Zernike polynomials explained

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, they play important roles in various optics branches such as beam optics and imaging.^{[1] [2]}

Definitions

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

Z ^{m} _{n(\rho,\varphi)} = R ^{m} _{n(\rho)\cos(m\varphi)}

(even function over the azimuthal angle

\varphi ), and the odd Zernike polynomials are defined asZ ^{- m} _{n(\rho,\varphi)} = R ^{m} _{n(\rho)\sin(m\varphi),}

(odd function over the azimuthal angle

\varphi ) where m and n are nonnegative integers with n ≥ m ≥ 0 (m = 0 for spherical Zernike polynomials), \varphi is the azimuthal angle, ρ is the radial distance 0 \le\rho\le 1 , and R ^{m} _{n} are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. |Z ^{m} _{n(\rho,\varphi)|} \le 1 . The radial polynomials R ^{m} _{n} are defined asR ^{m} _{n(\rho)} = \sum ^{\tfrac{n - m} _{k =0} { 2 }} (( -1 ) ^{k(n-k)!})/(k!\left (\tfrac{n + m) { 2 } - k \right )! \left (\tfrac{n - m}{ 2 } - k \right)!} \rho ^{n-2k}

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

R ^{m( 1 ) =1 .} _{n}

Other representations

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

R ^{\tfrac{n - m} _{k =0} { 2 }}( -1 ) ^{k} \binom{n - k}{k} \binom{n -2 k}{\tfrac{n - m}{ 2 } - k} \rho ^{n-2k} .

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

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

for n − m even.

The inverse relation expands

\rho ^{j} for fixed m\le j into R ^{m(\rho)} _{n} \rho ^{j} = \sum ^{j} _{n\equiv m \pmod 2} h _{j,n,m} R ^{m(\rho)} _{n} with rational coefficients h _{j,n,m} ^[3] h _{j,n,m} = (n +1)/(1+ (j + n)/(2)) (\binom{(j - m)/ 2)/((n - m)/ 2) }for even j - m =0 , 2 , 4 ,\ldots .

The factor

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

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.^[4] The table of this association

Z ^{l} _{n} → Z _{j} starts as follows .j = (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. The rule is the following.

• The even Zernike polynomials Z (with even azimuthal parts

\cos(m\varphi) , where m = l as l is a positive number) obtain even indices j.

• The odd Z obtains (with odd azimuthal parts

\sin(m\varphi) , where m = \left\vert l \right\vert as l is a negative number) odd indices j.

• Within a given n, a lower

\left\vert l \right\vert results in a lower j.

OSA/ANSI standard indices

OSA^[5] and ANSI single-index Zernike polynomials using:

j = (n(n +2 ) + l)/(2)

Fringe/University of Arizona indices

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.^{[6] [7]}

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

where

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

Wyant indices

James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1).^[8] This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

Rodrigues Formula

They satisfy the Rodrigues' formula

Z ^{m} _{n} (x) = (x ^{-m})/(\left( (n - m)/(2) \right) !) \left( (d)/(d\left( x ^{2} \right)) \right) ^{(n - m)/(2)} \left[ x ^{n+m} \left( x ^{2-1} \right) ^{(n - m)/(2)} \right] and can be related to the Jacobi polynomials asZ ^{m} _{n} (x) = x ^{m (P ^{( 0 ,m)} _{(n - m)/(2)} \left( 2 x ^{2-1} \right))/(P ^{( 0 ,m)} _{(n - m)/(2)} ( 1 ))} .

Properties

Orthogonality

The orthogonality in the radial part reads^[9]

\int ^{m} _{n'} (\rho)\rho d\rho = \delta _{n,n'}

or

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

Orthogonality in the angular part is represented by the elementary

\int ^{2 \pi} _{0} \cos(m\varphi)\cos(m'\varphi)d\varphi = \epsilon _{m\pi\delta} _{m,m'} , \int ^{2 \pi} _{0} \sin(m\varphi)\sin(m'\varphi)d\varphi = \pi\delta _{m,m'} ; m ≠ 0 , \int ^{2 \pi} _{0} \cos(m\varphi)\sin(m'\varphi)d\varphi =0 ,

where

\epsilon _{m} (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 if m =0 and 1 if m ≠ 0 . The product of the angular and radial parts establishes the orthogonality of the Zernike functions with respect to both indices if integrated over the unit disk,\int Z ^{l'} _{n'} (\rho,\varphi) d ^{2r} = (\epsilon _{l\pi})/(2 n +2) \delta _{n,n'} \delta _{l,l'} ,

