[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， [q_Yf!(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， DhB:8/J  
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？ |!&,etu  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 2t[inzn=E  
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function z = zernfun(n,m,r,theta,nflag) dJCu`34Y'|  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,=K!Y TeVl  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SDTX0v  
%   and angular frequency M, evaluated at positions (R,THETA) on the }g(aZ  
%   unit circle.  N is a vector of positive integers (including 0), and %OW[rbE.  
%   M is a vector with the same number of elements as N.  Each element Tk+\Biq   
%   k of M must be a positive integer, with possible values M(k) = -N(k) n>!E ]  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, b_
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%   and THETA is a vector of angles.  R and THETA must have the same 9}3W0F;  
%   length.  The output Z is a matrix with one column for every (N,M) zW+Y{^hf  
%   pair, and one row for every (R,THETA) pair. MA"iM+Ar  
% v "oO  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike a}e7Q<cGj  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qf7.Sh  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral "hQV\|!\  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {|>~#a49h  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized tT'd]  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %yptML9  
% W%Um:C\I  
%   The Zernike functions are an orthogonal basis on the unit circle. )5]z[sE  
%   They are used in disciplines such as astronomy, optics, and HlV3rYh  
%   optometry to describe functions on a circular domain. 36lIV,YnU  
% gR1X@j$_  
%   The following table lists the first 15 Zernike functions. BPi>SI0  
% u4Vc:n  
%       n    m    Zernike function           Normalization 8l)l9;4 6  
%       -------------------------------------------------- J"[OH,/_  
%       0    0    1                                 1 hRA.u'M  
%       1    1    r * cos(theta)                    2 B&L{/.v_z\  
%       1   -1    r * sin(theta)                    2 @#o$~'my  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) @W^g(I(w  
%       2    0    (2*r^2 - 1)                    sqrt(3) ydlH6>  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 4e*0kItC  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) uw]e$,x?  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) u5idH),<  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) SxQ|1:i%  
%       3    3    r^3 * sin(3*theta)             sqrt(8) #|$7. e  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 0<i~XN0g  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iY(hGlV  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Y*"%;e$tg  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +mxsjcq0  
%       4    4    r^4 * sin(4*theta)             sqrt(10) -=g`7^qa>  
%       -------------------------------------------------- Jl5<9x  
% rJNf&x%6  
%   Example 1: c#G(7.0MU  
% H|,{^b@9  
%       % Display the Zernike function Z(n=5,m=1) q F}5mUcZ4  
%       x = -1:0.01:1; N ~LR  
%       [X,Y] = meshgrid(x,x); JWxPH5L  
%       [theta,r] = cart2pol(X,Y); $p9XXZ"*  
%       idx = r<=1; ` D4J9;|;]  
%       z = nan(size(X)); <v{jJ7w  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); O|gb{  
%       figure /CZOO)n  
%       pcolor(x,x,z), shading interp
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%       axis square, colorbar +)7NWR\  
%       title('Zernike function Z_5^1(r,\theta)') bNL E=#ro  
% !`aodz*PO  
%   Example 2: 3a#!^G!~  
% />n0&~k[h  
%       % Display the first 10 Zernike functions 1,pg:=N9  
%       x = -1:0.01:1; oxad}Y  
%       [X,Y] = meshgrid(x,x); Kfj*#)SZ  
%       [theta,r] = cart2pol(X,Y); X^@d
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%       idx = r<=1; #b8/gRfS  
%       z = nan(size(X)); o/&:w z  
%       n = [0  1  1  2  2  2  3  3  3  3]; %/>_o{"hw  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; m+vwp\0  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; +osY iP5  
%       y = zernfun(n,m,r(idx),theta(idx)); 5-&P4  
%       figure('Units','normalized') :;|x'[JoE?  
%       for k = 1:10 / Sp+MB9  
%           z(idx) = y(:,k); Dxu)by  
%           subplot(4,7,Nplot(k)) n09|Jzv9  
%           pcolor(x,x,z), shading interp QeQbO  
%           set(gca,'XTick',[],'YTick',[]) #Io#OG<7b  
%           axis square N;D+]_;0|  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]_-$  
%       end A"P1B]  
% OPjscc5  
%   See also ZERNPOL, ZERNFUN2. p]aIMF_  
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%   Paul Fricker 11/13/2006 kW(8i}bg  
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% Check and prepare the inputs: m1`ln5(R  
