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 YkniiB[/
function z = zernfun(n,m,r,theta,nflag) ]+XYEv
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. clL2k8VS
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N g!?:Ye`5
% and angular frequency M, evaluated at positions (R,THETA) on the 1m:XR0 P
% unit circle. N is a vector of positive integers (including 0), and d%RC
% M is a vector with the same number of elements as N. Each element G
MX?
% k of M must be a positive integer, with possible values M(k) = -N(k) S+atn]eU@
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, BGD8w2
% and THETA is a vector of angles. R and THETA must have the same $Q96,rb}k;
% length. The output Z is a matrix with one column for every (N,M) [z`31F
% pair, and one row for every (R,THETA) pair. ||hb~%JK6
% El[)?+;D
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G~2jUyv
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ES.fOdx
% with delta(m,0) the Kronecker delta, is chosen so that the integral
-QM:
q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, K;>9ZZtl
% and theta=0 to theta=2*pi) is unity. For the non-normalized EN;}$jZ>47
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. j53*E
)d
% J'SZ
% The Zernike functions are an orthogonal basis on the unit circle. Gb#Cm]
% They are used in disciplines such as astronomy, optics, and b&~4t/Vq
% optometry to describe functions on a circular domain. `\gnl'
% l_P-j96WD
% The following table lists the first 15 Zernike functions. #fM#p+v
% \?0&0;5
% n m Zernike function Normalization / ';0H_
% -------------------------------------------------- yp KUkH/
% 0 0 1 1 w+#C-&z
% 1 1 r * cos(theta) 2 ;V*R*R
% 1 -1 r * sin(theta) 2 j9?}j#@
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]iN'x?Fo
% 2 0 (2*r^2 - 1) sqrt(3) )Dw,q~xgg0
% 2 2 r^2 * sin(2*theta) sqrt(6) .aAL]-Rj
% 3 -3 r^3 * cos(3*theta) sqrt(8) uxtWybv
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) tyXuG<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) )uj Ex7&c
% 3 3 r^3 * sin(3*theta) sqrt(8) Rzbj
% 4 -4 r^4 * cos(4*theta) sqrt(10) kP#B5K_U|
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &x[E;P*Fg
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) DnCP
aM4%
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *1p|5!4c
% 4 4 r^4 * sin(4*theta) sqrt(10) KIui(n#/
% -------------------------------------------------- Co (.:z~
% /y _O4
% Example 1: F(k.,0Nc
% U3T#6Rptl
% % Display the Zernike function Z(n=5,m=1) z=rT%lz6
% x = -1:0.01:1; Ir`eL
% [X,Y] = meshgrid(x,x); kbTm^y"
% [theta,r] = cart2pol(X,Y); -fwoTGlX
% idx = r<=1; 96 q_K84K
% z = nan(size(X)); {1V($aBl
% z(idx) = zernfun(5,1,r(idx),theta(idx)); QMa;Gy
% figure +Z7th7W/,
% pcolor(x,x,z), shading interp YQ+tDZY8`
% axis square, colorbar k9:{9wW
% title('Zernike function Z_5^1(r,\theta)') MBt9SXM
% (i34sqV$m
% Example 2: A+::O@_s
% u
[m
% % Display the first 10 Zernike functions y4*U6+ #.
% x = -1:0.01:1; N^HUijw<
% [X,Y] = meshgrid(x,x); GN ]cDik
% [theta,r] = cart2pol(X,Y); co~Pyj
% idx = r<=1; ?nofUD.
% z = nan(size(X)); #33fGmd[
% n = [0 1 1 2 2 2 3 3 3 3]; P_?gq>E8
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; |uqf:V`z:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; TD'L'm|2
% y = zernfun(n,m,r(idx),theta(idx)); T*#/^%HSG
% figure('Units','normalized') Bg&i63XL$$
% for k = 1:10 LQ(yScA@
% z(idx) = y(:,k); WFO4gB*
% subplot(4,7,Nplot(k)) O4r0R1VQM
% pcolor(x,x,z), shading interp {;N,t]>8M
% set(gca,'XTick',[],'YTick',[]) 9:ze{ c $
% axis square :rHJ4Tl
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) UUzYbuS>&l
% end g .onTFwN
% mz@T
% See also ZERNPOL, ZERNFUN2. J)`-+}7$v
73V|6tmgY
% Paul Fricker 11/13/2006 qQA}Z*(m
x^kp^
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]bj&bk#
% Check and prepare the inputs: B8B; y^b>i
% ----------------------------- ZAv,*5&<
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) u=/{cOJI6
error('zernfun:NMvectors','N and M must be vectors.') (yF:6$:#
end *pAV2V(!23
v%"|WV[N
if length(n)~=length(m) \^ZlG.
error('zernfun:NMlength','N and M must be the same length.') aa>xIW,u
end |?qquD 4=
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n = n(:); Svondc
4
m = m(:); 7NDr1Z#B6V
if any(mod(n-m,2)) r30 <