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, a-UD_|!
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, |~=?vw<W
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? ]VHdE_7)
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? D/!eov4"
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function z = zernfun(n,m,r,theta,nflag) &a2V-|G',
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,pGCgOG#}c
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )n3biQL_
% and angular frequency M, evaluated at positions (R,THETA) on the dTU.XgX)1^
% unit circle. N is a vector of positive integers (including 0), and Fm[?@Z&wP
% M is a vector with the same number of elements as N. Each element ek0;8Ds9
% k of M must be a positive integer, with possible values M(k) = -N(k) Jb)eC?6O
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, yW6[Fpw
% and THETA is a vector of angles. R and THETA must have the same Sj]T{3mi
% length. The output Z is a matrix with one column for every (N,M) ui#1 +p3G
% pair, and one row for every (R,THETA) pair. [jtj~]&mO
% 3Oig/KZ
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike NGb!7Mu9
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !tFU9Zt
% with delta(m,0) the Kronecker delta, is chosen so that the integral 1+PNy d
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RZ,<D I
% and theta=0 to theta=2*pi) is unity. For the non-normalized E6wST@r
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aBA#\eV
% W)Kpnb7
% The Zernike functions are an orthogonal basis on the unit circle. \SH D
% They are used in disciplines such as astronomy, optics, and n9-q5X^e>
% optometry to describe functions on a circular domain. o"+&^
% ZC\.};.
% The following table lists the first 15 Zernike functions. dO4U9{+
% nD?M;XN
% n m Zernike function Normalization &0<R:K ?>N
% -------------------------------------------------- w\8rh\Mvh
% 0 0 1 1 K&gc5L
% 1 1 r * cos(theta) 2 Ll E_{||h
% 1 -1 r * sin(theta) 2 !^"!fuoNC
% 2 -2 r^2 * cos(2*theta) sqrt(6) 2"{]A;@
% 2 0 (2*r^2 - 1) sqrt(3) DGuUI}|)
% 2 2 r^2 * sin(2*theta) sqrt(6) F#37Qv
% 3 -3 r^3 * cos(3*theta) sqrt(8) mLxwJ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) f!R^;'a
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &fNE9peQFa
% 3 3 r^3 * sin(3*theta) sqrt(8) BQfAen]
% 4 -4 r^4 * cos(4*theta) sqrt(10) u4*]jt;H
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uL2{v
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) XGup,7e9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3b[[2x_UU
% 4 4 r^4 * sin(4*theta) sqrt(10) $E@.G1T [
% -------------------------------------------------- H/la'f#o%
% a!J ow?(
% Example 1: Kd[`mkmS
% 02c.;ka3
% % Display the Zernike function Z(n=5,m=1) &+r
;>
% x = -1:0.01:1; Px?At5
% [X,Y] = meshgrid(x,x); AYQh=$)(
% [theta,r] = cart2pol(X,Y); \S@=zII_
% idx = r<=1; `::(jW.KO
% z = nan(size(X)); =`.5b:e
% z(idx) = zernfun(5,1,r(idx),theta(idx)); t:j07 ,1~
% figure ^)P5(fJ
% pcolor(x,x,z), shading interp <IkD=X
% axis square, colorbar D30Z9_^%:
% title('Zernike function Z_5^1(r,\theta)') u9~V2>r\
% wTAEJ{p
% Example 2: r
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% k49n9EX
% % Display the first 10 Zernike functions SVEA
% x = -1:0.01:1; lJQl$Wx^
% [X,Y] = meshgrid(x,x); @_:?N(%(
% [theta,r] = cart2pol(X,Y); hE`%1j2(
% idx = r<=1; 8 P y_Y>
% z = nan(size(X)); y42T.oK8c
% n = [0 1 1 2 2 2 3 3 3 3]; g:6}zHK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; nsw8[pk
% Nplot = [4 10 12 16 18 20 22 24 26 28]; aZCZ/
% y = zernfun(n,m,r(idx),theta(idx)); (IQ L`3f%
% figure('Units','normalized') ScmzbDu
% for k = 1:10 ,?N_67
% z(idx) = y(:,k); l{SPV8[i
% subplot(4,7,Nplot(k)) -EIMh^
% pcolor(x,x,z), shading interp w
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% set(gca,'XTick',[],'YTick',[]) 2X|jq4
% axis square -#z'A
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) P*=3$-`
% end zSufU2
% <y/AEY1
% See also ZERNPOL, ZERNFUN2. E0%Y%PQ**{
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% Paul Fricker 11/13/2006 $W]guG
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% Check and prepare the inputs: fe`G^hV
% ----------------------------- bH]!~[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %SFR.U0}yK
error('zernfun:NMvectors','N and M must be vectors.') -.3k
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end g5N<B+?!i
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if length(n)~=length(m) `%:(IGxz
error('zernfun:NMlength','N and M must be the same length.') 5Jd {Ev
end Fd.d(
T}x%=4<E
&jd<rs5}
n = n(:); m8A1^ R
m = m(:); xJ5!`#=
if any(mod(n-m,2)) j@\/]oL^We
error('zernfun:NMmultiplesof2', ... dp W%LXM_
'All N and M must differ by multiples of 2 (including 0).') vyy\^nL
end 6u3(G j@
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if any(m>n) qnV9TeU)
error('zernfun:MlessthanN', ... nECf2>Yp v
'Each M must be less than or equal to its corresponding N.') wA&)y>n-
end BkqW>[\5xm
%+J*oFwQu
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if any( r>1 | r<0 ) Oj\mkg
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @x
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end n:] 1^wX#
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) nL~
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error('zernfun:RTHvector','R and THETA must be vectors.') <OB~60h"
end Mc^7FWkw
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r = r(:); fbkjK`_q
theta = theta(:); Vtk|WV?>P+
length_r = length(r); 1"PE@!]
