(n+2)Z_n^m}$. Other self-adjoint differential operators can be constructed for which the Zernike polynomials form a spectrum, for example${\displaystyle \mathrm{\nabla }\cdot (1-{\rho }^2)\mathrm{\nabla }{Z}_n^m=\left(m^2-n(n+2)\right)Z_n^m}$ (relating to rough surfaceBRDFs^[6]), which differs from the above by a factor${\displaystyle {\mathrm{\partial }}_{\phi \phi }}$.

The Zernike polynomials satisfy the followingrecurrence relation:^[7]

${\displaystyle \begin{array}{r}{R}_n^m(\rho )+R_{n-2}^m(\rho )=\rho \left[R_{n-1}^{\left|m-1\right|}(\rho )+R_{n-1}^{m+1}(\rho )\right]\text{ .}\end{array}}$

From the definition of${\displaystyle R_n^m}$ it can be seen that${\displaystyle R_m^m(\rho )={\rho }^m}$ and${\displaystyle R_{m+2}^m(\rho )=((m+2){\rho }^2-(m+1)){\rho }^m}$. The following three-term recurrence relation^[8][9]then allows to calculate all other${\displaystyle R_n^m(\rho )}$:

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

The main use of these recurrences is to avoid cancellation of digits that occurs for large${\displaystyle n}$ in the accumulation of the oscillatory binomial terms in the power series notation^[10].^[11]

The above relation is also useful since the derivative of${\displaystyle R_n^m}$ can be calculated from two radial Zernike polynomials of adjacent degree:^[8]

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\rho }{R}_n^m(\rho )=\frac{(2nm({\rho }^2-1)+(n-m)(m+n(2{\rho }^2-1)))R_n^m(\rho )-(n+m)(n-m)R_{n-2}^m(\rho )}{2n\rho ({\rho }^2-1)}\text{ .}}$

Thedifferential equation of the Gaussian Hypergeometric Function is equivalent to

${\displaystyle {\rho }^2({\rho }^2-1)\frac{d^2}{d{\rho }^2}{R}_n^m(\rho )=[n(n+2){\rho }^2-m^2]R_n^m(\rho )+\rho (1-3{\rho }^2)\frac{d}{d\rho }{R}_n^m(\rho ).}$

Applications often involvelinear 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 indicesn andm to a single indexj has been introduced by Noll.^[12]

(In this section the${\displaystyle m}$ is the signed upper index of${\displaystyle Z}$ which may be positive or negative or zero.) The table of this association${\displaystyle Z_n^m\to Z_j}$ starts as follows (sequenceA176988 in theOEIS).

The rule is the following.

• The even Zernike polynomialsZ with${\displaystyle m>0}$ obtain even indices j.
• The oddZ where obtain odd indices j.
• Within a givenn, a lower${\displaystyle \left|m\right|}$ results in a lower j.

OSA^[13] andANSI single-index Zernike polynomials using:

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

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

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

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g.,photolithography.^[14][15]

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

where${\displaystyle \mathrm{sgn}l}$ is thesign 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).^[16] This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

The first few radial polynomials are:

${\displaystyle R_0^0(\rho )=1}$

${\displaystyle R_1^1(\rho )=\rho }$

${\displaystyle R_2^0(\rho )=2{\rho }^2-1}$

${\displaystyle R_2^2(\rho )={\rho }^2}$

${\displaystyle R_3^1(\rho )=3{\rho }^3-2\rho }$

${\displaystyle R_3^3(\rho )={\rho }^3}$

${\displaystyle R_4^0(\rho )=6{\rho }^4-6{\rho }^2+1}$

${\displaystyle R_4^2(\rho )=4{\rho }^4-3{\rho }^2}$

${\displaystyle R_4^4(\rho )={\rho }^4}$

${\displaystyle R_5^1(\rho )=10{\rho }^5-12{\rho }^3+3\rho }$

${\displaystyle R_5^3(\rho )=5{\rho }^5-4{\rho }^3}$

${\displaystyle R_5^5(\rho )={\rho }^5}$

${\displaystyle R_6^0(\rho )=20{\rho }^6-30{\rho }^4+12{\rho }^2-1}$

${\displaystyle R_6^2(\rho )=15{\rho }^6-20{\rho }^4+6{\rho }^2}$

${\displaystyle R_6^4(\rho )=6{\rho }^6-5{\rho }^4}$

${\displaystyle R_6^6(\rho )={\rho }^6.}$

The first few Zernike modes, at various indices,^[4][17][18] are shown below. In this table they are normalized differently than in the remaining sections:${\displaystyle {\int }_0^{2\pi }{\int }_0^1[Z_j(\rho ,\varphi ){]}^2\cdot \rho d\rho d\varphi =\pi }$, which is equivalent to${\displaystyle \mathrm{Var}(Z{)}_{\text{unit circle}}=1}$.

