[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， Sa.nUj{M=  
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， nDckT+eJ  
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？ D?*du#6  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ P$AHw;n[R  
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function z = zernfun(n,m,r,theta,nflag) 7hMh%d0d(_  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. lY,9bSF$  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,1<6=vL  
%   and angular frequency M, evaluated at positions (R,THETA) on the 4-'0# a  
%   unit circle.  N is a vector of positive integers (including 0), and sMJa4P>O@  
%   M is a vector with the same number of elements as N.  Each element "av/a  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ,5t_}d|3C=  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o2]Np~`g,  
%   and THETA is a vector of angles.  R and THETA must have the same -H_#et3&i  
%   length.  The output Z is a matrix with one column for every (N,M) z [u!C/  
%   pair, and one row for every (R,THETA) pair. X ) =-a  
% |6Iw\YU  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i \lr KA  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @&Yl'&pn-R  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral n36@&q+B&  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, P^lRJB<$Q  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized m#6p=E  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Xfg?\j/  
% XC/M:2$  
%   The Zernike functions are an orthogonal basis on the unit circle. !l.^]|  
%   They are used in disciplines such as astronomy, optics, and 7s:c
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%   optometry to describe functions on a circular domain. OMYbCy^  
% }J\7IsM&  
%   The following table lists the first 15 Zernike functions. ^tjM1uaZ5(  
% ^QHgc_oDm  
%       n    m    Zernike function           Normalization = 4'r+2[  
%       -------------------------------------------------- +f_3JL$  
%       0    0    1                                 1 H6$pA^  
%       1    1    r * cos(theta)                    2 irB}h!@  
%       1   -1    r * sin(theta)                    2 0PU
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%       2   -2    r^2 * cos(2*theta)             sqrt(6) vsI|HxpyC,  
%       2    0    (2*r^2 - 1)                    sqrt(3) 0\"]XYOH  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 5g9K|-  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) iy_3#x5>  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) YsTF10  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ._'.F'd  
%       3    3    r^3 * sin(3*theta)             sqrt(8) brF) %x`  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) "Vwk&~B%  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ah!RQ2hDrV  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) HXqG;Fds(  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q b5vyV `  
%       4    4    r^4 * sin(4*theta)             sqrt(10) H}1XK|K3#H  
%       -------------------------------------------------- N{!@M_C^%R  
%  /zir$  
%   Example 1: c+1<3)Q<  
% :pP l|"
 
%       % Display the Zernike function Z(n=5,m=1) = o1&.v2j  
%       x = -1:0.01:1; *zX^Sg-[  
%       [X,Y] = meshgrid(x,x); dFnu&u"  
%       [theta,r] = cart2pol(X,Y); Nb>C5TjR  
%       idx = r<=1; 5VLC\QgK^  
%       z = nan(size(X)); dJ{'b'#  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); U owbk:  
%       figure XJ7mvLM;  
%       pcolor(x,x,z), shading interp I
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%       axis square, colorbar $*+UX   
%       title('Zernike function Z_5^1(r,\theta)') @iYr<>iDZ  
% K7 tSSX<N  
%   Example 2: R&L^+?  
% "/\-?YJjw  
%       % Display the first 10 Zernike functions \!r,>P
 
