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 3 bGpK9M~
function z = zernfun(n,m,r,theta,nflag) cp[k[7XGD
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. KbSIKj
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N BLfoU _Z
% and angular frequency M, evaluated at positions (R,THETA) on the Cvq2UNz(R
% unit circle. N is a vector of positive integers (including 0), and U2!9Tl9".
% M is a vector with the same number of elements as N. Each element voCQ_~*)9
% k of M must be a positive integer, with possible values M(k) = -N(k) 3<?#*z4]_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, eFbr1IV
% and THETA is a vector of angles. R and THETA must have the same Zs)HzOP)9
% length. The output Z is a matrix with one column for every (N,M) RBiDU}j
% pair, and one row for every (R,THETA) pair. 3%'$AM}+s
% }F**!%4d
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 'R?;T[s%
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), h@/c76}f6p
% with delta(m,0) the Kronecker delta, is chosen so that the integral ->:G+<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f5^[`b3H
% and theta=0 to theta=2*pi) is unity. For the non-normalized l3-;z)SgH
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {B uh5U,
% Fn$EP:>
% The Zernike functions are an orthogonal basis on the unit circle. TDA+ rl
% They are used in disciplines such as astronomy, optics, and ,+%$vV
.g\
% optometry to describe functions on a circular domain. |5`z;u7V
% H 2\KI(
% The following table lists the first 15 Zernike functions. =((#k DrN
% E[^66(KR
% n m Zernike function Normalization ;E(%s=i
% -------------------------------------------------- StA5h+[m
% 0 0 1 1 *tO7A$LDT
% 1 1 r * cos(theta) 2
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% 1 -1 r * sin(theta) 2 4w\cS&X~C
% 2 -2 r^2 * cos(2*theta) sqrt(6) (Z;-u+ }.
% 2 0 (2*r^2 - 1) sqrt(3) 5q}680s9+
% 2 2 r^2 * sin(2*theta) sqrt(6) 1]m]b4]
% 3 -3 r^3 * cos(3*theta) sqrt(8) h)fi9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {088j?[hzk
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) do C8!
% 3 3 r^3 * sin(3*theta) sqrt(8) Mo0+"`
% 4 -4 r^4 * cos(4*theta) sqrt(10) Jah~h44&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *EvnN:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5L%A5C&|
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +m]$P,yMt
% 4 4 r^4 * sin(4*theta) sqrt(10) +t})tDPXw
% -------------------------------------------------- >y
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% mn)kd
% Example 1: C1Slx!}
% vn9_tL&
% % Display the Zernike function Z(n=5,m=1) ZV$qv=X
% x = -1:0.01:1; sTU`@}}
% [X,Y] = meshgrid(x,x); t[XxLG*
% [theta,r] = cart2pol(X,Y); _p<s!
% idx = r<=1; $RfM}!7?
% z = nan(size(X)); X~T"n<:a>
% z(idx) = zernfun(5,1,r(idx),theta(idx)); kTL{?-
% figure _t_X`
% pcolor(x,x,z), shading interp Nl"< $/
% axis square, colorbar .'saUcVg:
% title('Zernike function Z_5^1(r,\theta)') 5^~%10=
% Uo#%f+t
% Example 2: BC =U6>`/
% ri<E[8\
% % Display the first 10 Zernike functions 4N|^Joi
% x = -1:0.01:1; ]'3e#Cqeh
% [X,Y] = meshgrid(x,x); Y.$'<1
% [theta,r] = cart2pol(X,Y); s`B "qw
% idx = r<=1; }Zu>?U
% z = nan(size(X)); y2bL!Y<s9
% n = [0 1 1 2 2 2 3 3 3 3]; ^F"Q~?D)
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; yZE"t[q#O
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >gtKyn]
% y = zernfun(n,m,r(idx),theta(idx)); >jD,%yG
% figure('Units','normalized') QWp,(Mv:r
% for k = 1:10 SQ9s
% z(idx) = y(:,k); &'uFy0d,
% subplot(4,7,Nplot(k)) /p+ (_Y
% pcolor(x,x,z), shading interp Iww.Nd2
% set(gca,'XTick',[],'YTick',[]) -
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% axis square T0J"Wr>WY
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) u=JI 1
% end M^JRHpTn
% HS =qK
% See also ZERNPOL, ZERNFUN2. q{gt2OWqX
&=oW=g 2
% Paul Fricker 11/13/2006 S-&[Tp+N
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% Check and prepare the inputs: CZcnX8P'8
% ----------------------------- +P2f<~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z[[o u?c
error('zernfun:NMvectors','N and M must be vectors.') g!;k$`@{E'
end ]PJb 9$f2
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if length(n)~=length(m) @.T(\Dq^
error('zernfun:NMlength','N and M must be the same length.') .]}kOw:(#
end # Y/.%ch.
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n = n(:); MQ>.^]B]o
m = m(:); l=G=J( G
if any(mod(n-m,2)) UE33e(Q<
error('zernfun:NMmultiplesof2', ... L5=Tj4`
'All N and M must differ by multiples of 2 (including 0).') `@eo <6
end Ch8w_Jf1yx
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if any(m>n) f|G,pDLx
error('zernfun:MlessthanN', ... OoL#8R
'Each M must be less than or equal to its corresponding N.') H7bdL 8/
end 7714}%Z
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if any( r>1 | r<0 ) lX$6U|!
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ICwhqH&
end `oQ)qa_
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U9GmkXRix
error('zernfun:RTHvector','R and THETA must be vectors.') yG -1g0
end __<