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, $3uKw!z
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, p4i]7o@
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? _b.qkTWUB
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <_Q:'cx'
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function z = zernfun(n,m,r,theta,nflag) iVTGF<
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?Wt$6{)
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `8>Py~
% and angular frequency M, evaluated at positions (R,THETA) on the d[^~'V
% unit circle. N is a vector of positive integers (including 0), and >P $;79<
% M is a vector with the same number of elements as N. Each element w{90`
% k of M must be a positive integer, with possible values M(k) = -N(k) Cp]"1%M,
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, H0R&2#YD
% and THETA is a vector of angles. R and THETA must have the same +_xOLiu
% length. The output Z is a matrix with one column for every (N,M) 0}xFD6{X
% pair, and one row for every (R,THETA) pair. BQ2wnGc
% {TRsd
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lF?tQB/a
% with delta(m,0) the Kronecker delta, is chosen so that the integral {$^DMANDx
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3^~KB'RZ
% and theta=0 to theta=2*pi) is unity. For the non-normalized ?9=9C"&s
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2'<[7!
% ,SiY;(b=\
% The Zernike functions are an orthogonal basis on the unit circle. _fP&&}
% They are used in disciplines such as astronomy, optics, and ]a3iEA2 (
% optometry to describe functions on a circular domain. mA@Me7m}
% (q7
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% The following table lists the first 15 Zernike functions. ;/*6U
% I1>N4R-j
% n m Zernike function Normalization @*DyZB
% -------------------------------------------------- =.`qixN
% 0 0 1 1 Uyr3dN%*r
% 1 1 r * cos(theta) 2 k8uvNLA)a
% 1 -1 r * sin(theta) 2 gOK\%&S]
% 2 -2 r^2 * cos(2*theta) sqrt(6) ?cEskafb>
% 2 0 (2*r^2 - 1) sqrt(3) ed_FiQd
% 2 2 r^2 * sin(2*theta) sqrt(6) eBO@7F$
% 3 -3 r^3 * cos(3*theta) sqrt(8) :BGA.
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) RTu4@7XP
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) >xn}N6Rj2~
% 3 3 r^3 * sin(3*theta) sqrt(8) Z0>DNmH*
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4~OQhiJ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) hw~a:kD
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) lM[XS4/TRa
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HH>:g(bu
% 4 4 r^4 * sin(4*theta) sqrt(10) *cg(
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% -------------------------------------------------- *I0-O*Xr
% `3'0I /d"z
% Example 1: Iu35#j
% $eBX
% % Display the Zernike function Z(n=5,m=1) s{4 \xAS>
% x = -1:0.01:1; b]JI@=s?
% [X,Y] = meshgrid(x,x); W Qc>
% [theta,r] = cart2pol(X,Y); LR,7,DH$9'
% idx = r<=1; EIf~dOgH
% z = nan(size(X)); hwDbs[:
% z(idx) = zernfun(5,1,r(idx),theta(idx)); N9rBW
% figure Lh-`OmO0>F
% pcolor(x,x,z), shading interp %,*G[#*&
% axis square, colorbar `j9$T:`
% title('Zernike function Z_5^1(r,\theta)') 5]1h8PW!Y
%
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% Example 2:
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% X}6#II
% % Display the first 10 Zernike functions $8BE[u|H2
% x = -1:0.01:1; 2qO3XI
% [X,Y] = meshgrid(x,x); 6R29$D|HFO
% [theta,r] = cart2pol(X,Y); **[Z^$)u(
% idx = r<=1; (:+>#V)pZ
% z = nan(size(X)); )P>u9=?,=E
% n = [0 1 1 2 2 2 3 3 3 3]; ;*[9Q'lI*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; \ M/6m^zS
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ,s^<X85gp\
% y = zernfun(n,m,r(idx),theta(idx)); "XLe3n
% figure('Units','normalized') )2E%b+"
% for k = 1:10 #9LzY
% z(idx) = y(:,k); d'9:$!oz
% subplot(4,7,Nplot(k)) 9(!]NNf!
% pcolor(x,x,z), shading interp il:nXpM!
% set(gca,'XTick',[],'YTick',[]) gX?n4Csy'
% axis square d=]U_+
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]> Y/r-!
% end qYp$fmj
% vY*\R0/a
% See also ZERNPOL, ZERNFUN2. EC?Efc+O
[W,-1.$!dM
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% Paul Fricker 11/13/2006 XL}<1-}
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)-/gLZsx
% Check and prepare the inputs: |@o6NZ<9N
% ----------------------------- n`;R pr&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i3
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error('zernfun:NMvectors','N and M must be vectors.') - &[z\"T
end !|m9|
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if length(n)~=length(m) YJ2ro-X
error('zernfun:NMlength','N and M must be the same length.') pyW u9
end xUYow
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n = n(:); ZcryAm:I
m = m(:); M}.b"
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if any(mod(n-m,2)) rvwy~hO"
error('zernfun:NMmultiplesof2', ... s!6=|SS7
'All N and M must differ by multiples of 2 (including 0).') uiBTnG"
end 8kW /DcLE
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if any(m>n) s=EiH
error('zernfun:MlessthanN', ... hE!7RM+Y
'Each M must be less than or equal to its corresponding N.') GF--riyfB
end +CTmcbyOi
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if any( r>1 | r<0 ) (buw^
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;WI]vn
end mPmB6q%)]
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) IM-`<~(I#
error('zernfun:RTHvector','R and THETA must be vectors.') vg5NY =O
end mpef]9
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r = r(:); %N~;{!![p
theta = theta(:); c d%hW
length_r = length(r); KP~-$NR
if length_r~=length(theta) vO$ra5Z
error('zernfun:RTHlength', ... 9p>
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'The number of R- and THETA-values must be equal.') t]TyXAr~
end @
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xYR#%! M
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% Check normalization: cjg=nTsBA
% -------------------- jpO38H0)
if nargin==5 && ischar(nflag) OKQLv+q5K)
isnorm = strcmpi(nflag,'norm'); !s-/0ugZ
if ~isnorm I>((o`
error('zernfun:normalization','Unrecognized normalization flag.') _
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end >}F? <JB
else yH(V&T