非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 "7Qc:<ww
function z = zernfun(n,m,r,theta,nflag) ^]Mlkd:
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. %*d(1?\o
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >(>Fx\z}
% and angular frequency M, evaluated at positions (R,THETA) on the gHCk;dmq81
% unit circle. N is a vector of positive integers (including 0), and J*@(rb#G
% M is a vector with the same number of elements as N. Each element .CXe*Vbd
% k of M must be a positive integer, with possible values M(k) = -N(k) @mM])V
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, GMLDmTV
% and THETA is a vector of angles. R and THETA must have the same %*4Gx +b
% length. The output Z is a matrix with one column for every (N,M) 7|=*z
% pair, and one row for every (R,THETA) pair. L_$M9G|5n
% _ElA\L4g%
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ya$JX(aUe
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9D
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% with delta(m,0) the Kronecker delta, is chosen so that the integral b.Wf*I?
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LeY!A#j
% and theta=0 to theta=2*pi) is unity. For the non-normalized 4.@gV/U(|
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P=ARttT`(
% t%jB[w&,os
% The Zernike functions are an orthogonal basis on the unit circle. 8!e1T,:b
% They are used in disciplines such as astronomy, optics, and q r12"H
% optometry to describe functions on a circular domain. ?R2`RvQ
% 0:<dj:%M
% The following table lists the first 15 Zernike functions. G4Y]fzC
% P<@Yux#
% n m Zernike function Normalization \W73W_P&g
% -------------------------------------------------- pfCNFF*"
% 0 0 1 1 i,G )kt'H
% 1 1 r * cos(theta) 2 ;1`NsYI2
% 1 -1 r * sin(theta) 2 gB\
a
% 2 -2 r^2 * cos(2*theta) sqrt(6) F[ca4_lK
% 2 0 (2*r^2 - 1) sqrt(3) m*VM1k V
% 2 2 r^2 * sin(2*theta) sqrt(6) Oh9jr"Gm=
% 3 -3 r^3 * cos(3*theta) sqrt(8) e<