The Gavrielides system takes derivatives of phase w.r.t. radius and angle on a square grid and fits *real* Zernike polynomials over the given aperture up to the specified radial order using Gavrielides' method [Gavrielides, Optics Letters Vol 7 No 11 Nov 1982 pp526-8]. If a non-zero annulus diameter is specified, then the region within the annulus is excluded from the fit.

Output Coef is a vector of coefficients, one for each polynomial and a Zidx is vector twice the length, containing pairs of orders (radial & angular) for each polynomial. E.g., if maximum radial order n=3, output polynomials are two 1st order (tilt), three 2nd order (focus & astigmatism), & four 3rd order (third order coma & triangular astigmatism); 9 coefficients and 18 orders are output.

The output Zernike coefficients are orderered according to the 1-parameter scheme specified by the 'orderingScheme' parameter. The corresponding 2-parameter mode indecies are output in the vector Zidx. Either Noll (1) or Malacara (2) ordering should be selected; a deprecated ordering scheme (0) is the default due to backward compatibility concerns but this should not be used for new simulations. The Zernike coefficients are scaled such that the fit phase can be composed using the normalization scheme specified by the 'normalization' parameter.

The method of Zernike fitting is controlled by the 'tol' parameter. If tol is greater than or equal to one, then an overlap integral is always used; this is faster, but should only be used if you are certain that the Zernike polynomials are very near orthogonal over the aperture grid used. If tol is less than or equal to zero then least-squares fitting is always used; this is slower, but does not assume any independence of the fit modes. For values of tol between zero and one, the least-squares matrix is constructed; for an orthornal system it will be the identiy matrix. If any elements differ from the identity matrix by an amount more than tol then the matrix is inverted and least-squares fitting is used.

Parameters
apD	float	Aperture diameter (m)	Dap
annD	float	Annulus diameter (m)	0.0f
n	int	Maximum radial order of Zernike modes to fit (1=tilt, 2=focus & astigmatism, 3=3rd order coma & triangular astigmatism, etc.)	3
normalization	int	0-no normalization; 1-normalize to overlap integral=1/pi; 2-make the PV=1.0;3-Noll/Malacara	3
orderingScheme	int	0-deprecated;1-Noll;2-Malacara	0
tol	float	Deviation allowed from orthonormality before switching from overlap integral to least-squares	0.1f
Inputs
Dr	Grid<float>	Phase derivatives w.r.t. radius
Dt	Grid<float>	Phase derivatives w.r.t. angle
Outputs
Coef	Vector<float>	Coefficients of real Zernike polynomials
Zidx	Vector<int>	Pairs of radial & angular orders of Zernike polynomials
Subsystems
Last Saved: Mon Oct 20 13:06:34 MDT 2008 by TVE version 2009B