Zernike polynomials

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

${\displaystyle Z_{n}^{m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\cos(m\,\varphi )\!}$ 

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

${\displaystyle Z_{n}^{-m}(\rho ,\varphi )=R_{n}^{m}(\rho )\,\sin(m\,\varphi ),\!}$ 

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

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

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

${\displaystyle R_{n}^{m}(1)=1.}$ 

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

${\displaystyle R_{n}^{m}(\rho )=\sum _{k=0}^{\tfrac {n-m}{2}}(-1)^{k}{\binom {n-k}{k}}{\binom {n-2k}{{\tfrac {n-m}{2}}-k}}\rho ^{n-2k}}$ .

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.:

${\displaystyle {\begin{aligned}R_{n}^{m}(\rho )&=(-1)^{(n-m)/2}\rho ^{m}P_{(n-m)/2}^{(m,0)}(1-2\rho ^{2})\\&={\binom {n}{\tfrac {n+m}{2}}}\rho ^{n}\ {}_{2}F_{1}\left(-{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};-n;\rho ^{-2}\right)\\&=(-1)^{\tfrac {n-m}{2}}{\binom {\tfrac {n+m}{2}}{m}}\rho ^{m}\ {}_{2}F_{1}\left(1+{\tfrac {n+m}{2}},-{\tfrac {n-m}{2}};1+m;\rho ^{2}\right)\end{aligned}}}$ 

for n − m even.

The inverse relation expands ${\displaystyle \rho ^{j}}$  for fixed ${\displaystyle m\leq j}$  into ${\displaystyle R_{n}^{m}(\rho )}$ 

${\displaystyle \rho ^{j}=\sum _{n\equiv m{\pmod {2}}}^{j}h_{j,n,m}R_{n}^{m}(\rho )}$ 

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

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

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

The factor ${\displaystyle \rho ^{n-2k}}$  in the radial polynomial ${\displaystyle R_{n}^{m}(\rho )}$  may be expanded in a Bernstein 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 ${\displaystyle \lfloor n/2\rfloor -k\leq s\leq \lfloor n/2\rfloor }$ . The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:

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

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 ${\displaystyle Z_{n}^{l}\rightarrow Z_{j}}$  starts as follows (sequence A176988 in the OEIS). ${\displaystyle j={\frac {n(n+1)}{2}}+|l|+\left\{{\begin{array}{ll}0,&l>0\land n\equiv \{0,1\}{\pmod {4}};\\0,&l<0\land n\equiv \{2,3\}{\pmod {4}};\\1,&l\geq 0\land n\equiv \{2,3\}{\pmod {4}};\\1,&l\leq 0\land n\equiv \{0,1\}{\pmod {4}}.\end{array}}\right.}$ 

The rule is the following.

• The even Zernike polynomials Z (with even azimuthal parts ${\displaystyle \cos(m\varphi )}$ , where ${\displaystyle m=l}$  as ${\displaystyle l}$  is a positive number) obtain even indices j.
• The odd Z obtains (with odd azimuthal parts ${\displaystyle \sin(m\varphi )}$ , where ${\displaystyle m=\left\vert l\right\vert }$  as ${\displaystyle l}$  is a negative number) odd indices j.
• Within a given n, a lower ${\displaystyle \left\vert l\right\vert }$  results in a lower j.

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

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

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

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

where ${\displaystyle \operatorname {sgn} l}$  is the sign or signum function. The first 20 fringe numbers are listed below.

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

They satisfy the Rodrigues' formula

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

and can be related to the Jacobi polynomials as

${\displaystyle Z_{n}^{m}(x)=x^{m}{\frac {P_{\frac {n-m}{2}}^{(0,m)}\left(2x^{2}-1\right)}{P_{\frac {n-m}{2}}^{(0,m)}(1)}}}$ .

The orthogonality in the radial part reads^[9]

${\displaystyle \int _{0}^{1}{\sqrt {2n+2}}R_{n}^{m}(\rho )\,{\sqrt {2n'+2}}R_{n'}^{m}(\rho )\,\rho d\rho =\delta _{n,n'}}$ 

or

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

Orthogonality in the angular part is represented by the elementary

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

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

${\displaystyle \int _{0}^{2\pi }\cos(m\varphi )\sin(m'\varphi )\,d\varphi =0,}$ 

where ${\displaystyle \epsilon _{m}}$  (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 if ${\displaystyle m=0}$  and 1 if ${\displaystyle m\neq 0}$ . The product of the angular and radial parts establishes the orthogonality of the Zernike functions with respect to both indices if integrated over the unit disk,

