[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， fYBmW')  
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， cxn3e,d`  
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？ EBc_RpC/Z  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ (R5n
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function z = zernfun(n,m,r,theta,nflag) d0vn/k2I  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /}t>o* x  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ZCVwQ#Xe+  
%   and angular frequency M, evaluated at positions (R,THETA) on the LlKvi_z  
%   unit circle.  N is a vector of positive integers (including 0), and 4>x]v!d  
%   M is a vector with the same number of elements as N.  Each element Sc#B-4m  
%   k of M must be a positive integer, with possible values M(k) = -N(k) PT4Wox9U  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 2:3-mWE  
%   and THETA is a vector of angles.  R and THETA must have the same %&w 8E[  
%   length.  The output Z is a matrix with one column for every (N,M) z><uYO$  
%   pair, and one row for every (R,THETA) pair. &3~lZa;D  
% $R6iG\
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%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :Yeo*v9  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), mCah{~  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral >
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%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 2^RWGCEv  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized Vz_ac vfk^  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4I
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% `<_A#@  
%   The Zernike functions are an orthogonal basis on the unit circle. P5-1z&9O  
%   They are used in disciplines such as astronomy, optics, and $v5
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%   optometry to describe functions on a circular domain. OI/m_x
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% zB/#[~  
%   The following table lists the first 15 Zernike functions. jT/}5\  
% xgeDfpF'  
%       n    m    Zernike function           Normalization Lxz!>JO>  
%       -------------------------------------------------- vz$-KT4e^  
%       0    0    1                                 1 d+DdDr  
%       1    1    r * cos(theta)                    2 YNHQbsZUI,  
%       1   -1    r * sin(theta)                    2 Q5%$
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%       2   -2    r^2 * cos(2*theta)             sqrt(6) v_=xN^R  
%       2    0    (2*r^2 - 1)                    sqrt(3) ~hiJOaCzM  
%       2    2    r^2 * sin(2*theta)             sqrt(6) wMc/Og  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) b~$B0o)  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) $FR1^|P/G  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) X~+AaI:~K  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ,zXP,(x  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) cl2+,!:  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {p.D E  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) j<,Ho4v}_  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) e *9c33  
%       4    4    r^4 * sin(4*theta)             sqrt(10) '?$N.lj$d  
%       -------------------------------------------------- !W\Zq+^^J3  
% lSW6\jX  
%   Example 1: Jr2x`^aNO  
% b{+7sl  
%       % Display the Zernike function Z(n=5,m=1) CB!5>k+mC  
%       x = -1:0.01:1; Q5K<ECoPk  
%       [X,Y] = meshgrid(x,x); skSs|slp  
%       [theta,r] = cart2pol(X,Y); .
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%       idx = r<=1; sWtT"7>x  
%       z = nan(size(X)); Ku'OM6D<  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); b\ P6,s'(  
%       figure '>"
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%       pcolor(x,x,z), shading interp m%$GiNs}  
%       axis square, colorbar 0XgJCvMcB  
%       title('Zernike function Z_5^1(r,\theta)') 8,VX%CS#q  
% iwiHw  
%   Example 2: }8lv
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% 5Yxs_t4  
%       % Display the first 10 Zernike functions owR`Z`^h)  
%       x = -1:0.01:1; .
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%       [X,Y] = meshgrid(x,x); \+9~\eeXb  
%       [theta,r] = cart2pol(X,Y); @Yzdq\FI  
%       idx = r<=1; dx.,  
%       z = nan(size(X)); 6_rgj{L  
%       n = [0  1  1  2  2  2  3  3  3  3]; *- S/{ .&  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6cQ)*,Q  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; $4Vpl  
%       y = zernfun(n,m,r(idx),theta(idx)); QXaE2}}P  
%       figure('Units','normalized') [\_#n5  
%       for k = 1:10 3AQu\4+A  
%           z(idx) = y(:,k); K-<kp!v  
%           subplot(4,7,Nplot(k)) B)L=)N  
%           pcolor(x,x,z), shading interp o)B`K."  
%           set(gca,'XTick',[],'YTick',[]) *m>XtBw.  
%           axis square NT1"?Thx|  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U07G&?/  
%       end $E >)  
% _x'?igy  
%   See also ZERNPOL, ZERNFUN2. 03)R_A  
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%   Paul Fricker 11/13/2006 p?
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% Check and prepare the inputs: )ZHo7X  
% ----------------------------- [(81-j1v  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E0lro+'lS  
    error('zernfun:NMvectors','N and M must be vectors.') bMCy=5  
end <H]1 6  
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if length(n)~=length(m) $qp,7RW  
    error('zernfun:NMlength','N and M must be the same length.') Qzh`x-S  
end jmkVolz  
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n = n(:); Jx-dWfe  
m = m(:); f8AgTw,K8  
if any(mod(n-m,2)) BIK^<_?+ZU  
    error('zernfun:NMmultiplesof2', ... 9$iDK$%  
          'All N and M must differ by multiples of 2 (including 0).') .I1k+   
end s9\HjK*+  
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if any(m>n) $Y3mO~  
    error('zernfun:MlessthanN', ... %=G*{mK  
          'Each M must be less than or equal to its corresponding N.') s0/
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end nyRQ/.3  
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if any( r>1 | r<0 ) s.{nxk.  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') H%&e[PU  
end F?jFFwim  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) EZw<)Q  
    error('zernfun:RTHvector','R and THETA must be vectors.') o 00(\ -eb  
end xkPH_+4i8  
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r = r(:); g/_j"Nn  
theta = theta(:); Z<ABK`rEO  
length_r = length(r); {g@?\  
if length_r~=length(theta) &40# _>W7  
    error('zernfun:RTHlength', ... r,FPTf  
          'The number of R- and THETA-values must be equal.') ='U>P( R-  
end n72+X  
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% Check normalization: ?ZS/`P0}[  
% -------------------- M7x*LiKc2  
if nargin==5 && ischar(nflag) jVxX! V  
    isnorm = strcmpi(nflag,'norm'); BnwYyh  
    if ~isnorm ) Z^b)KAk  
        error('zernfun:normalization','Unrecognized normalization flag.') \YN(rD-  
    end =IC cN|  
else -
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    isnorm = false; O% }EpIP_  
end U1,f$McZs  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KO''B or  
% Compute the Zernike Polynomials t7; ^rk*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *F)+- BB  
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% Determine the required powers of r: )i$:iI >k  
% ----------------------------------- 7JL*y\'  
m_abs = abs(m); QH]G>+LI5  
rpowers = []; _O w]kP='  
for j = 1:length(n) "u=U@1 ^  
    rpowers = [rpowers m_abs(j):2:n(j)]; ?VCM@{9  
end 7LZA!3  
rpowers = unique(rpowers); 3{"MN=  
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Ak=|wY{  
% Pre-compute the values of r raised to the required powers, +`_Km5=  
% and compile them in a matrix: nbf w7u  
% ----------------------------- 6:$+"@ps  
if rpowers(1)==0 Q(0eq_X|6  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); zh60b{  
    rpowern = cat(2,rpowern{:}); [e.@Yx_}  
    rpowern = [ones(length_r,1) rpowern]; tg|7\Z7i  
else J\fu6Ti  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hxX-iQya  
    rpowern = cat(2,rpowern{:}); @Y| %  
end Duh[(r_  
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% Compute the values of the polynomials: 7!r)[2l  
% -------------------------------------- ~P@6fK/M  
y = zeros(length_r,length(n)); JA(M'&q4  
for j = 1:length(n) jDKL}x  
    s = 0:(n(j)-m_abs(j))/2; CgxGvM4  
    pows = n(j):-2:m_abs(j); iLR^V!  
    for k = length(s):-1:1 /GUbc
  