if length_r~=length(theta) nP5fh_/
error('zernfun:RTHlength', ... 3o^M%
'The number of R- and THETA-values must be equal.') |/Z)?
end #E)]7!_XG
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% Check normalization: LRs;>O
% -------------------- Jx?>1q=M
if nargin==5 && ischar(nflag) ,Yz+?SmSZ&
isnorm = strcmpi(nflag,'norm'); ``Rb-.Fq,
if ~isnorm >Sah\u`
error('zernfun:normalization','Unrecognized normalization flag.') !7?wd^C'f
end NQ=YTRU
else G"wQ(6J@
isnorm = false; `^{P,N>X
end zf u78
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lNAHn<ht
% Compute the Zernike Polynomials Wno5B/V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #IDCCD^1=
%Ski5q
4F!d V;"Z(
% Determine the required powers of r: ZZ7U^#RT
% ----------------------------------- ![%,pip2/&
m_abs = abs(m); ?>&Zm$5V
rpowers = []; DcHMiiVM
for j = 1:length(n) ry"zec
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rpowers = [rpowers m_abs(j):2:n(j)]; 1YL5 ![T
end F{tSfKy2
rpowers = unique(rpowers); n
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% Pre-compute the values of r raised to the required powers, q#Vf2U55m
% and compile them in a matrix: <X*8Xzmv
% ----------------------------- T(F8z5s5
if rpowers(1)==0 gZv<_0N
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;"z>p25=T
rpowern = cat(2,rpowern{:}); X3yr6J[ ^
rpowern = [ones(length_r,1) rpowern]; (=9&"UH
else B?Skw{&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (z7#KJ1+Aw
rpowern = cat(2,rpowern{:}); T:$_1I $
end +_Z/VQv
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% Compute the values of the polynomials: ri
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% -------------------------------------- g_U69
z
y = zeros(length_r,length(n)); 4^&vRD,
for j = 1:length(n) #C^m>o~R
s = 0:(n(j)-m_abs(j))/2; ig{5]wZ(
pows = n(j):-2:m_abs(j);
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for k = length(s):-1:1 )VC) }
p = (1-2*mod(s(k),2))* ... h;->i]
prod(2:(n(j)-s(k)))/ ... 8n? .w:Y/
prod(2:s(k))/ ... cx}-tj"m-
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... F04Etf
2k
prod(2:((n(j)+m_abs(j))/2-s(k))); E3!twR*Aw
idx = (pows(k)==rpowers); ,e2va7}3
y(:,j) = y(:,j) + p*rpowern(:,idx); CCV~nf
end }|,y`ui\
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if isnorm c3##:"wr
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); b3+PC$z2h
end j7&l&)5
end Fm"$W^H
% END: Compute the Zernike Polynomials +Sfv.6~v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'Nh^SbD+_|
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% Compute the Zernike functions: KXJHb{?
% ------------------------------ kN)ev?pQ[
idx_pos = m>0; (&(f`c@I
idx_neg = m<0; JFZ p^{
i weP3u##
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z = y; fvDwg
if any(idx_pos) D6w0Y:A{.
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `;;!>rm
end 9=|5-?^
if any(idx_neg) \IKr+wlN8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7F.,Xvw&@
end :"4~VDu
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% EOF zernfun }1V&(#H2