Zernike 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
, and the odd Zernike polynomials are defined as
where m and n are nonnegative integers with n ≥ m ≥ 0, is the azimuthal angle, ρ is the radial distance, and 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. if. The radial polynomials are defined as
for even n − m, while it is 0 for odd n − m. A special value is

Other representations

Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers:
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.:
for n − m even.
The inverse relation expands for fixed into
with rational coefficients
for even.
The factor in the radial polynomial may be expanded in a Bernstein basis of for even or times a function of for odd in the range. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:

Rodrigues Formula

The radial polynomials satisfy the Rodrigues' formula

Properties

Orthogonality

The orthogonality in the radial part reads
or
Orthogonality in the angular part is represented by the elementary
where is defined as 2 if and 1 if. 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,
where is the Jacobian of the circular coordinate system, and where and are both even.

Zernike transform

Any sufficiently smooth real-valued phase field over the unit disk can be represented in terms of its Zernike coefficients, just as periodic functions find an orthogonal representation with the Fourier series. We have
where the coefficients can be calculated using inner products. On the space of functions on the unit disk, there is an inner product defined by
The Zernike coefficients can then be expressed as follows:
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 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
The π shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates is
where could as well be written because as even numbers are only cases to get non-vanishing Zernike polynomials.
This property is sometimes used to categorize Zernike polynomials into even and odd polynomials in terms of their angular dependence.

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

The radial polynomials are also either even or odd, depending on the order n or the azimuthal index m:
These equalities are easily seen since with an odd m contains only odd powers to ρ.
The periodicity of the trigonometric functions results in invariance if rotated by multiples of radian around the center:

As eigenfunctions of a differential operator

The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation
self-adjoint over the unit disk, with negative eigenvalues. Other self-adjoint differential operators can be constructed for which the Zernike polynomials form a spectrum, for example, which differs from the above by a factor.

Recurrence relations

The Zernike polynomials satisfy the following recurrence relation:
From the definition of it can be seen that and. The following three-term recurrence relation
then allows to calculate all other :
The main use of these recurrences is to avoid cancellation of digits that occurs for large in the accumulation of the oscillatory binomial terms in the power series notation
The above relation is also useful since the derivative of can be calculated from two radial Zernike polynomials of adjacent degree:
The differential equation of the Gaussian Hypergeometric Function is equivalent to

Nomenclature

Noll's sequential indices

Applications often involve linear 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 indices n and m to a single index j has been introduced by Noll.

The table of this association starts as follows.

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 obtain even indices j.
The odd Z where obtain odd indices j.
Within a given n, a lower results in a lower j.

OSA/ANSI standard indices

OSA
and ANSI single-index Zernike polynomials using:

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

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

Fringe/University of Arizona indices

The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.
where is the sign or signum function. The first 20 fringe numbers are listed below.

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. This method is commonly used including interferogram analysis software in Zygo interferometers and the open source software DFTFringe.

Examples

Radial polynomials

The first few radial polynomials are:

Zernike polynomials

The first few Zernike modes, at various indices, are shown below. In this table they are normalized differently than in the remaining sections:, which is equivalent to.

OSA/ANSI index	Noll index	Wyant index	Fringe/UA index	Radial degree	Azimuthal degree	Classical name
0	1	0	1	0	0	Piston
1	3	2	3	1	−1	Tilt
2	2	1	2	1	+1	Tilt
3	5	5	6	2	−2	Oblique astigmatism
4	4	3	4	2	0	Defocus
5	6	4	5	2	+2	Vertical astigmatism
6	9	10	11	3	−3	Vertical trefoil
7	7	7	8	3	−1	Vertical coma
8	8	6	7	3	+1	Horizontal coma
9	10	9	10	3	+3	Oblique trefoil
10	15	17	18	4	−4	Oblique quadrafoil
11	13	12	13	4	−2	Oblique secondary astigmatism
12	11	8	9	4	0	Primary spherical
13	12	11	12	4	+2	Vertical secondary astigmatism
14	14	16	17	4	+4	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, 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, 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.

Higher dimensions

The concept translates to higher dimensions D if multinomials in Cartesian coordinates are converted to hyperspherical coordinates,, multiplied by a product of Jacobi polynomials of the angular variables. In dimensions, the angular variables are spherical harmonics, for example. Linear combinations of the powers define an orthogonal basis satisfying
The explicit representation is
for even, else identical to zero, with special case
Its differential equation for the Gaussian Hypergeometric Function is equivalent to
Kintner's recurrence for fixed and variable
quoted for above is in the general form
For this was proposed by Deng and Gwo.
For fixed and variable the recurrence is
The case for was published by Chong et al.