where

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

Zernike transform

Any sufficiently smooth real-valued phase field over the unit disk

G(\rho,\varphi) can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with the Fourier series. We haveG(\rho,\varphi) = \sum _{m,n} \left[ a _{m,n} Z ^{m} _{n(\rho,\varphi)} + b _{m,n} Z ^{- m} _{n(\rho,\varphi)} \right], where the coefficients can be calculated using inner products. On the space of L ^{2} functions on the unit disk, there is an inner product defined by\langle F, G \rangle : = \int F(\rho,\varphi)G(\rho,\varphi)\rho d\rho d\varphi. The Zernike coefficients can then be expressed as follows:\begin{align} a _{m,n} & = (2 n +2)/(\epsilon _{m\pi}) \left \langle G(\rho,\varphi),Z ^{m} _{n(\rho,\varphi)} \right \rangle, \\ b _{m,n} & = (2 n +2)/(\epsilon _{m\pi}) \left \langle G(\rho,\varphi),Z ^{- m} _{n(\rho,\varphi)} \right \rangle. \end{align}

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

Symmetries

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

Z ^{l} _{n} (\rho,\varphi) = Z ^{l} _{n} (\rho, - \varphi) for l ≥ 0,Z ^{l} _{n} (\rho,\varphi) =- Z ^{l} _{n} (\rho, - \varphi) for l < 0.The π shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates isZ ^{l(\rho,\varphi)} _{n} = ( -1 ) ^{l} Z ^{l(\rho,\varphi + \pi),} _{n} 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 ^{l(\rho,\varphi)} _{n} = Z ^{l(\rho,\varphi + \pi).} _{n}

• Angularly odd Zernike polynomials: Zernike polynomials with odd l so that

Z ^{l(\rho,\varphi)} _{n} = - Z ^{l(\rho,\varphi + \pi).} _{n}

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

R ^{m(\rho) = ( -1 )} _{n} ^{n} R ^{m( - \rho) = ( -1 )} _{n} ^{m} R ^{m( - \rho).} _{n}

These equalities are easily seen since

R ^{m(\rho)} _{n} with an odd (even) m contains only odd (even) powers to ρ (see examples of R ^{m(\rho)} _{n} below).

The periodicity of the trigonometric functions results in invariance if rotated by multiples of

2 \pi/l radian around the center:Z ^{l} _{n} \left (\rho, \varphi + \tfrac{ 2 \pi k}{l} \right ) = Z ^{l(\rho,\varphi),} _{n} k = 0 , \pm 1 ,\pm 2 , … .

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:^[10]

\begin{align} R ^{\left|m -1 \right|} _{n -1} (\rho) + R ^{m +1} _{n -1} (\rho)\right] . \end{align}

From the definition of

R ^{m} _{n} it can be seen that R ^{m(\rho)} _{m} = \rho ^{m} and R ^{m(\rho)} _{m +2} = ((m +2 )\rho ^{2} - (m +1 ))\rho ^{m} . The following three-term recurrence relation^[11] then allows to calculate all other R ^{m(\rho)} _{n} :R ^{m(\rho)} _{n} = (2 (n -1 )( 2 n(n -2 )\rho ^{2- m} ^{m(\rho)} ^{n -2} - n(n + m -2 )(n - m -2 )R ^{m(\rho)} _{n -4})/((n + m)(n - m)(n -2 )) .

The above relation is especially useful since the derivative of

R ^{m} _{n} can be calculated from two radial Zernike polynomials of adjacent degree:(\operatorname{d)/() } R_n^m(\rho) = \frac \text

The differential equation of the Gaussian Hypergeometric Function is equivalent to

\rho ^{2(\rho} ^{2-1)} (d ^{2})/(d\rho ^{2}) R ^{m(\rho)} _{n} = [n(n +2 )\rho ^{2-m} ^{m(\rho) + \rho( 1-3 \rho} ^{n} ^{2 ) (d)/(d\rho)} R ^{m(\rho).} _{n}

Examples

Radial polynomials

The first few radial polynomials are:

