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, Qr1e@ =B
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, 2L AYDaS
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? hYQ_45Z*?
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? \MxoZ
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function z = zernfun(n,m,r,theta,nflag) #jAqra._b
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. pLMRwgzr
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ladsw
% and angular frequency M, evaluated at positions (R,THETA) on the tb:L\A^:
% unit circle. N is a vector of positive integers (including 0), and 5XuT={o
% M is a vector with the same number of elements as N. Each element LlBN-9p
% k of M must be a positive integer, with possible values M(k) = -N(k) |F.)zC5{
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, T&86A\D\z
% and THETA is a vector of angles. R and THETA must have the same Z~A@o""F
% length. The output Z is a matrix with one column for every (N,M) gPAX4'
% pair, and one row for every (R,THETA) pair. 9]t[J_YM
% A2}Rl%+X]6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2+Px'U\
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #fj/~[Ajv
% with delta(m,0) the Kronecker delta, is chosen so that the integral qQ!1t>j+H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;q0uE:^S
% and theta=0 to theta=2*pi) is unity. For the non-normalized b':|uu*/
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Zo KcJA
% xEuN
% The Zernike functions are an orthogonal basis on the unit circle. 7PR#(ftz
% They are used in disciplines such as astronomy, optics, and *9)SmSs
% optometry to describe functions on a circular domain. 1 T130L
% JT
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% The following table lists the first 15 Zernike functions. >NB?&|
% X=8Y%
% n m Zernike function Normalization =8gHS[
% -------------------------------------------------- i{D=l7j|w
% 0 0 1 1 kE
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% 1 1 r * cos(theta) 2 `OymAyEYQ
% 1 -1 r * sin(theta) 2 @"T"7c?Cv
% 2 -2 r^2 * cos(2*theta) sqrt(6) l!#m&'16"
% 2 0 (2*r^2 - 1) sqrt(3) 8 6f2'o+
% 2 2 r^2 * sin(2*theta) sqrt(6) PSawMPw
% 3 -3 r^3 * cos(3*theta) sqrt(8) nA?Hxos
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) L6>pGx
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) k%y9aO
% 3 3 r^3 * sin(3*theta) sqrt(8) azjEq$<M
% 4 -4 r^4 * cos(4*theta) sqrt(10) '8Phxx|
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l"n{.aL
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) kt4d;4n
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =h(W4scgqX
% 4 4 r^4 * sin(4*theta) sqrt(10) 4@.|_zY
% -------------------------------------------------- : S$l"wrh\
% Yxv9
% Example 1: Knhp*V?
% iR$<$P5
% % Display the Zernike function Z(n=5,m=1) &'l>rD^o
% x = -1:0.01:1; zi~5l#I
% [X,Y] = meshgrid(x,x); $8l({:*q0
% [theta,r] = cart2pol(X,Y); `[zQf
% idx = r<=1; pf4 ^Bk}e
% z = nan(size(X)); _=
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); ::n;VY2&
% figure 4'KOpl
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% pcolor(x,x,z), shading interp o;b0m;~
% axis square, colorbar )Qm[[p nj
% title('Zernike function Z_5^1(r,\theta)') rQTr8DYH
% C0=9K@FCb
% Example 2: 5unG#szq
% Q4t(@0e}
% % Display the first 10 Zernike functions xUF_1hY
% x = -1:0.01:1; ;X ,1I
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); tweY'x.{
% idx = r<=1; iVB^,KQ@
% z = nan(size(X)); UZ8?[
% n = [0 1 1 2 2 2 3 3 3 3]; 0iCPi)B
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Gamr6I"K
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ,fEO>
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% y = zernfun(n,m,r(idx),theta(idx)); (]/9-\6(#
% figure('Units','normalized') n6F/Ac:
% for k = 1:10 C1T_9}L-A
% z(idx) = y(:,k); !~_zm*CqbZ
% subplot(4,7,Nplot(k)) }0,>2TTDN
% pcolor(x,x,z), shading interp uH3D{4
% set(gca,'XTick',[],'YTick',[]) 3cj3u4y
% axis square $ _8g8r}
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {;2i.m1
% end %iJ%{{f`
% 93[DAs
% See also ZERNPOL, ZERNFUN2. #6Xs.*b5C
PLM _#+R>
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% Paul Fricker 11/13/2006 R`F,aIJ]
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% Check and prepare the inputs: -{h
% ----------------------------- Bs`$ i ;&
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g%[n4
error('zernfun:NMvectors','N and M must be vectors.')
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end 4!`bZ`_Bw
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if length(n)~=length(m) [fu!AIQs
error('zernfun:NMlength','N and M must be the same length.') ctQbp~-
end wLuv6\E
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n = n(:); iq?#rb P#I
m = m(:); A`O <6
if any(mod(n-m,2)) a)*6gf<