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 I75>$"$<
function z = zernfun(n,m,r,theta,nflag) w\wS?E4G
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7P!<c/ E
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2xy
&mNx
% and angular frequency M, evaluated at positions (R,THETA) on the *xY}?vSs
% unit circle. N is a vector of positive integers (including 0), and s~OGlPK
% M is a vector with the same number of elements as N. Each element [k)xn3[
% k of M must be a positive integer, with possible values M(k) = -N(k) dN'2;X
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9I3vW]0x[
% and THETA is a vector of angles. R and THETA must have the same " sh%8
<N
% length. The output Z is a matrix with one column for every (N,M) :oRR1k
% pair, and one row for every (R,THETA) pair. @wa2Z
%
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "[W${q+0x
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), bvVEV
% with delta(m,0) the Kronecker delta, is chosen so that the integral #
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, )*!"6d)^
% and theta=0 to theta=2*pi) is unity. For the non-normalized Q4;eN w
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 70s.
% $$my,:nH
% The Zernike functions are an orthogonal basis on the unit circle. M'Fa[n*b?!
% They are used in disciplines such as astronomy, optics, and v/ dyu
% optometry to describe functions on a circular domain. d1MY>zq
% >,JLYz|</
% The following table lists the first 15 Zernike functions. 01bBZWX
% wNzALfS
% n m Zernike function Normalization .Pz( 0Y
% -------------------------------------------------- Ur^~fW1o
% 0 0 1 1 #Av6BGM|,
% 1 1 r * cos(theta) 2 f+*wDH
% 1 -1 r * sin(theta) 2 VKzY6
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]`[r=cG
% 2 0 (2*r^2 - 1) sqrt(3) sfLH[Q?
% 2 2 r^2 * sin(2*theta) sqrt(6) 'rWu}#Nb
% 3 -3 r^3 * cos(3*theta) sqrt(8)
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R
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Grot3a
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) kGaK(^w
% 3 3 r^3 * sin(3*theta) sqrt(8) "'389*-
% 4 -4 r^4 * cos(4*theta) sqrt(10) aI8k:FK"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z' cQ<
f
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ]#)1(ZE
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ARcPHV<(2
% 4 4 r^4 * sin(4*theta) sqrt(10) \SA"DT
% -------------------------------------------------- ^;on
% r3~~4Q4XI>
% Example 1: hN(sz
% /$]#L%
% % Display the Zernike function Z(n=5,m=1) Ww(($e!
% x = -1:0.01:1; Jptzc:~B
% [X,Y] = meshgrid(x,x); DyZe+,g;S
% [theta,r] = cart2pol(X,Y); &hciv\YT2W
% idx = r<=1; g~zz[F 8U
% z = nan(size(X)); qx#k()E.U
% z(idx) = zernfun(5,1,r(idx),theta(idx)); >FrF"u:kM
% figure &c;@u?:@S
% pcolor(x,x,z), shading interp eVRFb#EU0e
% axis square, colorbar h>s|MZQ:*
% title('Zernike function Z_5^1(r,\theta)') m(~5X0
% }zA
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% Example 2: #X~{p4Lr
% [A@K)A$f
% % Display the first 10 Zernike functions hXxgKi%
% x = -1:0.01:1; |~QHCg<
% [X,Y] = meshgrid(x,x); UkO L7M
% [theta,r] = cart2pol(X,Y); @I\&-Z ^
% idx = r<=1; axf 4N@
% z = nan(size(X)); #2N']VP
% n = [0 1 1 2 2 2 3 3 3 3]; mFL"h
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; desrKnY
% Nplot = [4 10 12 16 18 20 22 24 26 28]; N$Pi4
% y = zernfun(n,m,r(idx),theta(idx)); ifo^
M]v
% figure('Units','normalized') u!NY@$Wc
% for k = 1:10 ~d+.w%Z`
% z(idx) = y(:,k); yrp;G_
% subplot(4,7,Nplot(k)) 1e Wl:S}
% pcolor(x,x,z), shading interp 9XU"Ppv
% set(gca,'XTick',[],'YTick',[]) <r[5 S5y
% axis square _RzwE$+9
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) )v1y
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% end 7/p&]0w
% @-uV6X8|
% See also ZERNPOL, ZERNFUN2. fgmu*\x<