[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， _r>kR7A\{  
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， @km4qJZ  
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？ &4ndi=.#rg  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ q1v7(`O  
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function z = zernfun(n,m,r,theta,nflag) BS(jC  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cg_ " }]Y1  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `@ny!S|1/  
%   and angular frequency M, evaluated at positions (R,THETA) on the oW^>J-  
%   unit circle.  N is a vector of positive integers (including 0), and X ]W)D S  
%   M is a vector with the same number of elements as N.  Each element g#`}HuPoE  
%   k of M must be a positive integer, with possible values M(k) = -N(k) AN3oh1xe:  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, +*,!q7Gt  
%   and THETA is a vector of angles.  R and THETA must have the same bg|dV  
%   length.  The output Z is a matrix with one column for every (N,M) 41P0)o  
%   pair, and one row for every (R,THETA) pair. >'4$g7o,  
% 6lT< lzT  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Jg)( F|>o  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), $0vWC#.A]  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral %!eRR  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, g/}d> 6  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized v|KIVBkbT  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mG$N%
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% .)=*Yr M  
%   The Zernike functions are an orthogonal basis on the unit circle. \GQRpJ#h1  
%   They are used in disciplines such as astronomy, optics, and p3
Ozfk  
%   optometry to describe functions on a circular domain. QUaV;6 4  
% EV-sEl8ki  
%   The following table lists the first 15 Zernike functions. D+BiclJ  
% w]nt_xj  
%       n    m    Zernike function           Normalization }a#T\6rY  
%       -------------------------------------------------- 8:)[.  
%       0    0    1                                 1 9HEqB0|ZRu  
%       1    1    r * cos(theta)                    2 _`gkYu3R+  
%       1   -1    r * sin(theta)                    2 bRrSd:e  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ({@"{  
%       2    0    (2*r^2 - 1)                    sqrt(3)  JZ+6)R  
%       2    2    r^2 * sin(2*theta)             sqrt(6) w>8kBQ
?b  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) v9FR  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1zC
u1'Wv  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 'n>44_7L  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 4f~sRu
bK  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) EZ:? (|
h  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pJs`/  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 8EMBqhl  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IZm6.F  
%       4    4    r^4 * sin(4*theta)             sqrt(10) $_;rqTk]g  
%       -------------------------------------------------- U;IGV~oT  
% l1|*(%p?X  
%   Example 1: *xmC`oP  
% rk4KAX_[  
%       % Display the Zernike function Z(n=5,m=1) SvQ|SKE'
:  
%       x = -1:0.01:1; +H?g9v40  
%       [X,Y] = meshgrid(x,x); Z,SV9 ~M  
%       [theta,r] = cart2pol(X,Y); !.^x^OK%y  
%       idx = r<=1; j`q>YPp  
%       z = nan(size(X)); 2wnk~URj  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); #d3_7rI0V  
%       figure QH4m7M@ni  
%       pcolor(x,x,z), shading interp *0Z6H-Do,  
%       axis square, colorbar SXYwhID=  
%       title('Zernike function Z_5^1(r,\theta)') 1LSJy*yY  
% jnbR}a=fJ  
%   Example 2:  B9y5NX  
% XR9kxTuk  
%       % Display the first 10 Zernike functions `?.6}*4@_A  
%       x = -1:0.01:1; X Db%-  
%       [X,Y] = meshgrid(x,x); -,YI>!  
%       [theta,r] = cart2pol(X,Y); 0TA8#c  
%       idx = r<=1; 1Az&BZU[  
%       z = nan(size(X)); & wtE"w  
%       n = [0  1  1  2  2  2  3  3  3  3]; m1jEky(  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :RukW.MR  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 2;*G!rE&*`  
%       y = zernfun(n,m,r(idx),theta(idx)); re/u3\S  
%       figure('Units','normalized') A'7Y{oPHX  
%       for k = 1:10 p>\[[Md  
%           z(idx) = y(:,k); <*z'sUh+}  
%           subplot(4,7,Nplot(k)) -zMvpe-am&  
%           pcolor(x,x,z), shading interp u/wX7s  
%           set(gca,'XTick',[],'YTick',[]) a@&qdp  
%           axis square }Hg\ tj}i  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h$$JXf  
%       end x9\{a  
% xi.?@Lff
 