${\displaystyle \int Z_{n}^{l}(\rho ,\varphi )Z_{n'}^{l'}(\rho ,\varphi )\,d^{2}r={\frac {\epsilon _{l}\pi }{2n+2}}\delta _{n,n'}\delta _{l,l'},}$ 

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

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

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

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

${\displaystyle \langle F,G\rangle :=\int F(\rho ,\varphi )G(\rho ,\varphi )\rho d\rho d\varphi .}$ 

The Zernike coefficients can then be expressed as follows:

${\displaystyle {\begin{aligned}a_{m,n}&={\frac {2n+2}{\epsilon _{m}\pi }}\left\langle G(\rho ,\varphi ),Z_{n}^{m}(\rho ,\varphi )\right\rangle ,\\b_{m,n}&={\frac {2n+2}{\epsilon _{m}\pi }}\left\langle G(\rho ,\varphi ),Z_{n}^{-m}(\rho ,\varphi )\right\rangle .\end{aligned}}}$ 

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

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

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=Z_{n}^{l}(\rho ,-\varphi )}$  for l ≥ 0,

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=-Z_{n}^{l}(\rho ,-\varphi )}$  for l < 0.

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

${\displaystyle Z_{n}^{l}(\rho ,\varphi )=(-1)^{l}Z_{n}^{l}(\rho ,\varphi +\pi ),}$ 

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. (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 ${\displaystyle Z_{n}^{l}(\rho ,\varphi )=Z_{n}^{l}(\rho ,\varphi +\pi ).}$ 
• Angularly odd Zernike polynomials: Zernike polynomials with odd l so that ${\displaystyle Z_{n}^{l}(\rho ,\varphi )=-Z_{n}^{l}(\rho ,\varphi +\pi ).}$ 

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

${\displaystyle R_{n}^{m}(\rho )=(-1)^{n}R_{n}^{m}(-\rho )=(-1)^{m}R_{n}^{m}(-\rho ).}$ 

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

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

${\displaystyle Z_{n}^{l}\left(\rho ,\varphi +{\tfrac {2\pi k}{l}}\right)=Z_{n}^{l}(\rho ,\varphi ),\qquad k=0,\pm 1,\pm 2,\cdots .}$ 

The Zernike polynomials satisfy the following recurrence relation which depends neither on the degree nor on the azimuthal order of the radial polynomials:^[10]

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

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^[11] 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 above relation is especially useful since the derivative of ${\displaystyle R_{n}^{m}}$  can be calculated from two radial Zernike polynomials of adjacent degree:^[11]

${\displaystyle {\frac {\operatorname {d} }{\operatorname {d} \!\rho }}R_{n}^{m}(\rho )={\frac {(2nm(\rho ^{2}-1)+(n-m)(m+n(2\rho ^{2}-1)))R_{n}^{m}(\rho )-(n+m)(n-m)R_{n-2}^{m}(\rho )}{2n\rho (\rho ^{2}-1)}}{\text{ .}}}$ 

The differential equation of the Gaussian Hypergeometric Function is equivalent to

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

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, are shown below. They are normalized such that: ${\displaystyle \int _{0}^{2\pi }\int _{0}^{1}Z^{2}\cdot \rho \,d\rho \,d\phi =\pi }$ , which is equivalent to ${\displaystyle \operatorname {Var} (Z)_{\text{unit circle}}=1}$ .

