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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， |)72E[lL  
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， DS,FVh".|  
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？ 8#d1}Y  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ C^\*|=*\  
mC[U)` ey  
 !7"K>m<  
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function z = zernfun(n,m,r,theta,nflag) mB|mt+  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.   L@k;L  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N eV {FcJha  
%   and angular frequency M, evaluated at positions (R,THETA) on the ~&j`9jdOj  
%   unit circle.  N is a vector of positive integers (including 0), and ;KZtW  
%   M is a vector with the same number of elements as N.  Each element pJ*#aH[ySP  
%   k of M must be a positive integer, with possible values M(k) = -N(k) S'-`\%@7  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 1J{z}yPHc  
%   and THETA is a vector of angles.  R and THETA must have the same a+>W  
%   length.  The output Z is a matrix with one column for every (N,M) j~L1~@  
%   pair, and one row for every (R,THETA) pair. Ignv|TYG  
% $qUta< o2@  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D`~{[cv)\  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |n6 Q  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral (P>eWw\0  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3!oQmG_T  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized :rs\ydDUF  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <% 3SI.  
%  n wZr3r  
%   The Zernike functions are an orthogonal basis on the unit circle. | 8L`osg  
%   They are used in disciplines such as astronomy, optics, and sc $QbOc  
%   optometry to describe functions on a circular domain. +S5_J&~  
% #L IsL  
%   The following table lists the first 15 Zernike functions. @<TfA>*VJ  
% Z/05 wB  
%       n    m    Zernike function           Normalization 2eR+dT  
%       -------------------------------------------------- yDkDtO`K  
%       0    0    1                                 1 ^B!?;\4IM  
%       1    1    r * cos(theta)                    2 lKhh=Pc2  
%       1   -1    r * sin(theta)                    2 > v!c\  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) !E:Vn *k;  
%       2    0    (2*r^2 - 1)                    sqrt(3) {|J2clL  
%       2    2    r^2 * sin(2*theta)             sqrt(6) .iN*V|n  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) JTh =JHJ  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Nj-rZ%&  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) b;{"lJ:+Z  
%       3    3    r^3 * sin(3*theta)             sqrt(8) eZod}~J8  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) *JC{G^|Y  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) v4>"p!_C  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \;:@=9`  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HFx"fT  
%       4    4    r^4 * sin(4*theta)             sqrt(10) //u76nQ  
%       -------------------------------------------------- ^RyTK|SQ  
% rD U6 5j  
%   Example 1: (F '  
% w$;*~Qc  
%       % Display the Zernike function Z(n=5,m=1) Y7V&zF{  
%       x = -1:0.01:1; +ZA\ M:^b  
%       [X,Y] = meshgrid(x,x); BvW gH.OX  
%       [theta,r] = cart2pol(X,Y); l 75{JxZX  
%       idx = r<=1; y3[)zv  
%       z = nan(size(X)); 7C?mD75j  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); :+^$?[6]  
%       figure Cbg#Yz~/  
%       pcolor(x,x,z), shading interp 5m7Ax] \  
%       axis square, colorbar {i}Q}OgYq  
%       title('Zernike function Z_5^1(r,\theta)') G1^!ej  
% L8tLW09  
%   Example 2: <d&)|W  
% |21hY  
%       % Display the first 10 Zernike functions UG'U D"  
%       x = -1:0.01:1; H'\EA(v+  
%       [X,Y] = meshgrid(x,x); LP-Q'vb<=  
%       [theta,r] = cart2pol(X,Y); yW(+?7U  
%       idx = r<=1; zomNjy*  
%       z = nan(size(X)); X|1YGZJ  
%       n = [0  1  1  2  2  2  3  3  3  3]; @d^h/w  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; )9jQ_  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; /P"\ +Qp  
%       y = zernfun(n,m,r(idx),theta(idx)); <m:wuNEM  
%       figure('Units','normalized') h}&IlDG  
%       for k = 1:10 FY S83uq0  
%           z(idx) = y(:,k); OLup`~  
%           subplot(4,7,Nplot(k)) 6:tr8 X_  
%           pcolor(x,x,z), shading interp vl~   
%           set(gca,'XTick',[],'YTick',[]) ;L%~c4`l~m  
%           axis square F-,{+B66  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dTQvz9C  
%       end be%*0lr  
% *`.{K12T  
%   See also ZERNPOL, ZERNFUN2. B]F7t4Y!  
l4reG:uYG  
 R./6Q1  
%   Paul Fricker 11/13/2006 h:sG23@=  
kD7(}N8YR  
 iQ"F`C  
`#8R+c=$  
 k+M-D~@5H  
% Check and prepare the inputs: m e{SVG{  
% ----------------------------- O9)}:++T  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) '\\Cpc_g  
    error('zernfun:NMvectors','N and M must be vectors.') BQ0\+  
end ah9',((!  
%\&dFwb  
 xumv I{  
if length(n)~=length(m) *v/*_6f*  
    error('zernfun:NMlength','N and M must be the same length.') VVl-cU  
end q#3X*!)  
