[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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， Vm"{m/K0  
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， H\PY\O&cP  
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？ U65a_dakk  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ o8ERU($/  
n N_Ylw  
W,D$=Bg  
c %f'rj  
l&2pUv=  
function z = zernfun(n,m,r,theta,nflag) myvn@OsEw  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~%D=\iE  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GV"X) tGo  
%   and angular frequency M, evaluated at positions (R,THETA) on the te*|>NRS  
%   unit circle.  N is a vector of positive integers (including 0), and +
lNAog  
%   M is a vector with the same number of elements as N.  Each element ExW3LM9(  
%   k of M must be a positive integer, with possible values M(k) = -N(k) -*nd5(lY&  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, FSNzBN  
%   and THETA is a vector of angles.  R and THETA must have the same o-ee3j.  
%   length.  The output Z is a matrix with one column for every (N,M) dBeZx1Dy  
%   pair, and one row for every (R,THETA) pair. %"gV>E_u  
% &2Q0ii#Aa  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kw$*o k  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \Um &  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral wRCv?D`vV  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,UA-Pq3}  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized uJ:SN;  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (oG-h
"^/  
% 0{k*SCN#  
%   The Zernike functions are an orthogonal basis on the unit circle. 713)D4y}  
%   They are used in disciplines such as astronomy, optics, and `*ml/% \  
%   optometry to describe functions on a circular domain. >>I~v)a>w  
% m`lxQik  
%   The following table lists the first 15 Zernike functions. wc~k4B9"  
% lDf:~  
%       n    m    Zernike function           Normalization -udKGrT+  
%       -------------------------------------------------- |WUm;o4E`U  
%       0    0    1                                 1 ?E|be )  
%       1    1    r * cos(theta)                    2 Z]\IQDC  
%       1   -1    r * sin(theta)                    2 Z{p62|+Ck@  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) &`}8Jz=S  
%       2    0    (2*r^2 - 1)                    sqrt(3) a'prlXr\4  
%       2    2    r^2 * sin(2*theta)             sqrt(6) J12hjzk6@  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) H vezi>M  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |\#6?y[o  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,>vI|p,/G*  
%       3    3    r^3 * sin(3*theta)             sqrt(8) k4!z;Yq  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) +=JJ=F)  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) eI:;l];G9  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) zjlo3=FQX[  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7jtDhsVz  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ><r\ 5`  
%       --------------------------------------------------  1cvH  
% ]*\m@lWu  
%   Example 1: 9i`sSi8   
% lE 09Y  
%       % Display the Zernike function Z(n=5,m=1) AriW&E  
%       x = -1:0.01:1; [KT1.5M[  
%       [X,Y] = meshgrid(x,x); F =Zc_  
%       [theta,r] = cart2pol(X,Y); ]fb3>HOTJ  
%       idx = r<=1; @`S8d%6P  
%       z = nan(size(X)); m@#@7[6]o  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 'H|=]n0  
%       figure 1XD|H_JG<j  
%       pcolor(x,x,z), shading interp u^Ss8}d  
%       axis square, colorbar SGA!%=Lp  
%       title('Zernike function Z_5^1(r,\theta)') "U6:z M  
% U%zZw)  
%   Example 2: `a:L%Ex  
% h
np-x3  
%       % Display the first 10 Zernike functions `xm4?6  
%       x = -1:0.01:1; nApkK1?  
%       [X,Y] = meshgrid(x,x); 8Z1pQx-P2C  
%       [theta,r] = cart2pol(X,Y); 48t_?2>  
%       idx = r<=1; \UR/tlw+/  
%       z = nan(size(X)); _6/q.  
%       n = [0  1  1  2  2  2  3  3  3  3]; !ZC0n`  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; O%R*1 P9  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; GJB=5nE  
%       y = zernfun(n,m,r(idx),theta(idx)); f6O5k8n  
%       figure('Units','normalized') _=d X01  
%       for k = 1:10 1~_&XNb&  
%           z(idx) = y(:,k); HaiaDY)  
%           subplot(4,7,Nplot(k)) cPL]
WI0(  
%           pcolor(x,x,z), shading interp dXvp-oi  
%           set(gca,'XTick',[],'YTick',[]) n>["h2  
%           axis square 1-6[KBQ8  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :4'Fq;%C  
%       end )?qH#>mD6  
% *M^t@hl  
%   See also ZERNPOL, ZERNFUN2. 6~b]RZe7  
ocbNf'W;  
B6hd*f  
%   Paul Fricker 11/13/2006 wO&2S-;_K  
FY(C<fDRo{  
F>!gwmn~  
BMlu>,  
]Y%U5\$  
% Check and prepare the inputs: qZ79IX'y  
% ----------------------------- .)Af&+KT  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z.v2!u  
    error('zernfun:NMvectors','N and M must be vectors.') QTKN6P  
end z')zVoW,  
C6P
(86?  
v@KP~kp  
if length(n)~=length(m) SF#Rc>v  
    error('zernfun:NMlength','N and M must be the same length.') {6uhUb  
end XnCrxj  
:lGH31GG  
8fI&-uP{g  
n = n(:); HGJfj*JH  
m = m(:); qV`JZ\n  
if any(mod(n-m,2)) MaX:oG
F,  
    error('zernfun:NMmultiplesof2', ... sHwn,4|iY  
          'All N and M must differ by multiples of 2 (including 0).') {#Vck\&  
end o"5[~$O  
Q[U_ 0O,A9  
['l.]k-b}  
if any(m>n) *[MK{m  
    error('zernfun:MlessthanN', ... /Wqx@#  
          'Each M must be less than or equal to its corresponding N.') qp6*v&  
end Bt\z0*t=s  
NcVsQV  
yavoGk
 
