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 vT|`%~Be
function z = zernfun(n,m,r,theta,nflag) q:I$EpKf?Q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Wf#VA;d
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3jjMY
% and angular frequency M, evaluated at positions (R,THETA) on the lbQQtpEKO
% unit circle. N is a vector of positive integers (including 0), and O] Y v
% M is a vector with the same number of elements as N. Each element qve
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% k of M must be a positive integer, with possible values M(k) = -N(k)
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, C|MQ
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% and THETA is a vector of angles. R and THETA must have the same hoa7
% length. The output Z is a matrix with one column for every (N,M) Tc6:UF
% pair, and one row for every (R,THETA) pair. #B8*gFZB
% e ^Ds
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike b_TS<,
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Lxv6!?v|
% with delta(m,0) the Kronecker delta, is chosen so that the integral B
z^|SkEit
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, z-dFDtiA
% and theta=0 to theta=2*pi) is unity. For the non-normalized F.tfgW(A@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ,R?np9wc
% _]b3,%2
% The Zernike functions are an orthogonal basis on the unit circle. y%S1ZTScO
% They are used in disciplines such as astronomy, optics, and hfqqQ!,l!
% optometry to describe functions on a circular domain. :_ROJ
% !v|FT.
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% The following table lists the first 15 Zernike functions. 5;\gJf
% Wy4$*$
% n m Zernike function Normalization 8"%RCE
% --------------------------------------------------
%@Oma
% 0 0 1 1 w]u@G-e
% 1 1 r * cos(theta) 2 OoBCY-gj*
% 1 -1 r * sin(theta) 2 )[L^Dmd,
% 2 -2 r^2 * cos(2*theta) sqrt(6) G,f-.
% 2 0 (2*r^2 - 1) sqrt(3) "T2"]u<52
% 2 2 r^2 * sin(2*theta) sqrt(6) Q8D&tJg
% 3 -3 r^3 * cos(3*theta) sqrt(8) a2w T6jY
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 08s_v=cF
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iCj2"T4TN
% 3 3 r^3 * sin(3*theta) sqrt(8)
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% 4 -4 r^4 * cos(4*theta) sqrt(10) hB9Ee@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IvHh4DU3Z
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [kV;[c}
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /ASaB
% 4 4 r^4 * sin(4*theta) sqrt(10) \25/$Ae}c
% -------------------------------------------------- p5\]5bb
% :i~W
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% Example 1: JIm4vS
% G(iJi
% % Display the Zernike function Z(n=5,m=1)
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% x = -1:0.01:1; gU&%J4O
% [X,Y] = meshgrid(x,x); j}1zdA
% [theta,r] = cart2pol(X,Y); D&G"BZx|
% idx = r<=1; P 1XK*GZ
% z = nan(size(X)); G{Yz8]m
% z(idx) = zernfun(5,1,r(idx),theta(idx)); lg:
% figure 5cJ!"
% pcolor(x,x,z), shading interp %" D%:
% axis square, colorbar 6$U]9D
% title('Zernike function Z_5^1(r,\theta)') t5B7I59
% <TGn=>u
% Example 2: hR#-u1C
% e~l#4{w
% % Display the first 10 Zernike functions
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% x = -1:0.01:1; VU`OO$,W
% [X,Y] = meshgrid(x,x); oA] KE"T
% [theta,r] = cart2pol(X,Y); sRSz}]
% idx = r<=1; 7hP<f}xL
% z = nan(size(X)); k%s_0
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% n = [0 1 1 2 2 2 3 3 3 3]; =m89z}Ot
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; #Z+i~t{e(
% Nplot = [4 10 12 16 18 20 22 24 26 28]; r;BT,jiX
% y = zernfun(n,m,r(idx),theta(idx)); ~{hxR)x9
% figure('Units','normalized') E>b2+;Jv
% for k = 1:10 Zxr!:t7
% z(idx) = y(:,k); Vd^g9
% subplot(4,7,Nplot(k)) uvDzKMw~R
% pcolor(x,x,z), shading interp fmqb`%
% set(gca,'XTick',[],'YTick',[]) C+[%7vF1
% axis square ) J]9 lW&y
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [^CV>RuO
% end Y3.$G1{#0w
% q6Rr.A
% See also ZERNPOL, ZERNFUN2. :Z`:nq.a
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