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 "UYlC0 S\
function z = zernfun(n,m,r,theta,nflag) n:"0mWnL$y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. EQ [K
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ls({{34NF
% and angular frequency M, evaluated at positions (R,THETA) on the 0}mVP
% unit circle. N is a vector of positive integers (including 0), and g|Tkl
% M is a vector with the same number of elements as N. Each element \ gO!6
% k of M must be a positive integer, with possible values M(k) = -N(k) <sTY<i VR
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =Jax T90x
% and THETA is a vector of angles. R and THETA must have the same V7<w9MM
% length. The output Z is a matrix with one column for every (N,M) A$3ll|%j
% pair, and one row for every (R,THETA) pair. ]bP1gV(b-
% w,*#z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .QW@rV:T
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {ui{Y c
% with delta(m,0) the Kronecker delta, is chosen so that the integral qDS~|<Y5
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A'aY H`j
% and theta=0 to theta=2*pi) is unity. For the non-normalized 1lYQR`Uh
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P"{yV?CNg
% 3~%M4(
% The Zernike functions are an orthogonal basis on the unit circle. ku)/
8Z`$
% They are used in disciplines such as astronomy, optics, and zDf96eK
% optometry to describe functions on a circular domain. C1==a FD
% MX"M2>" pT
% The following table lists the first 15 Zernike functions. m1D,#=C,_
% ThY\K>@]
% n m Zernike function Normalization )YVs=0j
% -------------------------------------------------- Qk2*=BVh
% 0 0 1 1 d(YAH@
% 1 1 r * cos(theta) 2 *X!+wK-+
% 1 -1 r * sin(theta) 2 .npD<*
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]+5Y\~I
% 2 0 (2*r^2 - 1) sqrt(3) G0u
H6x?
% 2 2 r^2 * sin(2*theta) sqrt(6) [(; .D
% 3 -3 r^3 * cos(3*theta) sqrt(8) T"DG$R,Aj
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) |RH^|2:x9Q
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *7{{z%5Pu
% 3 3 r^3 * sin(3*theta) sqrt(8) NC3XJ
4
% 4 -4 r^4 * cos(4*theta) sqrt(10) +h? Gps
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "
1h~P,
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) )}J}d)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T"e"?JSRJ
% 4 4 r^4 * sin(4*theta) sqrt(10) RF
[81/w]
% -------------------------------------------------- 79uAsI2-Y
% ZEB,Q~
% Example 1: Jq:Wt+a
% TU1W!=Z
% % Display the Zernike function Z(n=5,m=1) Tdxc%'l
% x = -1:0.01:1; mUfANlQ:
% [X,Y] = meshgrid(x,x); IN@ =UAc&
% [theta,r] = cart2pol(X,Y); XzW\p8D^u
% idx = r<=1; %<>|cO
% z = nan(size(X)); &x3R+(H {
% z(idx) = zernfun(5,1,r(idx),theta(idx)); "`V:4uz
% figure ?NxaJ^
% pcolor(x,x,z), shading interp %~\I*v04
% axis square, colorbar 6RfS_
% title('Zernike function Z_5^1(r,\theta)') Hv*+HUc(:
% &r!jjT
% Example 2: ?s]?2>p
% m' eM&1Ba
% % Display the first 10 Zernike functions 82YZN5S3]3
% x = -1:0.01:1; L;U?s2&Y
% [X,Y] = meshgrid(x,x); =&mdxKoT0
% [theta,r] = cart2pol(X,Y); 0KN'\KE
% idx = r<=1; c^~R%Bx
% z = nan(size(X)); .X"\ Mg
% n = [0 1 1 2 2 2 3 3 3 3]; +hIMfhF
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ahR-^^'$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 1`9'.w+r
% y = zernfun(n,m,r(idx),theta(idx)); bLC+73BjC
% figure('Units','normalized') QSvgbjdE
% for k = 1:10 +
7nA; C
% z(idx) = y(:,k); eMjW^-RgE5
% subplot(4,7,Nplot(k)) iwfH~
% pcolor(x,x,z), shading interp Lw6}bB`}
% set(gca,'XTick',[],'YTick',[]) ]eI|_O^u
% axis square Gdr7d
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [ak[ZXC,
% end s-S|#5
% V7?Pv
Q
% See also ZERNPOL, ZERNFUN2. mW#p&{
J6J;
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% Paul Fricker 11/13/2006 1ifPc5j}
lmx'w
3Ol`i$
% Check and prepare the inputs: > M4QEv
% ----------------------------- !I Byv%m&\
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {+ WI>3
error('zernfun:NMvectors','N and M must be vectors.') @|}=W Q
end -P'c0I9z
KRh?{
if length(n)~=length(m) {&h=
error('zernfun:NMlength','N and M must be the same length.') G:;(,
end ;CA7\&L>
I z)~h>-F
n = n(:); &Fl*,
m = m(:); T0BM:ofx
if any(mod(n-m,2)) /pz(s+4=
error('zernfun:NMmultiplesof2', ... ]ChN]>o
'All N and M must differ by multiples of 2 (including 0).') tH9BC5+r}
end $1myf Z
=)2!qoE
if any(m>n) X-5&c$hv
error('zernfun:MlessthanN', ... +WSM<