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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， H^ds<I<)  
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， CWdpF>En  
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？ nQ+$  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ u!CcTE*  
2p %j@O  
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k^@dDLr"  
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function z = zernfun(n,m,r,theta,nflag) #=uV, dw  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. "UYlC0 S\  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TTagZI$  
%   and angular frequency M, evaluated at positions (R,THETA) on the &v)/mc7D  
%   unit circle.  N is a vector of positive integers (including 0), and .+) AeGh  
%   M is a vector with the same number of elements as N.  Each element qx5jaa3  
%   k of M must be a positive integer, with possible values M(k) = -N(k) gT_tR_g  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, */'j[uj  
%   and THETA is a vector of angles.  R and THETA must have the same N(J'h$E  
%   length.  The output Z is a matrix with one column for every (N,M) W:VX^8</  
%   pair, and one row for every (R,THETA) pair. !&adO,jN+=  
% Nr"gj$v  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {F=`IE3)w  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), VX:Kq<XwQ  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral sa?s[  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @rP#ktz]  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized j4wsDtmAU  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k 1l K`p  
% qm/#kPlM  
%   The Zernike functions are an orthogonal basis on the unit circle. Nt,:`o |  
%   They are used in disciplines such as astronomy, optics, and PO nF_FC  
%   optometry to describe functions on a circular domain. .4J7 ^l  
% ^U9b)KA  
%   The following table lists the first 15 Zernike functions. ;$vVYC  
% 3!op'X!  
%       n    m    Zernike function           Normalization %RX!Pi}5+g  
%       -------------------------------------------------- OUhlQq\  
%       0    0    1                                 1 GYrUB59  
%       1    1    r * cos(theta)                    2 Q k2*=BVh  
%       1   -1    r * sin(theta)                    2 d(YAH@  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) p`Ok(C_  
%       2    0    (2*r^2 - 1)                    sqrt(3) ;TKsAU  
%       2    2    r^2 * sin(2*theta)             sqrt(6) GdM|?u&s"  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) Gs/G_E(T  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 5mX"0a_Q  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) |RH^|2:x9Q  
%       3    3    r^3 * sin(3*theta)             sqrt(8) q{}U5(,{0  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) W0S\g#  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ip0`R+8  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) O[8wF86R  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =d$m@rc0r  
%       4    4    r^4 * sin(4*theta)             sqrt(10) z s\N)LyM  
%       -------------------------------------------------- 8&:dzS  
% !jR 1!i   
%   Example 1: <9[>+X  
% 62o nMY  
%       % Display the Zernike function Z(n=5,m=1)  GPrq(  
%       x = -1:0.01:1; =%S*h)}@  
%       [X,Y] = meshgrid(x,x); PKZMuEEy,  
%       [theta,r] = cart2pol(X,Y); 8CUl |I ~  
%       idx = r<=1; fskc'%x  
%       z = nan(size(X)); 1QbD]"=n  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2]5ux!Lqln  
%       figure 3 jghV?I{T  
%       pcolor(x,x,z), shading interp LYuMR,7E  
%       axis square, colorbar CN6b 982&  
%       title('Zernike function Z_5^1(r,\theta)') &r!jjT  
% ?s]?2>p  
%   Example 2: nFjaV`6`@  
% :m0 pm@  
%       % Display the first 10 Zernike functions brdY97s4  
%       x = -1:0.01:1; ${tBu#$-d  
%       [X,Y] = meshgrid(x,x); {tuGkRY2 ~  
%       [theta,r] = cart2pol(X,Y); 7-}/{o*,5  
%       idx = r<=1; ,Qt2?  
%       z = nan(size(X)); }0 Fu  
%       n = [0  1  1  2  2  2  3  3  3  3]; X CHN'l'  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; nc?Oj B  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; #Wt1Ph_;  
%       y = zernfun(n,m,r(idx),theta(idx)); k^%F4d3z@C  
%       figure('Units','normalized') H284 ]i  
%       for k = 1:10 qdZo cTf'  
%           z(idx) = y(:,k); Sr-!-eC  
%           subplot(4,7,Nplot(k)) |iVw7M:  
%           pcolor(x,x,z), shading interp V0*9Tnc  
%           set(gca,'XTick',[],'YTick',[]) `8D'r|=`Eh  
%           axis square H6Kt^s<6xu  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .c@,$z2M  
%       end :&m0eZZ%  
% npcL<$<6X  
%   See also ZERNPOL, ZERNFUN2. O*1la/~m  
9j1 tcT  
 WE&"W$0  
%   Paul Fricker 11/13/2006 y<)q;fI7  
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 +?Y(6$o  
b@[\+P] "  
 '.zr:l  
% Check and prepare the inputs: Gx-tPW}  
% ----------------------------- ;CA7\&L>  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I z)~h>-F  
    error('zernfun:NMvectors','N and M must be vectors.') ys9MV%*  
end SA.,Q~_T7  
ANd#m9(x  
 HNV"'p;  
if length(n)~=length(m) k5)e7Lb(  
    error('zernfun:NMlength','N and M must be the same length.') C6c]M@6  
end MU~nvs;:  
y_Nn%(j  
 0^3@>> ^  
n = n(:); ?N@p~ *x  
m = m(:); 6n'XRfQp)&  
if any(mod(n-m,2)) fg8U* 7  
    error('zernfun:NMmultiplesof2', ... ~[ x}  
          'All N and M must differ by multiples of 2 (including 0).') xkkW?[&  
end \Zo xJ&  
i)+2? <]  
 O\zGN/!  
