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， J"z8olV  
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， a([cuh.  
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？ N#UyAm<9  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ hxv/285B  
ul:jn]S*  
 ;Z8K3p  
r..Rh9v/=E  
 -Z#A}h  
function z = zernfun(n,m,r,theta,nflag) 3cs'Oz<w  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. , n+dB2\  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N sqkPC_;A  
%   and angular frequency M, evaluated at positions (R,THETA) on the OW6i2>Or  
%   unit circle.  N is a vector of positive integers (including 0), and {?t=*l\S{w  
%   M is a vector with the same number of elements as N.  Each element 5` Q#2  
%   k of M must be a positive integer, with possible values M(k) = -N(k) t<Og ?m}(  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Q!@" Y/  
%   and THETA is a vector of angles.  R and THETA must have the same Iz  ,C!c  
%   length.  The output Z is a matrix with one column for every (N,M) qEywExdiu  
%   pair, and one row for every (R,THETA) pair. Wx^L~[l  
% -ERDWY  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike tW 9vo-{+  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), UGAP$_j ]P  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral x=9drKIw>  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -()CgtSR  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized RCsd  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. a*o=,!  
% QupCr/Hs  
%   The Zernike functions are an orthogonal basis on the unit circle. 7q(RQQp  
%   They are used in disciplines such as astronomy, optics, and |J8c|h<  
%   optometry to describe functions on a circular domain. &6!x;RB  
% tNq~M  
%   The following table lists the first 15 Zernike functions. \`x$@s?  
% rFGbp8(2  
%       n    m    Zernike function           Normalization teET nz_L  
%       -------------------------------------------------- Es)Kw3^a  
%       0    0    1                                 1 @ UX'(W  
%       1    1    r * cos(theta)                    2 Yz[^?M%(D  
%       1   -1    r * sin(theta)                    2 yV_aza  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) -cOLg rmp  
%       2    0    (2*r^2 - 1)                    sqrt(3) N5o jXX!l%  
%       2    2    r^2 * sin(2*theta)             sqrt(6) f BukrPsV  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) !hEt UF  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 6=iz@C7r  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =r#of|`Q  
%       3    3    r^3 * sin(3*theta)             sqrt(8) "*<9)vQ6|  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 8I`>tY  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hh,q)(Wo  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s )\%%CM  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  \&"gCv#  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 4OC ^IS  
%       -------------------------------------------------- `cCsJm$V"  
% [>v.#:YM^  
%   Example 1: vDqmD{%4N  
% O*X ]oX  
%       % Display the Zernike function Z(n=5,m=1) AYfW}V"  
%       x = -1:0.01:1; ,d$V-~2,  
%       [X,Y] = meshgrid(x,x); ~CQsv `  
%       [theta,r] = cart2pol(X,Y); +CaPF  
%       idx = r<=1; 7gNJ}pLDx  
%       z = nan(size(X)); VPet1hAy  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); n~@;[=o?5  
%       figure t*iKkV^aE  
%       pcolor(x,x,z), shading interp MQ7N8@!t  
%       axis square, colorbar 9-5H~<}fF  
%       title('Zernike function Z_5^1(r,\theta)') lmf vT}$B  
% w9G (^jS6  
%   Example 2: jEo)#j];`<  
% !g'kWE[  
%       % Display the first 10 Zernike functions //xK v{3fI  
%       x = -1:0.01:1; VG_ PBG(  
%       [X,Y] = meshgrid(x,x); uD4on}  
%       [theta,r] = cart2pol(X,Y); *2P%731n5  
%       idx = r<=1; eVGO6 2|!  
%       z = nan(size(X)); )[oegfnn-  
%       n = [0  1  1  2  2  2  3  3  3  3]; \!x~FVA  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $SQ UN*/>  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 2<q>]G-nN  
%       y = zernfun(n,m,r(idx),theta(idx)); clV3x` z  
%       figure('Units','normalized') H`d595<=i;  
%       for k = 1:10 ^ ~'&K e  
%           z(idx) = y(:,k); ;SnpD)x@)  
%           subplot(4,7,Nplot(k)) oR}cE Sr  
%           pcolor(x,x,z), shading interp B]iPixA6  
%           set(gca,'XTick',[],'YTick',[]) 6Cfu19Dx  
%           axis square I&vD >a5#  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) z@Pv~"  
%       end M#Kke9%2  
% lJb1{\|.,  
%   See also ZERNPOL, ZERNFUN2. |Tv}leJF  
'guXdX]Gu  
 uGt}Hn  
%   Paul Fricker 11/13/2006 t/%{R.1MN  
2@~.FBby7@  
 4} .PQ{  
/<C}v~r  
 wIQ~a  
% Check and prepare the inputs: =>3wI'I  
% ----------------------------- ( f]@lNmx  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) E.LD1Pm0  
    error('zernfun:NMvectors','N and M must be vectors.') WVZ](D8Gc]  
end ~?#>QN\\c  
H?oBax:  
 O2{~Q{p  
if length(n)~=length(m) L)(JaZyV5  
    error('zernfun:NMlength','N and M must be the same length.') xbqFek$/r  
end /{Mo'.=Z  
27J!oin$  
 HKp|I%b]J  
n = n(:); `)y<X#[8  
m = m(:); nwS @r  
if any(mod(n-m,2)) `)~]3zmG  
    error('zernfun:NMmultiplesof2', ... }Wlm#t  
          'All N and M must differ by multiples of 2 (including 0).') QQI,$HId  
end \3"jW1Wb  
'Dn\.x^]1  
 WwUhwY1o!L  
if any(m>n) 0Wkk$0h9  
    error('zernfun:MlessthanN', ... FP=up#zl  
          'Each M must be less than or equal to its corresponding N.') %plu]^Vy  
end \<ko)I#%  
)fy-]Ky *  
 ES}V\k*}  
if any( r>1 | r<0 ) =e)t,YVm  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') (n{x"rLy/  
end $-=xG&fSz  
W{RZ@ 3ZY  
 &L[i"1a  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dl]pdg<  
    error('zernfun:RTHvector','R and THETA must be vectors.') ^%n]_[RUn4  
end *O|_)G  
'GI| t  
 o;TS69|D  
r = r(:); _lG|t6y  
theta = theta(:); ~] &yHzp2  
length_r = length(r); "hmLe(jo}  
if length_r~=length(theta) K<rv|bJ  
    error('zernfun:RTHlength', ... 9r. h^  
          'The number of R- and THETA-values must be equal.') H@xHkqan  
end v!`:{)2C  
yJK:4af;.  
