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 wd..{j0&
function z = zernfun(n,m,r,theta,nflag) Q&`$:h.~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9WtTUk
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N p8Lb*7W
% and angular frequency M, evaluated at positions (R,THETA) on the BI4p 3-
% unit circle. N is a vector of positive integers (including 0), and q/70fR7{v
% M is a vector with the same number of elements as N. Each element :ozHuHJ#
% k of M must be a positive integer, with possible values M(k) = -N(k) ?
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <^n9?[m*
% and THETA is a vector of angles. R and THETA must have the same ;#`Z(A}
% length. The output Z is a matrix with one column for every (N,M) Zp-
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% pair, and one row for every (R,THETA) pair. 'PV,c|f>
% {< jLfL1
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0/{-X[z
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *% Vd2jW/
% with delta(m,0) the Kronecker delta, is chosen so that the integral kj@#oLd%
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, k5g\s9n]
% and theta=0 to theta=2*pi) is unity. For the non-normalized )bi*y`UM]
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #mx;t3ja7
% <|
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% The Zernike functions are an orthogonal basis on the unit circle. 1MfRFv
% They are used in disciplines such as astronomy, optics, and N5%~~JRO
% optometry to describe functions on a circular domain. rIW`(IG_
% !G`w@E9M)
% The following table lists the first 15 Zernike functions. rZ:
% WDE_"Mm
% n m Zernike function Normalization ` mALx! `
% -------------------------------------------------- +vDT^|2SF
% 0 0 1 1 () b0Sh=
% 1 1 r * cos(theta) 2 ;)"r^M)):
% 1 -1 r * sin(theta) 2 lSXhHy
% 2 -2 r^2 * cos(2*theta) sqrt(6) CEqfsKrsxE
% 2 0 (2*r^2 - 1) sqrt(3) tQo"$ JN}
% 2 2 r^2 * sin(2*theta) sqrt(6)
F_YZV)q!W
% 3 -3 r^3 * cos(3*theta) sqrt(8) aH'^`]'_=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) (Clf]\_II
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~NU~jmT2
% 3 3 r^3 * sin(3*theta) sqrt(8) f=}u;^
% 4 -4 r^4 * cos(4*theta) sqrt(10) rAP+nh ans
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \ E[0KvN;O
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !Q#u
i[0q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =E4nNL?
% 4 4 r^4 * sin(4*theta) sqrt(10) Br\/7F
% -------------------------------------------------- O=c&
% K# _plpr
% Example 1: &/=xtO/Z{
% =k3QymA
% % Display the Zernike function Z(n=5,m=1) Vk0O^o
% x = -1:0.01:1; -?LSw
% [X,Y] = meshgrid(x,x); PcDPRX!@
% [theta,r] = cart2pol(X,Y); r8^1JJ~\
% idx = r<=1; 1;ZEuO
% z = nan(size(X)); {oBVb{<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 'Nfg%)-N
% figure bsqoR8
% pcolor(x,x,z), shading interp 8ne5 B4
% axis square, colorbar ,!sAr;Rk`
% title('Zernike function Z_5^1(r,\theta)') 2S!=2u+7
% pxDZ}4mOh
% Example 2: V!]e#QH;
% a`/[\K6
% % Display the first 10 Zernike functions kE6\G}zj
% x = -1:0.01:1; BtU,1`El5
% [X,Y] = meshgrid(x,x); u"C`S<c
% [theta,r] = cart2pol(X,Y);
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% idx = r<=1; R8":1 #&
% z = nan(size(X)); Z!LzyCVl
% n = [0 1 1 2 2 2 3 3 3 3]; Pw$'TE}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !B-&I E?
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Ix1ec^?f
% y = zernfun(n,m,r(idx),theta(idx)); z^Oiwzo
% figure('Units','normalized') _ogT(uYyr
% for k = 1:10 W=F?+KgL
% z(idx) = y(:,k); x%cKTpDh!
% subplot(4,7,Nplot(k)) ?;^_%XSQ*
% pcolor(x,x,z), shading interp Ai#W.
n
% set(gca,'XTick',[],'YTick',[]) +k8><_vr}
% axis square Dk]Y\:
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <`6-J `.
% end xvpS%MS
% g*Cs/w
% See also ZERNPOL, ZERNFUN2. Jc{zi^)(EN
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% Paul Fricker 11/13/2006 =_
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