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, w$jq2?l
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, #=#bv`
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? 0iVeM!bM
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? c!] yT0v&s
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function z = zernfun(n,m,r,theta,nflag) wc?YzXP+
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D'Uv7Mis
% and angular frequency M, evaluated at positions (R,THETA) on the ;upYam"
% unit circle. N is a vector of positive integers (including 0), and qm"AatA
% M is a vector with the same number of elements as N. Each element I|_U|H!`
% k of M must be a positive integer, with possible values M(k) = -N(k) spTIhZ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, GSVLZF'+
% and THETA is a vector of angles. R and THETA must have the same q1Ehl
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% length. The output Z is a matrix with one column for every (N,M) Y/qs\c+
% pair, and one row for every (R,THETA) pair. rvPmd%nk-
% QPKY9.Rvv
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #ib?6=sPC
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), wSG!.Ejc7
% with delta(m,0) the Kronecker delta, is chosen so that the integral bP7_QYQ6
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2bxW`.fa
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9''x'E=|
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nS]Ih 0(K
% a 9Kws[
% The Zernike functions are an orthogonal basis on the unit circle. T)MZ`dM
% They are used in disciplines such as astronomy, optics, and `}~NZ
% optometry to describe functions on a circular domain. q=;U(,Y
% Em/? 4&
% The following table lists the first 15 Zernike functions. 7&1dr
% ,AyQCUz{*?
% n m Zernike function Normalization -8z@FLUK-
% -------------------------------------------------- 7:n OAN}%
% 0 0 1 1 E*VOyH2[
% 1 1 r * cos(theta) 2 $pj;CoPm
% 1 -1 r * sin(theta) 2 OVEQ^\Q5D
% 2 -2 r^2 * cos(2*theta) sqrt(6) wPr!.:MF
% 2 0 (2*r^2 - 1) sqrt(3) L^??*XEUJ
% 2 2 r^2 * sin(2*theta) sqrt(6) '(SqHP|8&g
% 3 -3 r^3 * cos(3*theta) sqrt(8) -x+K#T0Z
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) yXCJ?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2(25IYMS8
% 3 3 r^3 * sin(3*theta) sqrt(8) g.COKA
% 4 -4 r^4 * cos(4*theta) sqrt(10) BZk0B?
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &cT@MV5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) no7Q%O9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C@rIyBj1g
% 4 4 r^4 * sin(4*theta) sqrt(10) \)2~oN
% -------------------------------------------------- sYd)r%%AU
% @c;:D`\p1C
% Example 1: B=|m._OL]n
% oe{,-<yck
% % Display the Zernike function Z(n=5,m=1) 077 wk
% x = -1:0.01:1; %dq|)r
% [X,Y] = meshgrid(x,x); :-e[$6}S
% [theta,r] = cart2pol(X,Y); 73kI%nNB
% idx = r<=1; xkfW^r
% z = nan(size(X)); >"pHk@AW K
% z(idx) = zernfun(5,1,r(idx),theta(idx)); \
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% figure u*8x.UE8C0
% pcolor(x,x,z), shading interp h&<>nK
% axis square, colorbar yu$xQ~ o
% title('Zernike function Z_5^1(r,\theta)') \Z$MH`_nu
% TH? wXd\
% Example 2: d5qGTT ~a
% XW BTBL
% % Display the first 10 Zernike functions o*:D/"gb
% x = -1:0.01:1; s@pIcNvx
% [X,Y] = meshgrid(x,x); "]x#kM
% [theta,r] = cart2pol(X,Y); 2\9OT>
% idx = r<=1; b^WF
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% z = nan(size(X)); qw}.
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% n = [0 1 1 2 2 2 3 3 3 3]; 52' 0l>
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; |^ J5YwCf
% Nplot = [4 10 12 16 18 20 22 24 26 28]; bs?&;R.5
% y = zernfun(n,m,r(idx),theta(idx)); J6g:.jsK!
% figure('Units','normalized') <L:}u!
% for k = 1:10 #oxP,LR
% z(idx) = y(:,k); K# BZ Jcb
% subplot(4,7,Nplot(k)) h:{^&d
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% pcolor(x,x,z), shading interp N.q0D5 :
% set(gca,'XTick',[],'YTick',[]) =|_k a8{?
