[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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， +8ib928E  
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， bCP2_h3*  
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？ @ *Jbp  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ .feB VRg  
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function z = zernfun(n,m,r,theta,nflag) *# <%04f  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. KiG
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%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \x$`/  
%   and angular frequency M, evaluated at positions (R,THETA) on the ?`OFn F,K  
%   unit circle.  N is a vector of positive integers (including 0), and w!*ZS~v/r  
%   M is a vector with the same number of elements as N.  Each element \<9aS Y'U  
%   k of M must be a positive integer, with possible values M(k) = -N(k) vg?(0Gasm*  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, aVHID{Gf Z  
%   and THETA is a vector of angles.  R and THETA must have the same U}HSL5v  
%   length.  The output Z is a matrix with one column for every (N,M) 7`~0j6FY  
%   pair, and one row for every (R,THETA) pair. u0)O Fz  
% ]LsT  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (8~mf$ zx,  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?v,c)  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral A]Hz?i  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^W}|1.uZ  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized <9H3d7%  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. s8:epcL`A  
% cl#XiyK>  
%   The Zernike functions are an orthogonal basis on the unit circle. Lm!]m\LRZD  
%   They are used in disciplines such as astronomy, optics, and _Cf:\
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%   optometry to describe functions on a circular domain. z( ^?xv  
% >~7XBb08  
%   The following table lists the first 15 Zernike functions. [x?9<#T  
% g#fn(A  
%       n    m    Zernike function           Normalization 'H`:c+KDG`  
%       -------------------------------------------------- )Dms9:  
%       0    0    1                                 1 ]?}pJ28  
%       1    1    r * cos(theta)                    2 G \a`F'Oo  
%       1   -1    r * sin(theta)                    2 VxOWv8}|  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ^Rl?)_)1HE  
%       2    0    (2*r^2 - 1)                    sqrt(3) GLub5GrxR  
%       2    2    r^2 * sin(2*theta)             sqrt(6) zGme}z;1@  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) &B:L9^  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) _nzT
d\L88  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) l'Li!u  
%       3    3    r^3 * sin(3*theta)             sqrt(8)  3bd`q $  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Mx0~^l
 
