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， iDD$pd,e\  
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， #K&Gp-  
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ 7$#u  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 4e  
+ai< q>+  
 ,)io5nZF  
b MBLXk  
 T4F/w|Q  
function z = zernfun(n,m,r,theta,nflag) =lSNs   
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Xc.`-J~Il  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {[F A#  
%   and angular frequency M, evaluated at positions (R,THETA) on the sRfcF`7  
%   unit circle.  N is a vector of positive integers (including 0), and <naz+QK'  
%   M is a vector with the same number of elements as N.  Each element 8EY:t zw  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ZC8wA;!z^  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 5R7DDJk  
%   and THETA is a vector of angles.  R and THETA must have the same &Qm@9Is  
%   length.  The output Z is a matrix with one column for every (N,M) 8k79&|  
%   pair, and one row for every (R,THETA) pair. Va8&Z  
% n@w%Zl  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .C(tMF]D,  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X?Au/  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral \NC3'G:Ii  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N g,j#  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 5 dg(e3T  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. adw2x pj  
% %)wjR/o  
%   The Zernike functions are an orthogonal basis on the unit circle. I@3MO0V^  
%   They are used in disciplines such as astronomy, optics, and ;rS{:  
%   optometry to describe functions on a circular domain. Js;h%  
% pJ{Y lS{  
%   The following table lists the first 15 Zernike functions. D,6:EV"sa  
% Dzbz)Zst  
%       n    m    Zernike function           Normalization uq{ beC  
%       -------------------------------------------------- -YE^zzh  
%       0    0    1                                 1 DI%saw  
%       1    1    r * cos(theta)                    2 -HuA \0J  
%       1   -1    r * sin(theta)                    2 Y$zSQ_k;U  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) P*o9a  
%       2    0    (2*r^2 - 1)                    sqrt(3) 5X+A"X ;C  
%       2    2    r^2 * sin(2*theta)             sqrt(6) =QsYXK7Mn4  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 5)E @F9N  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) [gB+C84%%  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =#\:}@J5I  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 8- i#8'/x  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) l^qI, M  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y0>y8U V  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) D]}G.v1  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rGO8!X 3d  
%       4    4    r^4 * sin(4*theta)             sqrt(10) fIF8%J ^3  
%       -------------------------------------------------- kP"9&R`E  
% "}!G!k:  
%   Example 1: 7L??ae  
% =Uh$&m  
%       % Display the Zernike function Z(n=5,m=1) m2o0y++TjW  
%       x = -1:0.01:1; g){<y~Mk  
%       [X,Y] = meshgrid(x,x); $?Wb}DU7_L  
%       [theta,r] = cart2pol(X,Y); l\mPHA23  
%       idx = r<=1; nlYNN/@"  
%       z = nan(size(X)); putrSSL}  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); vbNBLCwug  
%       figure G?ZXWu.  
%       pcolor(x,x,z), shading interp w@b)g  
%       axis square, colorbar q7!{?\T%  
%       title('Zernike function Z_5^1(r,\theta)') Fp:'M X  
% E3i4=!Y  
%   Example 2: w &(ag$p'  
% OnK4] S5  
%       % Display the first 10 Zernike functions <N)oS-m>  
%       x = -1:0.01:1; T |p"0b A  
%       [X,Y] = meshgrid(x,x); liZxBs :%i  
%       [theta,r] = cart2pol(X,Y); WM{=CD  
%       idx = r<=1; ^_6|X]tz1T  
%       z = nan(size(X)); g*Phv|kI  
%       n = [0  1  1  2  2  2  3  3  3  3]; O}P`P'Y|'  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; w@ pPcZ>z/  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; gSgr6TH0  
%       y = zernfun(n,m,r(idx),theta(idx)); ;,TFr}p`  
%       figure('Units','normalized') "z c l|@  
%       for k = 1:10 sS Mh`4'  
%           z(idx) = y(:,k); 0erNc'e  
%           subplot(4,7,Nplot(k)) nu^436MSOa  
%           pcolor(x,x,z), shading interp 6mE\OS-I  
%           set(gca,'XTick',[],'YTick',[]) |zU-KGO&  
%           axis square pJ=#zsE0  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) GeqPRah  
%       end @,}UWU  
% Bwrx*J  
%   See also ZERNPOL, ZERNFUN2. =vPj%oLp'a  
