[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， kg^VzNX
 
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， 8A ;)5!  
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？ p\ }Ep  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 3/i_?G  
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function z = zernfun(n,m,r,theta,nflag) m=hUHA,p4  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. O<o>/HH$  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H]x-s  
%   and angular frequency M, evaluated at positions (R,THETA) on the OmR)W'  
%   unit circle.  N is a vector of positive integers (including 0), and g VPtd[r  
%   M is a vector with the same number of elements as N.  Each element GF=rGn@,)`  
%   k of M must be a positive integer, with possible values M(k) = -N(k) R]! [h  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (6Tvu
5*4U  
%   and THETA is a vector of angles.  R and THETA must have the same _sGmkJi]  
%   length.  The output Z is a matrix with one column for every (N,M) ;0c -+,  
%   pair, and one row for every (R,THETA) pair. -FGQn |h4  
% :K)7_]y  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (Iz$_(  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),
1\aJ[t
 
%   with delta(m,0) the Kronecker delta, is chosen so that the integral 74p=uQ  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4fyds< f  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^I CSs]}1  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -E(0}\  
% #NZ#G~oeO  
%   The Zernike functions are an orthogonal basis on the unit circle. }@XokRk  
%   They are used in disciplines such as astronomy, optics, and ~>"m`Q&[  
%   optometry to describe functions on a circular domain. y k{8O.g  
% CVy\']  
%   The following table lists the first 15 Zernike functions. ?;0w1  
% O8Dav^\y?  
%       n    m    Zernike function           Normalization <Cbi5DtR  
%       -------------------------------------------------- u9zEhfg8  
%       0    0    1                                 1 s1GR!*z>  
%       1    1    r * cos(theta)                    2 4
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%       1   -1    r * sin(theta)                    2 iX|K4.Pz{  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) .;$Ub[
  
