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, uF9C-H@:
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, 0;AA/
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? '":lB]hS
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4'a=pnE$
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function z = zernfun(n,m,r,theta,nflag) i*Sqd a
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. LE9(fe) fe
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @,cowar*
% and angular frequency M, evaluated at positions (R,THETA) on the 7!EBH(,z
% unit circle. N is a vector of positive integers (including 0), and #t:S.A@
% M is a vector with the same number of elements as N. Each element &:dH,
% k of M must be a positive integer, with possible values M(k) = -N(k) 3L_\`Ia9
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }%'?p<^M
% and THETA is a vector of angles. R and THETA must have the same x7jC)M<k0
% length. The output Z is a matrix with one column for every (N,M) iS
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% pair, and one row for every (R,THETA) pair. s'E2P[:
% h?BFvbAt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i+S)
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]yx$(6_U
% with delta(m,0) the Kronecker delta, is chosen so that the integral Ay5i+)MD
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V )x$|!(
% and theta=0 to theta=2*pi) is unity. For the non-normalized t\{'F7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ktFhc3);!
% #Ssx!+q?
% The Zernike functions are an orthogonal basis on the unit circle. T|7}EAR=b
% They are used in disciplines such as astronomy, optics, and %_RQx2
% optometry to describe functions on a circular domain. Lvq>v0|
% s;S?;(QI
% The following table lists the first 15 Zernike functions. TarIPp
% 723bkJw
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% n m Zernike function Normalization T92UeG
% -------------------------------------------------- #h8Sq~0
% 0 0 1 1 v9w'!C)b
% 1 1 r * cos(theta) 2 s:#V(<J
% 1 -1 r * sin(theta) 2 h_:C+)13`x
% 2 -2 r^2 * cos(2*theta) sqrt(6) wsIW
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% 2 0 (2*r^2 - 1) sqrt(3) aT)BR?OYSJ
% 2 2 r^2 * sin(2*theta) sqrt(6) 4'`{H@]tb
% 3 -3 r^3 * cos(3*theta) sqrt(8) vY }A
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) bx{$Y_L+p
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) p?7v$ev_
% 3 3 r^3 * sin(3*theta) sqrt(8) Y^8C)p9r
% 4 -4 r^4 * cos(4*theta) sqrt(10) VY;{/.Sa
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =BSzsH7
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -@yh>8v
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Pe3@d|-,MU
% 4 4 r^4 * sin(4*theta) sqrt(10) x(etb<!jd
% -------------------------------------------------- wkA+j9.
% %Y<| ;0v
% Example 1: uxtWybv
% tyXuG<
% % Display the Zernike function Z(n=5,m=1) ?Z Rs\+{vG
% x = -1:0.01:1; \Xm,OE_v"
% [X,Y] = meshgrid(x,x); .S(TxksCz
% [theta,r] = cart2pol(X,Y); m?pstuUK(
% idx = r<=1; ,SynnE68
% z = nan(size(X)); 5][Ztx
% z(idx) = zernfun(5,1,r(idx),theta(idx)); -+ SF
% figure Cjqklb/
% pcolor(x,x,z), shading interp DoJ\ q+
% axis square, colorbar F(k.,0Nc
% title('Zernike function Z_5^1(r,\theta)') U3T#6Rptl
% z=rT%lz6
% Example 2: Ir`eL
% ,&jhlZ i
% % Display the first 10 Zernike functions ;1`fC@rI
% x = -1:0.01:1; @R/07&lBR
% [X,Y] = meshgrid(x,x); 8oUpQcim
% [theta,r] = cart2pol(X,Y); 4]G?G]lS>
% idx = r<=1; tBq
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% z = nan(size(X)); r=5{o1"
% n = [0 1 1 2 2 2 3 3 3 3];
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; o<<xY<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; BSYzC9h`
% y = zernfun(n,m,r(idx),theta(idx)); d
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% figure('Units','normalized') 4
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% for k = 1:10 U=DmsnD,
% z(idx) = y(:,k); C8[&S&<_<
% subplot(4,7,Nplot(k)) 9o;^[Ql-
% pcolor(x,x,z), shading interp 9xO#tu]
% set(gca,'XTick',[],'YTick',[]) i@P)a'W_
% axis square ]+|~cRQ9I
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |uqf:V`z:
% end GNXHM*~
% @ zs'Y8
% See also ZERNPOL, ZERNFUN2. /2UH=Q!x4E
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% Paul Fricker 11/13/2006 xIxn"^'
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% Check and prepare the inputs: 0'V5/W
% ----------------------------- RIb4!!',c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zo +nq%=
error('zernfun:NMvectors','N and M must be vectors.') q}~3C1
end JRSSn] pw
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if length(n)~=length(m) Xe<sJ.&Wf
error('zernfun:NMlength','N and M must be the same length.') lV1G<qP
end \@8+U;d
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n = n(:); [z9`)VIe
m = m(:); c0%"&a1]]V
if any(mod(n-m,2)) 1QLbf*zeIW
error('zernfun:NMmultiplesof2', ... FN\E*@>X=
'All N and M must differ by multiples of 2 (including 0).') A6:es_
end BFL`!^
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if any(m>n) B$b +Ymu
error('zernfun:MlessthanN', ... AtdlZ
'Each M must be less than or equal to its corresponding N.') k p<OJy
end 7w'wjX-
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if any( r>1 | r<0 ) o$w_Es]Ma
error('zernfun:Rlessthan1','All R must be between 0 and 1.') H*[M\gN$
end R{ a"Y$
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ezr:1 GJ
error('zernfun:RTHvector','R and THETA must be vectors.') H-~6Z",1
end ^:#D0[
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r = r(:); ,=Mt`aN
theta = theta(:); oW<5|FaN
length_r = length(r); x n5l0'2
if length_r~=length(theta) %kdEun
error('zernfun:RTHlength', ... "br,/Dk>MX
'The number of R- and THETA-values must be equal.') So0f)`A
end BsEF'h'Owh
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% Check normalization: DpUbzr41+k
% -------------------- z"0I>gl
if nargin==5 && ischar(nflag) 1UE6 4Kl:S
isnorm = strcmpi(nflag,'norm'); .ox8*OO<
if ~isnorm D'J0wT#
error('zernfun:normalization','Unrecognized normalization flag.') <$X3Hye
end P%#<I}0C
else O+]Ifm [
isnorm = false; }[4r4 1[
end 8=gjY\Dp
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]=>F.GE
% Compute the Zernike Polynomials 1IZ3=6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1`a5C.v
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% Determine the required powers of r: qYlhlHD
% ----------------------------------- go'-5in(
m_abs = abs(m); MM(xk
rpowers = []; %`&2+\`
for j = 1:length(n) =5kY6%E7c
rpowers = [rpowers m_abs(j):2:n(j)]; A1.7O
end
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rpowers = unique(rpowers); Q0&H#xgt
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% Pre-compute the values of r raised to the required powers, TrLu~4
% and compile them in a matrix: OH">b6>\
% ----------------------------- ][?G/*k
if rpowers(1)==0 oxz OA
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \lZf<