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, Z;> aW;Wt
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, Dqo:X`<bT
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? 0O9
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? AXv3jH,HF
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function z = zernfun(n,m,r,theta,nflag) -T="Ml&
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xVmUmftD
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '2B0D|r"a
% and angular frequency M, evaluated at positions (R,THETA) on the ZI:d&~1i1
% unit circle. N is a vector of positive integers (including 0), and ,2L,>?r6
% M is a vector with the same number of elements as N. Each element ri.|EmH2:D
% k of M must be a positive integer, with possible values M(k) = -N(k)
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, %li{VDb
% and THETA is a vector of angles. R and THETA must have the same %4g4 C#
% length. The output Z is a matrix with one column for every (N,M) dodz|5o%
% pair, and one row for every (R,THETA) pair. BqJrL/(
% ~#xs
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ZCq\Zk1O&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PyJblW
% with delta(m,0) the Kronecker delta, is chosen so that the integral |HIA[.q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'aSORVq^e[
% and theta=0 to theta=2*pi) is unity. For the non-normalized J +Y|# U
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. iO#xIl<
% lu(Omds+
% The Zernike functions are an orthogonal basis on the unit circle. )9P
% They are used in disciplines such as astronomy, optics, and 9#ay(g
% optometry to describe functions on a circular domain.
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% x-P_}}K 79
% The following table lists the first 15 Zernike functions. @n y{.s+
% wZolg~dg
% n m Zernike function Normalization !Kn+*' #
% -------------------------------------------------- u(Q(UuI
% 0 0 1 1 >?\ !k
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% 1 1 r * cos(theta) 2 7VD7di=D
% 1 -1 r * sin(theta) 2 k$mX81
% 2 -2 r^2 * cos(2*theta) sqrt(6) 8&AorYw[
% 2 0 (2*r^2 - 1) sqrt(3) kxiyF$
9
% 2 2 r^2 * sin(2*theta) sqrt(6) +c2>j8e6
% 3 -3 r^3 * cos(3*theta) sqrt(8) JC-yiORVr
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Gf$>!zXr
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) S 2` ;7
% 3 3 r^3 * sin(3*theta) sqrt(8) V'#u_`x"D)
% 4 -4 r^4 * cos(4*theta) sqrt(10) E&=?\KM
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -x5bdC(d
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'r3}= z4Y
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZI*A0_;L
% 4 4 r^4 * sin(4*theta) sqrt(10) DD3yl\#,
% -------------------------------------------------- MZ[g|o!)v
% , 0ja _
% Example 1: }|,\?7,
% AZP>\Dq
% % Display the Zernike function Z(n=5,m=1) w6Ny>(T/
% x = -1:0.01:1; k0=y_7
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% [X,Y] = meshgrid(x,x); aj~@r3E;
% [theta,r] = cart2pol(X,Y); / S^m!{
% idx = r<=1; xL#oP0d<e
% z = nan(size(X)); LA3,e (e
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0pG(+fN_9
% figure 7Et(p'
% pcolor(x,x,z), shading interp ~DS9{Y
% axis square, colorbar lJ2/xE ]
% title('Zernike function Z_5^1(r,\theta)') 5q*~h4=r7
% I!@`_Q9N
% Example 2: DEuW' .o>
% 1e%Xyqb
% % Display the first 10 Zernike functions uZI:Kt#
% x = -1:0.01:1; ?=Qg
% [X,Y] = meshgrid(x,x); UYLI>XSd
% [theta,r] = cart2pol(X,Y); %-1-J<<J
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% idx = r<=1; WWzns[$f
% z = nan(size(X)); 2o}FB\4^i
% n = [0 1 1 2 2 2 3 3 3 3]; ;\0RXirk
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !0_Y@>2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; &~i
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% y = zernfun(n,m,r(idx),theta(idx)); cMKh+r
% figure('Units','normalized') 'v5gg2
% for k = 1:10 61 |xv_/
% z(idx) = y(:,k); LLN^^>5|l
% subplot(4,7,Nplot(k)) N_}Im>;!
