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 dnH?@K
function z = zernfun(n,m,r,theta,nflag) yo3'\I
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. qHklu2_%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N //g~1(
% and angular frequency M, evaluated at positions (R,THETA) on the g?)9zJ9
% unit circle. N is a vector of positive integers (including 0), and y~jTI[kS
% M is a vector with the same number of elements as N. Each element c)+IX;q-C
% k of M must be a positive integer, with possible values M(k) = -N(k) PO1sVP.S
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, VQ2)qJ#l
% and THETA is a vector of angles. R and THETA must have the same Mvu!
% length. The output Z is a matrix with one column for every (N,M) %
?@PlQ
% pair, and one row for every (R,THETA) pair. S+7>Y? B!
% slXk <
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike v~9PS2
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^dld\t:tV7
% with delta(m,0) the Kronecker delta, is chosen so that the integral M5CFW >T
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b1R%JY7/S
% and theta=0 to theta=2*pi) is unity. For the non-normalized z1*8 5?
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d?.ewsC
% {b}Ri&oEOH
% The Zernike functions are an orthogonal basis on the unit circle. 9ssTG4Sa
% They are used in disciplines such as astronomy, optics, and ]W]o6uo7
% optometry to describe functions on a circular domain. 8 W79
% "o+<
\B~
% The following table lists the first 15 Zernike functions. %[l5){:05
% vg5i+ry<
% n m Zernike function Normalization
(0bvd
% -------------------------------------------------- )\8l6Gw
% 0 0 1 1 qn5e[Vn
% 1 1 r * cos(theta) 2 C5c@@ch :
% 1 -1 r * sin(theta) 2 Vr+X!DeY
% 2 -2 r^2 * cos(2*theta) sqrt(6) 7LbBS:@3z_
% 2 0 (2*r^2 - 1) sqrt(3) D37N*9}
% 2 2 r^2 * sin(2*theta) sqrt(6) @2na r<
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1kEXTs=,
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 4$oNh)+/h
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) |7LhE+E
% 3 3 r^3 * sin(3*theta) sqrt(8) |#^wYZO1U
% 4 -4 r^4 * cos(4*theta) sqrt(10) `A_CLVE
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Kc$j<MRtv
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 4V@raI-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d="Oge8
% 4 4 r^4 * sin(4*theta) sqrt(10) MqDz cB]
% -------------------------------------------------- <b.?G
% }6*+>?
% Example 1: G>&Ta p>
% 2~h! ouleY
% % Display the Zernike function Z(n=5,m=1) ry)g<OA
% x = -1:0.01:1; &@p _g8r#
% [X,Y] = meshgrid(x,x); % put=I
% [theta,r] = cart2pol(X,Y); ?%-VSL>$w=
% idx = r<=1; bFD
vCF
% z = nan(size(X)); M=:!d$c
% z(idx) = zernfun(5,1,r(idx),theta(idx)); "%ou'\}
% figure aDceOhfx
% pcolor(x,x,z), shading interp E!nEB(FD
% axis square, colorbar Vb yGr~t
% title('Zernike function Z_5^1(r,\theta)') .0+=#G>
% T#KF@8'-
% Example 2: 6Lj=%&
% O<[h
% % Display the first 10 Zernike functions xMsSZ{j%5
% x = -1:0.01:1; }-4@EC>
% [X,Y] = meshgrid(x,x); Xo[j*<=0
% [theta,r] = cart2pol(X,Y); 8S/SXyS
% idx = r<=1; #[ZToE4
% z = nan(size(X)); g^ .g9"
% n = [0 1 1 2 2 2 3 3 3 3]; 69/aP=
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; {,xI|u2R
% Nplot = [4 10 12 16 18 20 22 24 26 28]; tQ~vLPi$
% y = zernfun(n,m,r(idx),theta(idx)); 9j<qi\SSI
% figure('Units','normalized') u-qwG/$E
% for k = 1:10 mWEaUi)Zz
% z(idx) = y(:,k); l Oxz&m
% subplot(4,7,Nplot(k)) m03D+@F
% pcolor(x,x,z), shading interp Uao8#<CkvJ
% set(gca,'XTick',[],'YTick',[]) $.HZz
% axis square rG[iEY
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) X%JQ_Z
% end d?[gd(O
% tV.qdy/]}
% See also ZERNPOL, ZERNFUN2. ^V6cx2M
?|,dHqh{nM
% Paul Fricker 11/13/2006 W3Gg<!*Uo
/Q]6"nY
Hreu3N
% Check and prepare the inputs: t"# .I?S0
% ----------------------------- c+S<U*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) X;:qnnO
error('zernfun:NMvectors','N and M must be vectors.') j}s<Pn%4
end hSkI]%
({&