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 Q IQB
function z = zernfun(n,m,r,theta,nflag) >/;\{IG
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $SSE\+|3
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @;qC% +^
% and angular frequency M, evaluated at positions (R,THETA) on the 0 lXV+lj
% unit circle. N is a vector of positive integers (including 0), and \#1!qeF
% M is a vector with the same number of elements as N. Each element 6[$kEKOY=
% k of M must be a positive integer, with possible values M(k) = -N(k) `IOp*8
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, p^Ca-+R3
% and THETA is a vector of angles. R and THETA must have the same t>7t4>X
% length. The output Z is a matrix with one column for every (N,M) Hj
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% pair, and one row for every (R,THETA) pair. H$KE*Wwq
% \ 3n{%\_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Kv:U QdnU[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), z{d] ,M
% with delta(m,0) the Kronecker delta, is chosen so that the integral E$.|h;i]Q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FH)bE#4
% and theta=0 to theta=2*pi) is unity. For the non-normalized kuu9'Sqc'b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3:<+9X
% kMKI=>s+
% The Zernike functions are an orthogonal basis on the unit circle. )wP0U{7?v
% They are used in disciplines such as astronomy, optics, and Odxq ]HlbO
% optometry to describe functions on a circular domain. x,E#+
m
% :{h,0w'd
% The following table lists the first 15 Zernike functions. {.bLh0
% l~Kn-S{
% n m Zernike function Normalization 4U<'3~RN
% -------------------------------------------------- ?)<zrE5p
% 0 0 1 1 5IW^^<kiu
% 1 1 r * cos(theta) 2 %=/Y~ml?
% 1 -1 r * sin(theta) 2 '&Q_5\Tn
% 2 -2 r^2 * cos(2*theta) sqrt(6) ~^lQ[ x
% 2 0 (2*r^2 - 1) sqrt(3) +1Si>I
% 2 2 r^2 * sin(2*theta) sqrt(6) vF;6Y(h>
% 3 -3 r^3 * cos(3*theta) sqrt(8) )~_!u}+:(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) G\Hck=P[$3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) UYW%%5p?
% 3 3 r^3 * sin(3*theta) sqrt(8) CWE
jX-
% 4 -4 r^4 * cos(4*theta) sqrt(10) 1]A%lud4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -4,qAnuMx
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Ptzha?}OZ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lk \|EG
% 4 4 r^4 * sin(4*theta) sqrt(10) 3
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% -------------------------------------------------- <3P?rcd,5K
% 7$x@;%xd
% Example 1: 5U|f"3&8
% ZgtW
% % Display the Zernike function Z(n=5,m=1) yZxgUF&`
% x = -1:0.01:1; v 8{oXzyy
% [X,Y] = meshgrid(x,x); )jR:\fe
% [theta,r] = cart2pol(X,Y); MgHyKn'rL
% idx = r<=1; HGWwGd
% z = nan(size(X)); dmP*2
% z(idx) = zernfun(5,1,r(idx),theta(idx)); [H0jDbN
% figure ETH`.~%
% pcolor(x,x,z), shading interp rNU,(htS
% axis square, colorbar LAwX9q`
% title('Zernike function Z_5^1(r,\theta)') H b]
% dulW!&*No
% Example 2: (z2)<_bXJ
% cIl^5eE^Pq
% % Display the first 10 Zernike functions dT/Cn v=
% x = -1:0.01:1; q*DR~Ov
% [X,Y] = meshgrid(x,x); (d^pYPr{
% [theta,r] = cart2pol(X,Y); jA=uK6m
% idx = r<=1; ]!YzbvoR
% z = nan(size(X)); :b=`sUn<X+
% n = [0 1 1 2 2 2 3 3 3 3]; n+zXt?{u
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ?j8CkqX!
% Nplot = [4 10 12 16 18 20 22 24 26 28]; xw%?R=&L
% y = zernfun(n,m,r(idx),theta(idx)); rM [Ps=5
% figure('Units','normalized') *2MUG
h
% for k = 1:10 \5s!lv*&
% z(idx) = y(:,k); F__DPEAc_
% subplot(4,7,Nplot(k)) s<:"rw`
% pcolor(x,x,z), shading interp Fj 1/B0acS
% set(gca,'XTick',[],'YTick',[]) F`Q,pBl1p6
% axis square
H.Jcp|k[;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^%go\ C ;
% end p*Q"<@n
% .a=M@;p
% See also ZERNPOL, ZERNFUN2. 4$IPz7
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% Paul Fricker 11/13/2006 RF6(n8["MW
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