${\displaystyle Z_{n}^{l}}$	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	${\displaystyle 2\rho \sin \phi }$	Tilt (Y-Tilt, vertical tilt)
${\displaystyle Z_{1}^{1}}$	02	02	01	02	1	+1	${\displaystyle 2\rho \cos \phi }$	Tilt (X-Tilt, horizontal tilt)
${\displaystyle Z_{2}^{-2}}$	03	05	05	06	2	−2	${\displaystyle {\sqrt {6}}\rho ^{2}\sin 2\phi }$	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	${\displaystyle {\sqrt {6}}\rho ^{2}\cos 2\phi }$	Vertical astigmatism
${\displaystyle Z_{3}^{-3}}$	06	09	10	11	3	−3	${\displaystyle {\sqrt {8}}\rho ^{3}\sin 3\phi }$	Vertical trefoil
${\displaystyle Z_{3}^{-1}}$	07	07	07	08	3	−1	${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\sin \phi }$	Vertical coma
${\displaystyle Z_{3}^{1}}$	08	08	06	07	3	+1	${\displaystyle {\sqrt {8}}(3\rho ^{3}-2\rho )\cos \phi }$	Horizontal coma
${\displaystyle Z_{3}^{3}}$	09	10	09	10	3	+3	${\displaystyle {\sqrt {8}}\rho ^{3}\cos 3\phi }$	Oblique trefoil
${\displaystyle Z_{4}^{-4}}$	10	15	17	18	4	−4	${\displaystyle {\sqrt {10}}\rho ^{4}\sin 4\phi }$	Oblique quadrafoil
${\displaystyle Z_{4}^{-2}}$	11	13	12	13	4	−2	${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\sin 2\phi }$	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	${\displaystyle {\sqrt {10}}(4\rho ^{4}-3\rho ^{2})\cos 2\phi }$	Vertical secondary astigmatism
${\displaystyle Z_{4}^{4}}$	14	14	16	17	4	+4	${\displaystyle {\sqrt {10}}\rho ^{4}\cos 4\phi }$	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-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 ${\displaystyle \rho \approx 1}$ , 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]

The concept translates to higher dimensions D if multinomials ${\displaystyle x_{1}^{i}x_{2}^{j}\cdots x_{D}^{k}}$  in Cartesian coordinates are converted to hyperspherical coordinates, ${\displaystyle \rho ^{s},s\leq D}$ , multiplied by a product of Jacobi polynomials of the angular variables. In ${\displaystyle D=3}$  dimensions, the angular variables are spherical harmonics, for example. Linear combinations of the powers ${\displaystyle \rho ^{s}}$  define an orthogonal basis ${\displaystyle R_{n}^{(l)}(\rho )}$  satisfying

${\displaystyle \int _{0}^{1}\rho ^{D-1}R_{n}^{(l)}(\rho )R_{n'}^{(l)}(\rho )d\rho =\delta _{n,n'}}$ .

(Note that a factor ${\displaystyle {\sqrt {2n+D}}}$  is absorbed in the definition of R 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 the orthogonalization is involved.) The explicit representation is^[3]

${\displaystyle {\begin{aligned}R_{n}^{(l)}(\rho )&={\sqrt {2n+D}}\sum _{s=0}^{\tfrac {n-l}{2}}(-1)^{s}{{\tfrac {n-l}{2}} \choose s}{n-s-1+{\tfrac {D}{2}} \choose {\tfrac {n-l}{2}}}\rho ^{n-2s}\\&=(-1)^{\tfrac {n-l}{2}}{\sqrt {2n+D}}\sum _{s=0}^{\tfrac {n-l}{2}}(-1)^{s}{{\tfrac {n-l}{2}} \choose s}{s-1+{\tfrac {n+l+D}{2}} \choose {\tfrac {n-l}{2}}}\rho ^{2s+l}\\&=(-1)^{\tfrac {n-l}{2}}{\sqrt {2n+D}}{{\tfrac {n+l+D}{2}}-1 \choose {\tfrac {n-l}{2}}}\rho ^{l}\ {}_{2}F_{1}\left(-{\tfrac {n-l}{2}},{\tfrac {n+l+D}{2}};l+{\tfrac {D}{2}};\rho ^{2}\right)\end{aligned}}}$ 

for even ${\displaystyle n-l\geq 0}$ , else identical to zero.

