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,
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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, Cc0`Y lx~(
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? K7FuMB
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 9U>ID{
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function z = zernfun(n,m,r,theta,nflag) u6'vzLmM
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ms<^_\iPN
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 9bPQD{Qb
% and angular frequency M, evaluated at positions (R,THETA) on the (ivV [
% unit circle. N is a vector of positive integers (including 0), and s{NEP/QQJ
% M is a vector with the same number of elements as N. Each element zid?yuP
% k of M must be a positive integer, with possible values M(k) = -N(k) #StD]d
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, GD}3r:wDs
% and THETA is a vector of angles. R and THETA must have the same " 6~pTHT
% length. The output Z is a matrix with one column for every (N,M) = PqQJE}
% pair, and one row for every (R,THETA) pair. f62z9)`^
% 2xZg, \
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike B cX}[?c
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), b\7-u-
% with delta(m,0) the Kronecker delta, is chosen so that the integral z tHGY
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, K8pfk*NZ_@
% and theta=0 to theta=2*pi) is unity. For the non-normalized -3/:Dk`3
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. { Y|h;@j$
% Z_iu^Q
% The Zernike functions are an orthogonal basis on the unit circle. Q`7!~qV0=
% They are used in disciplines such as astronomy, optics, and [zm&}$nnN
% optometry to describe functions on a circular domain. MnO,Cd6{%d
% ":"QsS#*"#
% The following table lists the first 15 Zernike functions. H:`W\CP7_
% HyiuU`
% n m Zernike function Normalization Xf:CGR8_
% -------------------------------------------------- fNFdZ[qOd
% 0 0 1 1 Sr)/
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% 1 1 r * cos(theta) 2 C0jmjZ%w@
% 1 -1 r * sin(theta) 2 jm =E_86_
% 2 -2 r^2 * cos(2*theta) sqrt(6) V3$!`T}g4
% 2 0 (2*r^2 - 1) sqrt(3) 4(R O1VWsb
% 2 2 r^2 * sin(2*theta) sqrt(6) )*G3q/l1u6
% 3 -3 r^3 * cos(3*theta) sqrt(8) "}\2zub9
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @I]uK[qd
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) O*z x{a6
% 3 3 r^3 * sin(3*theta) sqrt(8) %bt2^
% 4 -4 r^4 * cos(4*theta) sqrt(10) _R1UEE3M
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;} gvBI2e
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'P)xY-15
% 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) teUCK(;23
% -------------------------------------------------- zek\AQN
% #dqZdj@
% Example 1: BtBo%t&
% )"m FlS<I
% % Display the Zernike function Z(n=5,m=1) y`E2IE2o
% x = -1:0.01:1; Z%`}
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% [X,Y] = meshgrid(x,x); Bo`fy/x#
% [theta,r] = cart2pol(X,Y); Ufv{6"sH
% idx = r<=1; ~r]ZD)
% z = nan(size(X)); J,;;`sf
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Fz?ON1\
% figure Y">Q16(
% pcolor(x,x,z), shading interp j~\\,fl=
% axis square, colorbar ~"gOq"y5p
% title('Zernike function Z_5^1(r,\theta)') $B~a*zZ7
% U@|{RP
% Example 2: 1;fs`k0p
% C0 .Xp
% % Display the first 10 Zernike functions q
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% x = -1:0.01:1; XC?H
% [X,Y] = meshgrid(x,x); A{>]M@QC2
% [theta,r] = cart2pol(X,Y); Fy`VQ\%7t
% idx = r<=1; E-X-LR{CC
% z = nan(size(X)); ^M,t`r{
% n = [0 1 1 2 2 2 3 3 3 3]; k|BY 7C
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; }C/}8<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3 V8SKBS
% y = zernfun(n,m,r(idx),theta(idx)); \z:p"eua z
% figure('Units','normalized') x)BG%{h
% for k = 1:10 csRba;Z[
% z(idx) = y(:,k); 7vNS@[8
% subplot(4,7,Nplot(k)) y3 LWh}~E
% pcolor(x,x,z), shading interp +O j28vR
% set(gca,'XTick',[],'YTick',[]) p J+>qy5
% axis square VEpIAC4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h6)hZ'zV
% end BR*""/3`
% !h?N)9e
% See also ZERNPOL, ZERNFUN2. #@2 `^1
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% Paul Fricker 11/13/2006 %- %/3
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% Check and prepare the inputs: |1J "r.K
% ----------------------------- DSd 5?
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g|)e3q{M
error('zernfun:NMvectors','N and M must be vectors.') {EW}Wd
end xqP0Z),Ow
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if length(n)~=length(m) r\_rnM)_xN
error('zernfun:NMlength','N and M must be the same length.') n0!S;HH-
end +ZizT.$&
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n = n(:); wKk
m = m(:); h=`rZC
if any(mod(n-m,2)) [0/ ?(i|
error('zernfun:NMmultiplesof2', ... )I1LBvfQ
'All N and M must differ by multiples of 2 (including 0).') o|^0DYb
end 86R}G/>>e
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if any(m>n) "q#(}1Zd
error('zernfun:MlessthanN', ... iW*0V3
'Each M must be less than or equal to its corresponding N.') =xG9a_^v
end `+"QhQ4w
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if any( r>1 | r<0 ) ^,U&v;
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;pAkdX&b