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jssylttc 2012-04-23 19:23

如何从zernike矩中提取出zernike系数啊

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， 'ROz|iJ  
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， 7(h@5  
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？ RJerx:]  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ RtHai[j  
NW`.7'aWT  
 h4|}BGO  
./Ek+p*96H  
 R#;xBBt8  
function z = zernfun(n,m,r,theta,nflag) {j]cL !Od  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @P75f5p}<  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ; YQB  
%   and angular frequency M, evaluated at positions (R,THETA) on the F!)[H["_  
%   unit circle.  N is a vector of positive integers (including 0), and d4\JM 65  
%   M is a vector with the same number of elements as N.  Each element un-%p#  
%   k of M must be a positive integer, with possible values M(k) = -N(k) uyB2   
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, &,jUaC5I  
%   and THETA is a vector of angles.  R and THETA must have the same &;P\e  
%   length.  The output Z is a matrix with one column for every (N,M) 'r%(,=L  
%   pair, and one row for every (R,THETA) pair. l/zv >  
% >Jx=k"Kv+  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6LGl]jHf  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), M57<e`m  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral W4d32+V  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %,02i@Fc  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized }s<;YC  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^GY^g-R  
% (Q% @]  
%   The Zernike functions are an orthogonal basis on the unit circle. h`N2M,  
%   They are used in disciplines such as astronomy, optics, and $o5i15Oy.  
%   optometry to describe functions on a circular domain. ZlMT) ~fM&  
% XL.f `N.O  
%   The following table lists the first 15 Zernike functions. W7 Iy_>  
% }k%6X@  
%       n    m    Zernike function           Normalization } f&=}  
%       -------------------------------------------------- $ [fqTh  
%       0    0    1                                 1 d!R+-Fp  
%       1    1    r * cos(theta)                    2 g*YA~J@  
%       1   -1    r * sin(theta)                    2 # M/n\em"X  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 7:uz{xPK6  
%       2    0    (2*r^2 - 1)                    sqrt(3) 7R:Ij[dV  
%       2    2    r^2 * sin(2*theta)             sqrt(6) M3@qhEf?vk  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) &k}B66  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Ul]7IUzsu  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) fv8x7l7  
%       3    3    r^3 * sin(3*theta)             sqrt(8) L@AFt)U  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Eq;w5;7s  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [ R+M .5  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) H OWpTu(  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CV"}(1T  
%       4    4    r^4 * sin(4*theta)             sqrt(10) q/I( e  
%       -------------------------------------------------- *|\bS "  
% 16 `M=R  
%   Example 1: 7JQ4*RM  
% b,~pwbHf  
%       % Display the Zernike function Z(n=5,m=1) MT>(d*0s  
%       x = -1:0.01:1; 1[Yl8W%pj  
%       [X,Y] = meshgrid(x,x); Y3:HQ0w`|  
%       [theta,r] = cart2pol(X,Y); ]9w)0iH  
%       idx = r<=1; _p0Yhju?  
%       z = nan(size(X)); qX-5/;n  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); e3CFW_p  
%       figure  eu$VKLY*  
%       pcolor(x,x,z), shading interp N^[ F+y  
%       axis square, colorbar #1'q'f:7 &  
%       title('Zernike function Z_5^1(r,\theta)') TCyev[(  
% TU~y;:OJ  
%   Example 2: N^oP,^+U  
% zi6J|u  
%       % Display the first 10 Zernike functions ^lV}![do!  
%       x = -1:0.01:1; # 2^H{7  
%       [X,Y] = meshgrid(x,x); \^dse  
%       [theta,r] = cart2pol(X,Y); }a5TY("d9H  
%       idx = r<=1; v; #y^O  
%       z = nan(size(X)); ` wEX;  
%       n = [0  1  1  2  2  2  3  3  3  3]; "0;WYw?  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k0V]<#h87  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Qv ~@  
%       y = zernfun(n,m,r(idx),theta(idx)); /fT"WaTEK  
%       figure('Units','normalized') /FXvrH(  
%       for k = 1:10 oz=ULPZ%  
%           z(idx) = y(:,k); _dk[k@5W{'  
%           subplot(4,7,Nplot(k)) BE@(| U  
%           pcolor(x,x,z), shading interp COHBju fmR  
%           set(gca,'XTick',[],'YTick',[]) (jU_lsG  
%           axis square Ss 5@n  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) '1b8>L  
%       end 8o|C43Q_  
% }1 qQ7}v  
%   See also ZERNPOL, ZERNFUN2. >uYQt ~s  
nsi? .c&0!  
 $nmt&lm  
%   Paul Fricker 11/13/2006 AH'c:w]~  
sv% E5@  
 nrev!h  
u=qK_$d4  
