首页 -> 登录 -> 注册 -> 回复主题 -> 发表主题
光行天下 -> ZEMAX,OpticStudio -> 如何从zernike矩中提取出zernike系数啊 [点此返回论坛查看本帖完整版本] [打印本页]

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， Xn:5pd;?B6  
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， *=G~26*!V  
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？ ,_SE!iL  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ ?a)X)#lQ  
9gWR djK:  
 S  ^5EG;[  
<HYK9{Q  
 Oe#*-  
function z = zernfun(n,m,r,theta,nflag) >.sdLA Si  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z]L_{=*  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z\1*g k  
%   and angular frequency M, evaluated at positions (R,THETA) on the cXcrb4IKD  
%   unit circle.  N is a vector of positive integers (including 0), and R\i8O^[  
%   M is a vector with the same number of elements as N.  Each element o<1a]M|  
%   k of M must be a positive integer, with possible values M(k) = -N(k) %P3|#0yg0  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9^yf'9S1  
%   and THETA is a vector of angles.  R and THETA must have the same 0cGO*G2Xr  
%   length.  The output Z is a matrix with one column for every (N,M) Z}X oWT2f  
%   pair, and one row for every (R,THETA) pair. <[*%d~92z  
% f&=WgITa  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Kivr)cIG  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NY(z 3G  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral *s=jKV#  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, G`;YB  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3bWYRW  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -'!K("  
% +ConK>;  
%   The Zernike functions are an orthogonal basis on the unit circle. a9f!f %9  
%   They are used in disciplines such as astronomy, optics, and MC'2;,  
%   optometry to describe functions on a circular domain. (ncm]W  
% Q4H(JD1f)  
%   The following table lists the first 15 Zernike functions. Xl/ SDm_p  
% vHydqFi9  
%       n    m    Zernike function           Normalization [ClDKswq  
%       -------------------------------------------------- lwVo%-  
%       0    0    1                                 1 XJ$mRh0`K  
%       1    1    r * cos(theta)                    2 hXAgT!ZD  
%       1   -1    r * sin(theta)                    2 `/e EdqT  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) sY- ] Q  
%       2    0    (2*r^2 - 1)                    sqrt(3) >$/<~j]  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 5YV3pFz$)  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) Bd++G'FZ  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) sxK|0i}6  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) og?>Q i Tr  
%       3    3    r^3 * sin(3*theta)             sqrt(8) l* ap$1'  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) a1 Kh  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :cE6-Fv  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Y^Y1re+}  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }EMds3<  
%       4    4    r^4 * sin(4*theta)             sqrt(10) xj!G9x<!  
%       -------------------------------------------------- ELvP<Ny}  
% qH=<8Iu  
%   Example 1: &s{" Vc9]  
%  /N8>>g  
%       % Display the Zernike function Z(n=5,m=1) #$dEg  
%       x = -1:0.01:1; Lu][0+-  
%       [X,Y] = meshgrid(x,x); w7d<Ky_C  
%       [theta,r] = cart2pol(X,Y); uHQf<R$:  
%       idx = r<=1; $bCN;yE  
%       z = nan(size(X)); h>a/3a$g  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); xwJ. cy  
%       figure G u4mP  
%       pcolor(x,x,z), shading interp Sb|9U8h  
%       axis square, colorbar au;ZAXM|  
%       title('Zernike function Z_5^1(r,\theta)') Ovhd%qV;Y  
% ^o8o  
%   Example 2: sX c|++  
% K2o\+t  
%       % Display the first 10 Zernike functions 6rll0c~  
%       x = -1:0.01:1; lP;X=X>  
%       [X,Y] = meshgrid(x,x); n5U-D0/Q  
%       [theta,r] = cart2pol(X,Y); [-p?gyl  
%       idx = r<=1; >D5WAQ>b  
%       z = nan(size(X)); \' Z^rjB  
%       n = [0  1  1  2  2  2  3  3  3  3]; !uc"|S?  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2FxrjA  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; DX b=Ku  
%       y = zernfun(n,m,r(idx),theta(idx)); L5RBe  
%       figure('Units','normalized') "q]r{0  
%       for k = 1:10 =U`9_]~1c@  
%           z(idx) = y(:,k); (Do](C  
%           subplot(4,7,Nplot(k)) ls,;ozU  
%           pcolor(x,x,z), shading interp y?-zQs0  
%           set(gca,'XTick',[],'YTick',[]) 3*C|"|lJ  
%           axis square LD gGVl  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ) OE!vA  
%       end *p.70,5,  
% *>,#'C2  
%   See also ZERNPOL, ZERNFUN2. Z[GeU>?P  
HxnWM\p  
 .Gcs/PN   
%   Paul Fricker 11/13/2006 ',l}$]y5  
&57s//PrX  
 g5[r!XO  
+\s&v!  
