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Zernike polynomials

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Zernike polynomials

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18 May 2005 (Updated )

Zernike polynomials and functions (orthogonal basis on the unit circle).

zernfun2(p,r,theta,nflag)
function z = zernfun2(p,r,theta,nflag)
%ZERNFUN2 Single-index Zernike functions on the unit circle.
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated
%   at positions (R,THETA) on the unit circle.  P is a vector of positive
%   integers between 0 and 35, R is a vector of numbers between 0 and 1,
%   and THETA is a vector of angles.  R and THETA must have the same
%   length.  The output Z is a matrix with one column for every P-value,
%   and one row for every (R,THETA) pair.
%
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2)
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi)
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1
%   for all p.
%
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36
%   Zernike functions (order N<=7).  In some disciplines it is 
%   traditional to label the first 36 functions using a single mode
%   number P instead of separate numbers for the order N and azimuthal
%   frequency M.
%
%   Example:
%
%       % Display the first 16 Zernike functions
%       x = -1:0.01:1;
%       [X,Y] = meshgrid(x,x);
%       [theta,r] = cart2pol(X,Y);
%       idx = r<=1;
%       p = 0:15;
%       z = nan(size(X));
%       y = zernfun2(p,r(idx),theta(idx));
%       figure('Units','normalized')
%       for k = 1:length(p)
%           z(idx) = y(:,k);
%           subplot(4,4,k)
%           pcolor(x,x,z), shading interp
%           set(gca,'XTick',[],'YTick',[])
%           axis square
%           title(['Z_{' num2str(p(k)) '}'])
%       end
%
%   See also ZERNPOL, ZERNFUN.

%   Paul Fricker 11/13/2006


% Check and prepare the inputs:
% -----------------------------
if min(size(p))~=1
    error('zernfun2:Pvector','Input P must be vector.')
end

if any(p)>35
    error('zernfun2:P36', ...
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ...
           '(P = 0 to 35).'])
end

% Get the order and frequency corresonding to the function number:
% ----------------------------------------------------------------
p = p(:);
n = ceil((-3+sqrt(9+8*p))/2);
m = 2*p - n.*(n+2);

% Pass the inputs to the function ZERNFUN:
% ----------------------------------------
switch nargin
    case 3
        z = zernfun(n,m,r,theta);
    case 4
        z = zernfun(n,m,r,theta,nflag);
    otherwise
        error('zernfun2:nargin','Incorrect number of inputs.')
end

% EOF zernfun2

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