% ----------------------------- :
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) XBeHyQp  
    error('zernfun:NMvectors','N and M must be vectors.') Uz62!)  
end v'iQLUgI  
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if length(n)~=length(m) 4 A  
    error('zernfun:NMlength','N and M must be the same length.') ?dTz?C.w  
end Lh. L~M1X  
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n = n(:);   ()SG  
m = m(:); 
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if any(mod(n-m,2)) g xLA1]>{  
    error('zernfun:NMmultiplesof2', ... O=+C Kx@  
          'All N and M must differ by multiples of 2 (including 0).') _R8-Hj E  
end r2hm`]\8M  
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b0&dpMgh:  
if any(m>n) D)!k  
    error('zernfun:MlessthanN', ... '~a!~F~>  
          'Each M must be less than or equal to its corresponding N.') xAoozDj  
end ]#J]f  
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if any( r>1 | r<0 ) 5\a5^FK~  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2[:`w),.  
end _mn4z+  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7aU*7!U  
    error('zernfun:RTHvector','R and THETA must be vectors.') 3*'!,gK~[  
end I)sCWC:Mq~  
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r = r(:); EGa}ml/G  
theta = theta(:); Uh7kB`2  
length_r = length(r); t[DXG2&  
if length_r~=length(theta) H-S28%.  
    error('zernfun:RTHlength', ... K1$Z=]a+  
          'The number of R- and THETA-values must be equal.') a1j6-p  
end &-{4JSII  
+^%F8GB  
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% Check normalization: aFbA=6  
% -------------------- d:j$!@o  
if nargin==5 && ischar(nflag) 'DKP-R"  
    isnorm = strcmpi(nflag,'norm'); q_I''L  
    if ~isnorm 9x:c"S*  
        error('zernfun:normalization','Unrecognized normalization flag.') `5gcc7b  
    end MbJV)*Q  
else muY4:F.C(  
    isnorm = false; b0 5h,  
end J M`uIVnNA  
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uEk$Y=p7!  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Kj}}O2  
% Compute the Zernike Polynomials i|2Q}$3t2  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /FQumqbnt  
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% Determine the required powers of r: jF'S"_/?  
% ----------------------------------- jd$lu^>I  
m_abs = abs(m); Yr0%ZYfN  
rpowers = []; z43H]  
for j = 1:length(n) x2tx{Z  
    rpowers = [rpowers m_abs(j):2:n(j)]; WJhI6lu  
end 4sG^bZ,  
rpowers = unique(rpowers); qf'uXH  
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% Pre-compute the values of r raised to the required powers, ,(]hykbXp  
% and compile them in a matrix: zfvl<"Rv  
% ----------------------------- yA6"8fr  
if rpowers(1)==0 F|Ou5WD  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fv}h;?C  
    rpowern = cat(2,rpowern{:}); (B[0BjU  
    rpowern = [ones(length_r,1) rpowern]; p6>3 p  
else ?-Oy/Y K  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Dd:Qotu  
    rpowern = cat(2,rpowern{:}); #N7@p}P  
end $n>.;CV  
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% Compute the values of the polynomials: M$|^?U>cm  
% -------------------------------------- S_1R]n1/  
y = zeros(length_r,length(n)); 
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for j = 1:length(n) m~%IHWO'  
    s = 0:(n(j)-m_abs(j))/2; z0doLb^!  
    pows = n(j):-2:m_abs(j); F
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    for k = length(s):-1:1 bluhiiATd  
        p = (1-2*mod(s(k),2))* ... ECQ>VeP  
                   prod(2:(n(j)-s(k)))/              ... Z^s&]  
                   prod(2:s(k))/                     ... sJMT _yt;  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Fvl_5l  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); >u~ l_?  
        idx = (pows(k)==rpowers); tP7l ;EX4  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 0~)cAKus  
    end L%I@HB9-Q0  
     n:'Mpux  
    if isnorm ..;}EFw5  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \M<C6m5  
    end F0])g  
end ?%#3p[  
% END: Compute the Zernike Polynomials xyBWV]Y  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .kyp5CD}4  
%^kBcId  
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% Compute the Zernike functions: SfEgmp-m  
% ------------------------------ 48W$,  
idx_pos = m>0; X\V1c$13CK  
idx_neg = m<0; ~#pQWa5  
bw&8"k>D?  
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z = y; Gdb6 U{  
if any(idx_pos) lN-vFna  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {p=`"H>  
end OXT 5 y)   
if any(idx_neg) NirG99kyo  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 2mRm.e9?  
end criOJ-  
W0R<^5_  
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% EOF zernfun \u9l4  

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)  =|LB,REN  

~e<^jhpJ
 
DDE还是手动输入的呢？ A1Es>NK[qW  
7TtDI=f  
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。