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 R9InUX"k
function z = zernfun(n,m,r,theta,nflag) dA$qzQ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'E %+ O
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7DIFJJE'
% and angular frequency M, evaluated at positions (R,THETA) on the =VF%Z[Gm
% unit circle. N is a vector of positive integers (including 0), and M(<.f}yZQ
% M is a vector with the same number of elements as N. Each element 9/6=[)
% k of M must be a positive integer, with possible values M(k) = -N(k) 8Oo16LPD
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ?9;r|G
% and THETA is a vector of angles. R and THETA must have the same YbuS[l8
% length. The output Z is a matrix with one column for every (N,M) 1^ y^b{
% pair, and one row for every (R,THETA) pair. Kl w9
% +D|E8sz8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike
~ P!%i9e_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), b!z kQ?h
% with delta(m,0) the Kronecker delta, is chosen so that the integral BS+=*3J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, S &F
% and theta=0 to theta=2*pi) is unity. For the non-normalized '<&rMn
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wQN/MYF[
% cGS7s 8U
% The Zernike functions are an orthogonal basis on the unit circle. 2g%p9-MO]I
% They are used in disciplines such as astronomy, optics, and w>6cc#>q
% optometry to describe functions on a circular domain. ;g_<i_*x#
% *hkNJ
% The following table lists the first 15 Zernike functions. GqD_6cdh
% Io7o*::6iw
% n m Zernike function Normalization +XL|bdK
% -------------------------------------------------- !Q5NV4gd+
% 0 0 1 1 Pe?b#
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% 1 1 r * cos(theta) 2 BVv{:m{w
% 1 -1 r * sin(theta) 2 1g_Dkv|D
% 2 -2 r^2 * cos(2*theta) sqrt(6) #\gx.2W7
% 2 0 (2*r^2 - 1) sqrt(3) gtRs||
% 2 2 r^2 * sin(2*theta) sqrt(6) yIma7H@=L
% 3 -3 r^3 * cos(3*theta) sqrt(8) CG[04y
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) %lS jC%Z'd
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 'Sjt*2blq
% 3 3 r^3 * sin(3*theta) sqrt(8) .@3bz
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0v'!(&m
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6}GcMhU<r
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Q a3+ 9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o/mGd~
% 4 4 r^4 * sin(4*theta) sqrt(10) bSS=<G9
% -------------------------------------------------- qp55U*
% El|Y]f
% Example 1: -)p| i~j^A
% lH%-#2]
% % Display the Zernike function Z(n=5,m=1) 287)\FU;3
% x = -1:0.01:1; .?*TU~S
% [X,Y] = meshgrid(x,x); #lO~n.+P
% [theta,r] = cart2pol(X,Y); lW3wmSWn%
% idx = r<=1; 6:qh%ZR
% z = nan(size(X)); 0'~Iv\s
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Yo[Pu< zR
% figure m$B)_WW
% pcolor(x,x,z), shading interp _/cL"Wf
% axis square, colorbar {V5eHn9/Q'
% title('Zernike function Z_5^1(r,\theta)') +<w\K*
% {qL}:ha?
% Example 2: dmk_xBy s|
% ($[)Tcq*~
% % Display the first 10 Zernike functions |!"qz$8fB
% x = -1:0.01:1; 5yQ\s[;o3
% [X,Y] = meshgrid(x,x); }+i~JK
% [theta,r] = cart2pol(X,Y); 9\KMU@Ne
% idx = r<=1; ~oE@y6Q
% z = nan(size(X)); Pm!/#PtX
% n = [0 1 1 2 2 2 3 3 3 3]; oO][X
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;'4HR+E"
% Nplot = [4 10 12 16 18 20 22 24 26 28]; =SLCG.
% y = zernfun(n,m,r(idx),theta(idx)); "D?:8!\!
% figure('Units','normalized') K#4Toc#=V
% for k = 1:10 d2(3 ,
% z(idx) = y(:,k); 6tv-PgZ
% subplot(4,7,Nplot(k)) Wd]MwDcO
% pcolor(x,x,z), shading interp fE,Io3
% set(gca,'XTick',[],'YTick',[]) <K
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% axis square zb& 3{,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^=7XA894
% end c`xgz#]v
% EN{o3@ O'
% See also ZERNPOL, ZERNFUN2. !\/J|~XZ
;eT+Ly|{
% Paul Fricker 11/13/2006 m$}Jw<