[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， -|V#U`mwF  
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(tJc}Mh+(  
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？ ,1I-%6L  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ wqG#jC!5  
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function z = zernfun(n,m,r,theta,nflag) Z`kVyuQ  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +(!/(2>~  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N u0W6u} 4;  
%   and angular frequency M, evaluated at positions (R,THETA) on the Z(q]rX5"  
%   unit circle.  N is a vector of positive integers (including 0), and y{M7kYWtHV  
%   M is a vector with the same number of elements as N.  Each element Kb]}p  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ;gL{*gR]S  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ;=joQWNDm  
%   and THETA is a vector of angles.  R and THETA must have the same u.A}&'H  
%   length.  The output Z is a matrix with one column for every (N,M) 6"_pCkn;c<  
%   pair, and one row for every (R,THETA) pair. ;8<HB1 &,  
% k9eyl)  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike f%PLR9Nh5@  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =R:O`qdC4e  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral @:im/SE  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +o@:8!IM1  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized `Ij EwKra  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. d%I7OBBx@
 
% |[~S&  
%   The Zernike functions are an orthogonal basis on the unit circle. fTpG>*{p  
%   They are used in disciplines such as astronomy, optics, and )&E]  
%   optometry to describe functions on a circular domain. F;_c x  
% ^zTe9:hz/\  
%   The following table lists the first 15 Zernike functions. r\QV%09R  
% iuj%.}  
%       n    m    Zernike function           Normalization |fyzb=Lg  
%       -------------------------------------------------- @|cH
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%       0    0    1                                 1 2c]751
  
%       1    1    r * cos(theta)                    2 8Dl(zYK;  
%       1   -1    r * sin(theta)                    2 ekY)?$v3  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) _#
Hd2h  
%       2    0    (2*r^2 - 1)                    sqrt(3) (Q*x"G#4>  
%       2    2    r^2 * sin(2*theta)             sqrt(6) r?u4[ Oe#  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) +8
xT}mX  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) n;Mk\*Cg  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 5=*i!c _m  
%       3    3    r^3 * sin(3*theta)             sqrt(8) eV%{XR?
y  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) onmpMU7w  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CF3x\6.q}  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) r<kgYU`  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j|8!gW  
%       4    4    r^4 * sin(4*theta)             sqrt(10) _N:$|O#  
%       -------------------------------------------------- v6G1y[Wl  
% sCJ|U6Q-  
%   Example 1: X9PbU1o;  
% 1?w=v|b:P)  
%       % Display the Zernike function Z(n=5,m=1)  #*rJI3  
%       x = -1:0.01:1; %7-(c  
%       [X,Y] = meshgrid(x,x); dLGHbeZ[(  
%       [theta,r] = cart2pol(X,Y); ogS
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%       idx = r<=1; .h4NG4FIF  
%       z = nan(size(X)); t{B@k[|  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); #qk=R7"Q  
%       figure MA_YMxP.'  
%       pcolor(x,x,z), shading interp ?f9M59(l  
%       axis square, colorbar Q_p&~PNy5  
%       title('Zernike function Z_5^1(r,\theta)') v6DjNyg<x  
% F3vywN1$,  
%   Example 2: O*/%zr  
% $aEv*{$y  
%       % Display the first 10 Zernike functions G1
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%       x = -1:0.01:1; U:99w  
%       [X,Y] = meshgrid(x,x); x]`F#5j  
%       [theta,r] = cart2pol(X,Y); Ohgu*5!o  
%       idx = r<=1; cQxUEY('+  
%       z = nan(size(X)); 66-\}8f8a  
%       n = [0  1  1  2  2  2  3  3  3  3]; "*/IP9?]  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Wm"q8-
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%       Nplot = [4 10 12 16 18 20 22 24 26 28]; vN v'%;L  
%       y = zernfun(n,m,r(idx),theta(idx)); FO(QsR=\s  
%       figure('Units','normalized') "5dke^yk0  
%       for k = 1:10 4Th?q
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%           z(idx) = y(:,k); _'Jjt9@S  
%           subplot(4,7,Nplot(k)) MCTJ^g"D  
%           pcolor(x,x,z), shading interp s>G]U)d<'  
%           set(gca,'XTick',[],'YTick',[]) ";`jS&"=  
%           axis square 1!V[fPJ  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ah<p_qe9|  
%       end |5`e
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% r[^.\&-  
%   See also ZERNPOL, ZERNFUN2. \z6UWZ  
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%   Paul Fricker 11/13/2006 2gklGDJD  
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% Check and prepare the inputs: 49>b]f,Vc  
% ----------------------------- Z5oDj|&l}  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C7R3W,  
    error('zernfun:NMvectors','N and M must be vectors.') {[:C_Up)f  
end t90M]EAV  
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rZ+4kf6S  
if length(n)~=length(m) Nz3+yxv1  
    error('zernfun:NMlength','N and M must be the same length.') #>KiX84  
end Fh
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n = n(:); YCStX)r  
m = m(:); Kyk{:UnI  
if any(mod(n-m,2)) ^/}4M'[w  
    error('zernfun:NMmultiplesof2', ... Qp[
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          'All N and M must differ by multiples of 2 (including 0).') [O^/"Qk  
end Q5dqn"?  
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+\%]<YO  
if any(m>n) NTX0vQG  
    error('zernfun:MlessthanN', ... %U}6(~  
          'Each M must be less than or equal to its corresponding N.') H;_Ce'oU(  
end t\QLj&h}E  
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if any( r>1 | r<0 ) \reVA$M[  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') zOMxg00  
end _IOUhMo  
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uUx7>algF  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1}c/l<d  
    error('zernfun:RTHvector','R and THETA must be vectors.') _2`b$/)-  
end Op9 ^Eu%n  
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r = r(:); k kD#Bb  
theta = theta(:); hTO2+F*  
length_r = length(r); ECM#J28D  
if length_r~=length(theta) q$yg^:]2  
    error('zernfun:RTHlength', ... nG5\vj,zB  
          'The number of R- and THETA-values must be equal.') Y~I>mc]  
end |[5;dt_U/  
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S}h d,"I  
% Check normalization: OW=3t#"7Kp  
% -------------------- XW8@c2jN\7  
if nargin==5 && ischar(nflag) ,KM%/;1Dm  
    isnorm = strcmpi(nflag,'norm'); b@4UR<  
    if ~isnorm .eVX/6,  
        error('zernfun:normalization','Unrecognized normalization flag.') eJ<P  
    end W\Scak>  
else ,vvfk=-  
    isnorm = false; '^WR5P<8c  
end G8w@C
 
