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， n5tsaU;  
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， ^E+fmY2a  
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？ Q `-Xx  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ h1f 05  
i]qxF&1  
  ]4K4Nh~  
sx'eu;S  
 3h A5"G+7  
function z = zernfun(n,m,r,theta,nflag) 3`NSSS  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Ya!PV&"Z  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Re*~C:  
%   and angular frequency M, evaluated at positions (R,THETA) on the ]bstkf}~u  
%   unit circle.  N is a vector of positive integers (including 0), and K\mFb  
%   M is a vector with the same number of elements as N.  Each element LIyb+rH#yg  
%   k of M must be a positive integer, with possible values M(k) = -N(k) $IdU  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, =gZA9@]W2  
%   and THETA is a vector of angles.  R and THETA must have the same ,TYFPulYcp  
%   length.  The output Z is a matrix with one column for every (N,M) BD]o+96qP  
%   pair, and one row for every (R,THETA) pair. K@:t6  
% )/2TU]//  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~I~lb/  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -uhVw_qq#  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral OO,EUOh-T:  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =hI;5KF  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized $)6M@S  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \pP1k.~UnC  
% Qx_N,1>S  
%   The Zernike functions are an orthogonal basis on the unit circle. pA_e{P/  
%   They are used in disciplines such as astronomy, optics, and z&jASL  
%   optometry to describe functions on a circular domain. O&VA79\UO  
% 8CH9&N5W5t  
%   The following table lists the first 15 Zernike functions. ' hdLQ\J  
% P=s3&NDD  
%       n    m    Zernike function           Normalization joNV4v"=`  
%       -------------------------------------------------- g?cxqC<  
%       0    0    1                                 1 DwWm(8&6;}  
%       1    1    r * cos(theta)                    2 K;>9K'n  
%       1   -1    r * sin(theta)                    2 Fw#1?/K~  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 0+.<BOcW5  
%       2    0    (2*r^2 - 1)                    sqrt(3) lB!M;2^)X  
%       2    2    r^2 * sin(2*theta)             sqrt(6) W[>qiYf^b  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) %:OX^ ^i;  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 5s>>] .%  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Rh)%;  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 9Q 7342  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) #wkSru&LS  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) LX=cx$K  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;3%Y@FS@  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vTL/% SJ8  
%       4    4    r^4 * sin(4*theta)             sqrt(10) u_FN'p=.  
%       -------------------------------------------------- r >sXvzv  
% #CW{y?=  
%   Example 1: $ % B  
% t%@u)bp  
%       % Display the Zernike function Z(n=5,m=1) Y)#,6\=U  
%       x = -1:0.01:1; !JVv`YN  
%       [X,Y] = meshgrid(x,x); 1{1mL-I;  
%       [theta,r] = cart2pol(X,Y); +I}!)$/  
%       idx = r<=1; `\/\C[Gg  
%       z = nan(size(X)); *nv ^s  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); p1T0FBV L  
%       figure 2+*o^`%4P  
%       pcolor(x,x,z), shading interp vN~joQ=d  
%       axis square, colorbar 43~v1pf{!  
%       title('Zernike function Z_5^1(r,\theta)') -M4VC^_  
% ~(=5`9  
%   Example 2: ID<[=es6  
% S,a:H*Hf  
%       % Display the first 10 Zernike functions EiyHZ  
%       x = -1:0.01:1; 90ov[|MkM  
%       [X,Y] = meshgrid(x,x); h${=gSJc  
%       [theta,r] = cart2pol(X,Y); (`R heEg@f  
%       idx = r<=1; -"XHN=H  
%       z = nan(size(X)); L=WKqRa>4  
%       n = [0  1  1  2  2  2  3  3  3  3]; fw a*|y;  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ofsLx6Po  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; sLSH`Xy?5  
%       y = zernfun(n,m,r(idx),theta(idx)); -MORd{GF  
%       figure('Units','normalized') /J(~NGT  
%       for k = 1:10 #vAqqAS`,  
%           z(idx) = y(:,k); =Q6JXp  
%           subplot(4,7,Nplot(k)) M-e|$'4u  
%           pcolor(x,x,z), shading interp 'aS: Azb  
%           set(gca,'XTick',[],'YTick',[]) 6_:KFqc W  
%           axis square _<l)4A3rS  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2(P<TP._E  
%       end ?=HoU3  
% Wq?vAnLbk  
%   See also ZERNPOL, ZERNFUN2. ,L-V?B(UQ  
zy|h1 .gd  
 B3I0H6O  
%   Paul Fricker 11/13/2006 O(:/ &`)  
oxJAI4{y 4  
 y(Em+YTD  
D8{f7{nY  
 !wZIXpeL  
% Check and prepare the inputs: X+ /^s)  
% ----------------------------- Pj5:=d8z(  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4E$d"D5]>p  
    error('zernfun:NMvectors','N and M must be vectors.') `1#Z9&bO  
end ']Z%6_WF  
7Jpq7;  
 " BU4\QF-  
if length(n)~=length(m) 162Dj$  
    error('zernfun:NMlength','N and M must be the same length.') R'oGsaPB2  
end `H9 !Z$7G  
>x ]{c b/m  
 sWi4+PAM0  
n = n(:); &4*f28 s  
m = m(:); 4.>y[_vu  
if any(mod(n-m,2)) i 1GQ=@  
    error('zernfun:NMmultiplesof2', ... #E#@6ZomT  
          'All N and M must differ by multiples of 2 (including 0).') f9O_M1=|lo  
end ^,J>=>,1\  
1k{H,p7  
 Fw^^sB  
if any(m>n) " ^!=e72  
    error('zernfun:MlessthanN', ... M7YbRl  
          'Each M must be less than or equal to its corresponding N.') G}aM~,v  
end Ml )<4@  
MFipXE!  
