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, '3o0J\cz
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, Xl6)&
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? &,Q{l$`X
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 2t { Cpw
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function z = zernfun(n,m,r,theta,nflag) uz4mHyS6
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?E2k]y6<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LM'` U-/e$
% and angular frequency M, evaluated at positions (R,THETA) on the }bznx[4?I
% unit circle. N is a vector of positive integers (including 0), and ;_i0@@J
% M is a vector with the same number of elements as N. Each element s/[i>`g/9
% k of M must be a positive integer, with possible values M(k) = -N(k) ^@L[0Z`
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, <nsl`C~6g0
% and THETA is a vector of angles. R and THETA must have the same 5?kA)!|UB
% length. The output Z is a matrix with one column for every (N,M) gE=~.P[ZX
% pair, and one row for every (R,THETA) pair. )C2d)(baEJ
% `Ik}Xw
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike savz>E&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7IJb$af:;
% with delta(m,0) the Kronecker delta, is chosen so that the integral M{kPEl&Z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |/fbU_d
% and theta=0 to theta=2*pi) is unity. For the non-normalized (&MSP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. GIVs)~/Eq
% ,P"R.A
% The Zernike functions are an orthogonal basis on the unit circle. r-YQsu&
% They are used in disciplines such as astronomy, optics, and 24N,Bo
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% optometry to describe functions on a circular domain. 3R#<9O
% !P Gow
% The following table lists the first 15 Zernike functions. $fKwJFr
% \S7OC
% n m Zernike function Normalization -N\{QX1Yd
% -------------------------------------------------- |>3a9]
% 0 0 1 1 G0s:Dum
% 1 1 r * cos(theta) 2 Bh' vr3|
% 1 -1 r * sin(theta) 2 ^/n[5@6H
% 2 -2 r^2 * cos(2*theta) sqrt(6) gy =`c MS@
% 2 0 (2*r^2 - 1) sqrt(3) .;KupQ;*
% 2 2 r^2 * sin(2*theta) sqrt(6) b"FsT
% 3 -3 r^3 * cos(3*theta) sqrt(8) ,O~2
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) peqFa._W
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Ic=V:
% 3 3 r^3 * sin(3*theta) sqrt(8) W=EO=}l#
% 4 -4 r^4 * cos(4*theta) sqrt(10) 8&C(0H]1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +~fu-%,k
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (Z"Xp{u
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VvF&E>fC
% 4 4 r^4 * sin(4*theta) sqrt(10) 93WYZNpX
% -------------------------------------------------- d}o1 j
% zRJy3/>
% Example 1: hE6tu'
% |(P;2q4>
% % Display the Zernike function Z(n=5,m=1) Ro1' L1:
% x = -1:0.01:1; I(<G;ft<}
% [X,Y] = meshgrid(x,x); 8&UuwZ6i-
% [theta,r] = cart2pol(X,Y); ,xh9,EpBk
% idx = r<=1; /3TorB~Y
% z = nan(size(X)); m~U{ V9;*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); f<;9q?0V F
% figure `2fuV]FW
% pcolor(x,x,z), shading interp blN1Q%m6
% axis square, colorbar ppnj.tLz;r
% title('Zernike function Z_5^1(r,\theta)') %@&)t?/=
% O(~Vvoq
% Example 2: _(z"l"l=$
% O^x t
% % Display the first 10 Zernike functions aXJe"IT.u
% x = -1:0.01:1; 7}x-({bqy
% [X,Y] = meshgrid(x,x); V]O
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% [theta,r] = cart2pol(X,Y); h@DJ/&;u@
% idx = r<=1; 2>!ykUw^O
% z = nan(size(X)); _[phs06A
% n = [0 1 1 2 2 2 3 3 3 3]; ;Pa(nUE@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Td F<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8
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% y = zernfun(n,m,r(idx),theta(idx)); "QF083$
% figure('Units','normalized') }6bLukv
% for k = 1:10
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% z(idx) = y(:,k); 1wgu%$|d
% subplot(4,7,Nplot(k)) tQ~B!j]
% pcolor(x,x,z), shading interp -&EmEXs%
% set(gca,'XTick',[],'YTick',[]) %pp+V1FH
% axis square (
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) op-#Ig$#
% end o/zCXZnw#
% 0hkuBQb\
% See also ZERNPOL, ZERNFUN2. u
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% Paul Fricker 11/13/2006 CN=&Je%I
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% Check and prepare the inputs: >&&xJ5
% ----------------------------- =eqI]rVj^
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }SV3PdE
error('zernfun:NMvectors','N and M must be vectors.') AVr!e
end rxK0<pWJhx
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if length(n)~=length(m) gHx-m2N
error('zernfun:NMlength','N and M must be the same length.') QVW6SY
end j1F+,
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n = n(:);
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m = m(:); pD# "8h
if any(mod(n-m,2)) :xPvEK[B7
error('zernfun:NMmultiplesof2', ... 6
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'All N and M must differ by multiples of 2 (including 0).') (g m^o{
end 4c=kT@=jX
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if any(m>n) g8+,wSE
error('zernfun:MlessthanN', ... U_- K6:tr
'Each M must be less than or equal to its corresponding N.') pYVy(]1I(3
end H040-Q;S'
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if any( r>1 | r<0 ) ^qx\ e$R
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z]TVH8%|k
end l _O~v?
