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, winner 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
(even function over the azimuthal angle formula_2), and the odd Zernike polynomials are defined as
(odd function over the azimuthal angle formula_2) where "m" and "n" are nonnegative integers with "n ≥ m ≥ 0" ("m" = 0 for even Zernike polynomials), "formula_2" is the azimuthal angle, "ρ" is the radial distance formula_6, and formula_7 are the radial polynomials defined below. Zernike polynomials have the property of being limited to a range of −1 to +1, i.e. formula_8. The radial polynomials formula_7 are defined as
for an even number of "n" − "m", while it is 0 for an odd number of "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 factor formula_14 in the radial polynomial formula_15 may be expanded in a Bernstein basis of formula_16 for even formula_17 or formula_18 times a function of formula_19 for odd formula_17 in the range formula_21. The radial polynomial may therefore be expressed by a finite number of Bernstein Polynomials with rational coefficients:
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 formula_23 starts as follows .
formula_24
The rule is the following.
OSA/ANSI standard indices.
OSA
 and ANSI single-index Zernike polynomials using:
Fringe/University of Arizona indices.
The Fringe indexing scheme is used in commercial optical design software and optical testing in, e.g., photolithography.
formula_33
where formula_34 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). 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
or
formula_36
Orthogonality in the angular part is represented by the elementary
where formula_40 (sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as "2" if formula_41 and "1" if formula_42. 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 formula_44 is the Jacobian of the circular coordinate system, and where formula_45 and formula_46 are both even.
Zernike transform.
Any sufficiently smooth real-valued phase field over the unit disk formula_47 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
where the coefficients can be calculated using inner products. On the space of formula_49 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 (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
The "π" shifts of trigonometric functions result that the parity with respect to point reflection at the center of coordinates is
where formula_55 could as well be written formula_56 because formula_45 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.)
The radial polynomials are also either even or odd, depending on order "n" or "m":
These equalities are easily seen since formula_15 with an odd (even) "m" contains only odd (even) powers to "ρ" (see examples of formula_15 below).
The periodicity of the trigonometric functions results in invariance if rotated by multiples of formula_63 radian around the center:
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:
From the definition of formula_66 it can be seen that formula_67 and formula_68. The following three-term recurrence relation then allows to calculate all other formula_15:
The above relation is especially useful since the derivative of formula_66 can be calculated from two radial Zernike polynomials of adjacent degree:
Examples.
Radial polynomials.
The first few radial polynomials are:
Zernike polynomials.
The first few Zernike modes, with OSA/ANSI and Noll single-indices, are shown below. They are normalized such that: formula_89.
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 formula_90, 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.
Higher dimensions.
The concept translates to higher dimensions "D" if multinomials formula_91 in Cartesian coordinates are converted to hyperspherical coordinates, formula_92, multiplied by a product of Jacobi polynomials of the angular variables. In formula_93 dimensions, the angular variables are spherical harmonics, for example. Linear combinations of the powers formula_94 define an orthogonal basis formula_95 satisfying
(Note that a factor formula_97 is absorbed in the definition of "R" here, whereas in formula_98 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
for even formula_100, else identical to zero.