R ^{0} _{0 (\rho)} = 1 R ^{1} _{1 (\rho)} = \rho R ^{0} _{2 (\rho)} = 2 \rho ^{2} - 1 R ^{2} _{2 (\rho)} = \rho ^{2} R ^{1} _{3 (\rho)} = 3 \rho ^{3} - 2 \rho R ^{3} _{3 (\rho)} = \rho ^{3} R ^{0} _{4 (\rho)} = 6 \rho ^{4} - 6 \rho ^{2} + 1 R ^{2} _{4 (\rho)} = 4 \rho ^{4} - 3 \rho ^{2} R ^{4} _{4 (\rho)} = \rho ^{4} R ^{1} _{5 (\rho)} = 10 \rho ^{5} - 12 \rho ^{3} + 3 \rho R ^{3} _{5 (\rho)} = 5 \rho ^{5} - 4 \rho ^{3} R ^{5} _{5 (\rho)} = \rho ^{5} R ^{0} _{6 (\rho)} = 20 \rho ^{6} - 30 \rho ^{4} + 12 \rho ^{2} - 1 R ^{2} _{6 (\rho)} = 15 \rho ^{6} - 20 \rho ^{4} + 6 \rho ^{2} R ^{4} _{6 (\rho)} = 6 \rho ^{6} - 5 \rho ^{4} R ^{6} _{6 (\rho)} = \rho ^{6.}

Zernike polynomials

The first few Zernike modes, at various indices, are shown below. They are normalized such that:

\int ^{2 \pi} _{0} \int ^{1} _{0} Z ^{2 ⋅ \rhod\rhod\phi} = \pi , which is equivalent to \operatorname{Var}(Z) _{unitcircle} = 1 .

Z ^{l} _{n}	OSA/ANSI index (j )	Noll index (j )	Wyant index (j )	Fringe/UA index (j ) !	Radial degree (n )	Azimuthal degree (l )	Z _{j}	Classical name
Z ^{0} _{0}	0	1	0	1	0	0	1	Piston (see, Wigner semicircle distribution)
Z ^{-1} _{1}	1	3	2	3	1	−1	2 \rho \sin \phi	Tilt (Y-Tilt, vertical tilt)
Z ^{1} _{1}	2	2	1	2	1	+1	2 \rho \cos \phi	Tilt (X-Tilt, horizontal tilt)
Z ^{-2} _{2}	3	5	5	6	2	−2	\sqrt{ 6 } \rho ^{2} \sin 2 \phi	Oblique astigmatism
Z ^{0} _{2}	4	4	3	4	2	0	\sqrt{ 3 } ( 2 \rho ^{2} - 1 )	Defocus (longitudinal position)
Z ^{2} _{2}	5	6	4	5	2	+2	\sqrt{ 6 } \rho ^{2} \cos 2 \phi	Vertical astigmatism
Z ^{-3} _{3}	6	9	10	11	3	−3	\sqrt{ 8 } \rho ^{3} \sin 3 \phi	Vertical trefoil
Z ^{-1} _{3}	7	7	7	8	3	−1	\sqrt{ 8 } ( 3 \rho ^{3} - 2 \rho) \sin \phi	Vertical coma
Z ^{1} _{3}	8	8	6	7	3	+1	\sqrt{ 8 } ( 3 \rho ^{3} - 2 \rho) \cos \phi	Horizontal coma
Z ^{3} _{3}	9	10	9	10	3	+3	\sqrt{ 8 } \rho ^{3} \cos 3 \phi	Oblique trefoil
Z ^{-4} _{4}	10	15	17	18	4	−4	\sqrt{ 10 } \rho ^{4} \sin 4 \phi	Oblique quadrafoil
Z ^{-2} _{4}	11	13	12	13	4	−2	\sqrt{ 10 } ( 4 \rho ^{4} - 3 \rho ^{2)} \sin 2 \phi	Oblique secondary astigmatism
Z ^{0} _{4}	12	11	8	9	4	0	\sqrt{ 5 } ( 6 \rho ^{4} - 6 \rho ^{2} +1 )	Primary spherical
Z ^{2} _{4}	13	12	11	12	4	+2	\sqrt{ 10 } ( 4 \rho ^{4} - 3 \rho ^{2)} \cos 2 \phi	Vertical secondary astigmatism
Z ^{4} _{4}	14	14	16	17	4	+4	\sqrt{ 10 } \rho ^{4} \cos 4 \phi	Vertical quadrafoil

Applications

The functions are a basis defined over the circular support area, typically the pupil planes in classical optical imaging at visible and infrared wavelengths through systems of lenses and mirrors of finite diameter. Their advantages are the simple analytical properties inherited from the simplicity of the radial functions and the factorization in radial and azimuthal functions; this leads, for example, to closed-form expressions of the two-dimensional Fourier transform in terms of Bessel functions.^{[12] [13]} Their disadvantage, in particular if high n are involved, is the unequal distribution of nodal lines over the unit disk, which introduces ringing effects near the perimeter

\rho ≈ 1 , which often leads attempts to define other orthogonal functions over the circular disk.^{[14] [15] [16]}

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.^[17] 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.^[18] 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^[19] or the surface of vibrating disks.^[20] Zernike Moments also have been used to quantify shape of osteosarcoma cancer cell lines in single cell level.^[21] 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.^[22]

