Zernike polynomials
Encyclopedia
In mathematics
, the Zernike polynomials are a sequence
of polynomial
s that are orthogonal
on the unit disk. Named after Frits Zernike
, they play an important role in beam optics
.
Zernike polynomials. The even ones are defined as
and the odd ones as
where m and n are nonnegative integer
s with n≥m, φ is the azimuthal angle
, and ρ is the radial distance
.
The radial polynomials Rmn are defined as
for n − m even, and are identically 0 for n − m odd.
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.
Applications often involve linear algebra, where integrals over products
of Zernike polynomials and some other factor build the 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
The rule is that the even Z (with azimuthal part
)
obtain even indices j, the odd Z odd indices j. Within a given n,
lower values of m obtain lower j.
.
Orthogonality in the angular part is represented by
,
,
,
where
(sometimes called the Neumann factor
because it frequently appears in conjunction with Bessel functions) 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.
A special value is
.
.
The parity with respect to point reflection at the center of coordinates is
,
where
could as well be written 
because
is even for the relevant, non-vanishing values.
The radial polynomials are also either even or odd:
.
The periodicity of the trigonometric functions implies invariance if rotated by multiples of
radian
around the center:
.
the pupil planes in classical optical imaging at optical and infrared wavelengths
through systems of lenses and mirrors
of finite diameter. Their advantage 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 order to achieve desired system performance.
In optometry
and ophthalmology
the Zernike polynomials are used to describe aberrations of the cornea
or lens from an ideal spherical shape, which result in refraction errors.
They are commonly used in adaptive optics
where they can be used to effectively cancel out atmospheric distortion. Obvious applications for this are IR or visual astronomy, and Satellite imagery
. For example, one of the zernike terms (for m = 0, n = 2) is called 'de-focus'. By coupling the output from this term to a control system, an automatic focus can be implemented.
Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of diffraction
and aberrations.
Zernike polynomials are widely used as basis functions of image moments
.
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
.
(Note that a factor
is absorbed in the definition of 
here,
whereas in
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
, else identical to zero.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Zernike polynomials are a sequence
Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s that are orthogonal
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
on the unit disk. Named after Frits Zernike
Frits Zernike
Frits Zernike was a Dutch physicist and winner of the Nobel prize for physics in 1953 for his invention of the phase contrast microscope, an instrument that permits the study of internal cell structure without the need to stain and thus kill the cells....
, they play an important role in beam optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
.
Definitions
There are even and oddEven and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...
Zernike polynomials. The even ones are defined as
and the odd ones as
where m and n are nonnegative integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s with n≥m, φ is the azimuthal angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
, and ρ is the radial distance
.The radial polynomials Rmn are defined as
for n − m even, and are identically 0 for n − m odd.
Other Representations
Rewriting the ratios of factorials in the radial part as products ofbinomials
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
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
Jacobi polynomials
In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
, to write down the differential equations, etc.:
for n − m even.
Applications often involve linear algebra, where integrals over products
of Zernike polynomials and some other factor build the 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 The rule is that the even Z (with azimuthal part
)obtain even indices j, the odd Z odd indices j. Within a given n,
lower values of m obtain lower j.
Orthogonality
The orthogonality in the radial part reads.Orthogonality in the angular part is represented by
,,,where
(sometimes called the Neumann factorCarl Neumann
Carl Gottfried Neumann was a German mathematician.Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann , who was professor of mineralogy and physics at Königsberg University...
because it frequently appears in conjunction with Bessel functions) 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 thecircular coordinate system, and where
and are both even.
A special value is
.Symmetries
The parity with respect to reflection along the x axis is.The parity with respect to point reflection at the center of coordinates is
,where
could as well be written because
is even for the relevant, non-vanishing values.The radial polynomials are also either even or odd:
.The periodicity of the trigonometric functions implies invariance if rotated by multiples of
radianaround the center:
.Examples
The first few Radial polynomials are:Applications
The functions are a basis defined over the circular support area, typicallythe pupil planes in classical optical imaging at optical and infrared wavelengths
through systems of lenses and mirrors
of finite diameter. Their advantage 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
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
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 otherorthogonal functions over the circular disk.
In precision optical manufacturing, Zernike polynomials are used to characterize higher-order errors observed in interferometric analyses, in order to achieve desired system performance.
In optometry
Optometry
Optometry is a health care profession concerned with eyes and related structures, as well as vision, visual systems, and vision information processing in humans. Optometrists, or Doctors of Optometry, are state licensed medical professionals trained to prescribe and fit lenses to improve vision,...
and ophthalmology
Ophthalmology
Ophthalmology is the branch of medicine that deals with the anatomy, physiology and diseases of the eye. An ophthalmologist is a specialist in medical and surgical eye problems...
the Zernike polynomials are used to describe aberrations of the cornea
Cornea
The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Together with the lens, the cornea refracts light, with the cornea accounting for approximately two-thirds of the eye's total optical power. In humans, the refractive power of the cornea is...
or lens from an ideal spherical shape, which result in refraction errors.
They are commonly used in adaptive optics
Adaptive optics
Adaptive optics is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, and in retinal imaging systems to reduce the...
where they can be used to effectively cancel out atmospheric distortion. Obvious applications for this are IR or visual astronomy, and Satellite imagery
Satellite imagery
Satellite imagery consists of photographs of Earth or other planets made by means of artificial satellites.- History :The first images from space were taken on sub-orbital flights. The U.S-launched V-2 flight on October 24, 1946 took one image every 1.5 seconds...
. For example, one of the zernike terms (for m = 0, n = 2) is called 'de-focus'. By coupling the output from this term to a control system, an automatic focus can be implemented.
Another application of the Zernike polynomials is found in the Extended Nijboer-Zernike (ENZ) theory of diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
and aberrations.
Zernike polynomials are widely used as basis functions of image moments
Image moments
In image processing, computer vision and related fields, an image moment is a certain particular weighted average of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation....
.
Higher Dimensions
The concept translates to higher dimensions if multinomials in Cartesian coordinates are converted to hyperspherical coordinates,, , multiplied by a product of Jacobi Polynomials of the angularvariables.
In
dimensions, the angular variables are Spherical harmonicsSpherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
, for example.
Linear combinations of the powers
define an orthogonalbasis
satisfying.(Note that a factor
is absorbed in the definition of here,whereas in
the normalization is chosen slightly differently. This is largelya 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
, else identical to zero.