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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， 'Z\{D*=V8  
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， oz%h)#;  
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？ OYBotk]{1  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ myD{sE2A  
-&trk  
 4$%`Qh>yA  
ewo*7j4*  
 %v]7BV^%6  
function z = zernfun(n,m,r,theta,nflag) :tKbz nd/  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. hgj CXl  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  \C|;F  
%   and angular frequency M, evaluated at positions (R,THETA) on the Ip>^O/}$1  
%   unit circle.  N is a vector of positive integers (including 0), and N%O[  
%   M is a vector with the same number of elements as N.  Each element Y.E?;iS  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 5+`=t07^et  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, gk"mr_03  
%   and THETA is a vector of angles.  R and THETA must have the same bKYY{V55  
%   length.  The output Z is a matrix with one column for every (N,M) GUKDhg,W  
%   pair, and one row for every (R,THETA) pair. 9^oo-,Su_  
% :ezA+=ENg  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike F\JLbY{x]  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), O-Dc[t%  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral !2(.$}E  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, O8gfiQqF&  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized NzAQ@E 2d:  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QS\Uq(Ja\  
% \ZqK\=  
%   The Zernike functions are an orthogonal basis on the unit circle. *h2`^Z  
%   They are used in disciplines such as astronomy, optics, and D^n xtuT*  
%   optometry to describe functions on a circular domain. [4Y[?)7  
% "rVU4F)  
%   The following table lists the first 15 Zernike functions. Y7-*2"!  
% 7 xUE,)?  
%       n    m    Zernike function           Normalization l7ZB3'  
%       -------------------------------------------------- M]1;  
%       0    0    1                                 1 fO #?k<p  
%       1    1    r * cos(theta)                    2 NJ<N%hcjK  
%       1   -1    r * sin(theta)                    2 D 0  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 3i6h"Wu`n  
%       2    0    (2*r^2 - 1)                    sqrt(3) =m~ruZ/  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 6>s=Ci ZB  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) F%}7cm2  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "i&fp:E0  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) "tKNlHBu'  
%       3    3    r^3 * sin(3*theta)             sqrt(8) F, zG;_  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) :v_w!+,/  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zb>;?et;)  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) C *a,<`  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #J\s%60pt  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ?;_H{/)m  
%       -------------------------------------------------- *(icR  
% -rO*7HO  
%   Example 1: X CzXS.  
% h<)yJh  
%       % Display the Zernike function Z(n=5,m=1) bTiBmS  
%       x = -1:0.01:1; >{b3>s~T  
%       [X,Y] = meshgrid(x,x); y=#j`MH{>  
%       [theta,r] = cart2pol(X,Y); .wD>Gs{sH[  
%       idx = r<=1; \KCWYi]  
%       z = nan(size(X)); z#RuwB+  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); T!(sZf  
%       figure 8]HY. $E  
%       pcolor(x,x,z), shading interp pD[pTMG@$  
%       axis square, colorbar (V!0'9c  
%       title('Zernike function Z_5^1(r,\theta)') ~~@y_e[N#l  
% -qfnUh  
%   Example 2: 2 ^oGwx @  
% pu#[pa  
%       % Display the first 10 Zernike functions fQO ""qh  
%       x = -1:0.01:1; *y?[ <2"$  
%       [X,Y] = meshgrid(x,x); t|_{;!^  
%       [theta,r] = cart2pol(X,Y); a5)JkC  
%       idx = r<=1; @gEr+O1K(  
%       z = nan(size(X)); &1l~&,,  
%       n = [0  1  1  2  2  2  3  3  3  3]; ~@@ Z|w  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; "Vx6 #u@}  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; a(f(R&-:$Y  
%       y = zernfun(n,m,r(idx),theta(idx)); \+9;!VWhl  
%       figure('Units','normalized') Td&d,;  
%       for k = 1:10 h"r!q[MN o  
%           z(idx) = y(:,k); =gs~\q  
%           subplot(4,7,Nplot(k)) KJX>DL 9\  
%           pcolor(x,x,z), shading interp 9>;} /*:H  
%           set(gca,'XTick',[],'YTick',[]) Zt_r9xs>  
%           axis square &//2eL  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) { m'AY)  
%       end ; [FLT:$  
% \ FXp*FbQ  
%   See also ZERNPOL, ZERNFUN2. T0Q51Q  
\C7q4p?8  
 7gr^z)${J  
%   Paul Fricker 11/13/2006 Q vJZkGX  
%4/xH 9  
 tpZ->)1  
*E7R(#,yC  
 -x5F;d}  
% Check and prepare the inputs: O+b6lg)q  
% ----------------------------- GMw)*  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) bS%C?8  
    error('zernfun:NMvectors','N and M must be vectors.') {Xv3:"E"O  
end fM2^MUp[=1  
x) 5LT}p  
 7f r>ZY^  
if length(n)~=length(m) c 6q/X*  
    error('zernfun:NMlength','N and M must be the same length.') wPgDy  
end QtlT&|$   
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 He<;4?:  
n = n(:); "k, K~@}  
m = m(:); #N9d$[R*  
if any(mod(n-m,2)) abSq2*5K  
    error('zernfun:NMmultiplesof2', ... 1$(  
          'All N and M must differ by multiples of 2 (including 0).') t}~UYG( h~  
end !-,Ww[G>  
x_W3sS]ej  
 .1{l[[= W  
if any(m>n) vpV$$=Qwp  
    error('zernfun:MlessthanN', ... ?;RD u[eD  
          'Each M must be less than or equal to its corresponding N.') EBwK 7c  
end TzY *;  
R\y'_S=#a  
 s~Gw  
if any( r>1 | r<0 ) I^:F)a:  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') v2=/[E@  
end [4\aYB9N  
6klD22b2$  
 n.ct]+L  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) nALnB1  
    error('zernfun:RTHvector','R and THETA must be vectors.') 1daL y  
end ~#4~_d.=L  
e96#2A5f  
 K:a3+k d  
r = r(:); 3a^)u-9,x  
theta = theta(:); ZRP y~wy>  
length_r = length(r); H`5Ct  
if length_r~=length(theta) ;j!UY.i  
    error('zernfun:RTHlength', ... ofK='G .  
