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 QL/(72K
function z = zernfun(n,m,r,theta,nflag) cZ*@$%_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. lFj]4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }6~hEc*/"
% and angular frequency M, evaluated at positions (R,THETA) on the Q\vpqE!9
% unit circle. N is a vector of positive integers (including 0), and :,7hWs
% M is a vector with the same number of elements as N. Each element Zl!kJ:0
% k of M must be a positive integer, with possible values M(k) = -N(k) 'oVx#w^mf
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, hE/cd1iJ$
% and THETA is a vector of angles. R and THETA must have the same v/plpNVp>
% length. The output Z is a matrix with one column for every (N,M) >|=ts
% pair, and one row for every (R,THETA) pair. 5;WH:XM
% Z\rwO>3
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike E&w7GZNt
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A{zN| S[
% with delta(m,0) the Kronecker delta, is chosen so that the integral gJ+'W1$/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2[yd> (`
% and theta=0 to theta=2*pi) is unity. For the non-normalized t}4,]ms
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wQf-sk#
% DCa^
u'f
% The Zernike functions are an orthogonal basis on the unit circle. = svN#q5s
% They are used in disciplines such as astronomy, optics, and H8jpxzXv
% optometry to describe functions on a circular domain. y.k~Y0
% 4_lrg|X1
% The following table lists the first 15 Zernike functions. wHLLu~m\
% TX/Xt7#R:
% n m Zernike function Normalization ejd(R+
% -------------------------------------------------- BlO<PMmhT&
% 0 0 1 1 29b9`NXt
% 1 1 r * cos(theta) 2 f~[7t:WD*
% 1 -1 r * sin(theta) 2 gJ{)-\
% 2 -2 r^2 * cos(2*theta) sqrt(6) 6MW{,N
% 2 0 (2*r^2 - 1) sqrt(3) OT*mO&Z
% 2 2 r^2 * sin(2*theta) sqrt(6) J;e2&gB
% 3 -3 r^3 * cos(3*theta) sqrt(8) i]4I [!
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) UkC!1Jy
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) =qIp2c}Rx
% 3 3 r^3 * sin(3*theta) sqrt(8) >=>2m2z=
% 4 -4 r^4 * cos(4*theta) sqrt(10) }.(B}/$u
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (t|Zn@uY
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "sCRdx]_
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xo&_bMO
% 4 4 r^4 * sin(4*theta) sqrt(10) <lPG=Xt
% -------------------------------------------------- q;CiV
% B9 uoVcW
% Example 1: @. l@\4m
% "S]TP$O D
% % Display the Zernike function Z(n=5,m=1) p
l0\2e)
% x = -1:0.01:1; xC TML!H
% [X,Y] = meshgrid(x,x); BU_nh+dF
% [theta,r] = cart2pol(X,Y); T^KKy0ZGM
% idx = r<=1; ^x,YW]AS}
% z = nan(size(X)); cT,sh~-x,
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 2zb"MEOS5
% figure Il'fL'3
% pcolor(x,x,z), shading interp ~
7s!VR
% axis square, colorbar SnfYT)Ph
% title('Zernike function Z_5^1(r,\theta)') W!(zT6#
% \b x$i*
% Example 2: "+s++@
z
% Hn"RH1Zy
% % Display the first 10 Zernike functions oc`H}Wvn
% x = -1:0.01:1; X"Swi&4
% [X,Y] = meshgrid(x,x); >bW#Zs,6
% [theta,r] = cart2pol(X,Y); oPM96
(
% idx = r<=1; CdQ!GS<'y
% z = nan(size(X)); KRzAy)8
% n = [0 1 1 2 2 2 3 3 3 3]; i.m^/0!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; D,feF9
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 0,")C5j
% y = zernfun(n,m,r(idx),theta(idx)); QWYJ*
% figure('Units','normalized') ~>|ziHx
% for k = 1:10 }}~ |!8
% z(idx) = y(:,k); }7Q% 6&IR
% subplot(4,7,Nplot(k)) e7 o.xR
% pcolor(x,x,z), shading interp L,!?Nt\
% set(gca,'XTick',[],'YTick',[]) L8B!u9%
% axis square 0l6.<-f{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) { <