 qZB}}pM#  
% Check and prepare the inputs: ><DXT nt'x  
% ----------------------------- tg"NWp6  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZQN%!2  
    error('zernfun:NMvectors','N and M must be vectors.') P/Zp3O H  
end py%_XL=w,  
m IYM+2p  
 %|o2d&i  
if length(n)~=length(m) =2&Sw(6j  
    error('zernfun:NMlength','N and M must be the same length.') 5`A^"}0  
end 8h$f6JE  
e .(  
 )Q5ja}-{V  
n = n(:); kNC]q,ljt5  
m = m(:); PCX X[N  
if any(mod(n-m,2)) oeA}b-Ct0  
    error('zernfun:NMmultiplesof2', ... X775j"<d  
          'All N and M must differ by multiples of 2 (including 0).') v[!ZRwk4w3  
end A[8vD</}_  
3aERfIJyE  
 SadffAvSA{  
if any(m>n) .?dYY;P  
    error('zernfun:MlessthanN', ... 3v,Bg4[i  
          'Each M must be less than or equal to its corresponding N.') ?>T (  
end 8I)}c1j`v  
)o N#%%SB<  
 ]MaD7q>+R  
if any( r>1 | r<0 ) (>M@Ukam:  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') "3e1 7dsY  
end 4H@K?b`  
~7O.}RP0  
 m5kt O^EU  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K['Gp>l  
    error('zernfun:RTHvector','R and THETA must be vectors.') ;Sw % t(@  
end cM;,nX%/  
5o2vj8::  
 i$"M'BG  
r = r(:); 4Tn97G7  
theta = theta(:); HE.YfD)  
length_r = length(r); yd~}CF  
if length_r~=length(theta) XFJz\'{  
    error('zernfun:RTHlength', ... 6[S-%|f  
          'The number of R- and THETA-values must be equal.') ;(0|2I'"  
end tJ9-8ZT*  
\\v1 \  
 =DLVWz/<  
% Check normalization: y25L`b  
% -------------------- ~ "stI   
if nargin==5 && ischar(nflag) p$!Q?&AV/  
    isnorm = strcmpi(nflag,'norm'); 8%#pv}  
    if ~isnorm g$9EI\a  
        error('zernfun:normalization','Unrecognized normalization flag.') c]>LL(R-7)  
    end /4 %ycr6  
else 6< @F  
    isnorm = false; w+G+&ak<  
end s%Irh;Bs  
L/7YI\C2  
 lm\~_ 4l1  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %}1v-z  
% Compute the Zernike Polynomials ?r/)s()ALf  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]^BgSC  
@; 0t+  
 VB&`g<  
% Determine the required powers of r: x) OJ?l  
% ----------------------------------- 7>!Rg~M  
m_abs = abs(m); -E,p[Sp  
rpowers = [];  \>l DM  
for j = 1:length(n) g#s hd~e  
    rpowers = [rpowers m_abs(j):2:n(j)]; MGfIA?u  
end z!>ml3  
rpowers = unique(rpowers); wgFAPZr  
#-9@*FFL,  
 0.lOSAq  
% Pre-compute the values of r raised to the required powers, U?&&yynK  
% and compile them in a matrix: .V.ga2+  
% ----------------------------- CaqqH`/E4  
if rpowers(1)==0 i2 7KuPjC  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); XI7:y4M  
    rpowern = cat(2,rpowern{:}); {~{</ g/  
    rpowern = [ones(length_r,1) rpowern]; _t.Ub:  
else " 'TEBkj|u  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;'P<#hM[$  
    rpowern = cat(2,rpowern{:}); cd:VFjT  
end Vk?US&1q}  
o7 1f<&1  
 E-"b":@:  
% Compute the values of the polynomials: B~7]x;8h  
% -------------------------------------- /6yH ,{(a  
y = zeros(length_r,length(n)); Q5>]f/LD  
for j = 1:length(n) 7kq6VS;p  
    s = 0:(n(j)-m_abs(j))/2; SJ).L.Cm6  
    pows = n(j):-2:m_abs(j); ,;~@t:!c  
    for k = length(s):-1:1 t^SND{[WcM  
        p = (1-2*mod(s(k),2))* ... `VD7VX,rp*  
                   prod(2:(n(j)-s(k)))/              ... *28:|blbL  
                   prod(2:s(k))/                     ... ]o2 Z 14  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... CN!~(1v  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); WN3]xw3  
        idx = (pows(k)==rpowers); 6'6,ySo]  
        y(:,j) = y(:,j) + p*rpowern(:,idx); BZ}_  
    end .y^T 3?}I  
     h[}e5A]}  
    if isnorm l$J2|\M6  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); YJMaIFt  
    end -,^Z5N#\|  
end ^Kfm(E  
% END: Compute the Zernike Polynomials Dhzm C  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i f!   
#Pe|}!)u  
 QrRnXlE M8  
% Compute the Zernike functions: =}m'qy  
% ------------------------------ ve@E.`  
idx_pos = m>0; `&5_~4T7  
idx_neg = m<0; C[Nh>V7=  
bPUldkB:  
 {^_K  
z = y; Be~In~~  
if any(idx_pos) RAAu3QKu  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); N`rz>6,k1  
end `i{p6-U3  
if any(idx_neg) &9Xn:<"`)  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $[{YE[a  
end iNv"!'|  
CjZIBMGc  
 WIYWql>*  
% EOF zernfun lCiRvh1K  
phoenixzqy 2012-04-23 20:38
慢慢研究，这个专业性很强的。用的人又少。
sansummer 2012-04-27 10:22
这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊
jssylttc 2012-05-14 11:28
sansummer:这个太牛了，我目前只能把zygo中的zernike的36项参数带入到zemax中，但是我目前对其结果的可信性表示质疑，以后多交流啊 (2012-04-27 10:22)  H/$oGhvl  
ASUL g{  
DDE还是手动输入的呢？ ?9Sc KN  
)}i|)^J  
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 --]

Copyright © 2005-2025 光行天下 蜀ICP备06003254号-1 网站统计