Higher dimensions

The concept translates to higher dimensions D if multinomials

x ^{j …} _{2} x ^{k} _{D} in Cartesian coordinates are converted to hyperspherical coordinates, \rho ^{s,} s\le D , multiplied by a product of Jacobi polynomials of the angular variables. In D =3 dimensions, the angular variables are spherical harmonics, for example. Linear combinations of the powers \rho ^{s} define an orthogonal basis R ^{(l)} _{n} (\rho) satisfying\int ^{1} _{0} \rho ^{D-1} R ^{(l)} _{n} (\rho)R ^{(l)} _{n'} (\rho)d\rho = \delta _{n,n'} .

(Note that a factor

\sqrt{ 2 n + D} is absorbed in the definition of R here, whereas in D =2 the normalization is chosen slightly differently. This is largely a matter of taste, depending on whether one wishes to maintain an integer set of coefficients or prefers tighter formulas if the orthogonalization is involved.) The explicit representation is\begin{align} R ^{(l)} _{n} (\rho) & = \sqrt{ 2 n + D}\sum ^{\tfrac{n - l} _{s =0} { 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{ 2 n + D} \sum ^{\tfrac{n - l} _{s =0} { 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{ 2 n + D} {\tfrac{n + l + D}{ 2 } -1 \choose \tfrac{n - l}{ 2 }} \rho ^{l} {} _{2F} _{1} \left ( - \tfrac{n - l}{ 2 },\tfrac{n + l + D}{ 2 }; l + \tfrac{D}{ 2 }; \rho ^{2} \right ) \end{align}

for even

n - l\ge 0 , else identical to zero.

See also

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

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• R. K.. Tyson. Conversion of Zernike aberration coefficients to Seidel and higher-order power series aberration coefficients. Opt. Lett.. 7. 1982. 10.1364/OL.7.000262. 1982OptL....7..262T. 6. 262–264. 19710893.
• J. Y.. Wang. D. E.. Silva. Wave-front interpretation with Zernike Polynomials. Appl. Opt.. 19. 1980. 10.1364/AO.19.001510. 20221066. 1980ApOpt..19.1510W. 9. 1510–8.
• R.. Barakat. Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials. J. Opt. Soc. Am.. 70. 1980. 739. 10.1364/JOSA.70.000739. 1980JOSA...70..739B. 6.
• T. A.. ten Brummelaar. Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike. Opt. Commun.. 132. 1996. 3–4. 329–342. 1996OptCo.132..329T. 10.1016/0030-4018(96)00407-5.
• Book: M.. Novotni. R.. Klein. Proceedings of the eighth ACM symposium on Solid modeling and applications. 3D zernike descriptors for content based shape retrieval. 216–225. 10.1145/781606.781639. 2003. 978-1581137064. 10.1.1.14.4970. 10514681.
• M.. Novotni. R.. Klein. Shape retrieval using 3D Zernike descriptors. Computer-Aided Design. 36. 11. 1047–1062. 10.1016/j.cad.2004.01.005. 2004. 10.1.1.71.8238.
• Book: Sajad. Farokhi. Siti Mariyam . Shamsuddin. U.U.. Sheikh. Jan. Flusser. The 8th International Conference on Robotic, Vision, Signal Processing & Power Applications. Near Infrared Face Recognition: A Comparison of Moment-Based Approaches. Lecture Notes in Electrical Engineering. https://www.researchgate.net/publication/261286249. 291. 129–135. 2014. 1. 10.1007/978-981-4585-42-2_15. 978-981-4585-41-5.
• Sajad. Farokhi. Siti Mariyam . Shamsuddin. Jan. Flusser. U.U.. Sheikh. Mohammad . Khansari. Kourosh. Jafari-Khouzani. Near infrared face recognition by combining Zernike moments and undecimated discrete wavelet transform. Digital Signal Processing. 31. 2014. 1. 10.1016/j.dsp.2014.04.008. 13–27.

