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, gHc0n0ZV
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, Yy]^_,r
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? 2@WF]*Z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? .z7%74p
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function z = zernfun(n,m,r,theta,nflag) A9^t$Ii
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. sp*_;h3'
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7N0V`&}T
% and angular frequency M, evaluated at positions (R,THETA) on the #xZ7%
% unit circle. N is a vector of positive integers (including 0), and |4NH}XVYJ>
% M is a vector with the same number of elements as N. Each element `PK1zSr
% k of M must be a positive integer, with possible values M(k) = -N(k) w7}m
T3p,)
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =w='qjh
% and THETA is a vector of angles. R and THETA must have the same cSy{*K{B
% length. The output Z is a matrix with one column for every (N,M) !U38aHG
% pair, and one row for every (R,THETA) pair. |~vo
% P wL]v. :
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >-fOkOWXy
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), t~m > \(&
% with delta(m,0) the Kronecker delta, is chosen so that the integral p<IMWe'tP
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /-BKdkBCpZ
% and theta=0 to theta=2*pi) is unity. For the non-normalized Nb/W+& y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Q $Y ]KV
% lrg3n[y-l
% The Zernike functions are an orthogonal basis on the unit circle. CC,_I>t
% They are used in disciplines such as astronomy, optics, and &`hx
% optometry to describe functions on a circular domain. Lk{ES$
% ^6Y4=
% The following table lists the first 15 Zernike functions. #T_m|LN7
% 7u/_3x1
% n m Zernike function Normalization Kp[ F@A#
% -------------------------------------------------- -Bymt[
% 0 0 1 1 mZLrU<)Y
% 1 1 r * cos(theta) 2 --*Jv"/0
% 1 -1 r * sin(theta) 2 8z\v|-%Z
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]A)`I
% 2 0 (2*r^2 - 1) sqrt(3) 9AQMB1D*v4
% 2 2 r^2 * sin(2*theta) sqrt(6) 8nn%wps
% 3 -3 r^3 * cos(3*theta) sqrt(8) c zTr_>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) U_!Wg|
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) >D20f<w(H
% 3 3 r^3 * sin(3*theta) sqrt(8) [:Kl0m7
% 4 -4 r^4 * cos(4*theta) sqrt(10) D90m..\w
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) }c/p+Wo
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LliOhr4
% 4 4 r^4 * sin(4*theta) sqrt(10) oJ}!qrrH
% -------------------------------------------------- 9 -7.4!]I
% 26n+v(re
% Example 1: yhF{
cK=
% 4t3Y/X
% % Display the Zernike function Z(n=5,m=1) -yKx"Q9F
% x = -1:0.01:1; BK._cDR
% [X,Y] = meshgrid(x,x); Ubtu?wRBW
% [theta,r] = cart2pol(X,Y); D*!p8J8Ku
% idx = r<=1; YCJc Dab
% z = nan(size(X)); x;d*?69f]
% z(idx) = zernfun(5,1,r(idx),theta(idx)); EXt?xiha?
% figure MVe:[=VOT|
% pcolor(x,x,z), shading interp v5J%
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% axis square, colorbar \ZSZ(p#1
% title('Zernike function Z_5^1(r,\theta)') :oRR1k
% |8,|>EyqK
% Example 2: "[W${q+0x
% -"m4 A0
% % Display the first 10 Zernike functions qfF/X"#0
% x = -1:0.01:1; 70s.