% axis square I4MZJAYk
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) / e]R0NI
% end i} ?\K>BWq
% P7
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% See also ZERNPOL, ZERNFUN2. <&iLMb:%
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% Paul Fricker 11/13/2006 VMJK9|JC[
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% Check and prepare the inputs: jYnP)xX;
% ----------------------------- |]tsf
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ![/ QW
error('zernfun:NMvectors','N and M must be vectors.') ZBXn&Gm
end RKwuvVI
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if length(n)~=length(m) LqoH]AcN
error('zernfun:NMlength','N and M must be the same length.') ]h}O&K/
end PvVn}i
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n = n(:); 1owe'7\J
m = m(:); E
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if any(mod(n-m,2)) [G7S
error('zernfun:NMmultiplesof2', ... '2v$xOh!y
'All N and M must differ by multiples of 2 (including 0).') AqjEz+TVt
end 7*g'4p-
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if any(m>n) GFgh{'|
error('zernfun:MlessthanN', ... [_zoJ
'Each M must be less than or equal to its corresponding N.') js)I%Z
end !E_RD,_
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if any( r>1 | r<0 ) }!@X(S!do
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;#S4$wISw`
end `bcCj~j
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) },vVc/
error('zernfun:RTHvector','R and THETA must be vectors.') XMm(D!6
end w"A%@<V3Ec
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r = r(:); `&>CK`%Xu
theta = theta(:); m'5rzZP
length_r = length(r); J3AS"+]
if length_r~=length(theta) 2jH&@g$cl;
error('zernfun:RTHlength', ... $jL+15^N0+
'The number of R- and THETA-values must be equal.') 0A.9<&Lod
end e(Ub7L#
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% Check normalization: $d%NFc&
% -------------------- &-4SA j
if nargin==5 && ischar(nflag) JsbH'l
isnorm = strcmpi(nflag,'norm'); D8wZC'7
if ~isnorm 1iIag}?p
error('zernfun:normalization','Unrecognized normalization flag.') LJmRa
end Ub<^;Du5
else ~6Df~uN
isnorm = false; mKhlYVn
end J7s\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Xsv^GmP+
% Compute the Zernike Polynomials * AjJf)o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (S
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% Determine the required powers of r: h,)UB1
% ----------------------------------- 1[H1l;
m_abs = abs(m); A_<1}8{L
rpowers = []; HLp'^
for j = 1:length(n) \z)` pno
rpowers = [rpowers m_abs(j):2:n(j)]; 7="I;
end iXFN|ml
rpowers = unique(rpowers); g'{hp:
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u4;#~##
% Pre-compute the values of r raised to the required powers, %[7<GcWl
% and compile them in a matrix: R|O."&CAB
% ----------------------------- hNGD`"U
if rpowers(1)==0 X1;ljX
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Z*Jp?[##
rpowern = cat(2,rpowern{:}); I85bzzZB
rpowern = [ones(length_r,1) rpowern]; {\zB'SNq
else x\2N
@*I:
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aO>Nev
rpowern = cat(2,rpowern{:}); osW"b"_f
end xyc`p[n&
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% Compute the values of the polynomials: 6I,4 6 XZ-
% -------------------------------------- /6a617?9J
y = zeros(length_r,length(n)); @F%_{6h
for j = 1:length(n) /E0/)@pDq
s = 0:(n(j)-m_abs(j))/2; [^GXHE=
pows = n(j):-2:m_abs(j); &Eqa y'
for k = length(s):-1:1 0R[onPU_vZ
p = (1-2*mod(s(k),2))* ... sFWH*kdP?
prod(2:(n(j)-s(k)))/ ... v^QUYsar
prod(2:s(k))/ ... Zfub+A
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]^
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prod(2:((n(j)+m_abs(j))/2-s(k))); v05B7^1@_
idx = (pows(k)==rpowers); %K|+4ZY3
y(:,j) = y(:,j) + p*rpowern(:,idx); ?v$kq}Rg
end VUE6M\&z>
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if isnorm CH3bpZv
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3D/<R|p
end p^ojhrr
end Zo(p6rku
% END: Compute the Zernike Polynomials
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &*3O+$L
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% Compute the Zernike functions: skRI\
% ------------------------------ >[|Y$$
idx_pos = m>0; TB
idx_neg = m<0; YoEL|r|
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z = y; ui#K`.dn
if any(idx_pos) 3om4q2R
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); a'm\6AW2)
end o#ajBOJ
if any(idx_neg) AD/7k3:
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +rA:/!b)Y
end K!a4>Du{
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% EOF zernfun 2%F!aeX