External links

• The Extended Nijboer-Zernike website
• MATLAB code for fast calculation of Zernike moments
• Python/NumPy library for calculating Zernike polynomials
• Zernike aberrations at Telescope Optics
• Example: using WolframAlpha to plot Zernike Polynomials
• orthopy, a Python package computing orthogonal polynomials (including Zernike polynomials)

Notes and References

1. Zernike . F. . 1934 . Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode . Physica . 1 . 8 . 10.1016/S0031-8914(34)80259-5. 1934Phy.....1..689Z . 689–704.
2. Book: Born, Max. Max Born. Wolf, Emil. Emil Wolf. amp. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. 1999. Cambridge University Press. Cambridge, UK. 9780521642224. 986. 7th. (see also at Google Books)
3. R. J.. Mathar. 10.2298/SAJ0979107M. Zernike Basis to Cartesian Transformations. Serbian Astronomical Journal. 179. 2009. 2009SerAJ.179..107M. 107–120. 179. 0809.2368 . 115159231.
4. R. J.. Noll. [ftp://ftp.bioeng.auckland.ac.nz/pub/pub/jtur044/references/fitting/NOLL1976.pdf Zernike polynomials and atmospheric turbulence ]. J. Opt. Soc. Am.. 66. 1976. 10.1364/JOSA.66.000207. 207. 1976JOSA...66..207N. 3.
5. L. N.. Thibos. R. A.. Applegate. J. T.. Schwiegerling. R.. Webb. Standards for reporting the optical aberrations of eyes. Journal of Refractive Surgery. 18. 5. 2002. S652-60. 10.3928/1081-597X-20020901-30. 12361175.
6. 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.
7. Proc SPIE . 4771. 276–286. 2002. 10.1117/12.482169. Orthogonality of Zernike polynomials . V. L. . Genberg . G. J. . Michels. K. B. . Doyle. Optomechanical design and Engineering 2002.
8. Book: Eric P. Goodwin . James C. Wyant . Field Guide to Interferometric Optical Testing . 2006 . 25 . 0-8194-6510-0.
9. V. . Lakshminarayanan. Andre . Fleck. 10.1080/09500340.2011.554896. J. Mod. Opt.. 2011. 2011JMOp...58..545L. 58. 7. 545–561. Zernike polynomials: a guide. 120905947.
10. 2013 . Recursive formula to compute Zernike radial polynomials . Opt. Lett. . 38 . 14. 2487–2489 . 10.1364/OL.38.002487 . Honarvar Shakibaei . Barmak. 23939089 . 2013OptL...38.2487H .
11. E. C.. Kintner. 10.1080/713819334. On the mathematical properties of the Zernike Polynomials. Opt. Acta. 23. 1976. 679–680. 1976AcOpt..23..679K. 8.
12. E. . Tatulli. 10.1364/JOSAA.30.000726 . 23595334. 2013. Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures. 2013JOSAA..30..726T. J. Opt. Soc. Am. A. 30. 4. 726–32. 1302.7106. 23491106.
13. A. J. E. M.. Janssen. New analytic results for the Zernike Circle Polynomials from a basic result in the Nijboer-Zernike diffraction theory. 10.2971/jeos.2011.11028. 2011. Journal of the European Optical Society: Rapid Publications. 6. 11028. 2011JEOS....6E1028J. free.
14. Richard. Barakat. Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials. 10.1364/JOSA.70.000739. 1980. J. Opt. Soc. Am.. 70. 6. 739–742. 1980JOSA...70..739B.
15. A. J. E. M.. Janssen. A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory. 1110.2369. 2011. math-ph.
16. R. J.. Mathar. Orthogonal basis function over the unit circle with the minimax property. 1802.09518. 2018. math.NA.
17. Akondi. Vyas. Dubra. Alfredo. 2020-06-22. Average gradient of Zernike polynomials over polygons. Optics Express. EN. 28. 13. 18876–18886. 10.1364/OE.393223. 32672177. 7340383. 2020OExpr..2818876A. 1094-4087. free.
18. A.. Tahmasbi. 2010. An Effective Breast Mass Diagnosis System using Zernike Moments. 17th Iranian Conf. on Biomedical Engineering (ICBME'2010). IEEE. Isfahan, Iran. 1–4. 10.1109/ICBME.2010.5704941.
19. Tahmasbi . A. . Saki . F. . Shokouhi . S.B. . 2011 . Classification of Benign and Malignant Masses Based on Zernike Moments . . 41 . 8 . 726–735 . 10.1016/j.compbiomed.2011.06.009. 21722886 .
20. W. P. . Rdzanek. 10.1016/j.jsv.2018.07.035. Sound radiation of a vibrating elastically supported circular plate embedded into a flat screen revisited using the Zernike circle polynomials. 2018. 434. 91–125. J. Sound Vib.. 2018JSV...434...92R. 125512636.
21. Alizadeh. Elaheh. Lyons. Samanthe M. Castle. Jordan M. Prasad. Ashok. Measuring systematic changes in invasive cancer cell shape using Zernike moments. Integrative Biology. 2016. 8. 11. 1183–1193. 10.1039/C6IB00100A. 27735002.
22. 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.