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 5 b,|6
function z = zernfun(n,m,r,theta,nflag) -`z%<)!Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Fo%`X[ ?
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `(P71T
% and angular frequency M, evaluated at positions (R,THETA) on the Uugq.'>
% unit circle. N is a vector of positive integers (including 0), and :J x%K
% M is a vector with the same number of elements as N. Each element *V+,X
% k of M must be a positive integer, with possible values M(k) = -N(k) \UM&|yk:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )Spa
F)N8
% and THETA is a vector of angles. R and THETA must have the same <}c7E3Uc
% length. The output Z is a matrix with one column for every (N,M) (JjxrZ+L
% pair, and one row for every (R,THETA) pair. HFFrS%
% 8r@GoG>
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -byaV;T?"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]c|JxgU
% with delta(m,0) the Kronecker delta, is chosen so that the integral SfrM|o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3fZoF`<a
% and theta=0 to theta=2*pi) is unity. For the non-normalized ` l'QAIo
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O7.eq524
% {x..>
4
% The Zernike functions are an orthogonal basis on the unit circle. pzQc UG
% They are used in disciplines such as astronomy, optics, and K)[\IJJM
% optometry to describe functions on a circular domain. fk1d iB
% ,+C?UW
% The following table lists the first 15 Zernike functions. mF4OLG3L0
% 0jxXUWO
% n m Zernike function Normalization ZJhI|wRwD
% -------------------------------------------------- []yIz1P=j
% 0 0 1 1 %Q.M& U
% 1 1 r * cos(theta) 2 'IVC!uL,%
% 1 -1 r * sin(theta) 2 Iue=\qUK^
% 2 -2 r^2 * cos(2*theta) sqrt(6) 2S[:mnK
% 2 0 (2*r^2 - 1) sqrt(3) >){}nlQf
% 2 2 r^2 * sin(2*theta) sqrt(6) z-"P raP
% 3 -3 r^3 * cos(3*theta) sqrt(8) 9a sA-'fZ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Al *yx_j
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) g1 y@z8Z{
% 3 3 r^3 * sin(3*theta) sqrt(8) Yb[)ETf^
% 4 -4 r^4 * cos(4*theta) sqrt(10) #hu`X6s"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *r9D+}Y(4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) T-7(3#&
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i*&b@.7N
% 4 4 r^4 * sin(4*theta) sqrt(10) FLkZZ\
% -------------------------------------------------- 3|)cT1ej
% 0lOan
% Example 1: )u]=^
% w_~tY*IwB
% % Display the Zernike function Z(n=5,m=1) !B9Yw/Ba
% x = -1:0.01:1; \FCPD.2s+
% [X,Y] = meshgrid(x,x); 1E4`&?
% [theta,r] = cart2pol(X,Y); +R{~%ZTK
% idx = r<=1; P+_1*lOG
% z = nan(size(X)); Wap\J7NY
% z(idx) = zernfun(5,1,r(idx),theta(idx)); XMxm2-%olP
% figure T0b/txS
% pcolor(x,x,z), shading interp P9S)7&+DL
% axis square, colorbar GlJOb|WOX
% title('Zernike function Z_5^1(r,\theta)') Su
+<mW
% 5UK}AkEe&x
% Example 2: KRP6b:+4L
% .]<gm9l
% % Display the first 10 Zernike functions jSdC1,wR
% x = -1:0.01:1; H3iYE~^#
% [X,Y] = meshgrid(x,x); XGYsTquSe
% [theta,r] = cart2pol(X,Y); oGbh*
% idx = r<=1; fmLDufx
% z = nan(size(X)); =t~]@?]1D
% n = [0 1 1 2 2 2 3 3 3 3]; [IHG9Xg
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 5dX0C
% Nplot = [4 10 12 16 18 20 22 24 26 28]; w=ufJRj
% y = zernfun(n,m,r(idx),theta(idx)); *`Ge8?qC
% figure('Units','normalized') hX-^h2eV
% for k = 1:10 'fzJw
% z(idx) = y(:,k); 'cK{FiIT
% subplot(4,7,Nplot(k)) $t5>1G1j7
% pcolor(x,x,z), shading interp ox";%|PP1
% set(gca,'XTick',[],'YTick',[]) oJE<}~_k
% axis square #a]\3X
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `J7@G]X;2
% end kaECjZ_&+
% "/taatcH
% See also ZERNPOL, ZERNFUN2. !SLfAFcS
,Vz-w;oDn
% Paul Fricker 11/13/2006 =4!m]*y
^0(D2:E
sYk#XNH
% Check and prepare the inputs: e%9zY{ABR%
% ----------------------------- /7.//klN
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) y^
st
T^
error('zernfun:NMvectors','N and M must be vectors.') Dj0D.}`~
end yVpru8+eD
d5=&