if any( r>1 | r<0 ) O*d4zBT  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'z}Hg *  
end [=xJh?*P  
rW&#Xw/a  
-<0PBl  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) M zbs#v0  
    error('zernfun:RTHvector','R and THETA must be vectors.') |0jmOcZF  
end w_sA8B  
%
^C.e*  
.}V&*-e
p  
r = r(:); =:;KYuTr  
theta = theta(:); 8%;K#,>  
length_r = length(r); 5v uB87`  
if length_r~=length(theta) ~il{6Z+#n  
    error('zernfun:RTHlength', ... lv*fK  
          'The number of R- and THETA-values must be equal.') @/m|T]'8  
end +z2+z  
O!c b-  
nvdo
|5  
% Check normalization: [v!TQwMU  
% -------------------- sMikTwR/^  
if nargin==5 && ischar(nflag) >(t_  
    isnorm = strcmpi(nflag,'norm'); {MaFv  
    if ~isnorm ZPISclSA+  
        error('zernfun:normalization','Unrecognized normalization flag.') TBzOz:k  
    end (Wm4JmX%  
else DG&[.dR+  
    isnorm = false; Jf,)Y>EI  
end 'xC83}!k  
gtBnP~zT\B  
8?S)>-mwv  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P2+Z^J`Y>  
% Compute the Zernike Polynomials 8jn
z;;|  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,;2x.We  
)/hb9+S  
"1U:qr2-H  
% Determine the required powers of r: 2eOde(K+  
% ----------------------------------- 'Si1r%'m#  
m_abs = abs(m); -[I}"Glz:  
rpowers = []; v=~=Q*\l  
for j = 1:length(n) #jja#PF]7  
    rpowers = [rpowers m_abs(j):2:n(j)]; 1f"LAs`%  
end ZL3aO,G2  
rpowers = unique(rpowers); :V%XEN)  
F_Q?0 Do0'  
c==` r C  
% Pre-compute the values of r raised to the required powers, l&z)Q/>?pZ  
% and compile them in a matrix: Nz,8NM]  
% ----------------------------- `+!GoXI  
if rpowers(1)==0 z'G~b[kG4n  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); I#]$H#}Av  
    rpowern = cat(2,rpowern{:}); ,AC+s"VS  
    rpowern = [ones(length_r,1) rpowern]; tsFwFB*  
else Ng6(2
Wt0e  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GYD`  
    rpowern = cat(2,rpowern{:}); 88dq8T4  
end \gh`PS-B  
{&'u1yR  
v;9VX   
% Compute the values of the polynomials: NC*h7  
% -------------------------------------- =Of!1TR(  
y = zeros(length_r,length(n)); cNW[i"  
for j = 1:length(n) 0aMw  
    s = 0:(n(j)-m_abs(j))/2; Ba$Ibq,r/  
    pows = n(j):-2:m_abs(j); GHMoT  
    for k = length(s):-1:1 g2=5IU<  
        p = (1-2*mod(s(k),2))* ... fR>(b?C  
                   prod(2:(n(j)-s(k)))/              ... ys5b34JN  
                   prod(2:s(k))/                     ... K#=)]qIk  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... He4sP
`&I  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ;P-xKRU!Xx  
        idx = (pows(k)==rpowers); f!`,!dZgkd  
        y(:,j) = y(:,j) + p*rpowern(:,idx); b@OL!?JP  
    end }ST9&wi~  
     (9N75uCa  
    if isnorm H4HWr6  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e,_b  
    end hi>sDU<x  
end Z=sCYLm  
% END: Compute the Zernike Polynomials xud  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z#wmEc.}C  
$vS`w4Y  
BfLh%XC  
% Compute the Zernike functions: =o5ZcC  
% ------------------------------ .)W'{2J-  
idx_pos = m>0; "+js7U-  
idx_neg = m<0; H)$-T1Wx4  
@`N)`u85[  
V~+{douq  
z = y; 8J:6uO c|  
if any(idx_pos) m8Q6ESg<*u  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dkuB{C,  
end vjI>TIy  
if any(idx_neg) ellj/u61bj  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); nn@"68]g  
end T!uK_  
l>RW&C&T  
8X`Gm!)  
% EOF zernfun Kc=&jCn  

支持一下，慢慢研究

顶顶·········

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)  #P$=P2o  

E^_P  
DDE还是手动输入的呢？ <#JJS}TLk  
?"\`u;  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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

慢慢研究，这个专业性很强的。用的人又少。