 T095]*Hm  
% Check normalization: %lk^(@+ T  
% -------------------- ,&~-Sq) ~  
if nargin==5 && ischar(nflag) mv,5Q6!  
    isnorm = strcmpi(nflag,'norm'); Wsb>3J  
    if ~isnorm =,b6yV+$D  
        error('zernfun:normalization','Unrecognized normalization flag.') 1oc@]0n  
    end AQ&vq$  
else "T$LJ1E  
    isnorm = false; EQWRfx?d  
end 5e3p9K`5  
#iKPp0`K*  
 })+iAxR  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wz..  
% Compute the Zernike Polynomials 2qdc$I&$  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tQ *?L  
c/7}5#Rs  
 P{L S +.  
% Determine the required powers of r: /X]gm\x7s  
% ----------------------------------- .-'_At4g  
m_abs = abs(m); l 88n*O  
rpowers = []; #<~oR5ddlb  
for j = 1:length(n) ;Fo7 -kK  
    rpowers = [rpowers m_abs(j):2:n(j)]; u6 QW*8b4  
end We++DWp  
rpowers = unique(rpowers); !1ZItJ74#  
H:EK&$sU  
 6j8\3H~  
% Pre-compute the values of r raised to the required powers, 2$o#b .  
% and compile them in a matrix: `$Um  
% ----------------------------- /d+v4GIB  
if rpowers(1)==0 h3bQ<?m  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7'OR ;b$  
    rpowern = cat(2,rpowern{:}); 7+8 8o:G9  
    rpowern = [ones(length_r,1) rpowern]; ?V}ub>J/=  
else ]x).C[^  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); yEMM@5W)8  
    rpowern = cat(2,rpowern{:}); Oq$-*N  
end >z~_s6#CP  
\K9.]PfbI  
 GSs?!BIC  
% Compute the values of the polynomials: ub!l Hl  
% -------------------------------------- );T&pm:C>  
y = zeros(length_r,length(n)); (t){o> l  
for j = 1:length(n) WJxcJE  
    s = 0:(n(j)-m_abs(j))/2; _M&n~ r  
    pows = n(j):-2:m_abs(j); 15VvZ![$V  
    for k = length(s):-1:1 mU(v9Jpf7  
        p = (1-2*mod(s(k),2))* ... z;?ztpa@  
                   prod(2:(n(j)-s(k)))/              ... 2}7_Y6RS*  
                   prod(2:s(k))/                     ... $}IG+ ,L  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ck%.D%=  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); [ #ih o(/  
        idx = (pows(k)==rpowers); }Ml BmD  
        y(:,j) = y(:,j) + p*rpowern(:,idx); H "Io!{aKU  
    end & kVa*O  
     DE7y\oO]  
    if isnorm $tF\7.e@  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }z#M!~  
    end b)x0;8<  
end } xA@3RT  
% END: Compute the Zernike Polynomials H5#]MOAP  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &^K(9"  
YcV^Fqi!  
 .%dGSDru  
% Compute the Zernike functions: `\|@w@f|;  
% ------------------------------ JU;`c>8=)  
idx_pos = m>0; Pwj|]0Y@  
idx_neg = m<0; r`"T{o\e   
FZ}^)u}o  
 *iY:R  
z = y; [3sZ=)G  
if any(idx_pos) 3=o4ncg(  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); cHVJ7yAZI  
end mR|5$1[b  
if any(idx_neg) x_I*6?  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); qou\4YZ  
end */JYP +  
Qd\='*:!  
 )t3`O$J  
% EOF zernfun ?{}P#sn  
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)  _aOs8#(X  
kppi>!6  
DDE还是手动输入的呢？ ^^7@kh mNl  
<uS/8MP{  
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