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
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function z = zernfun(n,m,r,theta,nflag) j7VaaA
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 2y9$ k\<xV
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N AxCFZf 5
% and angular frequency M, evaluated at positions (R,THETA) on the X>MDX.Z
% unit circle. N is a vector of positive integers (including 0), and _wZr`E)
% M is a vector with the same number of elements as N. Each element O+~@S~
% k of M must be a positive integer, with possible values M(k) = -N(k) cvV8;
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, YXGxE&!
% and THETA is a vector of angles. R and THETA must have the same h;J%Z!Rjw
% length. The output Z is a matrix with one column for every (N,M) $rQi$w/
% pair, and one row for every (R,THETA) pair. =jRC4]M})
% 7+P-MT
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qwd
T=H
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;O({|mpS\
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7t6TB*H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {=P}c:iW
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,WS{O6O7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Pm|S>r
% 4/&.N]
% The Zernike functions are an orthogonal basis on the unit circle. -L2%,.E>4
% They are used in disciplines such as astronomy, optics, and /I0}(;^y
% optometry to describe functions on a circular domain. WAb@d=H{+>
% AD"L>7
% The following table lists the first 15 Zernike functions. H$)otDOE
% .[vYT.LE
% n m Zernike function Normalization va;fT+k=
% -------------------------------------------------- K`kWfPwp
% 0 0 1 1 i0[mU,
% 1 1 r * cos(theta) 2 )AAPT7!U
% 1 -1 r * sin(theta) 2 6
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% 2 -2 r^2 * cos(2*theta) sqrt(6) }A7]bd
% 2 0 (2*r^2 - 1) sqrt(3) l>@){zxL
% 2 2 r^2 * sin(2*theta) sqrt(6) ztV%W6
% 3 -3 r^3 * cos(3*theta) sqrt(8) -qDL':
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) p+:MZP -%(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8s6^!e&
% 3 3 r^3 * sin(3*theta) sqrt(8) dijHi
% 4 -4 r^4 * cos(4*theta) sqrt(10) ?qczMck_
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;VPYWss
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5f_1 dn
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +Pb@@C&
% 4 4 r^4 * sin(4*theta) sqrt(10) [vcSt5R=
% -------------------------------------------------- iiV'-!3w
% WI\h@qSB
% Example 1: tL
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% w#RfD
% % Display the Zernike function Z(n=5,m=1) w;V+)r?w
% x = -1:0.01:1; ||rZ+<
% [X,Y] = meshgrid(x,x); G8OnNI
% [theta,r] = cart2pol(X,Y); p~Mw^SN'
% idx = r<=1; 4tFnZ2x
% z = nan(size(X)); Wvwjj~HP2}
% z(idx) = zernfun(5,1,r(idx),theta(idx)); biAa&
% figure 8,?*eYNjb
% pcolor(x,x,z), shading interp gqACIXR
% axis square, colorbar vA0f4W 8+
% title('Zernike function Z_5^1(r,\theta)') ag"Nf-o/Y
% sm;\;MP*yH
% Example 2: -|/*S]6kK
% m~vEandm
% % Display the first 10 Zernike functions !+ ??3-q
% x = -1:0.01:1; C'fQ Z,r-v
% [X,Y] = meshgrid(x,x); OG2&=~hOz-
% [theta,r] = cart2pol(X,Y); ?YhGW
% idx = r<=1; lgh+\pj
% z = nan(size(X)); 87:V-*8
% n = [0 1 1 2 2 2 3 3 3 3]; WlnS.P\+E
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; "$N 4S9U
% Nplot = [4 10 12 16 18 20 22 24 26 28]; oJVpJA0IA
% y = zernfun(n,m,r(idx),theta(idx)); 6g%~~hX
% figure('Units','normalized') k3r<']S^
% for k = 1:10 -^= JKd&p
% z(idx) = y(:,k); <|4L+?_(&
% subplot(4,7,Nplot(k)) ~X1<x4P\
% pcolor(x,x,z), shading interp %51HJB}C]
% set(gca,'XTick',[],'YTick',[]) 8DZ
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% axis square 2B=+p83<
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t$b{zv9C
% end ?
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% -5Ln3\ O@
% See also ZERNPOL, ZERNFUN2. MF.$E?_R
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% Paul Fricker 11/13/2006 %scQP{%aD
Mg=R**s1x%
teg[l-R"7z
% Check and prepare the inputs: e^Glgaf
% ----------------------------- uZ(,7>0
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (t2vt[A6ph
error('zernfun:NMvectors','N and M must be vectors.') TvwkeOS#}7
end A7sva@}W
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if length(n)~=length(m) :V:siIDn
error('zernfun:NMlength','N and M must be the same length.') @!2vS@f
end a
#Pr)H
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n = n(:); QF9$SCmv
m = m(:); ,(&