Zernike's Orthogonal Circle Polynomials

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.

Definitions

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

:$Z^_n(\rho,\varphi) = R^m_n(\rho)\,\cos(m\,\varphi) \!$

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

:$Z^_n(\rho,\varphi) = R^m_n(\rho)\,\sin(m\,\varphi), \!$

(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. $, Z^_n(\rho,\varphi), \le 1$. The radial polynomials $R^m_n$ are defined as

:$R^m_n(\rho) = \sum_^ \frac \;\rho^$

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

:$R_n^m(1)=1.$

Other representations

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

:$R_n^m(\rho)=\sum_^(-1)^k \binom \binom \rho^$.

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.:

:$\begin R_n^m(\rho) &= (-1)^ \rho^mP_^(1-2\rho^2) \\ &= \binom\rho^n \ _2F_\left(-\tfrac,-\tfrac;-n;\rho^\right) \\ &= (-1)^\binom\rho^m \ _2F_\left(1+\tfrac,-\tfrac;1+m;\rho^2\right) \end$

for ''n'' − ''m'' even.

The inverse relation expands $\rho ^j$ for fixed $m\le j$ into $R_n^m(\rho)$
:$\rho^j = \sum_^j h_R_n^m(\rho)$
with rational coefficients $h_$
:$h_= \frac \frac$
for even $j-m=0,2,4,\ldots$.

The factor $\rho^$ in the radial polynomial $R_n^m(\rho)$ may be expanded in a Bernstein basis of $b_(\rho^2)$ for even $n$ or $\rho$ times a function of $b_(\rho^2)$ 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:
:$R_n^m(\rho) = \frac \rho^ \sum_^ (-1)^ \binom\binom b_(\rho^2).$

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 ''m'' to a single index ''j'' has been introduced by Noll. The table of this association $Z_n^m \rightarrow Z_j$ starts as follows .
$j = \frac+, m, +\left\{\begin{array}{ll} 0, & m>0 \land n \equiv \{0,1\} \pmod 4;\\ 0, & m<0 \land n \equiv \{2,3\} \pmod 4;\\ 1, & m \ge 0 \land n \equiv \{2,3\} \pmod 4;\\ 1, & m \le 0 \land n \equiv \{0,1\} \pmod 4. \end{array}\right.$
{, class="wikitable"
!n,m
0,01,1 1,−1 2,0 2,−2 2,23,−1 3,1 3,−3 3,3
, -------
! j
12 3 4 5 6 7 8 9 10
, -----
!n,m
4,0 4,2 4,−24,44,−45,15,−15,3 5,−35,5
, -----
! j
11 12 13 141516 17 18 19 20

The rule is the following.

* The even Zernike polynomials ''Z'' with $m>0$ obtain even indices ''j.''
* The odd ''Z'' where $m< 0$odd indices ''j''.
* Within a given ''n'', a lower $\left\vert m \right\vert$ results in a lower ''j''.

OSA/ANSI standard indices

OSA
and ANSI single-index Zernike polynomials using:

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

{, class="wikitable"
!n,l
0,01,−1 1,1 2,−2 2,0 2,23,−3 3,−1 3,1 3,3
, -------
! j
0 1 2 3 4 5 6 7 8 9
, -----
!n,l
4,−4 4,−2 4,04,24,45,−55,−35,−1 5,15,3
, -----
! j
10 11 12 13 14 15 16 17 18 19

Fringe/University of Arizona indices

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.

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

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

{, class="wikitable"
!n,l
0,01,1 1,−1 2,0 2,2 2,−23,1 3,−1 4,0 3,3
, -------
! j
12 3 4 5 6 7 8 9 10
, -----
!n,l
3,−3 4,2 4,−25,15,−16,04,44,−4 5,35,−3
, -----
! j
11 12 13 141516 17 18 19 20

Wyant indices

James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1). 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_{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/math>
and can be related to the Jacobi polynomials as
:$Z_{n}^{m}(x)=x^m\frac{P_{\frac{n-m}{2^{(0,m)}\left( 2x^2-1 \right)}{P_{\frac{n-m}{2^{(0,m)}(1)}$.

Properties

Orthogonality

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

or

$\underset{0}{\overset{1}{\mathop \int \,R_{n}^{m}(\rho )R_{\operatorname{d}\! \rho} R_n^m(\rho) = \frac{(2 n m (\rho^2 - 1) + (n-m)(m + n(2\rho^2 - 1))) R_n^m(\rho) - (n+m)(n-m) R_{n-2}^m(\rho)}{2 n \rho (\rho^2 - 1)} \text{ .}$

The differential equation of the Gaussian Hypergeometric Function is equivalent to
:
\rho^2(\rho^2-1) \frac{d^2}{d\rho^2} R_n^m(\rho) = (n+2)\rho^2-m^2_n^m(\rho)+\rho(1-3\rho^2)\frac{d}{d\rho} R_n^m(\rho).

