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 K-@bwB7~s
function z = zernfun(n,m,r,theta,nflag) ua#K>sur.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. P(_(w
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #"r kuDO
% and angular frequency M, evaluated at positions (R,THETA) on the VkXn8J
% unit circle. N is a vector of positive integers (including 0), and q$ >_WF#||
% M is a vector with the same number of elements as N. Each element mQ,{=C=D
% k of M must be a positive integer, with possible values M(k) = -N(k) e^frVEV
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, DQ_ 2fX~)
% and THETA is a vector of angles. R and THETA must have the same .mt^m
% length. The output Z is a matrix with one column for every (N,M) ;1E_o
% pair, and one row for every (R,THETA) pair. 3^ ~Zj95M
% EXHR(t}e
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %UG/ak%z
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |[WL2<
% with delta(m,0) the Kronecker delta, is chosen so that the integral &;U|7l~vl
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <9N4"d!A
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;Jo*|pju
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 3 2y[
% yA}nPXrd
% The Zernike functions are an orthogonal basis on the unit circle. Rp4FXR jC
% They are used in disciplines such as astronomy, optics, and ,\>g
% optometry to describe functions on a circular domain. p">WK<N
% dqz1xQ1
% The following table lists the first 15 Zernike functions. BvJ\x)
% ~2 Oc
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% n m Zernike function Normalization *-7fa0<
% -------------------------------------------------- \E&th p
% 0 0 1 1 s((b"{fFb
% 1 1 r * cos(theta) 2 gix>DHq$k
% 1 -1 r * sin(theta) 2 @Yarz1
% 2 -2 r^2 * cos(2*theta) sqrt(6) J[o${^
% 2 0 (2*r^2 - 1) sqrt(3) &<t79d%{
% 2 2 r^2 * sin(2*theta) sqrt(6) `&,_xUA
% 3 -3 r^3 * cos(3*theta) sqrt(8) NYwGK|
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]:!8 s\#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) j]Ua\|t
% 3 3 r^3 * sin(3*theta) sqrt(8) %&2B
% 4 -4 r^4 * cos(4*theta) sqrt(10) SZE `J:w
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7YD\ !2b
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2{gwY85:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n4R]+&*
% 4 4 r^4 * sin(4*theta) sqrt(10) V^WQ6G1
% -------------------------------------------------- -G!6U2*#
% R/rcXX7%
% Example 1: *V<)p%l.
% <L%HG
% % Display the Zernike function Z(n=5,m=1) P;>!wU~*
% x = -1:0.01:1; &gJW6<
% [X,Y] = meshgrid(x,x); `U!(cDY
% [theta,r] = cart2pol(X,Y); G\uU- z$)
% idx = r<=1; Pgx+\;w"
% z = nan(size(X)); vj(@.uU)
% z(idx) = zernfun(5,1,r(idx),theta(idx)); WQTendS
% figure A` =]RJ
% pcolor(x,x,z), shading interp bsMC#xT
% axis square, colorbar nE^wxtY
% title('Zernike function Z_5^1(r,\theta)') Ho>p ^p
% ~6MMErSj
% Example 2: iPz1eUj
% JqQ3C}z
% % Display the first 10 Zernike functions O2$!'!hz
% x = -1:0.01:1; dZ-Ny_@&
% [X,Y] = meshgrid(x,x); t3K>\ :
% [theta,r] = cart2pol(X,Y); "wF*O"WQo
% idx = r<=1; PQQgDtiH
% z = nan(size(X)); Y'?Iznb
% n = [0 1 1 2 2 2 3 3 3 3]; VDPxue
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; v F]
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 5#P: "U
% y = zernfun(n,m,r(idx),theta(idx)); <kROH0+
% figure('Units','normalized') Fu#Y7)r
% for k = 1:10 8R&z3k;!t
% z(idx) = y(:,k); ~xP
Szf
% subplot(4,7,Nplot(k)) YdPlN];[
% pcolor(x,x,z), shading interp ^NcTWbs-T
% set(gca,'XTick',[],'YTick',[]) s!bHS_\e|
% axis square CCC4(v
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) "[Yip5
% end 7Zhli Y1
% LxIuxt=X|p
% See also ZERNPOL, ZERNFUN2. 9'D8[p%
;.L!%$0i#
% Paul Fricker 11/13/2006 NT'Ie]|
hcj{%^p
twAw01".
% Check and prepare the inputs: n})
% ----------------------------- CzK%x?~]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?exALv'B
error('zernfun:NMvectors','N and M must be vectors.') *
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end _P>1`IR
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if length(n)~=length(m) OX'/?B((
error('zernfun:NMlength','N and M must be the same length.') k&n\
=tKN
end y>?k<