Zernike's circle polynomials
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In mathematics, the Zernike polynomials are a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s that are orthogonal on the unit disk. Named after optical physicist
Frits Zernike Frits Zernike (; 16 July 1888 – 10 March 1966) was a Dutch physicist and winner of the Nobel Prize in Physics in 1953 for his invention of the phase-contrast microscope. Early life and education Frits Zernike was born on 16 July 1888 in Am ...
, winner of the 1953
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
in Physics and the inventor of
phase-contrast microscopy __NOTOC__ Phase-contrast microscopy (PCM) is an optical microscopy technique that converts phase shifts in light passing through a transparent specimen to brightness changes in the image. Phase shifts themselves are invisible, but become visible ...
, they play important roles in various optics branches such as beam
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s with ''n ≥ m ≥ 0'' (''m'' = 0 for even Zernike polynomials), ''\varphi'' is the
azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...
al
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
, ''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 an even number of ''n'' − ''m'', while it is 0 for an odd number of ''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 In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...
, to write down the differential equations, etc.: :\begin R_n^m(\rho) &= \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 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 ''l'' to a single index ''j'' has been introduced by Noll. The table of this association Z_n^l \rightarrow Z_j starts as follows . j = \frac+, 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. {, class="wikitable" !n,l 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,l 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 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 and
ANSI The American National Standards Institute (ANSI ) is a private non-profit organization that oversees the development of voluntary consensus standards for products, services, processes, systems, and personnel in the United States. The organi ...
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, + \frac{1-\sgn l}{2} 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.


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{ .}


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 The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...
) , - , 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 In optics, defocus is the aberration in optical systems, aberration in which an image is simply out of focus (optics), focus. This aberration is familiar to anyone who has used a camera, videocamera, microscope, telescope, or binoculars. Opti ...
(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 Ophthalmology ( ) is a surgical subspecialty within medicine that deals with the diagnosis and treatment of eye disorders. An ophthalmologist is a physician who undergoes subspecialty training in medical and surgical eye care. Following a medic ...
, Zernike polynomials are used to describe wavefront aberrations of the
cornea The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Along with the anterior chamber and lens, the cornea refracts light, accounting for approximately two-thirds of the eye's total optical ...
or
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements ...
from an ideal spherical shape, which result in
refraction error Refractive error, also known as refraction error, is a problem with focusing light accurately on the retina due to the shape of the eye and or cornea. The most common types of refractive error are near-sightedness, far-sightedness, astigmatism, ...
s. They are also commonly used in
adaptive optics Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effect of incoming wavefront distortions by deforming a mirror in order to compensate for the distortion. It is used in astronomical tele ...
, where they can be used to characterize
atmospheric distortion An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A ...
. 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 Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
of the object in a
region of interest A region of interest (often abbreviated ROI) is a sample within a data set identified for a particular purpose. The concept of a ROI is commonly used in many application areas. For example, in medical imaging, the boundaries of a tumor may be def ...
(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 In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
, 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.


See also

*
Jacobi polynomials In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...
* 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