%       x = -1:0.01:1; _w9:([_  
%       [X,Y] = meshgrid(x,x); 0VI[6t@  
%       [theta,r] = cart2pol(X,Y); a ,
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%       idx = r<=1; lhFv2.qR  
%       z = nan(size(X)); jsw0"d(  
%       n = [0  1  1  2  2  2  3  3  3  3]; %h=cwT6  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; nrz2f7d$  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; sYyya:ykxT  
%       y = zernfun(n,m,r(idx),theta(idx)); >=L<3W1  
%       figure('Units','normalized') H3BMN}K~  
%       for k = 1:10 t^<ki?*  
%           z(idx) = y(:,k); 7{u1ynt  
%           subplot(4,7,Nplot(k)) |%Ssb;M  
%           pcolor(x,x,z), shading interp D{, b|4  
%           set(gca,'XTick',[],'YTick',[]) /2]=.bLwz  
%           axis square X&|y|  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V#d8fRm  
%       end 77zDHq=  
% o~p%ODH  
%   See also ZERNPOL, ZERNFUN2. @-jI<g  
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%   Paul Fricker 11/13/2006 "@+r|x  
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% Check and prepare the inputs: K<D=QweOon  
% ----------------------------- 9]*hP](  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Zd[6-/-:  
    error('zernfun:NMvectors','N and M must be vectors.') aQ.mvuMa7'  
end bl`vT3  
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if length(n)~=length(m) ;,jms~ik  
    error('zernfun:NMlength','N and M must be the same length.') 4qLH3I[Y  
end )
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n = n(:); pe7R1{2Q_s  
m = m(:); G' a{;3  
if any(mod(n-m,2)) AU/L_hg  
    error('zernfun:NMmultiplesof2', ... }SF<. A  
          'All N and M must differ by multiples of 2 (including 0).') 3/
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end jMB&(r  
zD}2Zh]  
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if any(m>n) t"L-9kCM  
    error('zernfun:MlessthanN', ... ,aQ{  
          'Each M must be less than or equal to its corresponding N.') j`$d W H/2  
end >$iQDVh!  
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if any( r>1 | r<0 ) x$ oId{;  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') >`!Lh`n7_  
end h oL"
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) f6-OR]R5  
    error('zernfun:RTHvector','R and THETA must be vectors.') y72=d?]W  
end HOrD20  
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r = r(:); ;`MKi5g  
theta = theta(:); %nkP?gn"a  
length_r = length(r); Yr9!</;T  
if length_r~=length(theta) WTP~MJ#C  
    error('zernfun:RTHlength', ... BK16~Wl  
          'The number of R- and THETA-values must be equal.') E[N3`"  
end XrD@q  
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% Check normalization: BVk&TGa;[$  
% -------------------- S>s{t=AY~  
if nargin==5 && ischar(nflag) %uWq)D4r  
    isnorm = strcmpi(nflag,'norm'); P7`sJ("#  
    if ~isnorm %qf ?_2v  
        error('zernfun:normalization','Unrecognized normalization flag.') b _#r_`  
    end &6
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else ndU<,{r  
    isnorm = false;  "0( _  
end +vh 4I  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% > R5<D'cEN  
% Compute the Zernike Polynomials _
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `78:TU~5S  
#nOS7Q#uW  
R8U?s/*  
% Determine the required powers of r: fxKhe[;  
% ----------------------------------- ^YLk
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m_abs = abs(m); wZ_k]{J  
rpowers = []; -U"h3Ye^  
for j = 1:length(n) zJ2dPp~u  
    rpowers = [rpowers m_abs(j):2:n(j)]; Rt^~db  
end !C$bOhc  
rpowers = unique(rpowers); AQH\ ;L  
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% Pre-compute the values of r raised to the required powers, ogrh"  
% and compile them in a matrix: Fuuy_+p@G  
% ----------------------------- gLyE,1Z}u  
if rpowers(1)==0 O*8.kqlgt  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B]o5HA<k  
    rpowern = cat(2,rpowern{:}); GYq.!d@O  
    rpowern = [ones(length_r,1) rpowern]; k1
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else )@O80uOFh  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); uGxh}'&  
    rpowern = cat(2,rpowern{:}); u\9t+wi}<  
end im6Rx=}E{  
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% Compute the values of the polynomials: .(7end<  
% -------------------------------------- *)6:yn  
y = zeros(length_r,length(n)); LR5X=&k  
for j = 1:length(n) O"D0+BK79e  
    s = 0:(n(j)-m_abs(j))/2; hrRkam!y  
    pows = n(j):-2:m_abs(j); AP8YY8,  
    for k = length(s):-1:1 P'dH
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        p = (1-2*mod(s(k),2))* ... |H LU5=Y  
                   prod(2:(n(j)-s(k)))/              ... PSM~10l,  
                   prod(2:s(k))/                     ... Kn!n}GtR  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X<OOgC  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); $T)E
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        idx = (pows(k)==rpowers); E9]/sFA-]  
        y(:,j) = y(:,j) + p*rpowern(:,idx); |NsrO8H   
    end /R2K3E#  
     0KQDw  
    if isnorm tocZO  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sSM^net0  
    end _|!FhZ  
end eB$S d  
% END: Compute the Zernike Polynomials |N{?LKR %  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nsRZy0@$t  
@w H+,]xE  
:j}]nS  
% Compute the Zernike functions: `H:5D5]  
% ------------------------------ Z;nbnRz  
idx_pos = m>0; `H_.<``>  
idx_neg = m<0; [M7&  
] X9
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z = y; sC/5N  
if any(idx_pos) ?x*Ve2+]  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "o=*f/M  
end Y'75DE<BC  
if any(idx_neg) Vh.9/$xQ  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z`?Z1SBt  
end 80p?qe  
rW~hFSrV[o  
$[p<}o/6v]  
% EOF zernfun 9q##)  

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)  3GH@|id  

2"31k2H[  
DDE还是手动输入的呢？ q*^Y8s~3I  
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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。