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, za+)2/
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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, x4/{XRQ
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? vvG"rU
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 61b*uoq0w?
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function z = zernfun(n,m,r,theta,nflag) 4l{$dtKbI
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ak-agH
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B`t/21J
% and angular frequency M, evaluated at positions (R,THETA) on the <W>A }}q
% unit circle. N is a vector of positive integers (including 0), and &4+|{Zx0
% M is a vector with the same number of elements as N. Each element [V>s]c<4`o
% k of M must be a positive integer, with possible values M(k) = -N(k) ;aj;(Z.p)
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )t@9!V
% and THETA is a vector of angles. R and THETA must have the same *u:,@io7'G
% length. The output Z is a matrix with one column for every (N,M) G"m?2$^-A
% pair, and one row for every (R,THETA) pair. OR*JWW[]
% g$jT P#%b
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike f,F1k9-1!
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )0/*j]Kf
% with delta(m,0) the Kronecker delta, is chosen so that the integral K a&
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ] jY^*o[
% and theta=0 to theta=2*pi) is unity. For the non-normalized -EE'xh-zD
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |d&C<O;f
% 8sU5MQ5
% The Zernike functions are an orthogonal basis on the unit circle. jf- XVk5q
% They are used in disciplines such as astronomy, optics, and o&&`_"18
% optometry to describe functions on a circular domain. Yku6\/^
% [\#ANA"
% The following table lists the first 15 Zernike functions. .drY
% w/O'&],x
% n m Zernike function Normalization %8D>aS U
% -------------------------------------------------- 39hep8+
% 0 0 1 1 h]L.6G|hEN
% 1 1 r * cos(theta) 2 8nu!5 3
% 1 -1 r * sin(theta) 2 ,(a~vqNQW3
% 2 -2 r^2 * cos(2*theta) sqrt(6) [qW%H,_
% 2 0 (2*r^2 - 1) sqrt(3) vBOY[>=
% 2 2 r^2 * sin(2*theta) sqrt(6) J4"A6`O
% 3 -3 r^3 * cos(3*theta) sqrt(8) Y,GlAr s4
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8)
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ;l~a|KW0
% 3 3 r^3 * sin(3*theta) sqrt(8) z@,(^~C_
% 4 -4 r^4 * cos(4*theta) sqrt(10) u:lBFVqk
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6u #eLs
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %qz-b.
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T7"QwA
% 4 4 r^4 * sin(4*theta) sqrt(10) dqJ 8lU?
% -------------------------------------------------- i+qg*o$
% QNINn>2
% Example 1: W4&8
% ;Z"MO@9:
% % Display the Zernike function Z(n=5,m=1) Tx~w(A4:
% x = -1:0.01:1; @'}2xw[eU
% [X,Y] = meshgrid(x,x); =.;ib6M
% [theta,r] = cart2pol(X,Y); C4$P#DZT^
% idx = r<=1; xT_"` @
% z = nan(size(X)); .:f ao'
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 6WQN!H8+^
% figure =1,!EkG
% pcolor(x,x,z), shading interp qbsod
% axis square, colorbar yNXYS
% title('Zernike function Z_5^1(r,\theta)') $.pCoS]i
% <uv`)Q 9
% Example 2: 2w3LK2`ZL
% s|H7;.3gp
% % Display the first 10 Zernike functions "i(f+N,)
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); Zs79,*o+0M
% [theta,r] = cart2pol(X,Y); XJPIAN~l
% idx = r<=1; XWAIW=.
% z = nan(size(X)); |Vqm1.1/Zv
% n = [0 1 1 2 2 2 3 3 3 3]; uP%VL}%0
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @,eo*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2<5LQr
% y = zernfun(n,m,r(idx),theta(idx)); ?_d>-NC
% figure('Units','normalized') *X$qgSW
% for k = 1:10 M j[+h|e
% z(idx) = y(:,k); L!l?tM o
% subplot(4,7,Nplot(k)) H @k}
% pcolor(x,x,z), shading interp Z(tJd,
% set(gca,'XTick',[],'YTick',[]) Q2Ey RFT
% axis square -s2)!Iko&
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -XL?n/M
% end (^FMm1@T
% ?m2FN<S
% See also ZERNPOL, ZERNFUN2. d*Su
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% Paul Fricker 11/13/2006 yN{**?b
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% Check and prepare the inputs: 5)S;R,
% ----------------------------- Z{B [r;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *(q{k%/M
error('zernfun:NMvectors','N and M must be vectors.') uKXU.u*C
end 9NVtvBA
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if length(n)~=length(m) Q+K]:c
error('zernfun:NMlength','N and M must be the same length.') hlV(jz
end P;25F
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n = n(:); 1_LKqBgo
m = m(:); 7mi*#X}
if any(mod(n-m,2)) vFJ4`Gjw(
error('zernfun:NMmultiplesof2', ... Ja*,ht(5
'All N and M must differ by multiples of 2 (including 0).') mD +9/O!
end $aTo9{M ^
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if any(m>n) wjX0r7^@
error('zernfun:MlessthanN', ... ._x"b5C
'Each M must be less than or equal to its corresponding N.') sOWP0xY
end :jTbzDqQ
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if any( r>1 | r<0 ) E.:eO??g
error('zernfun:Rlessthan1','All R must be between 0 and 1.') MJe/ \
end Dy. |bUB!f
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) X
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error('zernfun:RTHvector','R and THETA must be vectors.') :k.>H.8+~
end u8A,f}D 3
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r = r(:); O[1Q#
theta = theta(:); K~UT@,CS60
length_r = length(r); 7[ kDc-
if length_r~=length(theta) UeBSt.
error('zernfun:RTHlength', ... :Oj!J&A
'The number of R- and THETA-values must be equal.') cru&nH*O^
end !h1|B7N
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% Check normalization: P6^\*xkMr
% -------------------- 9~f
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if nargin==5 && ischar(nflag) V^G+_#@,,
isnorm = strcmpi(nflag,'norm'); u`+kH8#
if ~isnorm K)`l >o1
error('zernfun:normalization','Unrecognized normalization flag.') %tkL<e
end K^AIqL8
else S|RUc}(
isnorm = false;
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end zBrqh9%8e
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= p2AK\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :NwFJc
% Compute the Zernike Polynomials y3'K+?4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4%jSqT@
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