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, IN/$b^Um
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, J%"5?)[z
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? [CEV&B
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? >8/Otg+h
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function z = zernfun(n,m,r,theta,nflag) pmP~1=3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. V(Pw|u"
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K\ Wzh;
% and angular frequency M, evaluated at positions (R,THETA) on the 5
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% unit circle. N is a vector of positive integers (including 0), and T"P}` mT
% M is a vector with the same number of elements as N. Each element 9X*Z\-
% k of M must be a positive integer, with possible values M(k) = -N(k) X_'tgP9
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, r??_2>Q
% and THETA is a vector of angles. R and THETA must have the same O^\:J2I(
% length. The output Z is a matrix with one column for every (N,M) /\Nc6Z/ L
% pair, and one row for every (R,THETA) pair. XcNL\fl1
% g5#LoGc
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gH7 +#/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DSHvBFQ
% with delta(m,0) the Kronecker delta, is chosen so that the integral n`^jNXE
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Xj]9/?B?
% and theta=0 to theta=2*pi) is unity. For the non-normalized /
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (P?|Bk[
% :sw5@JdJ
% The Zernike functions are an orthogonal basis on the unit circle. *i*\dl
% They are used in disciplines such as astronomy, optics, and *JImP9SE
% optometry to describe functions on a circular domain. 3]1 !g6
% +E9G"Z65iP
% The following table lists the first 15 Zernike functions. V^tD@N
% Oa:C'M
b
% n m Zernike function Normalization
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% -------------------------------------------------- ]?_~QE`
% 0 0 1 1 .}F
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% 1 1 r * cos(theta) 2 \
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% 1 -1 r * sin(theta) 2 nN$.^!;&
% 2 -2 r^2 * cos(2*theta) sqrt(6) L[44D6Vg
% 2 0 (2*r^2 - 1) sqrt(3) ~I N g9|
% 2 2 r^2 * sin(2*theta) sqrt(6) $|Ol?s
% 3 -3 r^3 * cos(3*theta) sqrt(8) [BdRx`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) o.Ww.F
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fwUvFK1G
% 3 3 r^3 * sin(3*theta) sqrt(8) j+'ua=T3
% 4 -4 r^4 * cos(4*theta) sqrt(10) M
p<r`PM2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [3=Y 9P:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ma\%uEgTD
% 4 4 r^4 * sin(4*theta) sqrt(10) q;R&valn
% -------------------------------------------------- b`%u}^B {
% 'r=2f6G>cP
% Example 1: Wk^{Tn/]
% {_W8Qm`.
% % Display the Zernike function Z(n=5,m=1) :!Z |_y{b
% x = -1:0.01:1; fph+05.%
% [X,Y] = meshgrid(x,x); nv0D4 t
% [theta,r] = cart2pol(X,Y); \aPH_sf,
% idx = r<=1; Gfx!.[Y
% z = nan(size(X)); bkR~>F]FAu
% z(idx) = zernfun(5,1,r(idx),theta(idx)); F%zMhX'AG
% figure P;(@"gD8z5
% pcolor(x,x,z), shading interp <9H3d7%
% axis square, colorbar s8:epcL`A
% title('Zernike function Z_5^1(r,\theta)') yU(}1ZID
% DNDzK
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% Example 2: _Cf:\Xs
m
% k"7ZA>5jk
% % Display the first 10 Zernike functions c{`!$Z'k<
% x = -1:0.01:1; kqZRg>1A
% [X,Y] = meshgrid(x,x); UazK0{t<f
% [theta,r] = cart2pol(X,Y); [e\IHakj
% idx = r<=1; )Dms9:
% z = nan(size(X)); @lM-+q(tl
% n = [0 1 1 2 2 2 3 3 3 3]; \aof
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; /qKor;x
% Nplot = [4 10 12 16 18 20 22 24 26 28]; rVhfj~Ts
% y = zernfun(n,m,r(idx),theta(idx)); `"'u
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% figure('Units','normalized') [V
/f{y~{
% for k = 1:10 ;L",K?6#
% z(idx) = y(:,k); i\Yd_
% subplot(4,7,Nplot(k)) +5-|6
% pcolor(x,x,z), shading interp F~fN7<9R
% set(gca,'XTick',[],'YTick',[]) S_*Gv O
% axis square _nzTd\L88
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) l'Li!u
% end 3bd`q
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% vx6lud0k}
% See also ZERNPOL, ZERNFUN2. vnf2Z,f%
O)R}|
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% Paul Fricker 11/13/2006 08AD~^^
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% Check and prepare the inputs: Y1AZ%{^0a
% ----------------------------- hb0)<^xu
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *E>R1bJ8
error('zernfun:NMvectors','N and M must be vectors.') y] 9/Xr/
end P1L+Vnfu
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if length(n)~=length(m) 1H%p|'FKA
error('zernfun:NMlength','N and M must be the same length.') ,[N(XstI
end Z9h4 pd
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n = n(:); /EHO(d!<
m = m(:); st.{AEv@
if any(mod(n-m,2)) 9 M?UPE
error('zernfun:NMmultiplesof2', ... ~[aV\r?