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l`6.(6  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~f[;(?39xZ  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3J8>r|u;1'  
%       4    4    r^4 * sin(4*theta)             sqrt(10) b'FTyi  
%       -------------------------------------------------- cJ?,\@uuP  
% 82)=#ye_P  
%   Example 1: (VkO[5j  
% H#X*OJ  
%       % Display the Zernike function Z(n=5,m=1) {]|<|vc;GI  
%       x = -1:0.01:1; a`9pHH:7Q  
%       [X,Y] = meshgrid(x,x); ~c+=$SL-=  
%       [theta,r] = cart2pol(X,Y); 2_bEo  
%       idx = r<=1; @ZYJY  
%       z = nan(size(X)); #CJET  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); S,|ZCl>+  
%       figure G{|"WaKW  
%       pcolor(x,x,z), shading interp %H_-`A`  
%       axis square, colorbar 8)s0$6
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%       title('Zernike function Z_5^1(r,\theta)') zSMM?g^T  
% }"RVUYU  
%   Example 2: c|'$3dB*  
% 37IHn6r\  
%       % Display the first 10 Zernike functions t0xE&#4  
%       x = -1:0.01:1; J pj[.Sq  
%       [X,Y] = meshgrid(x,x); :%28*fl  
%       [theta,r] = cart2pol(X,Y); Vnb@5W2\  
%       idx = r<=1; ze LI
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%       z = nan(size(X)); VqD_FS;E  
%       n = [0  1  1  2  2  2  3  3  3  3]; lHliMBSc  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 7c%dSs6  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Dbx zqd  
%       y = zernfun(n,m,r(idx),theta(idx)); B4zuWCE@  
%       figure('Units','normalized') \Lbwfd=  
%       for k = 1:10 rHybP6C<  
%           z(idx) = y(:,k); &eO.h%@  
%           subplot(4,7,Nplot(k)) j)nE!GKD(  
%           pcolor(x,x,z), shading interp KqBiF]Q  
%           set(gca,'XTick',[],'YTick',[]) } nIYNeP?D  
%           axis square a
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%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @^# 9N!Fj]  
%       end Xmb##:  
% >pol
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%   See also ZERNPOL, ZERNFUN2. ?J+*i d  
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%   Paul Fricker 11/13/2006 qauvwAMuX  
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% Check and prepare the inputs: c8_,S[W  
% ----------------------------- ht+wi5b  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BHkicb?   
    error('zernfun:NMvectors','N and M must be vectors.') u#V;  
end ;_ 1Rk&o!  
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if length(n)~=length(m) $j!:ET'V  
    error('zernfun:NMlength','N and M must be the same length.')  LR4W  
end ,WoB)V.{(  
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n = n(:); R
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m = m(:); wk=s3^  
if any(mod(n-m,2)) DU5rB\!.~  
    error('zernfun:NMmultiplesof2', ... ;?-{Uk  
          'All N and M must differ by multiples of 2 (including 0).') plzwk>b_  
end t`Xx\  
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if any(m>n) tPw7zFy6r  
    error('zernfun:MlessthanN', ... h-m0Ro?6  
          'Each M must be less than or equal to its corresponding N.') y#O/Xw  
end M%!j\}2A  
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if any( r>1 | r<0 ) 2@&"*1(Xu  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') D [v225  
end !l9#a{#6l  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) O=u.J
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    error('zernfun:RTHvector','R and THETA must be vectors.') )%: W;H  
end Z]OX6G  
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r = r(:); MGo`j:0  
theta = theta(:); /pGx!  
length_r = length(r); 7=
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if length_r~=length(theta) j{m{hVa  
    error('zernfun:RTHlength', ... LH
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          'The number of R- and THETA-values must be equal.') eW_EWVH  
end e.|t12)L "  
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% Check normalization: ,`$2  
% -------------------- UwDoueXs  
if nargin==5 && ischar(nflag) $BOIa  
    isnorm = strcmpi(nflag,'norm'); $K1
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    if ~isnorm n8&x=Z}Xs  
        error('zernfun:normalization','Unrecognized normalization flag.') >k2^A  
    end (Q|Y*yI  
else
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    isnorm = false; pTYV@5|  
end ;s-fYS6(>{  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r+MqjdXG  
% Compute the Zernike Polynomials (j}ed
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d^|r#"o[  
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% Determine the required powers of r: &([Gc+"5E.  
% ----------------------------------- ("J_< p  
m_abs = abs(m); DEenvS`,P  
rpowers = []; SsIN@  
for j = 1:length(n) qh wl  
    rpowers = [rpowers m_abs(j):2:n(j)]; ^cojETOv  
end 2-wgbC5  
rpowers = unique(rpowers); &$pA,Gjin\  
;BEX|wxn  
< 'r<MA<  
% Pre-compute the values of r raised to the required powers, jTok1k  
% and compile them in a matrix: I#CS;Yh95  
% ----------------------------- 95-%>?4  
if rpowers(1)==0 xT8!X5;  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xb^M33-y  
    rpowern = cat(2,rpowern{:}); GF*E+/ ;  
    rpowern = [ones(length_r,1) rpowern]; OKNGV,{`  
else X^.~f+d~  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MAG/7T5  
    rpowern = cat(2,rpowern{:}); 2!Pwg0%2  
end 7FP @ vng  
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% Compute the values of the polynomials: +hMF\@  
% -------------------------------------- A:,V)  
y = zeros(length_r,length(n)); k%({< ul  
for j = 1:length(n) ;DI"9  
    s = 0:(n(j)-m_abs(j))/2; -k@Uo(MB  
    pows = n(j):-2:m_abs(j); h,2?+}Fn  
    for k = length(s):-1:1 > ,P,{"  
        p = (1-2*mod(s(k),2))* ... x@<!#d+  
                   prod(2:(n(j)-s(k)))/              ... )$E'2|Gm/  
                   prod(2:s(k))/                     ... ?B:],aztf  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )0 i$Bo  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ;UWp0d%  
        idx = (pows(k)==rpowers); ._}Dqg$  
        y(:,j) = y(:,j) + p*rpowern(:,idx); IS }U2d,W  
    end \'Ca
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    if isnorm m\J"P'=  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 'J#uD|9)  
    end \&%y4=y<sE  
end A,GJ6qp3  
% END: Compute the Zernike Polynomials >qynd'eToR  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sy34doAZ  
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% Compute the Zernike functions: C=PV-Ul+  
% ------------------------------ hU
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idx_pos = m>0; fZJO}  
idx_neg = m<0; (`K~p Z  
azpXE  
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z = y; ;WsV.n  
if any(idx_pos) o|c"W}W  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3x~AaC.j  
end kpO+  
if any(idx_neg) 7:h_U9Za?$  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 1 ,4V8gp  
end C)qP9uW  
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tsqWnz=)  
% EOF zernfun :vy./83W  

支持一下，慢慢研究

顶顶·········

sansummer:这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊 (2012-04-27 10:22)  &0 VM <  

BV&}(9z  
DDE还是手动输入的呢？ Iy](?b  
5U[;T]{)e  
zygo和zemax的zernike系数，类型对应好就没问题了吧

这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊

慢慢研究，这个专业性很强的。用的人又少。