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, %p0b{P j_p
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, 0iYP
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? Ovvny$
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? j]pohxn$5
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function z = zernfun(n,m,r,theta,nflag) XwcMt r*
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |*:tyP%m^
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @l'G[jN5
% and angular frequency M, evaluated at positions (R,THETA) on the "H>.':c"+3
% unit circle. N is a vector of positive integers (including 0), and {3hqp*xl
% M is a vector with the same number of elements as N. Each element N~;*bvW{
% k of M must be a positive integer, with possible values M(k) = -N(k) eGLO!DdxZ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, vQUZVq5M
% and THETA is a vector of angles. R and THETA must have the same <eY%sFq,
% length. The output Z is a matrix with one column for every (N,M) ]nUR;8
% pair, and one row for every (R,THETA) pair. ##H;Yb
% wW-A b
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]/Vh{d|I&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [|4}~UV
% with delta(m,0) the Kronecker delta, is chosen so that the integral *z q .C
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tM)Iir*U#
% and theta=0 to theta=2*pi) is unity. For the non-normalized ~n
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ! .AhzU1%Y
% GuT6K}~|D
% The Zernike functions are an orthogonal basis on the unit circle. LfEvc2
v=g
% They are used in disciplines such as astronomy, optics, and z!$gVWG
% optometry to describe functions on a circular domain. 3:lDL2
% AH^e]<2-
% The following table lists the first 15 Zernike functions. ~\$=w10
% /}Yqf`CZy
% n m Zernike function Normalization F;u7A]H^
% -------------------------------------------------- 1Ao6y.S
% 0 0 1 1 , 9mgYp2
% 1 1 r * cos(theta) 2 `mzb(bE
% 1 -1 r * sin(theta) 2 ~Rs#|JWB2V
% 2 -2 r^2 * cos(2*theta) sqrt(6) ;hwzYXWF
% 2 0 (2*r^2 - 1) sqrt(3) bni)Qw
% 2 2 r^2 * sin(2*theta) sqrt(6) <FUon
% 3 -3 r^3 * cos(3*theta) sqrt(8) F.<L>
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) f ba&`
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &|b4\uj9
% 3 3 r^3 * sin(3*theta) sqrt(8) I5qM.@%zB
% 4 -4 r^4 * cos(4*theta) sqrt(10) .s2$al
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ca(U!T68
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1AF%-<`?s
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;1 |x
% 4 4 r^4 * sin(4*theta) sqrt(10) x?
N.WABr;
% -------------------------------------------------- Lz@$3(2
% HY ;9?KJ'
% Example 1: 9_ZBV{
% U&P{?>{u
% % Display the Zernike function Z(n=5,m=1) 8Atq,GcG
% x = -1:0.01:1; xvmt.> f
% [X,Y] = meshgrid(x,x); Q(Gyq:L=>
% [theta,r] = cart2pol(X,Y); qbiK^gR
% idx = r<=1; WULAty
% z = nan(size(X)); ZjD)?4
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ;F5"}x
% figure s\gp5MT
% pcolor(x,x,z), shading interp R4{-Qv#8
q
% axis square, colorbar @vRwzc\
% title('Zernike function Z_5^1(r,\theta)') 7?J3ci\
% >;4!O%F
% Example 2: <sX VW
% j13DJ.xu
% % Display the first 10 Zernike functions 0C$8g
Y*
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); $.rzc]s
% [theta,r] = cart2pol(X,Y); #DFp[\)1
% idx = r<=1; ~$<UE}qp
% z = nan(size(X)); |sIr?RL{C
% n = [0 1 1 2 2 2 3 3 3 3]; M:|8]y@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $6h*lT<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; `G!M>h@
% y = zernfun(n,m,r(idx),theta(idx)); A]o4Mf0>I
% figure('Units','normalized') (ChD]PWQ
% for k = 1:10 SV.z>p
% z(idx) = y(:,k); WO X}Sw"
% subplot(4,7,Nplot(k)) m#e*c[*G
% pcolor(x,x,z), shading interp <
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% set(gca,'XTick',[],'YTick',[]) :Q_3hK
% axis square %}3qR~;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6w|J-{2
% end lnh+a7a)
% NHm]`R,
% See also ZERNPOL, ZERNFUN2. };,/0Fu
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% Paul Fricker 11/13/2006 }Fa%%}
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% Check and prepare the inputs: DM{ 4@*]
% ----------------------------- e6E?t[hEeS
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;_O)p,p
error('zernfun:NMvectors','N and M must be vectors.') s?rBE.g@}
end 0w=R_C)s
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if length(n)~=length(m) % 0T+t.
error('zernfun:NMlength','N and M must be the same length.') F$V/K&&W
end ^oM|<";!?D
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n = n(:); ~" i0x
m = m(:); k*mt4~KLT8
if any(mod(n-m,2)) ! RW
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error('zernfun:NMmultiplesof2', ... pkgjTXR2b
'All N and M must differ by multiples of 2 (including 0).') ?jx1R^
end QDx$==Fo
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if any(m>n) P{Lf5V9# <
error('zernfun:MlessthanN', ... Ztr Cv?
'Each M must be less than or equal to its corresponding N.') vy9 w$ls
end 9$qm>,o
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if any( r>1 | r<0 ) #OO>rm$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') g%[c<l9
end `Ag{)
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [* ?Awf`
error('zernfun:RTHvector','R and THETA must be vectors.') Uu 8,@W+
end `-h8vj5uG
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r = r(:); pi<TFe@eG
theta = theta(:); G s+3e8
length_r = length(r); Zwz&