下面这个函数大家都不会陌生,计算zernike函数值的,并根据此可以还原出图像来, 9>zN 27
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, {4:En;
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? )?4m}
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]`u{^f
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function z = zernfun(n,m,r,theta,nflag) .# M5L
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. R]ppA=1*_l
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RRq*CLj
% and angular frequency M, evaluated at positions (R,THETA) on the D
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% unit circle. N is a vector of positive integers (including 0), and XY%8yII6
% M is a vector with the same number of elements as N. Each element ((X"D/F]
% k of M must be a positive integer, with possible values M(k) = -N(k) cYGZZC8 |K
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ifBJ$x(B.
% and THETA is a vector of angles. R and THETA must have the same s/A]&!`
% length. The output Z is a matrix with one column for every (N,M) |y=CmNG,
% pair, and one row for every (R,THETA) pair. UayRT#}]
% ;1eu8N8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rj{'X /
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), N ~LR
% with delta(m,0) the Kronecker delta, is chosen so that the integral iJsw:Nc
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |,yS>kjp
% and theta=0 to theta=2*pi) is unity. For the non-normalized i%\nJs*
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _q8s 7H
% /M'b137
% The Zernike functions are an orthogonal basis on the unit circle. 0@xuxm/i
% They are used in disciplines such as astronomy, optics, and t_j.@|/FZ
% optometry to describe functions on a circular domain. 8#oF7eE
% gW*ee
% The following table lists the first 15 Zernike functions. W<9GwMU
% %X.Q\T
% n m Zernike function Normalization +)7NWR\
% -------------------------------------------------- s&fU|Jk8
% 0 0 1 1 qi/%&)GZ
% 1 1 r * cos(theta) 2 zV2c`he%z
% 1 -1 r * sin(theta) 2 4CN8>J'-
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?X:RrZ:/
% 2 0 (2*r^2 - 1) sqrt(3) Q"Bgr&RJ
% 2 2 r^2 * sin(2*theta) sqrt(6) 3K#e]zoI
% 3 -3 r^3 * cos(3*theta) sqrt(8) [KjQW/sb'
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) uAJ_`o[
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) tKJ)'v?
% 3 3 r^3 * sin(3*theta) sqrt(8) |E?%Cj^W
% 4 -4 r^4 * cos(4*theta) sqrt(10) f0hi70\(X
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :ss9-
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) m\~[^H~g
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "=
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% 4 4 r^4 * sin(4*theta) sqrt(10)
=,?@p{g}
% -------------------------------------------------- 50'6l
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% tPp}/a%D
% Example 1: mKn[>M1
% tL
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% % Display the Zernike function Z(n=5,m=1) gjs-j{*
% x = -1:0.01:1; >>!+Ri\@
% [X,Y] = meshgrid(x,x); mybDK'EW
% [theta,r] = cart2pol(X,Y); K}$PI W
% idx = r<=1; {+`ep\.$&
% z = nan(size(X)); w]%r]PwU+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); g.9MPN
% figure A"P1B]
% pcolor(x,x,z), shading interp [jLx}\]
% axis square, colorbar |]B]0J#_
% title('Zernike function Z_5^1(r,\theta)') ({i|
% d&U;rMEv
% Example 2: "\V:W%23W{
% oiR`\uY
% % Display the first 10 Zernike functions _wqFKj
% x = -1:0.01:1; Y<M}'t
% [X,Y] = meshgrid(x,x); V5A7w
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% [theta,r] = cart2pol(X,Y); 9GQTe1[t4
% idx = r<=1; ^^ ?ECnpcU
% z = nan(size(X)); ;N,7#l|wi
% n = [0 1 1 2 2 2 3 3 3 3]; Dic(G[
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; L~;_R*Th
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2OZdj
% y = zernfun(n,m,r(idx),theta(idx)); 2
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% figure('Units','normalized') nDPfr\\
% for k = 1:10 fmSA.z
% z(idx) = y(:,k); FEP\5d>
% subplot(4,7,Nplot(k)) e~}+.B0
% pcolor(x,x,z), shading interp CP?\'a"Kt
% set(gca,'XTick',[],'YTick',[]) 0\i&