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, D#d8 ^U
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, nR o=J5tY
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? lj EB
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? chO'Q+pw
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function z = zernfun(n,m,r,theta,nflag) }0BL0N`_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. L{2b0Zh'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N C!7U<rI
% and angular frequency M, evaluated at positions (R,THETA) on the VR4E
2^
% unit circle. N is a vector of positive integers (including 0), and KP=D! l&q
% M is a vector with the same number of elements as N. Each element Mu'^OX82
% k of M must be a positive integer, with possible values M(k) = -N(k) X:G&5
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 7MO
% and THETA is a vector of angles. R and THETA must have the same U~{Sa+
% length. The output Z is a matrix with one column for every (N,M) .'5'0lR5
% pair, and one row for every (R,THETA) pair. l5=u3r9WYC
% O1?B{F/ e
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike n5bXQ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uX<+hG.n}
% with delta(m,0) the Kronecker delta, is chosen so that the integral (|g").L
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, C~ZE95g
% and theta=0 to theta=2*pi) is unity. For the non-normalized VLh%XoQx[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. t7|MkX1
% 9 m\)\/V
% The Zernike functions are an orthogonal basis on the unit circle. |.b%rVu
% They are used in disciplines such as astronomy, optics, and 0 W~.WkD
% optometry to describe functions on a circular domain. H\)gE>
% <#xrrRhm}
% The following table lists the first 15 Zernike functions. w::r?.9
% =<[7J]%
% n m Zernike function Normalization YO@hE>
% -------------------------------------------------- |x d@M-ln
% 0 0 1 1 v]WH8GI
% 1 1 r * cos(theta) 2 nU}~I)@V
% 1 -1 r * sin(theta) 2 %<aImR]
% 2 -2 r^2 * cos(2*theta) sqrt(6) AA))KBXq
% 2 0 (2*r^2 - 1) sqrt(3) kF+ZW%6N
% 2 2 r^2 * sin(2*theta) sqrt(6) 2;~KL-h0TK
% 3 -3 r^3 * cos(3*theta) sqrt(8) $Q8P@L)[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) '"`
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) D^,\cZbY
% 3 3 r^3 * sin(3*theta) sqrt(8) H9%l?r5
% 4 -4 r^4 * cos(4*theta) sqrt(10) tgO+*q5B
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) cwu$TP A>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [zY!'cz?
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6RH/V:YY
% 4 4 r^4 * sin(4*theta) sqrt(10) `0yb?Nk `:
% -------------------------------------------------- R]CZw;zS_
% 8W-]t1O%!
% Example 1: ?N4A9W9
% &bB6}H(
% % Display the Zernike function Z(n=5,m=1) \4OK!6LkI
% x = -1:0.01:1; n<{aPLQ
% [X,Y] = meshgrid(x,x); 54=}GnZN
% [theta,r] = cart2pol(X,Y); jZrY=f
% idx = r<=1; z8b
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% z = nan(size(X)); "3@KRb4f
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Ru)(dvk}S
% figure ZR1+
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% pcolor(x,x,z), shading interp HKpD2M
% axis square, colorbar [FB&4>V/
% title('Zernike function Z_5^1(r,\theta)') GSQfg
% c2/FHI0J;
% Example 2: --Oprl
% 0[lS(K
% % Display the first 10 Zernike functions bKYY{V55
% x = -1:0.01:1; PM@XtL7J
% [X,Y] = meshgrid(x,x); !{IC[g n
% [theta,r] = cart2pol(X,Y); /[0F6
% idx = r<=1; ?hKm&B;d
% z = nan(size(X));
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% n = [0 1 1 2 2 2 3 3 3 3]; iNt 4>
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;JYoW{2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ?3[tJreVj
% y = zernfun(n,m,r(idx),theta(idx)); ?;Un#6b
% figure('Units','normalized') \ZqK\=
% for k = 1:10 #
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% z(idx) = y(:,k); #`vVgGZ&
% subplot(4,7,Nplot(k)) ?J<T
% pcolor(x,x,z), shading interp mLJDxh'B
% set(gca,'XTick',[],'YTick',[]) .XIr?>G
% axis square 4*iHw+%mq
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mIRAS"Q!m
% end Ex6o=D2
% 'X54dXS?l
% See also ZERNPOL, ZERNFUN2. t{~@I
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% Paul Fricker 11/13/2006 R+s1[Z
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% Check and prepare the inputs: N^lAG"Jao[
% ----------------------------- u-kZW1wrQ
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _1P`]+K\D$
error('zernfun:NMvectors','N and M must be vectors.') x =h0Fq,T
end s*f1x N<
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if length(n)~=length(m) V(r`.75
error('zernfun:NMlength','N and M must be the same length.') b) Ux3PB
end %0lf
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n = n(:); Y_+
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m = m(:); kB
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if any(mod(n-m,2)) [-Cu4mff
error('zernfun:NMmultiplesof2', ... $]1qbE+
'All N and M must differ by multiples of 2 (including 0).') T RDxT
end %uua_)
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if any(m>n) >4~#%&
error('zernfun:MlessthanN', ... 3+%nn+m
'Each M must be less than or equal to its corresponding N.') dkpQZXi9%
end s@PLS5d"
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if any( r>1 | r<0 ) <W2ZoqaV
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8A!'I<S1
end wh*:\_!0\
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @gEr+O1K(
error('zernfun:RTHvector','R and THETA must be vectors.') &1l~&,,
end >P<'L4;
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r = r(:); PV(4$I}
theta = theta(:); SwX@I6huM
length_r = length(r); :xtT)w
if length_r~=length(theta) =gs~\q
error('zernfun:RTHlength', ... KJX>DL 9\
'The number of R- and THETA-values must be equal.') K'V 2FTJI
end 3
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% Check normalization: [83>T ,
% -------------------- f7
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if nargin==5 && ischar(nflag) HD H
isnorm = strcmpi(nflag,'norm'); !?b/-~o7S
if ~isnorm (2tH"I
error('zernfun:normalization','Unrecognized normalization flag.') \FXp*FbQ
end )P%4:P
else '-.wFB;
isnorm = false; {!r#f(?uT
end h;nQxmJ9
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q~M2:SN@X
% Compute the Zernike Polynomials F 3s?&T)[G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >:$"a
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% Determine the required powers of r: &oEyixe
% ----------------------------------- TL'0T,Jo
m_abs = abs(m); }^ ,q#'
rpowers = []; 5NFRPGYX
for j = 1:length(n) WL:0R>0
rpowers = [rpowers m_abs(j):2:n(j)]; -yl;3K]l
end #D0 ~{H
rpowers = unique(rpowers); UKj`_a6
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% Pre-compute the values of r raised to the required powers, #a'r_K=ch)
% and compile them in a matrix: JnHNkCaU
% ----------------------------- x,uBJ
if rpowers(1)==0 N|<bVq%
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ^)<