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 :8A@4vMS)?
function z = zernfun(n,m,r,theta,nflag) ?*~sx=mC
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ]L
k- -\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Y3H5}4QD
% and angular frequency M, evaluated at positions (R,THETA) on the R I:kp.V
% unit circle. N is a vector of positive integers (including 0), and Q $Sp'
% M is a vector with the same number of elements as N. Each element CSBDSz
% k of M must be a positive integer, with possible values M(k) = -N(k) 8\+DSA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u Vo"_c w
% and THETA is a vector of angles. R and THETA must have the same ,@zw
% length. The output Z is a matrix with one column for every (N,M) nPjK=o`KR
% pair, and one row for every (R,THETA) pair. 3sl6$NKo
% A~<cp)E
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1So`]N4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), mp*?GeV?M
% with delta(m,0) the Kronecker delta, is chosen so that the integral m;ju@5X
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -U|Z9sia
% and theta=0 to theta=2*pi) is unity. For the non-normalized 5+qdn|9%T
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 'oUTY *
% #6C<P!]V
% The Zernike functions are an orthogonal basis on the unit circle. !Yz
CK*av1
% They are used in disciplines such as astronomy, optics, and ONFx -U]
% optometry to describe functions on a circular domain. [i_evsUj?
% 6!([Hu#= *
% The following table lists the first 15 Zernike functions. XI,= W
% lWUQkS
% n m Zernike function Normalization .dwbJT
% -------------------------------------------------- #JN4K>_4
% 0 0 1 1 (#]9{C;
% 1 1 r * cos(theta) 2 *aGJ$ P0
% 1 -1 r * sin(theta) 2 ZWKvz3Wt
% 2 -2 r^2 * cos(2*theta) sqrt(6) U6YHq2<
% 2 0 (2*r^2 - 1) sqrt(3) uII! ?
% 2 2 r^2 * sin(2*theta) sqrt(6) *]!rT&E
% 3 -3 r^3 * cos(3*theta) sqrt(8) \~l"
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) j' b0sve|?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) R^"mGe\LL
% 3 3 r^3 * sin(3*theta) sqrt(8) d?V/V'T[
% 4 -4 r^4 * cos(4*theta) sqrt(10) Y&bO[(> 1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v4Kf{9q#
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Oc5f8uv
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $"MGu^0;1
% 4 4 r^4 * sin(4*theta) sqrt(10) >4os%T
% --------------------------------------------------
pQ7<\8s*
% SH O&:2
% Example 1: **.23<n^W
% 3 Zwhv+CP[
% % Display the Zernike function Z(n=5,m=1) D$E#:[
% x = -1:0.01:1; Zqb*-1Qw"*
% [X,Y] = meshgrid(x,x); MeAY\V%G=o
% [theta,r] = cart2pol(X,Y); cg9*+]rc
% idx = r<=1; *w}r:04F
% z = nan(size(X)); }ktK*4<k
% z(idx) = zernfun(5,1,r(idx),theta(idx)); KEf1GU6s
% figure NLUiNfCR
% pcolor(x,x,z), shading interp q_[`PYT
% axis square, colorbar [Mj5o<k;I
% title('Zernike function Z_5^1(r,\theta)') p(9[*0.};
% a %?v/Ku
% Example 2: 6P)D M
% *^CN2tm
% % Display the first 10 Zernike functions ~yA^6[a =
% x = -1:0.01:1; Bj\Us$cZ
% [X,Y] = meshgrid(x,x); "~Zdv}^xS
% [theta,r] = cart2pol(X,Y); AoK;6je`K^
% idx = r<=1; ]Rxrt~ ZB
% z = nan(size(X)); ?[%.4i;-h
% n = [0 1 1 2 2 2 3 3 3 3]; r>.l^U9hJ
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; G&4D0f
% Nplot = [4 10 12 16 18 20 22 24 26 28]; K??jV&Xor
% y = zernfun(n,m,r(idx),theta(idx)); _Ih"*~ r/&
% figure('Units','normalized') fB'Jo<C
% for k = 1:10 15%6;K?b
% z(idx) = y(:,k); ]cMZ7V^
% subplot(4,7,Nplot(k)) LLoV]~dvUu
% pcolor(x,x,z), shading interp Cu<' b'%;
% set(gca,'XTick',[],'YTick',[]) U!YoZ?
% axis square !)05,6WQ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,wy;7T>ODd
% end `,4YPjk^
% 7Q,<h8N\5
% See also ZERNPOL, ZERNFUN2. @moaa} 1
a.ijc>K
% Paul Fricker 11/13/2006 G;USVF-'K
4w]<1V
ad=7FhnIa3
% Check and prepare the inputs: "#iO{uMWb
% ----------------------------- ZVit]3hd
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /nEK|.j
error('zernfun:NMvectors','N and M must be vectors.') 8cRc5X
end ?9?o8!
Ok}e|b[D
if length(n)~=length(m) n7zM;@{7
error('zernfun:NMlength','N and M must be the same length.') "chf\-!$
end K9K.mGYc
i.7$~}
n = n(:); L:Faq1MG
m = m(:); +aqQa~}r
if any(mod(n-m,2)) S(rnVsW%Ki
error('zernfun:NMmultiplesof2', ... ~4c,'k@
'All N and M must differ by multiples of 2 (including 0).')
0BAZWm
end [FBc&HN
y{XNB}E
if any(m>n) /gn\7&