1^^D :tt  
 S]=Vr%irX  
n = n(:); M)^9e?  
m = m(:); 1u+ (rVQN  
if any(mod(n-m,2)) H5 hUY'O  
    error('zernfun:NMmultiplesof2', ... Yb{t!KL  
          'All N and M must differ by multiples of 2 (including 0).') Hvo27THLo  
end z5vI0 N$  
_u2  
 $Xc<K_Z  
if any(m>n) ye2Oh7  
    error('zernfun:MlessthanN', ... y<d#sv(s  
          'Each M must be less than or equal to its corresponding N.') w/6@R 4)p  
end 'FFc"lqj  
(SA*9%  
 htym4\Z=  
if any( r>1 | r<0 ) *=@pdQkR  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') lXKZNCL  
end _/ZY&5N  
 U~t(YT  
 kq0m^`  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :%MWbnVSC,  
    error('zernfun:RTHvector','R and THETA must be vectors.') b.;}Hq>  
end qG]PUc>j  
_I4sy=tYXK  
 B{ "<\g  
r = r(:); Ngnjr7Q={T  
theta = theta(:); =LnAMl#9  
length_r = length(r); )aSkUytg"  
if length_r~=length(theta) ayp}TYh*  
    error('zernfun:RTHlength', ... \]%U?`A  
          'The number of R- and THETA-values must be equal.') 3/FB>w gt  
end e*H$c?7NL  
0{F.DDiNT  
 e r$'c  
% Check normalization: */E{s?  
% -------------------- Lai"D[N  
if nargin==5 && ischar(nflag) --kK<9J7  
    isnorm = strcmpi(nflag,'norm'); ^&HYnwk  
    if ~isnorm I r~X#$Upc  
        error('zernfun:normalization','Unrecognized normalization flag.') KL4/"$l]  
    end sXu+F2O  
else W$S.?[X  
    isnorm = false; X2v'9 x  
end 8F1!9W7  
mM.&c5U  
 =w-H )  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F}>`3//u  
% Compute the Zernike Polynomials (xL=X%6a  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |=s3a5sl  
$ cSZX#\  
 J~.kb k  
% Determine the required powers of r: Ji q[VeLe  
% ----------------------------------- %R"Fx$tQ  
m_abs = abs(m); ez{&Y>n  
rpowers = []; Lt_]3g o  
for j = 1:length(n) y e'5 A   
    rpowers = [rpowers m_abs(j):2:n(j)]; }R$%MU5::  
end ;rgsPVbVf  
rpowers = unique(rpowers); YP l{5 =  
gp=0;#4 4  
 o@. !Z8  
% Pre-compute the values of r raised to the required powers, X;h~s:LM  
% and compile them in a matrix: '! (`?  
% ----------------------------- 1~ Nz6  
if rpowers(1)==0 "Q1hP9xV  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); z/b*]"g,  
    rpowern = cat(2,rpowern{:}); =xoTH3/,>  
    rpowern = [ones(length_r,1) rpowern]; 14RL++  
else -eTGRr  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rtm28|0H'  
    rpowern = cat(2,rpowern{:}); 16vfIUtb  
end GcuZPIN%D  
Lrq&k40y  
 AI2CfH#:C  
% Compute the values of the polynomials: 71_N9ub@z  
% -------------------------------------- 0W> ",2|z  
y = zeros(length_r,length(n)); X}$S|1CjO  
for j = 1:length(n) F <(Y  
    s = 0:(n(j)-m_abs(j))/2; 6F2}|c  
    pows = n(j):-2:m_abs(j); ( C&f~U  
    for k = length(s):-1:1 ,P^"X5$   
        p = (1-2*mod(s(k),2))* ... !?{5ET,gtN  
                   prod(2:(n(j)-s(k)))/              ... _EP}el  
                   prod(2:s(k))/                     ... zw?6E8$h  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +Ji dP  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); bGZy0.  
        idx = (pows(k)==rpowers); NQmDm!-4  
        y(:,j) = y(:,j) + p*rpowern(:,idx); /1*\*<cs  
    end F~EriO  
     dSbV{*B;>  
    if isnorm Mtu8zm  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qcqf9g  
    end A.n1|Q#  
end ;I>`!|mT  
% END: Compute the Zernike Polynomials f4q-wX_1  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 945psG@|  
b> | oU  
 9 wc=B(a|  
% Compute the Zernike functions: "LYob}_z  
% ------------------------------ XZhhr1-<a  
idx_pos = m>0; ; ?!sU  
idx_neg = m<0; \2YhI0skW  
<T['J]k%  
 ;Bm{_$hf=  
z = y; Rzp-Q5@M Y  
if any(idx_pos) m9/a!|fBE  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); q_!3<.sf  
end F XbNmBXF  
if any(idx_neg) sB $!X@  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); fI6F};I5}T  
end @I%m}>4Jm  
DGcd|>q  
 }+z}vb  
% EOF zernfun q;rU}hAzG0  
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)  vuHqOAFNs  
,2"-G";!f\  
DDE还是手动输入的呢？ lOql(ZH`w  
QHK$2xtq|  
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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