 hb>uHUb&  
if any( r>1 | r<0 ) c4bvJy8  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') V}JBv$+ko  
end ]1I-e2Q-J  
X9rao n  
 (R9"0WeF  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (aB:P03  
    error('zernfun:RTHvector','R and THETA must be vectors.') L:.Rv0XT  
end SjcX|=S  
 \7e4t  
 j_b/66JyN  
r = r(:); 4I.)>+8V  
theta = theta(:);  }s8xr>  
length_r = length(r); ('.I)n  
if length_r~=length(theta) C\0,D9  
    error('zernfun:RTHlength', ... jPg[LZQ'  
          'The number of R- and THETA-values must be equal.') $-\%%n0>6  
end |:`)sx3@#  
ciW;sK8  
 <Rz[G+0S=  
% Check normalization: X @7:FzU9  
% -------------------- nd5.Py$  
if nargin==5 && ischar(nflag) 6}*4co  
    isnorm = strcmpi(nflag,'norm'); CM t$ )  
    if ~isnorm 8A 'SMJi  
        error('zernfun:normalization','Unrecognized normalization flag.') U?Vik  
    end t{Ks}9B  
else SXV2Y-  
    isnorm = false; C\ 34R  
end )[=C@U  
eUD 5 V  
 qr~zTBT] E  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y"H(F,(N  
% Compute the Zernike Polynomials A>*#Nw5L  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q{ { =  
EV;"]lC9  
 d^"|ESQEU  
% Determine the required powers of r: 4UN|`'c  
% ----------------------------------- &C#?&AQ  
m_abs = abs(m); ~b[5}_L=>  
rpowers = []; kAW2vh  
for j = 1:length(n) Ze?H  
    rpowers = [rpowers m_abs(j):2:n(j)]; %xC}#RDf  
end zXe]P(p<  
rpowers = unique(rpowers); tNAmA  
`J;g~#/k  
 nr&9\lG]G  
% Pre-compute the values of r raised to the required powers, '1Ex{$Yk  
% and compile them in a matrix: 9q2x}  
% ----------------------------- /KlSI<T@  
if rpowers(1)==0 mw83pU6  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [9y y<Z5  
    rpowern = cat(2,rpowern{:}); M0]J `fL@  
    rpowern = [ones(length_r,1) rpowern]; mlz|KI~\F;  
else hF1Lj=x  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z0M|Bv9_  
    rpowern = cat(2,rpowern{:}); l?HC-_Pbh  
end r4]hcoU  
C`8.8  
 Lv#}Gm  
% Compute the values of the polynomials: d[p?B-7%  
% -------------------------------------- j% '~l#nw  
y = zeros(length_r,length(n)); JRDIGS_~  
for j = 1:length(n) $;)A:*e  
    s = 0:(n(j)-m_abs(j))/2; Zy>y7O(,  
    pows = n(j):-2:m_abs(j); ~hT(uxU/  
    for k = length(s):-1:1 x. /WP~I  
        p = (1-2*mod(s(k),2))* ... o\nFSG kn  
                   prod(2:(n(j)-s(k)))/              ... gi8f)MNP?~  
                   prod(2:s(k))/                     ... (?P\;yDG  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1Q$/L+uJ5  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 2HDWlUTNVO  
        idx = (pows(k)==rpowers); .gCun_td#  
        y(:,j) = y(:,j) + p*rpowern(:,idx); = @ 1{LF;  
    end t;t;+M|W  
     Iz!]LW  
    if isnorm |doG}C  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )t$-/8  
    end gbFHH,@  
end M4t:)!dji?  
% END: Compute the Zernike Polynomials '.7ER  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZD<e$PxxCd  
RZ?abE8  
 *tl;0<n  
% Compute the Zernike functions: t1!>EI`  
% ------------------------------ Wyb+K)Tg  
idx_pos = m>0; u_Xp\RJ  
idx_neg = m<0; w|-m*v .  
*~zB{  
 W O'nW  
z = y; 0 .ck!"h}  
if any(idx_pos) _I0=a@3  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R vd'uIJ  
end [u^~ND'  
if any(idx_neg) 8,)<,g-/=  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p,3}A( >  
end ~8)l/I=`);  
pzT`.#N:M  
 zuJ@@\75  
% EOF zernfun :bq UA(k  
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)  ~oRT@E  
wif1|!aL  
DDE还是手动输入的呢？ A%~t[ H  
5qkyi]/U8  
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