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, ~YviXSW
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, cSG(kFQ
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? k][{4~z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ZGCp[2$
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function z = zernfun(n,m,r,theta,nflag) JPj/+f
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. M;KeY[u
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \X]I: 0^j
% and angular frequency M, evaluated at positions (R,THETA) on the Pmr'W\aIR
% unit circle. N is a vector of positive integers (including 0), and q1r-xsjV=
% M is a vector with the same number of elements as N. Each element +x4*T
% k of M must be a positive integer, with possible values M(k) = -N(k) ,5 3`t
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ('d,Sh
% and THETA is a vector of angles. R and THETA must have the same ,MHF
% length. The output Z is a matrix with one column for every (N,M) /!/Pk'p=/
% pair, and one row for every (R,THETA) pair. B/hQvA;(
% `7V1 F.\
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike d$?+>t/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A
L|,\s
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0 EA3>$;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, G[OJ<px
% and theta=0 to theta=2*pi) is unity. For the non-normalized "tpD ->
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. X;vUz
% Lc-WfzT
% The Zernike functions are an orthogonal basis on the unit circle. I_`NjJ;61
% They are used in disciplines such as astronomy, optics, and jgkY^l
% optometry to describe functions on a circular domain. X"HVK+
% { W5
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% The following table lists the first 15 Zernike functions. |&bucG=
% eU]I !pI<
% n m Zernike function Normalization mOLz(0
% -------------------------------------------------- +#X+QG
% 0 0 1 1 7v.O Lp
% 1 1 r * cos(theta) 2 x&EMg!
% 1 -1 r * sin(theta) 2 b 1."mT!p
% 2 -2 r^2 * cos(2*theta) sqrt(6) o{mVXidE
% 2 0 (2*r^2 - 1) sqrt(3) k@[[vj|W
% 2 2 r^2 * sin(2*theta) sqrt(6) X?`mYoe
% 3 -3 r^3 * cos(3*theta) sqrt(8) wp:Zur5Y
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) O\%0D.HEz
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) v`6vc)>8
% 3 3 r^3 * sin(3*theta) sqrt(8) OsYZa`$,
% 4 -4 r^4 * cos(4*theta) sqrt(10) 2IkyC`
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &{q'$oF
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) yaHkWkl
=
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '?X?'_3
% 4 4 r^4 * sin(4*theta) sqrt(10) 8N<mV^|}
% -------------------------------------------------- sdgI ,
% 4"^W/Zo
% Example 1: 7.kH="@
% ?1eu9; q\*
% % Display the Zernike function Z(n=5,m=1) Dx9k%G)!
% x = -1:0.01:1; L,,*8
% [X,Y] = meshgrid(x,x); 7WmY:g#s
% [theta,r] = cart2pol(X,Y); plNw>rFa
% idx = r<=1; +p]@ b
% z = nan(size(X));
!rG-[7K
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 'I2[}>mj2
% figure vEt+^3=
% pcolor(x,x,z), shading interp }C~9?Y
% axis square, colorbar KT*"Sbh
% title('Zernike function Z_5^1(r,\theta)') CT<z1)#@^
% lhBAT%U\
% Example 2: iqsR]mab
% GQE7P()
% % Display the first 10 Zernike functions ]RF(0;
% x = -1:0.01:1; &oFgZ .
% [X,Y] = meshgrid(x,x); 5T'viG}%
% [theta,r] = cart2pol(X,Y); 2B_+5
% idx = r<=1; 7Q&S [])
% z = nan(size(X)); #!r>3W&
% n = [0 1 1 2 2 2 3 3 3 3]; VZ9`Kbu
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 4#ifm#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; N)KN!!
