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 ~gbq^
function z = zernfun(n,m,r,theta,nflag) L5>.ku=T
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;Q8rAsf9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <+7-^o_
% and angular frequency M, evaluated at positions (R,THETA) on the !P* z=
% unit circle. N is a vector of positive integers (including 0), and SJI+$L\'
% M is a vector with the same number of elements as N. Each element cW, 6MAQo
% k of M must be a positive integer, with possible values M(k) = -N(k) b"#|0d0
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Qte'f+
% and THETA is a vector of angles. R and THETA must have the same D\GP+Ota
% length. The output Z is a matrix with one column for every (N,M) Y]1b39O
% pair, and one row for every (R,THETA) pair. r \]iw v
% tB{O6=q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike n&uD=-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), R*psL&N
% with delta(m,0) the Kronecker delta, is chosen so that the integral itIzs99j
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s@bo df&
% and theta=0 to theta=2*pi) is unity. For the non-normalized >sE{c>R%
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -J*jW
N!
% (%EhkTb
% The Zernike functions are an orthogonal basis on the unit circle. h3Z0NJ=xM
% They are used in disciplines such as astronomy, optics, and 3YPoObY
% optometry to describe functions on a circular domain. G8oOFBQD
% U ()36
% The following table lists the first 15 Zernike functions. sHulaX{
% as6YjE.Yy
% n m Zernike function Normalization 8CKI9
% -------------------------------------------------- w;Na9tR
% 0 0 1 1 [Y]\sF;J
% 1 1 r * cos(theta) 2 0dgp<
% 1 -1 r * sin(theta) 2 u=h/l!lR
% 2 -2 r^2 * cos(2*theta) sqrt(6) hpJi,4r.d
% 2 0 (2*r^2 - 1) sqrt(3) ;M"JN:J8
% 2 2 r^2 * sin(2*theta) sqrt(6) HGpj(U:`c
% 3 -3 r^3 * cos(3*theta) sqrt(8) q\g|K3V)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) f=Rx8I
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) J@c)SK%2h
% 3 3 r^3 * sin(3*theta) sqrt(8) ']ussFaQ
% 4 -4 r^4 * cos(4*theta) sqrt(10)
( XoL,lJ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;u0MY
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z@3t>k|K
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %g4G&My@J
% 4 4 r^4 * sin(4*theta) sqrt(10) o4CgtqRs
% -------------------------------------------------- lclSzC9
% )xuvY3BPB?
% Example 1: P p[?E.]P
% Ojf.D6nY
% % Display the Zernike function Z(n=5,m=1) g2v0!
% x = -1:0.01:1; @<O
Bt d
% [X,Y] = meshgrid(x,x); 0XBv8fg
% [theta,r] = cart2pol(X,Y); wQX,a;Br
% idx = r<=1; UmSy p\i
% z = nan(size(X));
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); /}\EMP
% figure J
;=~QYn[
% pcolor(x,x,z), shading interp ch}t++`l]
% axis square, colorbar j ,'$i[F'
% title('Zernike function Z_5^1(r,\theta)') Ph'P<h:V
% Vs)Pg\B?
% Example 2: {re<S<j&
% Oozt&* F
% % Display the first 10 Zernike functions %(,Kj
~0
% x = -1:0.01:1; ;{79d8/=
% [X,Y] = meshgrid(x,x); Yp1;5Bbp
% [theta,r] = cart2pol(X,Y); I]|X6
% idx = r<=1; "RH pj3 si
% z = nan(size(X)); Pvq74?an`
% n = [0 1 1 2 2 2 3 3 3 3]; |<l
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; lU0'5!3R,
% Nplot = [4 10 12 16 18 20 22 24 26 28]; i"~J -{d}
% y = zernfun(n,m,r(idx),theta(idx)); |gW>D=rkj
% figure('Units','normalized') lr:rQw9
% for k = 1:10 ^#T@NN0T
% z(idx) = y(:,k); #MbkU])
% subplot(4,7,Nplot(k)) VFj}{Y
% pcolor(x,x,z), shading interp Qx-/t 9`!Z
% set(gca,'XTick',[],'YTick',[]) |^^'GZ%a
% axis square TzT(aWP"
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /*)zQ?N
% end K]{Y >w
% J|-X?V;ZW
% See also ZERNPOL, ZERNFUN2. *"\QR>n
(,wIbwa
% Paul Fricker 11/13/2006 EIqe|a+
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|kPjjVGF{
% Check and prepare the inputs: nm)H\i
% ----------------------------- ]o18oY(
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rz%8Vigb
error('zernfun:NMvectors','N and M must be vectors.') B8){
end .tv'`
K}e%E&|>
if length(n)~=length(m) 'O%itCy)
error('zernfun:NMlength','N and M must be the same length.') j\kT
H
end 1 ]Q;fe
WZ\bm$
n = n(:); R_IUuz$e
m = m(:); N?Byp&rqI<
if any(mod(n-m,2)) V(hM@ztN
error('zernfun:NMmultiplesof2', ... v]UT1d=_T
'All N and M must differ by multiples of 2 (including 0).') i^SuVca
end iI|mFc|V
I!FIV^}Z(
if any(m>n) .E H&GX
error('zernfun:MlessthanN', ... AgEX,SPP
'Each M must be less than or equal to its corresponding N.') 0!<qfT
a
end )k)HQcfjD
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if any( r>1 | r<0 ) 3q'["SS
error('zernfun:Rlessthan1','All R must be between 0 and 1.') lyY\P6
X
end 77KB-l2
T?vM\o%i3
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 00jW s@K
error('zernfun:RTHvector','R and THETA must be vectors.') GtR!a
end BbU%p
7+_TdDBYs
r = r(:); #0HZ"n
theta = theta(:); BC: d@
length_r = length(r); nHAET
if length_r~=length(theta) Blw AD
error('zernfun:RTHlength', ... LqNt.d @
'The number of R- and THETA-values must be equal.') O+iNR9O
end ?4k/V6n@y
WP*xu-(:
% Check normalization: %rE:5)
% -------------------- _C`&(?}
if nargin==5 && ischar(nflag) ;Gc,-BDFw
isnorm = strcmpi(nflag,'norm'); #`Af
if ~isnorm ( *~ '#k
error('zernfun:normalization','Unrecognized normalization flag.') tx` Z?K[
end /b&ka&|t
else ,7HlYPec
isnorm = false; z):LF<
end O*Gg57a
W&g