% [X,Y] = meshgrid(x,x); ;+pS-Zb
6
% [theta,r] = cart2pol(X,Y); d4'*K1m
% idx = r<=1; VRr_s:CWK
% z = nan(size(X)); R+}x#
% n = [0 1 1 2 2 2 3 3 3 3]; x\/N09
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; t^N
92$|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *=+m;%]_
% y = zernfun(n,m,r(idx),theta(idx)); RZwjc<T
% figure('Units','normalized') i7@qfe$fR
% for k = 1:10 `U-i{i
% z(idx) = y(:,k); 8=XfwwWHy<
% subplot(4,7,Nplot(k)) `$V7AqX (
% pcolor(x,x,z), shading interp xc|pl!ns
% set(gca,'XTick',[],'YTick',[]) T )QZ9a
% axis square '3B\I#
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) wD(1Sr5n
% end ARcPHV<(2
% bwHl}3
% See also ZERNPOL, ZERNFUN2. ED9uKp<Wbv
#9HQW:On
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% Paul Fricker 11/13/2006 6Xu^cbD
:wlX`YW+e
j Wa%vA
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% Check and prepare the inputs: HH_w!_f
% ----------------------------- 3$cIm+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5f.G^A: _X
error('zernfun:NMvectors','N and M must be vectors.') o;.6Y `-fJ
end >G4EiJS
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if length(n)~=length(m) q(tdBd'o6
error('zernfun:NMlength','N and M must be the same length.') Vfm (K
end ql
Z()
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n = n(:); a(!_3i@
m = m(:); kpxWi=y
if any(mod(n-m,2)) !8cS1(a
error('zernfun:NMmultiplesof2', ... D{b*,F:&@)
'All N and M must differ by multiples of 2 (including 0).') aSu6SU
end BQ&G7V
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if any(m>n) B@R3j
error('zernfun:MlessthanN', ... "O`{QVg:
'Each M must be less than or equal to its corresponding N.') J;?#Zt]`L
end KY1(yni&8[
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if any( r>1 | r<0 ) SONv]));
error('zernfun:Rlessthan1','All R must be between 0 and 1.') T]&%
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end )3W`>7>
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) NGA8JV/U
error('zernfun:RTHvector','R and THETA must be vectors.') -\Y"MwIED
end Z/y&;N4
=Gka;,n
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r = r(:); 2`D1cX
theta = theta(:); dNQR<v\IL
length_r = length(r); D..dGh.MY
if length_r~=length(theta) fjc8@S5x9j
error('zernfun:RTHlength', ... .XD.'S
'The number of R- and THETA-values must be equal.') ck=x_HB1
end QkEIV<T&)l
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5>E]C=maD
% Check normalization: 8T:?C~"
% -------------------- Z0Tpz2m
if nargin==5 && ischar(nflag) MfX1&/Z+
isnorm = strcmpi(nflag,'norm'); +<\)b(
if ~isnorm 4|?y
[j6
error('zernfun:normalization','Unrecognized normalization flag.') Ec44JD
end n^H Kf^]
else P;A9t #\
isnorm = false; QD<GXPu?N
end *]L(,_:"
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j. @CB`
% Compute the Zernike Polynomials Ya%-/u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% byT@O:f L
W @X/Z8.(
Y|*a,H"_
% Determine the required powers of r: aan)yP
% ----------------------------------- ?3f-"K_r
m_abs = abs(m); f!|$!r*q
rpowers = []; 7W)*IJ
for j = 1:length(n) Ia>07av
rpowers = [rpowers m_abs(j):2:n(j)]; kOu C@~,
end %OI4}!z@l
rpowers = unique(rpowers); *,hg+?lZ
S<
TUZ
/;
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% Pre-compute the values of r raised to the required powers, %K9 9_Cl3
% and compile them in a matrix: <)D)j[
% ----------------------------- X9|={ng)g#
if rpowers(1)==0 Q1cM{$}M
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }^ g6Y3\
rpowern = cat(2,rpowern{:}); bgi
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rpowern = [ones(length_r,1) rpowern]; nfL-E:n=
else 5<Lal^c D
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); RM5$O+"
rpowern = cat(2,rpowern{:}); J)*7JX
end 4#dS.UfI
O]|T !
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% Compute the values of the polynomials: #i t)
% -------------------------------------- .B9i`)0
y = zeros(length_r,length(n)); T5:p^;?g
for j = 1:length(n) q6A"+w,N
s = 0:(n(j)-m_abs(j))/2; C0RnBu
pows = n(j):-2:m_abs(j); <-Hw@g
for k = length(s):-1:1 <WWn1k_
p = (1-2*mod(s(k),2))* ... V2v}F=
prod(2:(n(j)-s(k)))/ ... \dB)G<_
prod(2:s(k))/ ... >[$j(k^
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... {.,-lFb\
prod(2:((n(j)+m_abs(j))/2-s(k))); Hzn6H4Rc
idx = (pows(k)==rpowers); Cyn_UE
y(:,j) = y(:,j) + p*rpowern(:,idx); ['`Vg=O.{
end ,MmX(O0
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if isnorm <w9<G
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); T@{}!
end xE0'eC5n^
end @xqjAcfg
% END: Compute the Zernike Polynomials r_p4pxs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (^U
8wit/
,;
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W%&[gDp
% Compute the Zernike functions: Omkpjr(1
% ------------------------------ `S&.gPE2
idx_pos = m>0; n
_H]*~4F
idx_neg = m<0; Klv~#9Si
UHaY|I${U
mO?yrM *
z = y; ]b&"](A
if any(idx_pos) IXz)xdP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ah8xiABa
end JKJ+RkXf3
if any(idx_neg) JvI6+[
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9 M<3m
end J+]W*?m
$+%eLx*
i*3*)l y
% EOF zernfun +~Lt;xNFk