        p = (1-2*mod(s(k),2))* ... ckCb)r_  
                   prod(2:(n(j)-s(k)))/              ... hOH D
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                   prod(2:s(k))/                     ... R.rxpJ+kU  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... j 5{"j  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 8*\PWl  
        idx = (pows(k)==rpowers); o%1dbbh  
        y(:,j) = y(:,j) + p*rpowern(:,idx); T>e4Og"?  
    end }p$@.+  
     n)6mfoe  
    if isnorm '"~ 2xiin  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); @Q#<-/  
    end ,{rm<M.)  
end !y 7SCz g  
% END: Compute the Zernike Polynomials )cUFb:D*"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y-vLEIX=  
=bDy :yY}  
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% Compute the Zernike functions: :^92B?q  
% ------------------------------ Dqh rg;  
idx_pos = m>0; 8O='Q-&8  
idx_neg = m<0; uU;]/  
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z = y; ),53(=/hl  
if any(idx_pos) +D&aE$<  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ImZ!8#  
end Qe,aIh
 
if any(idx_neg) W2 p&LP  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yWkg4  
end lf?dTPrD  
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% EOF zernfun F
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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)  K@DK4{  

0 jszZ_  
DDE还是手动输入的呢？ )C0dN>Gb  
CG -^}xE:  
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。