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, gsa@ci
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, $o$WFV+h
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? 6zNWDUf
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? O2 + K
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function z = zernfun(n,m,r,theta,nflag) ,Sy&?t}`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. lHTr7uF(
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }ALli0n`V)
% and angular frequency M, evaluated at positions (R,THETA) on the FDGG$z?>m
% unit circle. N is a vector of positive integers (including 0), and BTG_c_?]e
% M is a vector with the same number of elements as N. Each element m9&%A0
% k of M must be a positive integer, with possible values M(k) = -N(k) jWh)bsqI!
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Zp<#( OIu
% and THETA is a vector of angles. R and THETA must have the same X*5N&AJ
% length. The output Z is a matrix with one column for every (N,M) f4+wP/n&
% pair, and one row for every (R,THETA) pair. W_3BL]^=
% bH'2iG
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike eU e, P
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), co^h2b
% with delta(m,0) the Kronecker delta, is chosen so that the integral 8?: 2<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 87!m l
% and theta=0 to theta=2*pi) is unity. For the non-normalized Z ZCm438
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. V*Xr}FE
% $}z/BV1I
% The Zernike functions are an orthogonal basis on the unit circle. h5-yhG
% They are used in disciplines such as astronomy, optics, and h9iQn<lp4.
% optometry to describe functions on a circular domain. F8Mf,jnPs
% m!P<#
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% The following table lists the first 15 Zernike functions. X{ 6a
% elpTak@
% n m Zernike function Normalization sdyNJh7Jr
% -------------------------------------------------- v*<rNZI
% 0 0 1 1 `s Pk:cNz~
% 1 1 r * cos(theta) 2 ~3f|-%Z
% 1 -1 r * sin(theta) 2 734n1-F?I%
% 2 -2 r^2 * cos(2*theta) sqrt(6) y}|E)
% 2 0 (2*r^2 - 1) sqrt(3) )/~o'M3
% 2 2 r^2 * sin(2*theta) sqrt(6) ucU7
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% 3 -3 r^3 * cos(3*theta) sqrt(8) ue'dI
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :$PrlE
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) -"H0Qafm
% 3 3 r^3 * sin(3*theta) sqrt(8) R(cg`8
% 4 -4 r^4 * cos(4*theta) sqrt(10) eQn[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KU+\fwYpnk
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Z5)v
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &}pF6eIar
% 4 4 r^4 * sin(4*theta) sqrt(10) u&UmI-}
% -------------------------------------------------- VEn3b
% KtH^k&z.f
% Example 1: #5'@at'1
% Fpeokr"i
% % Display the Zernike function Z(n=5,m=1) |3Oyg ?2
% x = -1:0.01:1; LXhR"PWZM\
% [X,Y] = meshgrid(x,x); 8ZM#.yBB
% [theta,r] = cart2pol(X,Y); *rHz/& ,
% idx = r<=1; v9S=$Aj
% z = nan(size(X)); C8|#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); x#zj0vI-8
% figure ,tg(aL
% pcolor(x,x,z), shading interp ;$gV$KB:xA
% axis square, colorbar #M+_Lk3
% title('Zernike function Z_5^1(r,\theta)') t*A[v
% IA[:-2_
% Example 2: n~}[/ly
% 9&`";dg
% % Display the first 10 Zernike functions ;FF+uK
% x = -1:0.01:1; $ Y^0l
% [X,Y] = meshgrid(x,x); #d/T7c#
% [theta,r] = cart2pol(X,Y); e#mqerpJ
% idx = r<=1; 5;XYF0
% z = nan(size(X)); p|mFF0SL
% n = [0 1 1 2 2 2 3 3 3 3]; ]*lZFP~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 6akI5\b
% Nplot = [4 10 12 16 18 20 22 24 26 28]; dC-~=}HR^
% y = zernfun(n,m,r(idx),theta(idx)); [{[m)Z^
% figure('Units','normalized') 8~s0%%{,M
% for k = 1:10 y@1QVt04
% z(idx) = y(:,k); J:&.[
% subplot(4,7,Nplot(k)) ]7yxXg
% pcolor(x,x,z), shading interp 748:*
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% set(gca,'XTick',[],'YTick',[]) pL`Q+}c}
% axis square J[hmY= ,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vTK8t:JQ~
% end bGK*1FlH
% \)wch P_0
% See also ZERNPOL, ZERNFUN2. w\eC{,00:
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% Paul Fricker 11/13/2006 j2u'5kJ
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% Check and prepare the inputs: C7* YZe
% -----------------------------
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cRD;a?0/6s
error('zernfun:NMvectors','N and M must be vectors.') ?*+U[*M
end xE^G*<mj:
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if length(n)~=length(m) !t"/w6X1I
error('zernfun:NMlength','N and M must be the same length.') oq!\100
end jl(D;JnF
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n = n(:); zuW4gJ
m = m(:); -s`Wd4AP
if any(mod(n-m,2)) L[Z^4l_!
error('zernfun:NMmultiplesof2', ... jQ%1lQ#R)
'All N and M must differ by multiples of 2 (including 0).') CrL9|78
end xR&:]M[Vg
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if any(m>n) lnyq%T[^
error('zernfun:MlessthanN', ... 3'` &D/n
'Each M must be less than or equal to its corresponding N.') eF.nNu
end ?hc=w 2Ci
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if any( r>1 | r<0 ) 7sOAaWx
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \ moLQ
end "U4c'iW
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ot`%5<E^
error('zernfun:RTHvector','R and THETA must be vectors.') h'=)dFw7
end o4EY2
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r = r(:); 0[%{YmI{W
theta = theta(:); VV/T)qEe7>
length_r = length(r); )z@
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if length_r~=length(theta) SH =S>
error('zernfun:RTHlength', ... @YH>|{S&
'The number of R- and THETA-values must be equal.') iBbaHU*V
end =0Y0o_
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% Check normalization: 3MHpP5C
% -------------------- zx=eqN@!@
if nargin==5 && ischar(nflag) a]V8F&)g#
isnorm = strcmpi(nflag,'norm'); <_|@~^u
if ~isnorm >h#juO"
error('zernfun:normalization','Unrecognized normalization flag.')
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end x/M$_E<G
else h;+O96V4.
isnorm = false; Bl6I@w
end 2SD
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% db_?da;!`
% Compute the Zernike Polynomials xPUukmG:B
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t855|
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% Determine the required powers of r: "'Fvt-<^S7
% ----------------------------------- 1<#D3CXK
m_abs = abs(m); W?4:sLC#3
rpowers = []; z,m3U(
for j = 1:length(n) qtZzJ>Y
rpowers = [rpowers m_abs(j):2:n(j)]; Khi6z&