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

• Weisstein, Eric W. "Zernike Polynomial". MathWorld.
• Andersen, Torben B. (2018). "Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates". Opt. Express. 26 (15): 18878–18896. Bibcode:2018OExpr..2618878A. doi:10.1364/OE.26.018878. PMID 30114148.
• Bhatia, A. B.; Wolf, E. (1952). "The Zernike circle polynomials occurring in diffraction theory". Proc. Phys. Soc. B. 65 (11): 909–910. Bibcode:1952PPSB...65..909B. doi:10.1088/0370-1301/65/11/112.
• Callahan, P. G.; De Graef, M. (2012). "Precipitate shape fitting and reconstruction by means of 3D Zernike functions". Modelling and Simulation in Materials Science and Engineering. 20 (1): 015003. Bibcode:2012MSMSE..20a5003C. doi:10.1088/0965-0393/20/1/015003. S2CID 121700658.
• Campbell, C. E. (2003). "Matrix method to find a new set of Zernike coefficients form an original set when the aperture radius is changed". J. Opt. Soc. Am. A. 20 (2): 209–217. Bibcode:2003JOSAA..20..209C. doi:10.1364/JOSAA.20.000209. PMID 12570287.
• Cerjan, C. (2007). "The Zernike-Bessel representation and its application to Hankel transforms". J. Opt. Soc. Am. A. 24 (6): 1609–16. Bibcode:2007JOSAA..24.1609C. doi:10.1364/JOSAA.24.001609. PMID 17491628.
• Comastri, S. A.; Perez, L. I.; Perez, G. D.; Martin, G.; Bastida Cerjan, K. (2007). "Zernike expansion coefficients: rescaling and decentering for different pupils and evaluation of corneal aberrations". J. Opt. Soc. Am. A. 9 (3): 209–221. Bibcode:2007JOptA...9..209C. doi:10.1088/1464-4258/9/3/001.
• Conforti, G. (1983). "Zernike aberration coefficients from Seidel and higher-order power-series coefficients". Opt. Lett. 8 (7): 407–408. Bibcode:1983OptL....8..407C. doi:10.1364/OL.8.000407. PMID 19718130.
• Dai, G-m.; Mahajan, V. N. (2007). "Zernike annular polynomials and atmospheric turbulence". J. Opt. Soc. Am. A. 24 (1): 139–155. Bibcode:2007JOSAA..24..139D. doi:10.1364/JOSAA.24.000139. PMID 17164852.
• Dai, G-m. (2006). "Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula". J. Opt. Soc. Am. A. 23 (3): 539–543. Bibcode:2006JOSAA..23..539D. doi:10.1364/JOSAA.23.000539. PMID 16539048.
• Díaz, J. A.; Fernández-Dorado, J.; Pizarro, C.; Arasa, J. (2009). "Zernike Coefficients for Concentric, Circular, Scaled Pupils: An Equivalent Expression". Journal of Modern Optics. 56 (1): 149–155. Bibcode:2009JMOp...56..131D. doi:10.1080/09500340802531224. S2CID 122620015.
• Díaz, J. A.; Fernández-Dorado, J. "Zernike Coefficients for Concentric, Circular, Scaled Pupils". from The Wolfram Demonstrations Project.
• Farokhi, Sajad; Shamsuddin, Siti Mariyam; Flusser, Jan; Sheikh, U.U.; Khansari, Mohammad; Jafari-Khouzani, Kourosh (2013). "Rotation and noise invariant near-infrared face recognition by means of Zernike moments and spectral regression discriminant analysis". Journal of Electronic Imaging. 22 (1): 013030. Bibcode:2013JEI....22a3030F. doi:10.1117/1.JEI.22.1.013030. S2CID 16758261.
• Gu, J.; Shu, H. Z.; Toumoulin, C.; Luo, L. M. (2002). "A novel algorithm for fast computation of Zernike moments". Pattern Recognition. 35 (12): 2905–2911. Bibcode:2002PatRe..35.2905G. doi:10.1016/S0031-3203(01)00194-7.
• Herrmann, J. (1981). "Cross coupling and aliasing in modal wave-front estimation". J. Opt. Soc. Am. 71 (8): 989. Bibcode:1981JOSA...71..989H. doi:10.1364/JOSA.71.000989.
• Hu, P. H.; Stone, J.; Stanley, T. (1989). "Application of Zernike polynomials to atmospheric propagation problems". J. Opt. Soc. Am. A. 6 (10): 1595. Bibcode:1989JOSAA...6.1595H. doi:10.1364/JOSAA.6.001595.
• Kintner, E. C. (1976). "On the mathematical properties of the Zernike Polynomials". Opt. Acta. 23 (8): 679–680. Bibcode:1976AcOpt..23..679K. doi:10.1080/713819334.
• Lawrence, G. N.; Chow, W. W. (1984). "Wave-front tomography by Zernike Polynomial decomposition". Opt. Lett. 9 (7): 267–269. Bibcode:1984OptL....9..267L. doi:10.1364/OL.9.000267. PMID 19721566.