~@!bsLSMU  
 %)|s1B'd  
%   Paul Fricker 11/13/2006 yX5\gO6G  
B[}6-2<>?C  
 N;R^h? '  
*I+Q~4  
 LscGTs,  
% Check and prepare the inputs: 4 :v=pZ  
% ----------------------------- fOHxtHM  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  bLL2  
    error('zernfun:NMvectors','N and M must be vectors.') @d_M@\r=j  
end B:<VA=  
u ?"Vm  
 YQ} o?Q$z  
if length(n)~=length(m) _M1%Z~  
    error('zernfun:NMlength','N and M must be the same length.') *v`eUQ:  
end jo7\`#(Q  
0"R|..l/  
 :]"V-1#}  
n = n(:); ]iWRo'  
m = m(:); @ZJS&23E  
if any(mod(n-m,2)) FwK] $4*  
    error('zernfun:NMmultiplesof2', ... *Ly6`HZ9  
          'All N and M must differ by multiples of 2 (including 0).') [7-?7mp!B  
end lYIH/:T  
l}h!B_P'  
 2eogY#  
if any(m>n) e.%nRhSs3  
    error('zernfun:MlessthanN', ... rOYx b }1  
          'Each M must be less than or equal to its corresponding N.') yauvXosX  
end ]|@^1we  
54,er$$V  
 / 1RpM]d  
if any( r>1 | r<0 ) bD^owa  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') CITc2v3a  
end !Cs_F&l"j  
x^ni1=kU  
 `^vE9nW 7  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) mDWG7Asp  
    error('zernfun:RTHvector','R and THETA must be vectors.') im8CmQ  
end `L zPotz  
=I<R!ZSN  
 SM '|+ d  
r = r(:); 0Gk<l{o?^  
theta = theta(:); baasGa3}s  
length_r = length(r); |)&%A%m  
if length_r~=length(theta) 4*L_)z&4;  
    error('zernfun:RTHlength', ... D9df=lv mD  
          'The number of R- and THETA-values must be equal.') H\ %7%  
end J,hCvm  
' QG?nu  
 u, ff>/1  
% Check normalization: _$'ashF  
% -------------------- Z;i:](  
if nargin==5 && ischar(nflag) ^~dWU>  
    isnorm = strcmpi(nflag,'norm'); w xH7?tsf  
    if ~isnorm 5R-6ji  
        error('zernfun:normalization','Unrecognized normalization flag.') RN1_S  
    end Hz~zu{;{J  
else :h$$J lP  
    isnorm = false; eRYK3W  
end )4OxY[2J  
1x)J[fyId  
 +0&/g&a\R  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +TJCLZ..  
% Compute the Zernike Polynomials 2iOV/=+  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |=w@H]r  
uT{q9=w  
 7r!x1  
% Determine the required powers of r: Wri<h:1  
% ----------------------------------- )UR7i8]!0  
m_abs = abs(m); A<{{iBEI`  
rpowers = []; WY/}1X9.%  
for j = 1:length(n) 2:kH[#  
    rpowers = [rpowers m_abs(j):2:n(j)]; %A`+WYeuX  
end vt8By@]:  
rpowers = unique(rpowers); l;Wj]  
X, n:,'  
 JI}'dU>*U:  
% Pre-compute the values of r raised to the required powers, }j%5t ~Qa  
% and compile them in a matrix: Y|n"dMrL  
% ----------------------------- p'%s=TGwv  
if rpowers(1)==0 AKC`TA*E  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yAt ^;  
    rpowern = cat(2,rpowern{:}); 3n _htgcv  
    rpowern = [ones(length_r,1) rpowern]; ,prf;|e?  
else Xhm c6?  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); M  >u_4AY  
    rpowern = cat(2,rpowern{:}); az$FnVNn=  
end CrLrw T  
r; {.%s7  
 .]^?<bG  
% Compute the values of the polynomials: ;+%rw2Z,B  
% -------------------------------------- icgfB-1|i  
y = zeros(length_r,length(n)); O-^Ma- }  
for j = 1:length(n) se)TzI^]b@  
    s = 0:(n(j)-m_abs(j))/2; \D4:Nt#  
    pows = n(j):-2:m_abs(j); Hka2  
    for k = length(s):-1:1 mt .sucT  
        p = (1-2*mod(s(k),2))* ... s AkdMo  
                   prod(2:(n(j)-s(k)))/              ... g#bRT*,L  
                   prod(2:s(k))/                     ... iTwm3V P  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y4-t7UlS;  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); +>,I1{u%&  
        idx = (pows(k)==rpowers); s[jTP(d)8  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ax`o>_)  
    end R_C)  
     zPO9!?7|  
    if isnorm HN"Z]/ 5j  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); &~CI<\o P  
    end ]kSGR  
end Vr}'.\$  
% END: Compute the Zernike Polynomials tw;}jh  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *@5@,=d  
=bOW~0Z1  
 6Mf0`K  
% Compute the Zernike functions: 1zv'.uu.,  
% ------------------------------ 4RO}<$Nx}  
idx_pos = m>0; ?`s8 pPc4  
idx_neg = m<0; 9{l}bu/u  
G{}VPcrbC  
 RZLq]8pM  
z = y; lA]8&+,ZM  
if any(idx_pos) Ml_^ `vn  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?s01@f#  
end uRvP hkqm  
if any(idx_neg) u 9e@a9c  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @ Y+oiB~Y  
end ^qs $v06  
SUiOJ[5,  
 D*jM1w_`  
% EOF zernfun 4?kcv59  
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)  Wwo0%<2y  
!WlH'y-I  
DDE还是手动输入的呢？  *CMx-_  
kZ~~/?B  
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