%       2    0    (2*r^2 - 1)                    sqrt(3) TF1,7Qd  
%       2    2    r^2 * sin(2*theta)             sqrt(6) aVvma=  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) F!_8?=|  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) rijavZS6  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g]Jt (aYK  
%       3    3    r^3 * sin(3*theta)             sqrt(8)
@?vC4+'  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) $~+(si2  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )p^" J|  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) x=M%QFe  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?bH&F  
%       4    4    r^4 * sin(4*theta)             sqrt(10) tSVWO]<  
%       -------------------------------------------------- 6|G&d>G$_  
% Db`SNk=  
%   Example 1: d2a*xDkv  
% n(h9I'V8)F  
%       % Display the Zernike function Z(n=5,m=1) j"F?^0aR,Q  
%       x = -1:0.01:1; h4#y'E!,Z  
%       [X,Y] = meshgrid(x,x); v6C$Y+5~  
%       [theta,r] = cart2pol(X,Y); |ns B'Q  
%       idx = r<=1; 1]A$  
%       z = nan(size(X)); C==yl"w  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); .mbqsb]&Y  
%       figure -^aJ}[uaI  
%       pcolor(x,x,z), shading interp ('k9XcTPP  
%       axis square, colorbar akA7))Q  
%       title('Zernike function Z_5^1(r,\theta)') 4OaU1Y[  
% hGy[L3{  
%   Example 2: T!7B0_  
% lsaA    
%       % Display the first 10 Zernike functions r@a]fTf  
%       x = -1:0.01:1; ~NMx:PP  
%       [X,Y] = meshgrid(x,x); QdrZi.qKH  
%       [theta,r] = cart2pol(X,Y); 2{Y~jYt{h  
%       idx = r<=1; XkPE%m_5D  
%       z = nan(size(X)); :N^+!,i  
%       n = [0  1  1  2  2  2  3  3  3  3]; p9>1a j2a  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; uc>":V  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Vak\N)=u  
%       y = zernfun(n,m,r(idx),theta(idx)); \(A A|;  
%       figure('Units','normalized') $<QrV,T  
%       for k = 1:10 8c\\-{  
%           z(idx) = y(:,k); ~].?8C.>*  
%           subplot(4,7,Nplot(k)) [=BccT:b  
%           pcolor(x,x,z), shading interp o(k{Ed  
%           set(gca,'XTick',[],'YTick',[]) 45?%D}  
%           axis square ,v%'2[}  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) uOO\!Hqq  
%       end lvsj4cT  
% E<l/o5<nC  
%   See also ZERNPOL, ZERNFUN2. U`aB&[=$  
[{$%9lm  
s IFE:/1,  
%   Paul Fricker 11/13/2006 3K=%I+G(4  
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b'W.l1]<-  
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% Check and prepare the inputs: P)l_ :;&  
% ----------------------------- !:PiQ19 'u  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rz0)S py6  
    error('zernfun:NMvectors','N and M must be vectors.') hvGD`  
end ?P}bl_  
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if length(n)~=length(m) j4<K0-?  
    error('zernfun:NMlength','N and M must be the same length.') D4b-Y[/"  
end &7i&"TNptP  
Z5E; FGPb  
P6&%`$  
n = n(:); 1uO2I&B  
m = m(:); ! ,bQ;p3g|  
if any(mod(n-m,2)) ftG3!}  
    error('zernfun:NMmultiplesof2', ... ;=7K*npT  
          'All N and M must differ by multiples of 2 (including 0).') &s(J:P$!  
end eP,bFc  
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if any(m>n) |his8\C+x  
    error('zernfun:MlessthanN', ... L R\LC6kM  
          'Each M must be less than or equal to its corresponding N.') Cs_&BSs  
end ?!K6")SE  
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if any( r>1 | r<0 ) PcBD;[cn  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') j.ucv  
end hLbWqF  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m`l9d4p w?  
    error('zernfun:RTHvector','R and THETA must be vectors.') *5 +GJWKN  
end 5Yhcnwdm!  
f"KrPx!^b  
s-_D,$ |  
r = r(:); (c>g7d<>n  
theta = theta(:); UrHndnqM  
length_r = length(r); 4 sax  
if length_r~=length(theta) 0:`|T jf_  
    error('zernfun:RTHlength', ... )Xh}N  
          'The number of R- and THETA-values must be equal.') HeO:=OE~>  
end 4;I\%qes  
g_1#if&  
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% Check normalization: !(rAI  
% -------------------- 4WJY+)  
if nargin==5 && ischar(nflag) >UMxlvTg&  
    isnorm = strcmpi(nflag,'norm'); "bIb?e2h9G  
    if ~isnorm Bz<hP*.O  
        error('zernfun:normalization','Unrecognized normalization flag.') +?Ii=*7n  
    end ?0_<u4  
else 7IkPi?&{  
    isnorm = false; Oj;*Gi9E  
end +bS\iw+  
$uZmIu9Bi+  
MzD1sWmK  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G;3%k.{  
% Compute the Zernike Polynomials @^<odmM  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )"S%'myj  
E=N$JM  
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% Determine the required powers of r: Qz,2PO  
% ----------------------------------- st;i
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m_abs = abs(m); BS*79heY  
rpowers = []; UPI- j#yc  
for j = 1:length(n) 3)y1q>CQf  
    rpowers = [rpowers m_abs(j):2:n(j)]; b3^:
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end 0+e=s0s.  
rpowers = unique(rpowers); s`jlE|jtN  
/)6T>/  
 px<psR5  
% Pre-compute the values of r raised to the required powers, pM?~AYWb  
% and compile them in a matrix: &{V|%u}v  
% ----------------------------- J, -.5  
if rpowers(1)==0 t,?,T~#9  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LUbj^iQ9  
    rpowern = cat(2,rpowern{:}); `qc"JB  
    rpowern = [ones(length_r,1) rpowern]; 
u]Ku96!  
else uQIPnd(V  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >$JE!.p%o  
    rpowern = cat(2,rpowern{:}); HG;;M6  
end hSm?Z!+  
w$:\!FImx  
C2}y#AI  
% Compute the values of the polynomials: +})QTFV  
% -------------------------------------- c!ZZMCs  
y = zeros(length_r,length(n)); S=_u3OH0  
for j = 1:length(n) <= o<lRU  
    s = 0:(n(j)-m_abs(j))/2; A|\A|8=b  
    pows = n(j):-2:m_abs(j); wa8jr5/k"  
    for k = length(s):-1:1 7|5kak>=  
        p = (1-2*mod(s(k),2))* ... ,VS\mG/}s  
                   prod(2:(n(j)-s(k)))/              ... itYoR-XJ  
                   prod(2:s(k))/                     ... qWhW4$7x  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... J\y^T3Z  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ^2~ZOP$A  
        idx = (pows(k)==rpowers); #<xFO^TB  
        y(:,j) = y(:,j) + p*rpowern(:,idx); .;4N:*hY  
    end 6Qkjr</  
     ,{PN6B  
    if isnorm O2Qmz=%  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p(n0(}eVC'  
    end <=NnrZOF  
end 1kvX#h&V  
% END: Compute the Zernike Polynomials K1 "HJsj  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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3gUY13C}:p  
% Compute the Zernike functions: z\m$>C|  
% ------------------------------ cb^IJA9}  
idx_pos = m>0; kH eD(Ea  
idx_neg = m<0; Qn$'bK2V  
3N_
KNW  
#&'S-XE+  
z = y; LO_Xrj  
if any(idx_pos) PEI$1,z  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); zDdo RK@  
end H1k)ya x4_  
if any(idx_neg) ww{k_'RRJ  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); LA6XTgcu  
end N/o?\q8  
CH4Nz'X2  
-dM~3'  
% EOF zernfun (2;Aqx5i  

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)  G=e'H-  

,H
|K3nh  
DDE还是手动输入的呢？  de8xl  
ZkibfVwe  
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。