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6wB>-/'Y
error('zernfun:RTHvector','R and THETA must be vectors.') *'YNRM\}
end f#kevf9zc
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r = r(:); K=x1mM+RK
theta = theta(:); +)JqEwCrq
length_r = length(r); rp#*uV9;
if length_r~=length(theta) +~Lzsh"
error('zernfun:RTHlength', ... `_U0>Bfg;
'The number of R- and THETA-values must be equal.') ' 1'1T5x~
end $pfe2(8
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% Check normalization: 6r=)V$K<
% -------------------- j' KobyX<
if nargin==5 && ischar(nflag) k^5Rf
isnorm = strcmpi(nflag,'norm'); ~|{)h^]@
if ~isnorm q;../h]Ne
error('zernfun:normalization','Unrecognized normalization flag.') SE)j}go
end l;}7A,u
else [y[v]'
isnorm = false; (l8r>V
end [RFK-E
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $;i$k2n:
% Compute the Zernike Polynomials }t
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z*(!`,.bB
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% Determine the required powers of r:
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% ----------------------------------- *DCNu{6
m_abs = abs(m); O;BMwg_7
rpowers = []; !BQ ELB$0
for j = 1:length(n) 0S:!Gv+
rpowers = [rpowers m_abs(j):2:n(j)]; mz$Wo *FB
end a^\- }4yR
rpowers = unique(rpowers); *_/eAi/WG
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% Pre-compute the values of r raised to the required powers, , HI%Xn
% and compile them in a matrix: Hv gK_'
% ----------------------------- JeTrMa 2
if rpowers(1)==0 _l!U[{l*d
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aU.0dsq
rpowern = cat(2,rpowern{:}); tct5*.|
rpowern = [ones(length_r,1) rpowern]; D*T$ v
else F `pyhc>1;
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); BRU9LS
rpowern = cat(2,rpowern{:}); b8{h[YJL2
end ?^48Zq6wM
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% Compute the values of the polynomials: O{ %A&Ui
% -------------------------------------- F{*9[jY
y = zeros(length_r,length(n)); OU.9 #|q U
for j = 1:length(n) r6`^>c
s = 0:(n(j)-m_abs(j))/2; "Eok;io
pows = n(j):-2:m_abs(j); H&yFSz}6a
for k = length(s):-1:1 =Mu'+,dT
p = (1-2*mod(s(k),2))* ... U8QR*"GmT
prod(2:(n(j)-s(k)))/ ... 1_j<%1{sZ
prod(2:s(k))/ ... -4y)qGb*?
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Sp`fh7d.(
prod(2:((n(j)+m_abs(j))/2-s(k))); <7FP"YU
idx = (pows(k)==rpowers); }OP%p/eY
y(:,j) = y(:,j) + p*rpowern(:,idx); 0'%+X|
end g"Q}h
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if isnorm 76IALJ00V
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *`g-gk
end *<.WL"Qhl
end )kL`&+#>
% END: Compute the Zernike Polynomials Mdlt zy=)L
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }W@#S_-e8
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% Compute the Zernike functions: EN-8uY.
% ------------------------------ ~aqT~TL_
idx_pos = m>0; 36^C0uNdX
idx_neg = m<0; mHI4wS>()+
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z = y; nv_m!JG7
if any(idx_pos) zO).<xIq+
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FU]8.)`G
end 6cQeL$,SQ
if any(idx_neg) GLaZN4`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~.4W,QLuD
end \'It,PN
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% EOF zernfun 75\RG+kQ