 1ds4C:M+<  
% Check and prepare the inputs: l59\Lo:  
% ----------------------------- }W 5ks-L6  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) oc,I, v  
    error('zernfun:NMvectors','N and M must be vectors.') [UzacXt  
end c8mh#T bl  
#p*uk  
 o[Qb/ 7  
if length(n)~=length(m) #OM'2@  
    error('zernfun:NMlength','N and M must be the same length.') Q+Q"JU  
end qQ)1+^  
=6ru%.8U,  
 Ip7#${f5M  
n = n(:); LRu*%3xx  
m = m(:); [5IbR9_  
if any(mod(n-m,2)) ELnUpmv\  
    error('zernfun:NMmultiplesof2', ... /Lr`Aka5  
          'All N and M must differ by multiples of 2 (including 0).') +i!HMyM  
end !`Kg&t [&V  
8 f~x\.  
 2C:u)}R7D  
if any(m>n) !YGHJwW:  
    error('zernfun:MlessthanN', ... dO z|CfUhI  
          'Each M must be less than or equal to its corresponding N.') {~9HJDcM  
end |0}Xb|+  
Ot47.z  
 G.L}VpopM  
if any( r>1 | r<0 ) 5%9Uh'y#  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') Iv 3O8 GU  
end y[l{ UBue:  
9jf9 u0  
 Pi5MFw'v  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ly34aD/p~,  
    error('zernfun:RTHvector','R and THETA must be vectors.') .F@Lx45  
end Xux[  
0|-}>>qb\  
 iB W:t  
r = r(:); U`3?bhzua  
theta = theta(:); '"7b;%EN'  
length_r = length(r); `. JW_F)1  
if length_r~=length(theta) T5}3Y3G,6  
    error('zernfun:RTHlength', ... YC 4c-M  
          'The number of R- and THETA-values must be equal.') ]8}2  
end ,_(=w.F   
bf.+Ewb(  
 V*s\~h)  
% Check normalization: jEQ_#KKYJ  
% -------------------- W^^0Rh_  
if nargin==5 && ischar(nflag) =/'>.p3/S  
    isnorm = strcmpi(nflag,'norm'); a"xRc  
    if ~isnorm 3 $%#n*  
        error('zernfun:normalization','Unrecognized normalization flag.') VFZyWX@#u  
    end dL4VcUS.  
else  gh[q*%#  
    isnorm = false; X:`=\D  
end X4:84  
5s^vC2$)  
 $H3C/|  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GjW(&p$&  
% Compute the Zernike Polynomials ]!S#[Wt {k  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0&NM=~  
y~]D402Cx  
 V}<<?_  
% Determine the required powers of r:  Rh6CV  
% ----------------------------------- d!<>Fh^6,  
m_abs = abs(m); c %Y *XJ'  
rpowers = []; [V?HK_~  
for j = 1:length(n) r%=a:GdAg  
    rpowers = [rpowers m_abs(j):2:n(j)]; L=Aj+  
end ] g9SUFM  
rpowers = unique(rpowers); BR@gJ(2  
vxPr)"Vvz  
 rr`_\ut  
% Pre-compute the values of r raised to the required powers, %jj-\Gz!  
% and compile them in a matrix: _G-6G=q  
% ----------------------------- ;9)nG,P3  
if rpowers(1)==0 [?<v|k  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3nhQ^zqf  
    rpowern = cat(2,rpowern{:}); 7`L]aRS[  
    rpowern = [ones(length_r,1) rpowern]; @~$=96^  
else /-lW$.+{?  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); lws.;abm%n  
    rpowern = cat(2,rpowern{:}); .XK3o .ZhW  
end U<XfO'XJ  
aW|=|K  
 j3w~2q"r  
% Compute the values of the polynomials: &~.|9P/45  
% -------------------------------------- `3[W~Cq  
y = zeros(length_r,length(n)); Z[z" v  
for j = 1:length(n) G DBV  
    s = 0:(n(j)-m_abs(j))/2; A;ZluQ  
    pows = n(j):-2:m_abs(j); 0#yH<h$   
    for k = length(s):-1:1 *R4=4e2#S  
        p = (1-2*mod(s(k),2))* ... h8M}}   
                   prod(2:(n(j)-s(k)))/              ... Tp~Qg{%Og  
                   prod(2:s(k))/                     ... m#Z9wf] F  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... po]<sB  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); *pS3xit~  
        idx = (pows(k)==rpowers); Ls|)SiXrY  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ~9!@BL\  
    end %|/\Qu  
     j";L{  
    if isnorm ^Bw"+6d  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); H_Hr=_8}-  
    end _8`S&[E?  
end 60|m3|0o  
% END: Compute the Zernike Polynomials rwwyYIlEg  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6eB~S)Ko  
A #pH$s  
 b4KNIP7E  
% Compute the Zernike functions: CIwI1VR^  
% ------------------------------ %ID48_>*  
idx_pos = m>0; I L&PN`#  
idx_neg = m<0; E'+z.~+  
)WEOqaR]  
 -DZ5nx  
z = y; VL\Ah3+  
if any(idx_pos) 6]!Jo)BF  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');  HsG3s?*  
end 0"sZP\<p  
if any(idx_neg) ,*L3  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); iy8Ln,4z(  
end j6*e^ B  
&u2m6 r>W  
 ]REF1<)4z  
% EOF zernfun ~-yq,x  
phoenixzqy 2012-04-23 20:38
慢慢研究，这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
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
jssylttc 2012-05-14 11:28
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)  s<LYSrd  
N/B-u)?\:  
DDE还是手动输入的呢？ <)oW  
u-wj\BU  
zygo和zemax的zernike系数，类型对应好就没问题了吧
jssylttc 2012-05-14 11:37
顶顶·········
18257342135 2016-12-13 10:03
支持一下，慢慢研究
查看本帖完整版本: [-- 如何从zernike矩中提取出zernike系数啊 --] [-- top --]

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