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 rJj~cPwL"
function z = zernfun(n,m,r,theta,nflag) POs~xaZ`H
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Rj=Om
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N fdHxrH>*
% and angular frequency M, evaluated at positions (R,THETA) on the g+*[CKO{
% unit circle. N is a vector of positive integers (including 0), and 6[7k}9`alz
% M is a vector with the same number of elements as N. Each element d69VgLg
% k of M must be a positive integer, with possible values M(k) = -N(k) wB"Gw` D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;Nij*-U4~
% and THETA is a vector of angles. R and THETA must have the same y$NG ..S
% length. The output Z is a matrix with one column for every (N,M) ;wB3H
% pair, and one row for every (R,THETA) pair. :E*U*#h/
% &|] ^ u/
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike mr.DP~O:9p
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4/_|Qy
% with delta(m,0) the Kronecker delta, is chosen so that the integral pBLO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Gjr2]t;E
% and theta=0 to theta=2*pi) is unity. For the non-normalized yK3z3"1M?
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. lNAHn<ht
% r U5'hK
% The Zernike functions are an orthogonal basis on the unit circle. }C}_
I:=C
% They are used in disciplines such as astronomy, optics, and %Ski5q
% optometry to describe functions on a circular domain. ZZ7U^#RT
% ![%,pip2/&
% The following table lists the first 15 Zernike functions. G> >_G<x
% W -&5
v
% n m Zernike function Normalization l0)uu4|
% -------------------------------------------------- HskN(Ho
% 0 0 1 1 HbVLL`06*
% 1 1 r * cos(theta) 2 7i/Cax
% 1 -1 r * sin(theta) 2 l[ k$O$jo
% 2 -2 r^2 * cos(2*theta) sqrt(6) O2f2Fb$B7
% 2 0 (2*r^2 - 1) sqrt(3) {c;3$
% 2 2 r^2 * sin(2*theta) sqrt(6) Ymom 0g+f
% 3 -3 r^3 * cos(3*theta) sqrt(8) 37Y]sJrs$
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =ndKG5
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Hc9pWr"N
% 3 3 r^3 * sin(3*theta) sqrt(8) ]9Hy
"#Fz
% 4 -4 r^4 * cos(4*theta) sqrt(10) W[s>TDc`v
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g (k|"g`*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) /G ;yxdb
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) cK&oC$[r-
% 4 4 r^4 * sin(4*theta) sqrt(10) 0
HmRl
% -------------------------------------------------- ,jmG!qJb
% lH.2H
% Example 1: $EF@x}h:A
% _(foJRr
% % Display the Zernike function Z(n=5,m=1) v!Z 9T
% x = -1:0.01:1; _!7o
% [X,Y] = meshgrid(x,x); 9j`-fs@:
% [theta,r] = cart2pol(X,Y); @@jdF-Utj;
% idx = r<=1; 605|*(
% z = nan(size(X)); q0wVV
% z(idx) = zernfun(5,1,r(idx),theta(idx));
2X_ef
% figure >.|gmo>b
% pcolor(x,x,z), shading interp hLRQ)
% axis square, colorbar xJCpWU3wM
% title('Zernike function Z_5^1(r,\theta)') /&yT2p
% t=AR>M!w~
% Example 2: tUQ)q
% CggEAi~
% % Display the first 10 Zernike functions #eYVZ=E
% x = -1:0.01:1; }^muAr
% [X,Y] = meshgrid(x,x); Sls>
OIc
% [theta,r] = cart2pol(X,Y); Pp2)P7
% idx = r<=1; Npqb xb
% z = nan(size(X)); VM[8w`
% n = [0 1 1 2 2 2 3 3 3 3]; *rLs!/[Z_
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; pC6_
jIZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; $$a"A(Y
% y = zernfun(n,m,r(idx),theta(idx)); s><co]
% figure('Units','normalized') e 3K
% for k = 1:10 Cp%|Q.?
% z(idx) = y(:,k); 8{C3ijR
% subplot(4,7,Nplot(k)) $4&Ql
% pcolor(x,x,z), shading interp q<VhP2R
% set(gca,'XTick',[],'YTick',[]) |wDCIHzQ
% axis square ry'(mM
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :&m(W Z\
% end =>G A_
% ,v"A}g0"
% See also ZERNPOL, ZERNFUN2. Ty=}A MMyE
S4w/
kml3
% Paul Fricker 11/13/2006 =R05H2hs
amRtFrc|
|($pXVLH`
% Check and prepare the inputs: Q *he%@w
% ----------------------------- k;sUD mrO
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) YdFC YSiS
error('zernfun:NMvectors','N and M must be vectors.') V;"'!dVX
end ^|Y!NHYH$Z
X_lNnk
if length(n)~=length(m) DxlX-
error('zernfun:NMlength','N and M must be the same length.') ]9' \<uR
end SZ_hG D 0
<