${\displaystyle Z_n^l}$	OSA/ANSI index (${\displaystyle j}$)	Noll index (${\displaystyle j}$)	Wyant index (${\displaystyle j}$)	Fringe/UA index (${\displaystyle j}$)	Radial degree (${\displaystyle n}$)	Azimuthal degree (${\displaystyle l}$)	${\displaystyle Z_j}$	Classical name
${\displaystyle Z_0^0}$	00	01	00	01	0	00	${\displaystyle 1}$	Piston (see,Wigner semicircle distribution)
${\displaystyle Z_1^{-1}}$	01	03	02	03	1	−1	${\displaystyle 2\rho \sin \varphi }$	Tilt (Y-Tilt, vertical tilt)
${\displaystyle Z_1^1}$	02	02	01	02	1	+1	${\displaystyle 2\rho \cos \varphi }$	Tilt (X-Tilt, horizontal tilt)
${\displaystyle Z_2^{-2}}$	03	05	05	06	2	−2	${\displaystyle \sqrt{6}{\rho }^2\sin 2\varphi }$	Oblique astigmatism
${\displaystyle Z_2^0}$	04	04	03	04	2	00	${\displaystyle \sqrt{3}(2{\rho }^2-1)}$	Defocus (longitudinal position)
${\displaystyle Z_2^2}$	05	06	04	05	2	+2	${\displaystyle \sqrt{6}{\rho }^2\cos 2\varphi }$	Vertical astigmatism
${\displaystyle Z_3^{-3}}$	06	09	10	11	3	−3	${\displaystyle \sqrt{8}{\rho }^3\sin 3\varphi }$	Vertical trefoil
${\displaystyle Z_3^{-1}}$	07	07	07	08	3	−1	${\displaystyle \sqrt{8}(3{\rho }^3-2\rho )\sin \varphi }$	Vertical coma
${\displaystyle Z_3^1}$	08	08	06	07	3	+1	${\displaystyle \sqrt{8}(3{\rho }^3-2\rho )\cos \varphi }$	Horizontal coma
${\displaystyle Z_3^3}$	09	10	09	10	3	+3	${\displaystyle \sqrt{8}{\rho }^3\cos 3\varphi }$	Oblique trefoil
${\displaystyle Z_4^{-4}}$	10	15	17	18	4	−4	${\displaystyle \sqrt{10}{\rho }^4\sin 4\varphi }$	Oblique quadrafoil
${\displaystyle Z_4^{-2}}$	11	13	12	13	4	−2	${\displaystyle \sqrt{10}(4{\rho }^4-3{\rho }^2)\sin 2\varphi }$	Oblique secondary astigmatism
${\displaystyle Z_4^0}$	12	11	08	09	4	00	${\displaystyle \sqrt{5}(6{\rho }^4-6{\rho }^2+1)}$	Primary spherical
${\displaystyle Z_4^2}$	13	12	11	12	4	+2	${\displaystyle \sqrt{10}(4{\rho }^4-3{\rho }^2)\cos 2\varphi }$	Vertical secondary astigmatism
${\displaystyle Z_4^4}$	14	14	16	17	4	+4	${\displaystyle \sqrt{10}{\rho }^4\cos 4\varphi }$	Vertical quadrafoil

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-dimensionalFourier transform in terms of Bessel functions.^[19][20] Their disadvantage, in particular if highn are involved, is the unequal distribution of nodal lines over the unit disk, which introduces ringing effects near the perimeter${\displaystyle \rho \approx 1}$, which often leads attempts to define otherorthogonal functions over the circular disk.^[21][22][23]

In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses. In wavefront slope sensors like theShack-Hartmann, Zernike coefficients of the wavefront can be obtained by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures.^[24] Inoptometry andophthalmology, Zernike polynomials are used to describewavefront aberrations of thecornea orlens from an ideal spherical shape, which result inrefraction errors. They are also commonly used inadaptive optics, where they can be used to characterizeatmospheric distortion. Obvious applications for this are IR or visual astronomy andsatellite imagery.

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

Zernike polynomials are widely used as basis functions ofimage moments. Since Zernike polynomials areorthogonal 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 thescaling and thetranslation of the object in aregion of interest (ROI), theirmagnitudes are independent of the rotation angle of the object.^[25] Thus, they can be utilized to extractfeatures from images that describe the shape characteristics of an object. For instance, Zernike moments are utilized as shape descriptors to classify benign and malignantbreast masses^[26] or the surface of vibrating disks.^[27] Zernike Moments also have been used to quantify shape of osteosarcoma cancer cell lines in single cell level.^[28] 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.^[29]

The concept translates to higher dimensionsD if multinomials${\displaystyle x_1^i{x}_2^j\cdots x_D^k}$ in Cartesian coordinates are converted tohyperspherical coordinates,${\displaystyle {\rho }^s,s\le D}$, multiplied by a product of Jacobi polynomials of the angular variables. In${\displaystyle D=3}$ dimensions, the angular variables arespherical harmonics, for example. Linear combinations of the powers${\displaystyle {\rho }^s}$ define an orthogonal basis${\displaystyle R_n^{(l)}(\rho )}$ satisfying

${\displaystyle {\int }_0^1{\rho }^{D-1}{R}_n^{(l)}(\rho )R_{n^{\prime }}^{(l)}(\rho )d\rho ={\delta }_{n,n^{\prime }}}$.

(Note that a factor${\displaystyle \sqrt{2n+D}}$ is absorbed in the definition ofR here, whereas in${\displaystyle D=2}$ 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 theorthogonalization is involved.) The explicit representation is^[3]

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

Its differential equation for the Gaussian Hypergeometric Function is equivalent to

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

Kintner's recurrence for fixed${\displaystyle l}$ and variable${\displaystyle n\pm 2}$quoted for${\displaystyle D=2}$ above is in the general form^[30]

For${\displaystyle D=3}$ this was proposed by Deng and Gwo.^[31]

For fixed${\displaystyle n}$ and variable${\displaystyle l\pm 2}$ the recurrence is^[32]

The case for${\displaystyle D=2}$ was published by Chong et al.^[11]

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

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Wikimedia Commons has media related toZernike polynomials.

• Braat, Joseph (2026)."The Extended Nijboer-Zernike website".
• Tahmasbi, Amir (2014)."Zernike Moments". Matlab codeArchived 1 August 2015 at theWayback Machine
• Python/NumPy library for calculating Zernike polynomials
• Zernike aberrations atTelescope Optics
• Example: using WolframAlpha to plot Zernike Polynomials
• orthopy, a Python package computing orthogonal polynomials (including Zernike polynomials)

• Orthogonal polynomials

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