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, )fXxkOd
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, nUY)LnI
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? H94_a e
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? C&F%
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function z = zernfun(n,m,r,theta,nflag) Jv(E'"H
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. [:,|g;=Y}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K[SzE{5=P
% and angular frequency M, evaluated at positions (R,THETA) on the d+Mogku2
% unit circle. N is a vector of positive integers (including 0), and &WCVdZK:
% M is a vector with the same number of elements as N. Each element B'Nvl#
% k of M must be a positive integer, with possible values M(k) = -N(k) ^`-Hg= d
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _2k<MiqCD[
% and THETA is a vector of angles. R and THETA must have the same b5p;)#
% length. The output Z is a matrix with one column for every (N,M) [)IaXa
% pair, and one row for every (R,THETA) pair. ;J?fK69%
% +vFqHfmP
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike NgGpLdaC2v
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), kPEU }Kv
% with delta(m,0) the Kronecker delta, is chosen so that the integral cLp9|y0r
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, GNG.N)q#C
% and theta=0 to theta=2*pi) is unity. For the non-normalized Q2|6W E
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?h7[^sxJ
% )W @
% The Zernike functions are an orthogonal basis on the unit circle. ),<h6$
% They are used in disciplines such as astronomy, optics, and Q1h v2*/U
% optometry to describe functions on a circular domain. HDo=W qG
% F&/}x15
% The following table lists the first 15 Zernike functions. {YzpYc1
% }k1[Fc|
% n m Zernike function Normalization 7|m{hSc
% -------------------------------------------------- EY1L5Ba.
% 0 0 1 1 6{Bvl[mhI
% 1 1 r * cos(theta) 2 Y5>'(A>
% 1 -1 r * sin(theta) 2 6yaWxpW
% 2 -2 r^2 * cos(2*theta) sqrt(6) [woxCfSA
% 2 0 (2*r^2 - 1) sqrt(3) X bV?=
% 2 2 r^2 * sin(2*theta) sqrt(6) z ISy\uka
% 3 -3 r^3 * cos(3*theta) sqrt(8) a.<!>o<t:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) I7ySm12}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) lZ+1A0e
% 3 3 r^3 * sin(3*theta) sqrt(8) WsM/-P1Y
% 4 -4 r^4 * cos(4*theta) sqrt(10) :Ea]baM"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Dx3Sf}G
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "MT{t><
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (w 'k\y
% 4 4 r^4 * sin(4*theta) sqrt(10) . Vq_O
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% -------------------------------------------------- is-{U?-
% M+Y^ A7
% Example 1: iL IKrU+`
% /3vj`#jD
% % Display the Zernike function Z(n=5,m=1) j%0g*YI
% x = -1:0.01:1; 9e1KH'
% [X,Y] = meshgrid(x,x); B415{
% [theta,r] = cart2pol(X,Y); ,wra f#UdP
% idx = r<=1; 2wh{[Q2f
% z = nan(size(X)); y/?;s]>b
% z(idx) = zernfun(5,1,r(idx),theta(idx)); an?g'8! r:
% figure gtP;Qw'
% pcolor(x,x,z), shading interp p4zV<qZ>e
% axis square, colorbar
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% title('Zernike function Z_5^1(r,\theta)') r(=3yd/G$
% I."4u~[
% Example 2: M#>f:_`<
% UUql"$q
% % Display the first 10 Zernike functions +%RXV~
% x = -1:0.01:1; hta$k%2
% [X,Y] = meshgrid(x,x); ?9H.JR2s%
% [theta,r] = cart2pol(X,Y); z[, `
% idx = r<=1; =mk7'A>l
% z = nan(size(X)); Y-,1&$&
% n = [0 1 1 2 2 2 3 3 3 3]; :!EOg4%i
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; kjW`k?'s
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ZGOI8M]@
% y = zernfun(n,m,r(idx),theta(idx)); mk~Lkwl
% figure('Units','normalized') Ec]cCLB
% for k = 1:10 @{n2R3)k
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% z(idx) = y(:,k); xF{<-b
% subplot(4,7,Nplot(k)) xH8nn3U
% pcolor(x,x,z), shading interp :XZ
% set(gca,'XTick',[],'YTick',[]) m;LeaD}0
% axis square LNU9M>
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) BO ^T
:
% end o' DXd[y
% P8s'e_t
% See also ZERNPOL, ZERNFUN2. \h"QgHzp
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% Paul Fricker 11/13/2006 :FdV$E]]<
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% Check and prepare the inputs: ODM<$Yo:d
% ----------------------------- DI!l.w5P_
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'u.`!w '|L
error('zernfun:NMvectors','N and M must be vectors.') mvxg|<
end : C;=<$
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if length(n)~=length(m) R16"lG
error('zernfun:NMlength','N and M must be the same length.') ?z60b=f8
end 4 ITSDx
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n = n(:); 4oOe
m = m(:); hD l+
if any(mod(n-m,2)) $0K9OF9$
error('zernfun:NMmultiplesof2', ... :h3
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'All N and M must differ by multiples of 2 (including 0).') Md[nlz
end d8
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if any(m>n) 9!FV.yp%F
error('zernfun:MlessthanN', ... yZ+o7?(2p
'Each M must be less than or equal to its corresponding N.') A0WQZt!FEN
end f=J#mmHw$
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if any( r>1 | r<0 ) DjL(-7'p
error('zernfun:Rlessthan1','All R must be between 0 and 1.') K#";!
