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 8Vy/n^3)
function z = zernfun(n,m,r,theta,nflag) f?TS#jG4}
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xwj{4fzpk{
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +UiJWO
% and angular frequency M, evaluated at positions (R,THETA) on the </b_Rar
% unit circle. N is a vector of positive integers (including 0), and z'*{V\
% M is a vector with the same number of elements as N. Each element PbfgWGr
% k of M must be a positive integer, with possible values M(k) = -N(k) kG5Uc83#G
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, X<H{
% and THETA is a vector of angles. R and THETA must have the same FD5OO;$
% length. The output Z is a matrix with one column for every (N,M) {{AZW
% pair, and one row for every (R,THETA) pair. (w vU;u
% C=bQ2t=Z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 39d$B'"<1
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), C}ASVywc,1
% with delta(m,0) the Kronecker delta, is chosen so that the integral h+S]C#X,}
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "N)InPR-
% and theta=0 to theta=2*pi) is unity. For the non-normalized RY1-Zjlb<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. NN5G
'|i
% 9m<%+S5&
% The Zernike functions are an orthogonal basis on the unit circle. i(*fv(z
% They are used in disciplines such as astronomy, optics, and N,.awA{
% optometry to describe functions on a circular domain. ,!X:wY}dW
% DR]4Tc z#
% The following table lists the first 15 Zernike functions. vQj{yJ\l1
% Gyrc~m[$
% n m Zernike function Normalization ,c
0]r;u!
% -------------------------------------------------- HBs
6:[q
% 0 0 1 1 B1]FB|0's
% 1 1 r * cos(theta) 2 1^ iLs
% 1 -1 r * sin(theta) 2 ",' Zr<T
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]:m4~0^#-(
% 2 0 (2*r^2 - 1) sqrt(3) DiZ;FHnaG?
% 2 2 r^2 * sin(2*theta) sqrt(6) Z-yoJZi
% 3 -3 r^3 * cos(3*theta) sqrt(8) g4{0
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0_,un^
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 1:_}`x=hM
% 3 3 r^3 * sin(3*theta) sqrt(8) 4q(,uk&R[
% 4 -4 r^4 * cos(4*theta) sqrt(10) uo*lW2&U
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M:L-j{?y_
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ,b?G]WQrHs
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tK
`A_hC
% 4 4 r^4 * sin(4*theta) sqrt(10) ~#)9Kl7<X
% -------------------------------------------------- 9$}>O]
% b@sq}8YD|z
% Example 1: +UX}
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% ~}SQLYy7Z
% % Display the Zernike function Z(n=5,m=1) 9>ZX@1]m_
% x = -1:0.01:1; #-{ljjMQI
% [X,Y] = meshgrid(x,x); SRU#Y8Xv|
% [theta,r] = cart2pol(X,Y); XhN?E-WywQ
% idx = r<=1; ]BTISaL-R
% z = nan(size(X)); R;uP^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?,C'\8'
% figure " LhXR
% pcolor(x,x,z), shading interp ^K
9jJS9K
% axis square, colorbar Ye^xV,U@
% title('Zernike function Z_5^1(r,\theta)') @&9<)1F
% 3M'Y'Szm
% Example 2: [|YJg]i-
% 1{
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% % Display the first 10 Zernike functions 4 XGEw9`3
% x = -1:0.01:1; Nov
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% [X,Y] = meshgrid(x,x); 8^R~qpg%
% [theta,r] = cart2pol(X,Y); n@S|^cH
% idx = r<=1; &yqk96z
% z = nan(size(X)); Ie8SPNY-H
% n = [0 1 1 2 2 2 3 3 3 3]; |>-0q~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 6+C]rEY/o
% Nplot = [4 10 12 16 18 20 22 24 26 28]; F$9+WS`c
% y = zernfun(n,m,r(idx),theta(idx)); u!b0<E
% figure('Units','normalized') n:Dr< q.
% for k = 1:10 tMo=q7ig
% z(idx) = y(:,k); a`Q-5*\;z
% subplot(4,7,Nplot(k)) 6c}nP[6|
% pcolor(x,x,z), shading interp '[bw7T
% set(gca,'XTick',[],'YTick',[]) 5 L-6@@/
% axis square y@Td]6|f
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [kPl7[OL
% end w2K>k/v{-
% '%a:L^a?
% See also ZERNPOL, ZERNFUN2. 1z@ ncqe
59?$9}ob
% Paul Fricker 11/13/2006 Yof]
"<"s&ws;k
ffaMF~+
% Check and prepare the inputs: ;3Q3!+%j
% ----------------------------- cQ0+kX<
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5)gC<
error('zernfun:NMvectors','N and M must be vectors.') W@~a#~1O
end V<d`.9*}
nNRc@9Lt
if length(n)~=length(m) kQrby\F(<
error('zernfun:NMlength','N and M must be the same length.')
"b`3
end vnX~OVz2
mrlhj8W?!
n = n(:); xJFxrG'c
m = m(:); CR-2>,*a9
if any(mod(n-m,2)) }jg,[jw_"X
error('zernfun:NMmultiplesof2', ... Qaiqx"x3
'All N and M must differ by multiples of 2 (including 0).') *bi;mQ
end T`Xz*\}Zb
kB-<17
if any(m>n) i"{znKz vD
error('zernfun:MlessthanN', ... q]y{
4"=5
'Each M must be less than or equal to its corresponding N.') >a: 6umY
end hP
jL
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if any( r>1 | r<0 ) Lvn+EM
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =8DS~J{
end U#4>GO;A
nB%[\LtZ?
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $u,`bX
error('zernfun:RTHvector','R and THETA must be vectors.') Kq:vTz&<
end '^.3}N{Fo
*(nu0
r = r(:); CbT ;#0
theta = theta(:); s18A
length_r = length(r); bWMb@zm
if length_r~=length(theta) Qs_]U
error('zernfun:RTHlength', ... L#/<y{
'The number of R- and THETA-values must be equal.') gE6{R+sp
end #LG<o3An
A)nE+ec1
% Check normalization: !GoHCe[10
% -------------------- {)-3g~
if nargin==5 && ischar(nflag) ABhQ7
x|
isnorm = strcmpi(nflag,'norm'); GUsJF;;V
if ~isnorm zHvW@A'F
error('zernfun:normalization','Unrecognized normalization flag.') /ASpAl[J
end (}CA?/
else ZZW%6 -B
isnorm = false; YU1z\pK
end #M:Vwn
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