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 aM
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function z = zernfun(n,m,r,theta,nflag) u|Oc+qA(
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ::+;PRy_E
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z^}[CQ&Am
% and angular frequency M, evaluated at positions (R,THETA) on the (t\U5-w
% unit circle. N is a vector of positive integers (including 0), and fdWqc_
% M is a vector with the same number of elements as N. Each element \>>P%EU,
% k of M must be a positive integer, with possible values M(k) = -N(k) piH0_7qr
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pGfGGY>i%
% and THETA is a vector of angles. R and THETA must have the same dF09_nw
% length. The output Z is a matrix with one column for every (N,M) ,2rfN"o
% pair, and one row for every (R,THETA) pair. Ozhn`9L+1!
% z@J>A![m
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike K@JaN/OM
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [KFCc_:
% with delta(m,0) the Kronecker delta, is chosen so that the integral ByuBZ!m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RJUIB
% and theta=0 to theta=2*pi) is unity. For the non-normalized D)pTE?@W'
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }zS5o
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% j:qexhtho
% The Zernike functions are an orthogonal basis on the unit circle. Mo<q(_ZeRP
% They are used in disciplines such as astronomy, optics, and sa&`CEa
% optometry to describe functions on a circular domain. WF-jy7+
% _si 5z
% The following table lists the first 15 Zernike functions. -%]1q#C>@
% +Z2XP76(4A
% n m Zernike function Normalization =E>P,"D
% -------------------------------------------------- Y8^WuN$
% 0 0 1 1 A^p{Cq@E
% 1 1 r * cos(theta) 2 ^-Ygh[x
% 1 -1 r * sin(theta) 2 K9.Gjw
% 2 -2 r^2 * cos(2*theta) sqrt(6) :s&dn%5N"
% 2 0 (2*r^2 - 1) sqrt(3) _9t1aP5
% 2 2 r^2 * sin(2*theta) sqrt(6) F~qZIggD
% 3 -3 r^3 * cos(3*theta) sqrt(8) )`(]jx!
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~bm'i%$k
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) oPF]]Imu
% 3 3 r^3 * sin(3*theta) sqrt(8) jDqG9]
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,~&HL7v
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7)Vbp--b#
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Z\U r F0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b>8TH-1t~
% 4 4 r^4 * sin(4*theta) sqrt(10) @&EIH,c
% -------------------------------------------------- xp'Q>%v
% !zx8I7e4
% Example 1: ;Vc|3
% uDXV@;6<
% % Display the Zernike function Z(n=5,m=1) 4bp})>}jB
% x = -1:0.01:1; \lm]G7h
% [X,Y] = meshgrid(x,x); fqY'Uq$=
% [theta,r] = cart2pol(X,Y); ,c^nW
% idx = r<=1; qljsoDG
% z = nan(size(X)); $,]U~7S
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 9(q(;|;Hp
% figure d23=WNn
% pcolor(x,x,z), shading interp nD XEm6|e
% axis square, colorbar TwI s_r:
% title('Zernike function Z_5^1(r,\theta)') Y I;iG[T,&
% TEY~E*=}$
% Example 2: 'sH_^{V2
% {QylNC9
% % Display the first 10 Zernike functions OqDP{X:
% x = -1:0.01:1; 7L6L{~8
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% [X,Y] = meshgrid(x,x); mICEJ\`x
% [theta,r] = cart2pol(X,Y); R'zi#FeP
% idx = r<=1; HnKgD:
% z = nan(size(X)); Wh| T3&
% n = [0 1 1 2 2 2 3 3 3 3]; j}",+Hv
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;m#4Q6k)V?
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ;>jEeIlT
% y = zernfun(n,m,r(idx),theta(idx)); ;h+~xxu=X
% figure('Units','normalized') sH;_U)ssH
% for k = 1:10 ?#xm6oe#aH
% z(idx) = y(:,k); \>Rfa+
% subplot(4,7,Nplot(k)) =WW5H\?
% pcolor(x,x,z), shading interp p> >H$t
% set(gca,'XTick',[],'YTick',[]) RU4X#gP4Vh
% axis square o.A:29KoU
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sAqy(oy#M
% end J](NCD
% 6(d6Uwc`
% See also ZERNPOL, ZERNFUN2. 4Ex&A