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: $\int_0^{2\pi} \int_0^1 Z^2\cdot\rho\,d\rho\,d\phi = \pi$, which is equivalent to $\operatorname{Var}(Z)_\text{unit circle} = 1$.
{, class="wikitable sortable"
, -
! $Z_n^l$, , 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
, -
, $Z_2^0$ , , 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
, -
, $Z_3^{-1}$ , , 7 , , 7 , , 7 , , 8 , , 3 , , −1 , , $\sqrt{8} (3 \rho^3 - 2\rho) \sin \phi$ , , Vertical coma
, -
, $Z_3^1$ , , 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
, -
, $Z_4^{-2}$ , , 11 , , 13 , , 12 , , 13 , , 4 , , −2 , , $\sqrt{10} (4 \rho^4 - 3\rho^2) \sin 2 \phi$ , , Oblique secondary astigmatism
, -
, $Z_4^0$ , , 12 , , 11 , , 8 , , 9 , , 4 , , 0 , , $\sqrt{5} (6 \rho^4 - 6 \rho^2 +1)$ , , Primary spherical
, -
, $Z_4^2$ , , 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. 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 $\rho\approx 1$, which often leads attempts to define other orthogonal functions over the circular disk.

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.
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. 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 or the surface of vibrating disks. Zernike Moments also have been used to quantify shape of osteosarcoma cancer cell lines in single cell level. 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.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.

Higher dimensions

The concept translates to higher dimensions ''D'' if multinomials $x_1^ix_2^j\cdots x_D^k$ 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 $R_n^{(l)}(\rho)$ satisfying

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

(Note that a factor $\sqrt{2n+D}$ 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

:$\begin{align} R_n^{(l)}(\rho) &= \sqrt{2n+D}\sum_{s=0}^{\tfrac{n-l}{2 (-1)^s {\tfrac{n-l}{2} \choose s}{n-s-1+\tfrac{D}{2} \choose \tfrac{n-l}{2\rho^{n-2s} \\ &=(-1)^{\tfrac{n-l}{2 \sqrt{2n+D} \sum_{s=0}^{\tfrac{n-l}{2 (-1)^s {\tfrac{n-l}{2} \choose s} {s-1+\tfrac{n+l+D}{2} \choose \tfrac{n-l}{2 \rho^{2s+l} \\ &=(-1)^{\tfrac{n-l}{2 \sqrt{2n+D} {\tfrac{n+l+D}{2}-1 \choose \tfrac{n-l}{2 \rho^l \ {}_2F_1 \left ( -\tfrac{n-l}{2},\tfrac{n+l+D}{2}; l+\tfrac{D}{2}; \rho^2 \right ) \end{align}$

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

Its differential equation for the Gaussian Hypergeometric Function is equivalent to

\rho^2(\rho^2-1)\frac{d^2}{d\rho^2}R_n^{(l)}(\rho)
=
\left \rho^2(n+D)-l(D-2+l)\right_n^{(l)}(\rho)
+
\rho\left -1-(D+1)\rho^2\right\frac{d}{d\rho}R_n^{(l)}(\rho).

See also

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

References

*
*
*
*
*
*
*
*
*
*
*
* from The Wolfram Demonstrations Project.
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* {{cite journal
, first1=Sajad
, last1=Farokhi
, first2=Siti Mariyam
, last2=Shamsuddin
, first3=Jan
, last3=Flusser
, first4=U.U.
, last4=Sheikh
, first5=Mohammad
, last5=Khansari
, first6=Kourosh
, last6=Jafari-Khouzani
, title=Near infrared face recognition by combining Zernike moments and undecimated discrete wavelet transform
, journal=Digital Signal Processing
, volume=31
, year=2014
, issue=1
, doi=10.1016/j.dsp.2014.04.008
, pages=13–27

External links

The Extended Nijboer-Zernike website

MATLAB code for fast calculation of Zernike moments

Python/NumPy library for calculating Zernike polynomials


a
Telescope Optics

Example: using WolframAlpha to plot Zernike Polynomials

orthopy, a Python package computing orthogonal polynomials (including Zernike polynomials)

Orthogonal polynomials