if any(m>n) ,7izrf8  
    error('zernfun:MlessthanN', ... {1]Of'x'  
          'Each M must be less than or equal to its corresponding N.') 6Er%td)f  
end VK?c='zg  
2bt2h.a  
 qGKQrb,K  
if any( r>1 | r<0 ) uM9RlI5  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') xZ(VvINL'  
end Fd@:*ER  
1|%C66f^  
 ]0)=0pc]E  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,a?$F1Z-  
    error('zernfun:RTHvector','R and THETA must be vectors.') R(F+Xg je  
end ""^.fh  
9oJ=:E~CP  
 *dm?,~f%<  
r = r(:); "6 fTZ<  
theta = theta(:); '}T6e1#JV  
length_r = length(r); ;&G8e* bM2  
if length_r~=length(theta) zq&,KZ  
    error('zernfun:RTHlength', ... 6z80Y*|eJ  
          'The number of R- and THETA-values must be equal.') p*Hbc|?{Q&  
end Z CS{D  
5x; y{qT  
 x?MSHOia`P  
% Check normalization: ckPI^0A!  
% -------------------- _<1uO=km6  
if nargin==5 && ischar(nflag) {9C+=v?  
    isnorm = strcmpi(nflag,'norm'); ['rqz1DL5  
    if ~isnorm =e$6o2!'}  
        error('zernfun:normalization','Unrecognized normalization flag.') fdRw:K8  
    end F,-S&d  
else ghd*EXrF H  
    isnorm = false; &r Lg/UEV-  
end |KxFi H  
B!cg)Y?.bd  
 uM<6][^`  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -O-qEQd  
% Compute the Zernike Polynomials O[[#\BL  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yPqZ ,  
0cm+:  
 px1{=~V/  
% Determine the required powers of r: ;/8oP ;X2  
% ----------------------------------- r&t)%R@q  
m_abs = abs(m); <H)I06];  
rpowers = []; #}rv)  
for j = 1:length(n) j7)Xm,wI8  
    rpowers = [rpowers m_abs(j):2:n(j)]; |Skk1 #  
end a}+7MEUmZ/  
rpowers = unique(rpowers); UldG0+1d  
&[hq !v  
 PDnwaK   
% Pre-compute the values of r raised to the required powers, }#/,nJm'  
% and compile them in a matrix: 1MCHwX3/  
% ----------------------------- P]@m0f  
if rpowers(1)==0 'e4  ;,m  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \e/'d~F  
    rpowern = cat(2,rpowern{:}); \=yx~c_$L  
    rpowern = [ones(length_r,1) rpowern]; %:eep G|  
else 5Q.bwl:  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4#z@B1Jx  
    rpowern = cat(2,rpowern{:}); :>.~"uWo{  
end Et=N`k _gO  
qxsK-8KT<  
 F-k3F80=  
% Compute the values of the polynomials: .1.Bf26}d  
% -------------------------------------- _tg&_P+kV  
y = zeros(length_r,length(n)); ?[\(i)]  
for j = 1:length(n) -VESe}c:nQ  
    s = 0:(n(j)-m_abs(j))/2; }7Si2S  
    pows = n(j):-2:m_abs(j); Cr,UP8MO  
    for k = length(s):-1:1 '[fo  
        p = (1-2*mod(s(k),2))* ... | 1Fy  
                   prod(2:(n(j)-s(k)))/              ... dE+xU(\, w  
                   prod(2:s(k))/                     ... byYdX'd.  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... tVZj tGz=  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); @L8('8~d  
        idx = (pows(k)==rpowers); 9tVA.:FOZ  
        y(:,j) = y(:,j) + p*rpowern(:,idx); N4%q-fi  
    end 4425,AR  
     g(\FG  
    if isnorm Nkt(1?:-'  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ch`XwLY9  
    end d* Y&V$?zl  
end 'Pudy\Ab  
% END: Compute the Zernike Polynomials Nw}y_Qf{  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dlC)&Ai  
;$G.?r  
 )k0P' zGb  
% Compute the Zernike functions: ;f7(d\=y  
% ------------------------------ H'E >QT  
idx_pos = m>0; > _1*/o JO  
idx_neg = m<0; <h2WM (n  
0+0+%#?  
 bIP{DxKS  
z = y; #]i*u1  
if any(idx_pos) :luVsQ  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #N%xr'H  
end $ +h~VC  
if any(idx_neg) k~u$&a  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #J]u3*T n|  
end lkK+Fm  
BF2,E<^A  
 K vC`6  
% EOF zernfun udDhJ?  
phoenixzqy 2012-04-23 20:38
慢慢研究，这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊 (2012-04-27 10:22)  .`& ($W  
Cfu]umZLn  
DDE还是手动输入的呢？ mL`,v WL/`  
`:!mPNW#  
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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