end Ar=pzQ<Z{
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #ZGWU_l}
error('zernfun:RTHvector','R and THETA must be vectors.') ;Fuxj!gF
end sbNCviKP
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r = r(:); g[@]OsX
theta = theta(:); F=^vu7rf
length_r = length(r); Jp5~iC2d
if length_r~=length(theta) {q8V
error('zernfun:RTHlength', ... ~Cj+6CrT
'The number of R- and THETA-values must be equal.') <6n(a)L1
end }
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% Check normalization: )Hlr 09t=]
% -------------------- 0R*
if nargin==5 && ischar(nflag) ~"}-cl,
isnorm = strcmpi(nflag,'norm'); jPA?0h
if ~isnorm eB!0:nHN
error('zernfun:normalization','Unrecognized normalization flag.') AytHnp\H
end I#S6k%-'
else p#J}@a
isnorm = false; xp>ra2A
end t91v%L
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZZFa<AK4
% Compute the Zernike Polynomials cy/;qd+!M
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `<Zp!Hl(j
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% Determine the required powers of r: MWsBZJRr
% ----------------------------------- vVZ@/D6w
m_abs = abs(m); pt|u?T_+
rpowers = []; xk.\IrB_
for j = 1:length(n) @;d(>_n
rpowers = [rpowers m_abs(j):2:n(j)]; H-0A&oG
end ;9 XM
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rpowers = unique(rpowers); Y R#_<o
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% Pre-compute the values of r raised to the required powers, NBE)DL
% and compile them in a matrix: cq
%=DZ
% ----------------------------- "i4@'`r
if rpowers(1)==0 2Wq)y1R<T
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <q%buyQna
rpowern = cat(2,rpowern{:}); 0D:J d6\
rpowern = [ones(length_r,1) rpowern]; 8KT|ixs
else Sep}{`u
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); HDA!;&NRS
rpowern = cat(2,rpowern{:}); ~0t]`<y=
end Nm:nSqc
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% Compute the values of the polynomials: W9{6?,]
% -------------------------------------- ^ ,U9N
y = zeros(length_r,length(n)); )DfmO
for j = 1:length(n) a` 95eL}
s = 0:(n(j)-m_abs(j))/2; EM;]dLh
pows = n(j):-2:m_abs(j); =?0o5|u]
for k = length(s):-1:1 -`FTWH
p = (1-2*mod(s(k),2))* ... I%b,
H`
prod(2:(n(j)-s(k)))/ ... ZVVK:dDgt
prod(2:s(k))/ ... GmPNzHDb
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'X"@C;q
prod(2:((n(j)+m_abs(j))/2-s(k))); 9] Uvy|
idx = (pows(k)==rpowers); w,;CrW T2t
y(:,j) = y(:,j) + p*rpowern(:,idx); * pyi;
end 2zh?]if
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if isnorm W$v5o9\Px
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <<@$0RW
end 'h.{fKG]ME
end {(Drw~/@
% END: Compute the Zernike Polynomials | ?~-k[|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /
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% Compute the Zernike functions: IFH%R>={
% ------------------------------ tb/bEy^
idx_pos = m>0; A+69_?B
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idx_neg = m<0; /J<?2T9G
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; mu9;ixZ
z = y; *Ny^XQ_ X
if any(idx_pos) wt? 8-_
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,[D,G
end Uz>5!_
if any(idx_neg) fF;Oz"I{\
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -z-58FLlO
end L L7a20
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% EOF zernfun hT>h