% pcolor(x,x,z), shading interp 7t/SZm
% set(gca,'XTick',[],'YTick',[]) ^DJU99
% axis square Ee| y[y,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) SpQ6A]M gm
% end x$4'a~E
% p8bTR!rvz
% See also ZERNPOL, ZERNFUN2. S}yb~uc,
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% Paul Fricker 11/13/2006 cE?J]5#^
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% Check and prepare the inputs: #{PNdINoU
% ----------------------------- -hfY:W`Dz
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $80/ub:R
error('zernfun:NMvectors','N and M must be vectors.') J>&GP#7}
end "=O)2}
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if length(n)~=length(m) j$Co-b1
error('zernfun:NMlength','N and M must be the same length.') M3;B]iRQD
end v.J#d>tvf
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n = n(:); YTA&G
m = m(:); uLht;-`{n
if any(mod(n-m,2)) Nq3P?I(<
error('zernfun:NMmultiplesof2', ... \v_(*
'All N and M must differ by multiples of 2 (including 0).') ~CscctD{;
end chbs9y0
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if any(m>n) Mf"B!WU>]B
error('zernfun:MlessthanN', ... )i>KgX
'Each M must be less than or equal to its corresponding N.') ^~$
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end e)8iPu ..
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if any( r>1 | r<0 ) 8B5%IgA
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7085&\9
end h
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) gsAO<Fy
error('zernfun:RTHvector','R and THETA must be vectors.') ~gD'up@$/
end AseY.0
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r = r(:); "lt[)3*
theta = theta(:); iD~s,
length_r = length(r); 2I
if length_r~=length(theta) {lA@I*_lj
error('zernfun:RTHlength', ... lHU$A;
'The number of R- and THETA-values must be equal.') `N0E;=g
end Q2o:wXvj
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% Check normalization: QK\z-'&n
% -------------------- KK}&4^q
if nargin==5 && ischar(nflag) l;ugrAo?
isnorm = strcmpi(nflag,'norm'); gQ[4{+DSf
if ~isnorm ,>Q,0bVhH0
error('zernfun:normalization','Unrecognized normalization flag.') *4bV8T>0Z
end l`k3!EZDS
else //(c 1/s
isnorm = false; D+U^ pl-
end ME.LS2'n
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z,z^[Jz
% Compute the Zernike Polynomials !Kis,e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W*0KAC`m
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% Determine the required powers of r: $FoNEr&q
% ----------------------------------- :MpCj<<[
m_abs = abs(m); 8dv1#F|
rpowers = []; 8[k-8h|
for j = 1:length(n) 86i =N_
rpowers = [rpowers m_abs(j):2:n(j)]; bFpwq#PDW>
end KLk37IY2\
rpowers = unique(rpowers); LakP'P6`E
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1DX=\BWp
% Pre-compute the values of r raised to the required powers, c09 uCito
% and compile them in a matrix: q#Bdq8
% ----------------------------- xc!"?&\*
if rpowers(1)==0 ;tHF$1!J
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *(rq AB0~
rpowern = cat(2,rpowern{:}); #pZ3xa3R
rpowern = [ones(length_r,1) rpowern]; /N$T[
else f-Sb:O!V
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (efH>oY[
rpowern = cat(2,rpowern{:}); MKbW^:
end ;3w W)gL1
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% Compute the values of the polynomials: )XD_Yq@E
% -------------------------------------- X/Ae-1!
y = zeros(length_r,length(n)); z:w7e0
for j = 1:length(n) O_E[FE:+
s = 0:(n(j)-m_abs(j))/2; (qaY,>je]D
pows = n(j):-2:m_abs(j); PKP(:3|
for k = length(s):-1:1 yEH30zSt
p = (1-2*mod(s(k),2))* ... 5yry$w$G)
prod(2:(n(j)-s(k)))/ ... $+tkBM
prod(2:s(k))/ ... [P^ .=F
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &ha39&I
prod(2:((n(j)+m_abs(j))/2-s(k))); rA9"CN
idx = (pows(k)==rpowers); Agl[Z>Q
y(:,j) = y(:,j) + p*rpowern(:,idx); rn(T
Z}
end (*|hlD~
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if isnorm "2 Kh2[K
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); O:1YG$uKa
end o/Z?/alt4
end smSUo/
% END: Compute the Zernike Polynomials wL:3RZB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P?>p+dM
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% Compute the Zernike functions: P5[.2y_qM
% ------------------------------ /
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idx_pos = m>0; A;h~Fx6s
idx_neg = m<0; 291v
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z = y; !bQ5CB
if any(idx_pos) )jnxR${M
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Yk:\oM
end 9] l7j\L
if any(idx_neg) IXg0g<JZ
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); CT/`Kg_
end a6[bF
#\fApRL
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% EOF zernfun wL~
dZ!,J