% y = zernfun(n,m,r(idx),theta(idx)); )2:U]d%pk
% figure('Units','normalized') :"Rx$;a
% for k = 1:10 `vf]C'
% z(idx) = y(:,k); V.ae 5@;
% subplot(4,7,Nplot(k)) UyDq`@h
% pcolor(x,x,z), shading interp U\[b qw
% set(gca,'XTick',[],'YTick',[]) OY!WEP$F-C
% axis square o$}$Z&LK
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;iUO1t)^
% end ykxAm\O
% $]^Io)}f@
% See also ZERNPOL, ZERNFUN2. u|Ng>lU
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% Paul Fricker 11/13/2006 iU#"G" &
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% Check and prepare the inputs: JDOn`7!w
% ----------------------------- ?rdWhF]
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) R~RE21kAc
error('zernfun:NMvectors','N and M must be vectors.') F$O$Y[
end >#Bu [nD%
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+.xK`_[M
if length(n)~=length(m) =n8M'
error('zernfun:NMlength','N and M must be the same length.') : T qeVf
end nM99AW
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n = n(:); _gl1Qtv@rf
m = m(:); ++=jh6
if any(mod(n-m,2)) =RofC9,
error('zernfun:NMmultiplesof2', ... U8<C4
'All N and M must differ by multiples of 2 (including 0).') Z55C4F5v
end ]Z*B17//
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if any(m>n) x,GLGGi}_x
error('zernfun:MlessthanN', ... z<9Llew^e
'Each M must be less than or equal to its corresponding N.') [?2,(X0yh1
end O5*3
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if any( r>1 | r<0 ) <%=<9~e
error('zernfun:Rlessthan1','All R must be between 0 and 1.') U/h@Q\~U
end Z,8t!Y
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J.c
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error('zernfun:RTHvector','R and THETA must be vectors.') +HG*T[%/
end }|Bs|$q
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r = r(:); ? ^`fPH=
theta = theta(:); -_Kw3x
length_r = length(r); S[N9/2
if length_r~=length(theta) Epm8S}6K
error('zernfun:RTHlength', ... v'r)d-T
'The number of R- and THETA-values must be equal.') (,cG+3r]
end $\PU Y8
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% Check normalization: *<X*)A{C
% -------------------- #RHt;SFx
if nargin==5 && ischar(nflag) sFsf~|
isnorm = strcmpi(nflag,'norm'); 9q\_UbF
if ~isnorm 6.6?Rp".
error('zernfun:normalization','Unrecognized normalization flag.') 2)-4?uz~
end NnaO!QW%
else wNmC1HOh
isnorm = false; d;{k,rP6
end Bi>]s%zp
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uJ0Wb$%
% Compute the Zernike Polynomials g2A#BMe'.$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rgl cd
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% Determine the required powers of r: t=fP^bJ
% ----------------------------------- @|e
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m_abs = abs(m); 3jHg9M23[^
rpowers = []; '~1Zr uO
for j = 1:length(n) 6E.[F\u
rpowers = [rpowers m_abs(j):2:n(j)]; (*AJ6BQWa
end lr@w1*
rpowers = unique(rpowers); `g0^W/j
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% Pre-compute the values of r raised to the required powers, g1(5QWb
% and compile them in a matrix: >P//]nn
% ----------------------------- [ 6Sk>j
if rpowers(1)==0 kfZ(:3W$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <2~DI0pp(
rpowern = cat(2,rpowern{:}); *vq75k$7
rpowern = [ones(length_r,1) rpowern]; m!=5Q S3Z
else 1qBE|PwBp
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); q+cD
rpowern = cat(2,rpowern{:}); G\^<MR|
end WZh_z^rwn
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% Compute the values of the polynomials: nq~fH(QY
% -------------------------------------- cvhwd\
y = zeros(length_r,length(n)); UT[{NltH
for j = 1:length(n) [(&aVHUj
s = 0:(n(j)-m_abs(j))/2; l}&2A*c.
pows = n(j):-2:m_abs(j); S\!vDtD@
for k = length(s):-1:1 VN'\c3;
p = (1-2*mod(s(k),2))* ... KVUub'k
prod(2:(n(j)-s(k)))/ ... < RtyW
prod(2:s(k))/ ... 16 \)C/*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2 )3oX
prod(2:((n(j)+m_abs(j))/2-s(k))); kE|x'(x
idx = (pows(k)==rpowers); 7>0u
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y(:,j) = y(:,j) + p*rpowern(:,idx); yO,Jgn
end 0Ng?U+6
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if isnorm oK$Krrs0&
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); VT=gb/W6)a
end 5?([jAOf
end w.#z>4#3-
% END: Compute the Zernike Polynomials k8%@PC$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _6'@#DN
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% Compute the Zernike functions: UBk:B
% ------------------------------ OK YbEn#
idx_pos = m>0; jicH 94#(]
idx_neg = m<0; \u))1zRd
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z = y;
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if any(idx_pos) GlRjbNW?Q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); b;#_?2c
end ?mSZQF:d@
if any(idx_neg) %[M0TE=J
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^:}C,lIrG
end n&zEYCSI
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% EOF zernfun 1i_%1Oip