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 "B9zQ,[Q
function z = zernfun(n,m,r,theta,nflag) OaY]}4tI$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3wQ\L=
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s !II}'Je
% and angular frequency M, evaluated at positions (R,THETA) on the M&e=LV
% unit circle. N is a vector of positive integers (including 0), and SQN{/")T
% M is a vector with the same number of elements as N. Each element C;ME"4,(
% k of M must be a positive integer, with possible values M(k) = -N(k) ?P}) Qa
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, xXJzE|)1h!
% and THETA is a vector of angles. R and THETA must have the same fT<3~Z>m
% length. The output Z is a matrix with one column for every (N,M) $4kbOqn4
% pair, and one row for every (R,THETA) pair. sosIu
% p*JP='p
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }:*?w>=
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c~d*SDca
% with delta(m,0) the Kronecker delta, is chosen so that the integral >b~Q%{1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ES#q/yab5
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]SN5&S
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. V'9OGn2v
% 1yeD-M"w
% The Zernike functions are an orthogonal basis on the unit circle. J3'0^JP*
% They are used in disciplines such as astronomy, optics, and 89W8cJ$yW
% optometry to describe functions on a circular domain. T,B%iZ gCh
% @[1,i~H
% The following table lists the first 15 Zernike functions. \2Kl]G(w%y
% yK mHTjX=
% n m Zernike function Normalization s}DNu<"g
% -------------------------------------------------- [7[$P.MS{
% 0 0 1 1 d8WEsQ+)A
% 1 1 r * cos(theta) 2 R^.c
% 1 -1 r * sin(theta) 2 .:(gg
% 2 -2 r^2 * cos(2*theta) sqrt(6) <!X]$kvG
% 2 0 (2*r^2 - 1) sqrt(3) e)i-$0L"
% 2 2 r^2 * sin(2*theta) sqrt(6) u@zT~\ h*
% 3 -3 r^3 * cos(3*theta) sqrt(8) Iapzh y2l
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 3 >^B%qg6
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) $:?=A5ttuo
% 3 3 r^3 * sin(3*theta) sqrt(8) ON"V`_dq+M
% 4 -4 r^4 * cos(4*theta) sqrt(10) 2XeN E[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y1BxRd?D
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (e3?--~b6
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /FcwsD\=$
% 4 4 r^4 * sin(4*theta) sqrt(10) " j:15m5
% -------------------------------------------------- \d w ["k
% /!y3ZzL
% Example 1: Tn$|
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% =.9tRq
% % Display the Zernike function Z(n=5,m=1) <q|eG\01S
% x = -1:0.01:1; +:z%#D
% [X,Y] = meshgrid(x,x); S7CD#Y[s
% [theta,r] = cart2pol(X,Y); &<C&(g{Z
% idx = r<=1; ^Ux*"\/Es
% z = nan(size(X)); _3gF~qr
% z(idx) = zernfun(5,1,r(idx),theta(idx)); b~K-mjJI
% figure 1$"wN z
% pcolor(x,x,z), shading interp ,Nev7X[0
% axis square, colorbar eBW]hwhKzM
% title('Zernike function Z_5^1(r,\theta)') BFn}~\wzK
% utw@5
% Example 2: ;fv/s]X86I
% ;giT[KK
% % Display the first 10 Zernike functions dr4 m}v.
% x = -1:0.01:1; Uq2 Qh@B
% [X,Y] = meshgrid(x,x); [_p&,$z8[
% [theta,r] = cart2pol(X,Y); ' @j8tK
% idx = r<=1; l,Ixz1S3e
% z = nan(size(X)); N37#Vs
% n = [0 1 1 2 2 2 3 3 3 3]; 3.
g-V
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;'| t>'0_
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }@g#S@o
% y = zernfun(n,m,r(idx),theta(idx)); vu)V:y
% figure('Units','normalized') sT"{ e7;F;
% for k = 1:10 m*TJ@gI*t
% z(idx) = y(:,k); i)d'l<RA
% subplot(4,7,Nplot(k)) C#.d
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% pcolor(x,x,z), shading interp 71Mk!E=1
% set(gca,'XTick',[],'YTick',[]) \"A~ks~
% axis square =7U8`]WA
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7ZgFCK,8m,
% end F}#=qBa[
% <1E*wPm8
% See also ZERNPOL, ZERNFUN2. f.u[!T
{I"d"'h
% Paul Fricker 11/13/2006 a7l-kG=R;
6.GIUM%D
[Uu!:SZ
% Check and prepare the inputs: 0CUUgwA/
% ----------------------------- L+"5g@
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i52:<<