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, n<47#-
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, )i@j``P
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? JD1IL` ta;
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ^gx`@^su
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function z = zernfun(n,m,r,theta,nflag) R$qp3I
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. YU! SdT$
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N %\OG#36
% and angular frequency M, evaluated at positions (R,THETA) on the QR4!r@*=
% unit circle. N is a vector of positive integers (including 0), and ox9$aBjJ
% M is a vector with the same number of elements as N. Each element 'r_{T=
% k of M must be a positive integer, with possible values M(k) = -N(k) }T([gc7~
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, B?d^JWTZ
% and THETA is a vector of angles. R and THETA must have the same w6ZyMR,T
% length. The output Z is a matrix with one column for every (N,M) `uL^!-
% pair, and one row for every (R,THETA) pair. ;{@ [ek6
% _?]E)i'RI
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7q_B`$ata
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zq ;YE
% with delta(m,0) the Kronecker delta, is chosen so that the integral -58
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3q7Z?1'o
% and theta=0 to theta=2*pi) is unity. For the non-normalized AWkXWl}
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aKi&2>c5>
% *fs'%"w-
% The Zernike functions are an orthogonal basis on the unit circle. rybs9:_}
% They are used in disciplines such as astronomy, optics, and o/Z
% optometry to describe functions on a circular domain. K/)*P4C-
% t+C9QXY
% The following table lists the first 15 Zernike functions. |l5ol@2*
% vFuf{ @P
% n m Zernike function Normalization qfF/X"#0
% -------------------------------------------------- Qoa gy L
% 0 0 1 1 j*2Q{ik>J
% 1 1 r * cos(theta) 2 ,+`1 /
% 1 -1 r * sin(theta) 2 [QC<u1/"K
% 2 -2 r^2 * cos(2*theta) sqrt(6) 5\hJ&
% 2 0 (2*r^2 - 1) sqrt(3) (J!FW(Ma|=
% 2 2 r^2 * sin(2*theta) sqrt(6) VRr_s:CWK
% 3 -3 r^3 * cos(3*theta) sqrt(8) C*O648yz[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ;IklS*p]
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) p'#
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% 3 3 r^3 * sin(3*theta) sqrt(8) 3]Jl\<0
% 4 -4 r^4 * cos(4*theta) sqrt(10) y*i_Ec\h
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) k
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Vaj4p""\F
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Cso!VdCX
% 4 4 r^4 * sin(4*theta) sqrt(10) 0#K?SuY.eN
% -------------------------------------------------- cL/6p0S
% 3aMfZa<=
% Example 1: +n#kpi'T
% mc{gcZIm
% % Display the Zernike function Z(n=5,m=1) \_H-TbU8
% x = -1:0.01:1; 0UV5}/2rP
% [X,Y] = meshgrid(x,x); cY &SKV#
% [theta,r] = cart2pol(X,Y); RPH]@
% idx = r<=1; A\{dq:
% z = nan(size(X)); G8Hj<3`
% z(idx) = zernfun(5,1,r(idx),theta(idx)); rgth2y]
% figure tCkKJ)m
% pcolor(x,x,z), shading interp if|j)h&
% axis square, colorbar "S#}iYp
% title('Zernike function Z_5^1(r,\theta)') [=Qv?am
% Y\CR*om!W
% Example 2: 0I|IL]JL
%
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% % Display the first 10 Zernike functions *{3&?pxx
% x = -1:0.01:1; ;W ZA
% [X,Y] = meshgrid(x,x); %O9kq
% [theta,r] = cart2pol(X,Y); \\<waU''
% idx = r<=1; TDvUiJm
% z = nan(size(X)); o;.6Y `-fJ
% n = [0 1 1 2 2 2 3 3 3 3]; >G4EiJS
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 'g3!SdaLF
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [A@K)A$f
% y = zernfun(n,m,r(idx),theta(idx)); hXxgKi%
% figure('Units','normalized') |~QHCg<
% for k = 1:10 UkO L7M
% z(idx) = y(:,k); MjGeH>c
% subplot(4,7,Nplot(k)) 4';~@IBf
% pcolor(x,x,z), shading interp cP >MsUZWl
% set(gca,'XTick',[],'YTick',[]) {|Ew]Wq
% axis square Mi|PhDXMh
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t:pgw[UJ
% end K YSyz)M}
% z|';Y!kQ
% See also ZERNPOL, ZERNFUN2. U g'y
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% Paul Fricker 11/13/2006 E=Z;T
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% Check and prepare the inputs: fgmu*\x<