Zernikepolynomier

Zernikepolynomier, or Zernike polynomials, are a sequence of orthogonal polynomials defined on the unit disk. Named after the Dutch physicist Frits Zernike, who introduced them in 1934, these polynomials are widely used in optics, engineering, and image analysis to describe wavefront distortions, surface irregularities, and aberrations.

The Zernike polynomials are typically expressed in polar coordinates (r, θ), where r is the radial distance from the origin (0 ≤ r ≤ 1) and θ is the azimuthal angle (0 ≤ θ < 2π). They are characterized by two integer indices: the radial order n (non-negative) and the azimuthal frequency m, which can be positive, negative, or zero, with the condition that n ≥ |m|. The polynomials are defined as the product of a radial polynomial R_n^m(r) and a angular function (cos mθ or sin mθ) depending on m's sign.

These polynomials form a complete orthogonal set over the unit disk, allowing any sufficiently smooth function defined on this domain to be expanded as a series of Zernike polynomials. This property makes them particularly effective for modeling wavefront aberrations in optical systems, including telescopes and microscopes.

The Zernike polynomials are also used in adaptive optics, ophthalmology for corneal topography, and image processing for shape description. Their mathematical properties facilitate efficient decomposition and analysis of complex surface and wavefront data, advancing applications where precise optical modeling is essential.