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, fq1w <e
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, l[ko)%7V
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? -0uGzd+m*
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Zn1((J7
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function z = zernfun(n,m,r,theta,nflag) }P=FMme{F(
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. D~qi6@Ga
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .aL%}`8l?
% and angular frequency M, evaluated at positions (R,THETA) on the =Q"thsR
% unit circle. N is a vector of positive integers (including 0), and q2k}bb +
% M is a vector with the same number of elements as N. Each element [/ CB1//Y
% k of M must be a positive integer, with possible values M(k) = -N(k) 2C0j.Ib
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, [3irr0D7l
% and THETA is a vector of angles. R and THETA must have the same Pf8_6 z_
% length. The output Z is a matrix with one column for every (N,M) i Q3wi
% pair, and one row for every (R,THETA) pair. mj9|q8v{+
% HH*,Oe
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike B'Nvl#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^`-Hg= d
% with delta(m,0) the Kronecker delta, is chosen so that the integral _2k<MiqCD[
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 6o#J
% and theta=0 to theta=2*pi) is unity. For the non-normalized )p!")
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b%VBSNZ
% KW0KXO06a
% The Zernike functions are an orthogonal basis on the unit circle. WbFCj0
% They are used in disciplines such as astronomy, optics, and v&sp;%I6=
% optometry to describe functions on a circular domain. 823y;
% }zo-%#
% The following table lists the first 15 Zernike functions. Jx3a7CpX
% uPFbKSJj
% n m Zernike function Normalization 'o_ RC{k2"
% -------------------------------------------------- ),<h6$
% 0 0 1 1 Q1h v2*/U
% 1 1 r * cos(theta) 2 HDo=W qG
% 1 -1 r * sin(theta) 2 F&/}x15
% 2 -2 r^2 * cos(2*theta) sqrt(6) {YzpYc1
% 2 0 (2*r^2 - 1) sqrt(3) Z\-Gr
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% 2 2 r^2 * sin(2*theta) sqrt(6) #.j:P#
% 3 -3 r^3 * cos(3*theta) sqrt(8) $~EY:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) I
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ht:L
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% 3 3 r^3 * sin(3*theta) sqrt(8) esTK4z]
% 4 -4 r^4 * cos(4*theta) sqrt(10) F7p`zf@O]
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8W.-Y|[5?
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) fQU_A
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) RvW>kATb_F
% 4 4 r^4 * sin(4*theta) sqrt(10) ^-}3+YA
% -------------------------------------------------- E;a9RV|
% 7
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% Example 1: 5OFb9YX
% Z${@;lgP
% % Display the Zernike function Z(n=5,m=1) KbRKPA`
% x = -1:0.01:1; ht)KS9Xu
% [X,Y] = meshgrid(x,x); Z}O0DfT;
% [theta,r] = cart2pol(X,Y); Io;26F""
% idx = r<=1; lce~6}
% z = nan(size(X)); "%t !+E>nr
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Bsz;GnD|r
% figure Bq:: 5,v
% pcolor(x,x,z), shading interp \AR3DDm
% axis square, colorbar H%c{ }F
% title('Zernike function Z_5^1(r,\theta)') 0xutG/-&N
% 5a l44[
% Example 2: an?g'8! r:
% gtP;Qw'
% % Display the first 10 Zernike functions p4zV<qZ>e
% x = -1:0.01:1;
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% [X,Y] = meshgrid(x,x); r(=3yd/G$
% [theta,r] = cart2pol(X,Y); "Zicac@N
% idx = r<=1; K[|d7e
% z = nan(size(X)); v3jx2Z
% n = [0 1 1 2 2 2 3 3 3 3]; t#J
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Y) 4D$9:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; (aH_K07
% y = zernfun(n,m,r(idx),theta(idx)); )6zwprH!
% figure('Units','normalized') ~Urj:l
% for k = 1:10 jZY9Lx8o
% z(idx) = y(:,k); P(r}<SM
% subplot(4,7,Nplot(k)) Z.0^:rVp~
% pcolor(x,x,z), shading interp My'6yQL
% set(gca,'XTick',[],'YTick',[]) ?3i-wpzMp
% axis square hAZ"M:f
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]pA}h.R#-
% end Ec]cCLB
% hMx/}Tw wt
% See also ZERNPOL, ZERNFUN2. <BN)>NqM
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