%   See also ZERNPOL, ZERNFUN2. U&:-Vf~&  
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^Ss<<  
%   Paul Fricker 11/13/2006 j DEym&-  
RA!m,"RM  
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% Check and prepare the inputs: BR?DW~7J j  
% ----------------------------- )'g4Ty  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +h/OQ]`/m  
    error('zernfun:NMvectors','N and M must be vectors.') p=eSJ*  
end RrrlfFms  
SeS ZMv  
% q!i  
if length(n)~=length(m) )BI%cD  
    error('zernfun:NMlength','N and M must be the same length.') IcQpbF0  
end *P7n YjG  
n} !')r  
Y>FLc* h  
n = n(:); !,Gavt7f  
m = m(:); 2Hx*kh2  
if any(mod(n-m,2)) QD^=;!  
    error('zernfun:NMmultiplesof2', ... 5>CeFy  
          'All N and M must differ by multiples of 2 (including 0).') RT'5i$q
[  
end v,
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i@WO>+iB  
if any(m>n) !@Vj&>mH$  
    error('zernfun:MlessthanN', ... ak3WER|f#  
          'Each M must be less than or equal to its corresponding N.') qkc,93B3  
end S\sy^Kt~4:  
&1=,?s]&  
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if any( r>1 | r<0 ) K)`R?CZ:s  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') .3Smqwm=Y  
end Gv8Z  
\!uf*=d  
n]5Pfg|a  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I 6<LKI/  
    error('zernfun:RTHvector','R and THETA must be vectors.') #3?"#),q
 
end L:lnm9<  
s:,fXg25J  
=yqg,w&Q  
r = r(:); 9S'\&mRl  
theta = theta(:); Ly, ];  
length_r = length(r); 4U)%JK.ta  
if length_r~=length(theta) }c4F}Cy  
    error('zernfun:RTHlength', ... C5Fq%y{$.  
          'The number of R- and THETA-values must be equal.') 93w$ck},?G  
end 4T&Jlu?:  
2e ~RM2PQ  
p"cY/2w:j  
% Check normalization: sZqi)lo-s  
% -------------------- \[+':o`LH  
if nargin==5 && ischar(nflag) G8^b9xoA+.  
    isnorm = strcmpi(nflag,'norm'); :t+LuH g  
    if ~isnorm )0;O<G] d  
        error('zernfun:normalization','Unrecognized normalization flag.') flBJO.2  
    end lu1T+@t  
else Ja\B%f  
    isnorm = false; {=R vFA  
end X/Fip0i  
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*;t_VlaZ  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !a5e
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% Compute the Zernike Polynomials #]}G{ P  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =`gFwH<   
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% Determine the required powers of r: _1$+S0G;  
% ----------------------------------- Qej<(:J5  
m_abs = abs(m); OW> >6zM  
rpowers = []; {`L,F
 
for j = 1:length(n) jJ_6_8#  
    rpowers = [rpowers m_abs(j):2:n(j)]; \nVoBW(  
end .8|5;!`WB  
rpowers = unique(rpowers); <("P5@cExU  
,?GAFgK:  
.M\0+,%/  
% Pre-compute the values of r raised to the required powers, ,}Ic($To  
% and compile them in a matrix: IifH=%2Y  
% ----------------------------- R*O6Z"h  
if rpowers(1)==0 <jVk}gi)Jp  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); o`ODz[04  
    rpowern = cat(2,rpowern{:}); JlH5 <:#PN  
    rpowern = [ones(length_r,1) rpowern]; -f(<2i  
else jin?;v  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `jDmbD +=  
    rpowern = cat(2,rpowern{:}); -32.g\]  
end :4238J8  
T=cb:PD{%  
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% Compute the values of the polynomials: gi\UNT9x  
% -------------------------------------- EmcwX4|  
y = zeros(length_r,length(n)); zhwajc  
for j = 1:length(n) X@B,w
_b  
    s = 0:(n(j)-m_abs(j))/2; MWc{7,  
    pows = n(j):-2:m_abs(j); FEg&EYI  
    for k = length(s):-1:1 UC9w T  
        p = (1-2*mod(s(k),2))* ... 0`e- ;  
                   prod(2:(n(j)-s(k)))/              ... ';x5 $5k'  
                   prod(2:s(k))/                     ... g\,HiKBXd  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... W3i X;-Z  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); \RTXfe-`  
        idx = (pows(k)==rpowers); N &vQis  
        y(:,j) = y(:,j) + p*rpowern(:,idx); y `w5u.'  
    end  qZP>h4  
     <H!;/p/S  
    if isnorm gLv";"4S  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3sGe#s%  
    end 4,R1}.?BzJ  
end ^S`c-N  
% END: Compute the Zernike Polynomials C[(Exe  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %OsV(7  
[k
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i  
,t>/_pI+=  
% Compute the Zernike functions: E|^~R}z)  
% ------------------------------ I#hzU8Cc  
idx_pos = m>0; ~4~>;e  
idx_neg = m<0; mh`VZQ@  
-n$fh::^  
0IjQqI  
z = y; =u`^QE  
if any(idx_pos) Y3I+TI>x  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -T-h~5  
end ;zvg]  %  
if any(idx_neg) WAcQRa~C  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M3dNG]3E  
end G@QZmuj&KH  
Tp/+{|~  
$ V"7UA22  
% EOF zernfun R|AGN*.  

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)  izow=}  

=x9zy]  
DDE还是手动输入的呢？ N%:QaCZKw  
PygaW&9Z|d  
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。