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 juR
function z = zernfun(n,m,r,theta,nflag) K7+^Yv\YQx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. P&h/IBA_
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N JE/l#Q!
% and angular frequency M, evaluated at positions (R,THETA) on the jt/l,=9YK
% unit circle. N is a vector of positive integers (including 0), and zz[g{[SN
% M is a vector with the same number of elements as N. Each element t&8<k+m
% k of M must be a positive integer, with possible values M(k) = -N(k) UP5%C;
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, \l>qY(gu
% and THETA is a vector of angles. R and THETA must have the same 4{g:^?1=
% length. The output Z is a matrix with one column for every (N,M) C5BzWgK
% pair, and one row for every (R,THETA) pair. *1R##9\jU7
% "^18&>^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fp,1qzU[k
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gD,A9a(3
% with delta(m,0) the Kronecker delta, is chosen so that the integral 9UB??049z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, $>nkGb%Kp
% and theta=0 to theta=2*pi) is unity. For the non-normalized M^Q&A R'F
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [8xeQKp4
% 5?D1][
% The Zernike functions are an orthogonal basis on the unit circle. t%0r"bTi
% They are used in disciplines such as astronomy, optics, and Hf!9`R[
% optometry to describe functions on a circular domain. WNCM|VUl
% u XaL
% The following table lists the first 15 Zernike functions. fmatc#G
% ^)(G(=-Rf
% n m Zernike function Normalization ~+7a d$
% -------------------------------------------------- YK(XS"Kl
% 0 0 1 1 FZM
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% 1 1 r * cos(theta) 2 D!81(}p
% 1 -1 r * sin(theta) 2 Kc%tnVyGh:
% 2 -2 r^2 * cos(2*theta) sqrt(6) *2w_oKE'+5
% 2 0 (2*r^2 - 1) sqrt(3) i!s~kk
% 2 2 r^2 * sin(2*theta) sqrt(6) `;zu1o
% 3 -3 r^3 * cos(3*theta) sqrt(8) XfD
z
#
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) u>JqFw1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m$j
n5:
% 3 3 r^3 * sin(3*theta) sqrt(8) ^yzo!`)fso
% 4 -4 r^4 * cos(4*theta) sqrt(10) =d:R/Z%,
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;9 =}_h)]
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tf.q~@Pi
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >#Grf)@"6
% 4 4 r^4 * sin(4*theta) sqrt(10) Ak<IHp^Q
% -------------------------------------------------- CpBQ>!CW
% !7kAJG g
% Example 1: N]3-L`t
% ?CcR
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% % Display the Zernike function Z(n=5,m=1) &!H~bzg
% x = -1:0.01:1; ?,A}E|jZ
% [X,Y] = meshgrid(x,x); HV#?6,U}
% [theta,r] = cart2pol(X,Y); SSSDl$}'t
% idx = r<=1; P wt ?9I
% z = nan(size(X)); V{7lltu
% z(idx) = zernfun(5,1,r(idx),theta(idx)); :)^#
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% figure 0KWy?6 X
% pcolor(x,x,z), shading interp ;EE{~
% axis square, colorbar ?NL&x
% title('Zernike function Z_5^1(r,\theta)') I@y2HxM
% =woqHTR
% Example 2: a PcGI
% y<I Z|f
% % Display the first 10 Zernike functions ,}xpYq_/
% x = -1:0.01:1; A>&>6O4
% [X,Y] = meshgrid(x,x); m!FM+kge
% [theta,r] = cart2pol(X,Y); 0+VncL)u
% idx = r<=1; 7cOg(6N
% z = nan(size(X)); _oMs
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% n = [0 1 1 2 2 2 3 3 3 3]; u"Hd55"&
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; oHc-0$eMKY
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Y]`lEq%
% y = zernfun(n,m,r(idx),theta(idx)); '6dD^0dZ
% figure('Units','normalized') `-9*@_-=M
% for k = 1:10 #J<`p
% z(idx) = y(:,k); s)`1Rf
% subplot(4,7,Nplot(k))
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% pcolor(x,x,z), shading interp J*^,l`C/
% set(gca,'XTick',[],'YTick',[]) SSA%1l2!
% axis square ],fwZd[t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) r(?'Y y
% end Fw_bY/WN{
% V5(tf'
% See also ZERNPOL, ZERNFUN2. &t9XK 8S
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% Paul Fricker 11/13/2006 {Vt^Xc
mBD!:V'
)N%1%bg^-
% Check and prepare the inputs: tnKpn-LPA
% ----------------------------- l/y
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) F]dd>#
error('zernfun:NMvectors','N and M must be vectors.') JQ{zWJlt
end an[3vKb
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M
if length(n)~=length(m) ]y)R C-N
error('zernfun:NMlength','N and M must be the same length.') >X\s[d&(
end j4
&
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n = n(:); AT'_0>x8
m = m(:); y9re17{
X
if any(mod(n-m,2)) R>YMGUH~w
error('zernfun:NMmultiplesof2', ... "k_n+cH%
'All N and M must differ by multiples of 2 (including 0).') ixI5Xd<
end ,nu7r1}
X~Rk ,d3
if any(m>n) nV,{w4t+
error('zernfun:MlessthanN', ... O>"r. sR
'Each M must be less than or equal to its corresponding N.') ,$zSJzS
end "DcueU#!
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if any( r>1 | r<0 ) wEHrer
error('zernfun:Rlessthan1','All R must be between 0 and 1.') O(
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end ]cGz~TN~
J9$]]\52s.
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;o)`9<es!2
error('zernfun:RTHvector','R and THETA must be vectors.') @qr3v>3X<
end [&O:qaD^
%]:vT&M
r = r(:); [:hy
theta = theta(:); ?/|@ #&
length_r = length(r); dnWt\>6&
2
if length_r~=length(theta) >!v,`O1
error('zernfun:RTHlength', ... @)juP- o%
'The number of R- and THETA-values must be equal.') HTtGpTsF
end (=3&