'All N and M must differ by multiples of 2 (including 0).') x~m$(LT
end eC 2~&:$L
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if any(m>n) m'G=WO*%
error('zernfun:MlessthanN', ... uARkf'
'Each M must be less than or equal to its corresponding N.') fMHw=wJQ
end EHl~y=9
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if any( r>1 | r<0 ) g rI#' x
error('zernfun:Rlessthan1','All R must be between 0 and 1.') l7<VH z0b
end &_@M
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "9X(.v0ze
error('zernfun:RTHvector','R and THETA must be vectors.') DP *$@5
end .;U?%t_7
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r = r(:); *Br
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theta = theta(:); s/3sOb}sA
length_r = length(r); q)@;8Z=_c
if length_r~=length(theta) Gw6Odj
error('zernfun:RTHlength', ... 'UGgY3
'The number of R- and THETA-values must be equal.') wsR\qq
end -nD}k
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% Check normalization: JvCy&xrE;
% -------------------- F7=\*U
if nargin==5 && ischar(nflag) tmeg=U7
isnorm = strcmpi(nflag,'norm'); !6#.%"{-
if ~isnorm 9Ns%<FRO@
error('zernfun:normalization','Unrecognized normalization flag.') zT!.5qd
end ?}uvpB1}
else *y+K{ fM1
isnorm = false; PVN`k, 4
end HFYe@ 2r
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5yW}#W>
% Compute the Zernike Polynomials gId
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,>kXn1 ,
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% Determine the required powers of r: gA5DEit
% ----------------------------------- e-xT.RnQ
m_abs = abs(m); O|9Nl*rXz
rpowers = []; xkkG#n)
for j = 1:length(n) 96gaun J
rpowers = [rpowers m_abs(j):2:n(j)]; O!F"w!5@
end ^Y8G}Z|
rpowers = unique(rpowers); i!<(R$Lo
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% Pre-compute the values of r raised to the required powers, -l"8L;`
% and compile them in a matrix: (f*r
% ----------------------------- q #8z%/~k
if rpowers(1)==0 sI#h&V,9
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?Qpi(Czbpq
rpowern = cat(2,rpowern{:}); S!iDPl~
rpowern = [ones(length_r,1) rpowern]; {-|El}.M
else #TgP:t]p
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5["n] i
rpowern = cat(2,rpowern{:}); N B8Yn\{B
end &k(tDP
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% Compute the values of the polynomials: \x_$Pu
% -------------------------------------- UyMlk
y = zeros(length_r,length(n)); K$$%j "s
for j = 1:length(n) ]go.IfH
s = 0:(n(j)-m_abs(j))/2; 'E\qqE[;
pows = n(j):-2:m_abs(j); tU8aPiUl
for k = length(s):-1:1 X(]J\?n'
p = (1-2*mod(s(k),2))* ... E_xk8X~
prod(2:(n(j)-s(k)))/ ... xfqu=z8X
prod(2:s(k))/ ... s21)*d
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Gl T/JZ9
prod(2:((n(j)+m_abs(j))/2-s(k))); /DSy/p0%
idx = (pows(k)==rpowers); 7l'1
y(:,j) = y(:,j) + p*rpowern(:,idx); kPnuU!
end Z~"8C Kz
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if isnorm s%N6^}N
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pTYV@5|
end ;s-fYS6(>{
end A&Q!W)=
% END: Compute the Zernike Polynomials S.owVMQ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r+MqjdXG
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% Compute the Zernike functions: DH4|lb}
% ------------------------------ m&Y?]nbq
idx_pos = m>0; &([Gc+"5E.
idx_neg = m<0; ("J_< p
%S%0/
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z = y; Z
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if any(idx_pos) NB&zBJ#
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); TyaK_XW
end
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if any(idx_neg) JaJyH%+$!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PO0/C q)
end Sr6?^>A@t
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% EOF zernfun u$1^=