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, L0%hnA@
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, XI Jlc~2
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? L5IbExjV
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? \uG`|Dn
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function z = zernfun(n,m,r,theta,nflag) fWGOP~0
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. S>q>K"j^!
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1\L[i];L8
% and angular frequency M, evaluated at positions (R,THETA) on the pWE `x|J
% unit circle. N is a vector of positive integers (including 0), and |DF9cd^
% M is a vector with the same number of elements as N. Each element -V %gVI[
% k of M must be a positive integer, with possible values M(k) = -N(k) 'z=:[#b
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 'j%F]CK
% and THETA is a vector of angles. R and THETA must have the same V2|3i}V"
% length. The output Z is a matrix with one column for every (N,M) M!M!Ni
% pair, and one row for every (R,THETA) pair. BsZ{|,oQnZ
% qJR!$?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~9Cz6yF
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1on'^8]0
% with delta(m,0) the Kronecker delta, is chosen so that the integral +~sd"v6
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p3^jGj@
% and theta=0 to theta=2*pi) is unity. For the non-normalized '[ P}&<ie,
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )<4_:
% ']ood!
% The Zernike functions are an orthogonal basis on the unit circle. qu6DQ@
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% They are used in disciplines such as astronomy, optics, and 7yI@"c#O
% optometry to describe functions on a circular domain. ! o,5h|\
% pL1s@KR
% The following table lists the first 15 Zernike functions. tZWrz
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% ;%q39U}
% n m Zernike function Normalization FdOFE.l
% -------------------------------------------------- (3,.3)%`
% 0 0 1 1 j%Y\A~DV
% 1 1 r * cos(theta) 2 )wzV
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% 1 -1 r * sin(theta) 2 g+zJ?
% 2 -2 r^2 * cos(2*theta) sqrt(6) $<)Yyi>6E
% 2 0 (2*r^2 - 1) sqrt(3) }b["Jk\2
% 2 2 r^2 * sin(2*theta) sqrt(6) 5iFV;W
% 3 -3 r^3 * cos(3*theta) sqrt(8) Y\/gU8w/
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?T:
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) oholt/gb+0
% 3 3 r^3 * sin(3*theta) sqrt(8) q$ghLGz
% 4 -4 r^4 * cos(4*theta) sqrt(10) QYE7p\
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j;P+_Hfe/E
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) j,%EW+j$
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) eQX`,9:5
% 4 4 r^4 * sin(4*theta) sqrt(10) YwT-T,oD
% -------------------------------------------------- W,hWOO
% P]<= ! F
% Example 1: wod/&!)]A
% M7y|EB))
% % Display the Zernike function Z(n=5,m=1) hY\{|
% x = -1:0.01:1; yDd[e]zS`
% [X,Y] = meshgrid(x,x); Db03Nk>#
% [theta,r] = cart2pol(X,Y); u"VS* hSH
% idx = r<=1; -
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% z = nan(size(X)); Bmr<O!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); +GNWF%
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% figure )q?$p9
% pcolor(x,x,z), shading interp \%_ZV9cKF
% axis square, colorbar jD<pIHau
% title('Zernike function Z_5^1(r,\theta)') ~5#)N{GbY
% 9fVj
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% Example 2: } ~enEZ
% oFg'wAO.
% % Display the first 10 Zernike functions #+sF`qR,
% x = -1:0.01:1; jq oPLbxT
% [X,Y] = meshgrid(x,x); mA{#]Yvf1
% [theta,r] = cart2pol(X,Y); iK}v`xq
% idx = r<=1; 0o/B{|rv
% z = nan(size(X)); 2*[Un(
% n = [0 1 1 2 2 2 3 3 3 3]; ,Q2N[Jwd$
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; CI^|k/
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ,?b78_,2
% y = zernfun(n,m,r(idx),theta(idx)); -Ds|qzrN%
% figure('Units','normalized') ;~tsF.=
% for k = 1:10 IKm&xzV-
% z(idx) = y(:,k); Yw"P)Zp
% subplot(4,7,Nplot(k)) ckwF|:e7*
% pcolor(x,x,z), shading interp ?n*fy
% set(gca,'XTick',[],'YTick',[]) hLA;Bl
% axis square !UNNjBBP7
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .ewZV9P)t
% end VO9f~>`(
% R7aXR\ R
% See also ZERNPOL, ZERNFUN2. x0x $ 9
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% Paul Fricker 11/13/2006 F.KrZ3%4iB
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% Check and prepare the inputs: i_Ar<9a~
% ----------------------------- =J.EH|
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f'_S1\
error('zernfun:NMvectors','N and M must be vectors.') 8eww7k^R
end ,P{HE8.
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if length(n)~=length(m) Y\8+}g;KR
error('zernfun:NMlength','N and M must be the same length.') C"No5r'K3
end Y(z}[`2
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n = n(:); M71R -B`-
m = m(:); *f*f&l