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZGX"Vn|YL  
% Compute the Zernike Polynomials _nzq(m1@  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [#\OCdb*3  
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&,PA+#  
% Determine the required powers of r: dn])6Xl;i  
% ----------------------------------- TBJ?8W(  
m_abs = abs(m); 7=X6_AD  
rpowers = []; x4g6Qze  
for j = 1:length(n) OA9P"*  
    rpowers = [rpowers m_abs(j):2:n(j)]; YZOwr72VL  
end FVP,$  
rpowers = unique(rpowers); &Q"vXs6Gt  
3I}AA.h'00  
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% Pre-compute the values of r raised to the required powers, RaTNA W)v>  
% and compile them in a matrix:
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% ----------------------------- 4%q
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if rpowers(1)==0 7|dm"%@  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4mp)v*z  
    rpowern = cat(2,rpowern{:}); (ESFR0  
    rpowern = [ones(length_r,1) rpowern]; _'Vo3b  
else t'W6Fmwkx  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )q4n
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    rpowern = cat(2,rpowern{:}); AriV4 +  
end ]P7gEBi  
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%j17QD8  
% Compute the values of the polynomials: a}VR>!b  
% -------------------------------------- 8,+T[S  
y = zeros(length_r,length(n)); hF^JSCDz l  
for j = 1:length(n) x2I|iA=  
    s = 0:(n(j)-m_abs(j))/2; r/ATZAgHP  
    pows = n(j):-2:m_abs(j); 9dszn^]T  
    for k = length(s):-1:1 V?^qW#AG  
        p = (1-2*mod(s(k),2))* ... og+Vr
d  
                   prod(2:(n(j)-s(k)))/              ... ?Y\WSI?i  
                   prod(2:s(k))/                     ... Jr2>D=  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 6z~ [Ay  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); (kK8 OxfF  
        idx = (pows(k)==rpowers); ';v2ld 9  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Mx93D   
    end oliVaavj  
     ;2fzA<RkK  
    if isnorm ~/SLGyu  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PeEaF@#k  
    end c??m9=OX1  
end ;VCFDE{K=  
% END: Compute the Zernike Polynomials *Y53bZ  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (1er?4  
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at*DYZBjDB  
% Compute the Zernike functions: v/]xdP^Z  
% ------------------------------ n.5M6i/~a  
idx_pos = m>0; Avljrds+7  
idx_neg = m<0;
5f@&XwD9  
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Bgsi$2hI  
z = y; /N/jwLr  
if any(idx_pos) v)K|{x  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #z_.!E  
end 7I(QTc)*  
if any(idx_neg) 8h}1t4k  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T|YMU?4  
end MbTmdRf  
B=_5gZ4Y  
?5pZp~  
% EOF zernfun 1Nv qtVC  

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)  U0dhr;l  

[F^j(
qTR  
DDE还是手动输入的呢？ &jF[f4:7  
~qb-uT\(99  
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。