Zernike polynomials
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 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.
James C. Wyant uses the "Fringe" indexing scheme except it starts at 0 instead of 1 (subtract 1).
They satisfy the Rodrigues' formula and can be related to the Jacobi polynomials as The orthogonality in the radial part reads[9] or
(sometimes called the Neumann factor because it frequently appears in conjunction with Bessel functions) is defined as 2 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
Any sufficiently smooth real-valued phase field over the unit disk
can be represented in terms of its Zernike coefficients (odd and even), just as periodic functions find an orthogonal representation with the Fourier series.
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.
The radial polynomials are also either even or odd, depending on order n or m: These equalities are easily seen since
The periodicity of the trigonometric functions results in invariance if rotated by multiples of
radian around the center: The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation[10] 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
can be calculated from two radial Zernike polynomials of adjacent degree:[13] The differential equation of the Gaussian Hypergeometric Function is equivalent to The first few radial polynomials are: The first few Zernike modes, at various indices, are shown below.
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.
[14][15] 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.
[16][17][18] 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.
[19] 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.
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.
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.
[20] 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[21] or the surface of vibrating disks.
[23] 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.
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.)