• Liu, Haiguang; Morris, Richard J.; Hexemer, A.; Grandison, Scott; Zwart, Peter H. (2012). "Computation of small-angle scattering profiles with three-dimensional Zernike polynomials". Acta Crystallogr. A. 68 (2): 278–285. doi:10.1107/S010876731104788X. PMID 22338662.
• Lundström, L.; Unsbo, P. (2007). "Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils". J. Opt. Soc. Am. A. 24 (3): 569–77. Bibcode:2007JOSAA..24..569L. doi:10.1364/JOSAA.24.000569. PMID 17301846.
• Mahajan, V. N. (1981). "Zernike annular polynomials for imaging systems with annular pupils". J. Opt. Soc. Am. 71: 75. Bibcode:1981JOSA...71...75M. doi:10.1364/JOSA.71.000075.
• Prata Jr, A.; Rusch, W. V. T. (1989). "Algorithm for computation of Zernike polynomials expansion coefficients". Appl. Opt. 28 (4): 749–54. Bibcode:1989ApOpt..28..749P. doi:10.1364/AO.28.000749. PMID 20548554.
• Schwiegerling, J. (2002). "Scaling Zernike expansion coefficients to different pupil sizes". J. Opt. Soc. Am. A. 19 (10): 1937–45. Bibcode:2002JOSAA..19.1937S. doi:10.1364/JOSAA.19.001937. PMID 12365613.
• Sheppard, C. J. R.; Campbell, S.; Hirschhorn, M. D. (2004). "Zernike expansion of separable functions in Cartesian coordinates". Appl. Opt. 43 (20): 3963–6. Bibcode:2004ApOpt..43.3963S. doi:10.1364/AO.43.003963. PMID 15285082.
• Shu, H.; Luo, L.; Han, G.; Coatrieux, J.-L. (2006). "General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes". J. Opt. Soc. Am. A. 23 (8): 1960–1966. Bibcode:2006JOSAA..23.1960S. doi:10.1364/JOSAA.23.001960. PMC 1961626. PMID 16835654.
• Swantner, W.; Chow, W. W. (1994). "Gram-Schmidt orthogonalization of Zernike polynomials for general aperture shapes". Appl. Opt. 33 (10): 1832–7. Bibcode:1994ApOpt..33.1832S. doi:10.1364/AO.33.001832. PMID 20885515.
• Tango, W. J. (1977). "The circle polynomials of Zernike and their application in optics". Appl. Phys. A. 13 (4): 327–332. Bibcode:1977ApPhy..13..327T. doi:10.1007/BF00882606. S2CID 120469275.
• Tyson, R. K. (1982). "Conversion of Zernike aberration coefficients to Seidel and higher-order power series aberration coefficients". Opt. Lett. 7 (6): 262–264. Bibcode:1982OptL....7..262T. doi:10.1364/OL.7.000262. PMID 19710893.
• Wang, J. Y.; Silva, D. E. (1980). "Wave-front interpretation with Zernike Polynomials". Appl. Opt. 19 (9): 1510–8. Bibcode:1980ApOpt..19.1510W. doi:10.1364/AO.19.001510. PMID 20221066.
• Barakat, R. (1980). "Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials". J. Opt. Soc. Am. 70 (6): 739. Bibcode:1980JOSA...70..739B. doi:10.1364/JOSA.70.000739.
• ten Brummelaar, T. A. (1996). "Modeling atmospheric wave aberrations and astronomical instrumentation using the polynomials of Zernike". Opt. Commun. 132 (3–4): 329–342. Bibcode:1996OptCo.132..329T. doi:10.1016/0030-4018(96)00407-5.
• Novotni, M.; Klein, R. (2003). "3D zernike descriptors for content based shape retrieval". Proceedings of the eighth ACM symposium on Solid modeling and applications (PDF). pp. 216–225. CiteSeerX 10.1.1.14.4970. doi:10.1145/781606.781639. ISBN 978-1581137064. S2CID 10514681.
• Novotni, M.; Klein, R. (2004). "Shape retrieval using 3D Zernike descriptors" (PDF). Computer-Aided Design. 36 (11): 1047–1062. CiteSeerX 10.1.1.71.8238. doi:10.1016/j.cad.2004.01.005.
• Farokhi, Sajad; Shamsuddin, Siti Mariyam; Sheikh, U.U.; Flusser, Jan (2014). "Near Infrared Face Recognition: A Comparison of Moment-Based Approaches". The 8th International Conference on Robotic, Vision, Signal Processing & Power Applications. Lecture Notes in Electrical Engineering. Vol. 291. pp. 129–135. doi:10.1007/978-981-4585-42-2_15. ISBN 978-981-4585-41-5.
• Farokhi, Sajad; Shamsuddin, Siti Mariyam; Flusser, Jan; Sheikh, U.U.; Khansari, Mohammad; Jafari-Khouzani, Kourosh (2014). "Near infrared face recognition by combining Zernike moments and undecimated discrete wavelet transform". Digital Signal Processing. 31 (1): 13–27. doi:10.1016/j.dsp.2014.04.008.

• 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)