          'The number of R- and THETA-values must be equal.') .l=p[BI  
end 8JP6M!F#  
z{/LX \  
 2qXo{C3  
% Check normalization: q<EEb  
% -------------------- vcM~i^24)  
if nargin==5 && ischar(nflag) \OA L Or  
    isnorm = strcmpi(nflag,'norm'); Wp`C:H  
    if ~isnorm #h{Nz/h+  
        error('zernfun:normalization','Unrecognized normalization flag.') sJv`fjf%8  
    end ftaBilkjp  
else IX.sy  
    isnorm = false; Q.1XP  
end ]LVnt-q  
I{EIHD<  
 l#ZyB|  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SBz/VQ  
% Compute the Zernike Polynomials RAwk7F3qn  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *l {4lu  
d EI a=e|  
 phmVkV2a;#  
% Determine the required powers of r: >Fp&8p`am  
% ----------------------------------- 1\IZcJ {  
m_abs = abs(m); 7j T#BWt  
rpowers = []; ng 9NE8F  
for j = 1:length(n) |eF.ZC)QWh  
    rpowers = [rpowers m_abs(j):2:n(j)]; P8IRH#ED  
end nNu[c[V  
rpowers = unique(rpowers); L&QtHSzy  
"vsjen.K>  
 H>Ks6V)RL4  
% Pre-compute the values of r raised to the required powers, AoIc9E lEX  
% and compile them in a matrix: Y&:/~&'  
% ----------------------------- @2?=3Wf  
if rpowers(1)==0 RyE_|]I62u  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); W#V fX!~  
    rpowern = cat(2,rpowern{:}); F'XlJ M  
    rpowern = [ones(length_r,1) rpowern]; nSS}%&a:LX  
else ?;fv!'?%  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5c)wZ  
    rpowern = cat(2,rpowern{:}); Yx!n*+:J  
end m EFWo  
FbuKZp+  
 R@&?i=gk  
% Compute the values of the polynomials: [d\#[l_  
% -------------------------------------- h~:H?pj3g  
y = zeros(length_r,length(n)); g&P9UW>qS  
for j = 1:length(n) TD%WJ9K\  
    s = 0:(n(j)-m_abs(j))/2; i6FP[6H1  
    pows = n(j):-2:m_abs(j); a0 qj[+  
    for k = length(s):-1:1 4 #G3ew  
        p = (1-2*mod(s(k),2))* ... F; 0Dp  
                   prod(2:(n(j)-s(k)))/              ... pIL`WE1'  
                   prod(2:s(k))/                     ... Wz5d| b  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $sM]BE:  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); m8p4U-*j  
        idx = (pows(k)==rpowers); |]I#CdO  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ^T=5zqRD  
    end %$^$'6\77  
     7;~ 2e  
    if isnorm > d^r">!,  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %bN"bxv^  
    end =g1D;  
end m[{nm95QZ  
% END: Compute the Zernike Polynomials 3EO#EYAHiM  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;B(;2.<"J  
WI%zr2T  
 oA[2)BU  
% Compute the Zernike functions: Y/#:)(&@  
% ------------------------------ H9/XW6W,"w  
idx_pos = m>0; YK7gd|LR]  
idx_neg = m<0; &MBOAHhze  
9  4 "f  
 8}]l9"q(  
z = y; A=y24m  
if any(idx_pos) 'w :tq  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); iFY]0@yt  
end MI?]8+l  
if any(idx_neg) J*/$ywI  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TZ)(ZKX*R  
end VNr!|bp5  
 j@s=ER  
 NWaI[P  
% EOF zernfun ~y( ,EO  
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)  V{/)RZ/  
8VBkIYgb  
DDE还是手动输入的呢？ f D]An<  
'Kt4O9=p  
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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