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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， A9Kt^HR  
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， ? R!Pf: t  
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？ wNvq['P  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ *{x8@|K8  
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function z = zernfun(n,m,r,theta,nflag) BK]5g[   
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. aO &!Y\=@  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aE(DNeG-H  
%   and angular frequency M, evaluated at positions (R,THETA) on the {Ri6975  
%   unit circle.  N is a vector of positive integers (including 0), and jI/#NCKE  
%   M is a vector with the same number of elements as N.  Each element t9~Y ?  
%   k of M must be a positive integer, with possible values M(k) = -N(k) cB0"vbdO  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, fL(_V/p^  
%   and THETA is a vector of angles.  R and THETA must have the same w5<&b1:  
%   length.  The output Z is a matrix with one column for every (N,M) k 5gvo  
%   pair, and one row for every (R,THETA) pair. Rt.2]eZEJ  
% W %<,GV  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike y/Ui6D  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `|p8zV  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral C23Gp3_0/  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, LkyT4HC8n  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized %6Y\4Fe  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QC Jf   
% E3V_qT8  
%   The Zernike functions are an orthogonal basis on the unit circle. R+# g_"1@p  
%   They are used in disciplines such as astronomy, optics, and ]u|5ZCv0  
%   optometry to describe functions on a circular domain. tq=7HM  
% >)t-Zh:n  
%   The following table lists the first 15 Zernike functions. 9}T(m(WQVu  
% ]|QA`5=$  
%       n    m    Zernike function           Normalization &SMM<^P.  
%       -------------------------------------------------- RPw1i*  
%       0    0    1                                 1 +2#pP  
%       1    1    r * cos(theta)                    2 8a;;MJ)  
%       1   -1    r * sin(theta)                    2 $C t(M)  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ra F+Bt`  
%       2    0    (2*r^2 - 1)                    sqrt(3) 6m0- he~  
%       2    2    r^2 * sin(2*theta)             sqrt(6) R^6]v`j;  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) xf3;:soC  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ~? n)/i("  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) k $E{'Dv  
%       3    3    r^3 * sin(3*theta)             sqrt(8) _iH:>2p5R  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) :gM_v?sy  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ssd7]G+n:  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) UYH&x:WEd  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {# N,&?[  
%       4    4    r^4 * sin(4*theta)             sqrt(10) Mk/ZEyq^  
%       -------------------------------------------------- 5Z_aN|Xn  
% `svOPB4C'  
%   Example 1: 0Wb3M"#9<  
% mW)C=X%  
%       % Display the Zernike function Z(n=5,m=1) P EMuIYm$  
%       x = -1:0.01:1; u vyvy  
%       [X,Y] = meshgrid(x,x); ;y.<I&  
%       [theta,r] = cart2pol(X,Y); <3 I0$?xL  
%       idx = r<=1; i9^m;Y)^I  
%       z = nan(size(X)); }g"K\x:Z  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ;HmQRiCg  
%       figure K7},X01^  
%       pcolor(x,x,z), shading interp  z:d+RMA  
%       axis square, colorbar /mQ9} E4X  
%       title('Zernike function Z_5^1(r,\theta)') ^,#MfF6  
% 6oLZH6fG  
%   Example 2:  s x)x7  
% @rlL'|&X*  
%       % Display the first 10 Zernike functions -^,wQW:o)  
%       x = -1:0.01:1;  WYW@%t  
%       [X,Y] = meshgrid(x,x); Fv3:J~Yf  
%       [theta,r] = cart2pol(X,Y); ?mh0^G  
%       idx = r<=1; kOV6O?h  
%       z = nan(size(X)); =UV=F/Af^  
%       n = [0  1  1  2  2  2  3  3  3  3]; 71G00@&w9D  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :Oc&{z?q  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 5wI j:s  
%       y = zernfun(n,m,r(idx),theta(idx)); h5ZxxtGU  
%       figure('Units','normalized') S%7%@Qs"%  
%       for k = 1:10 3&Fqd  
%           z(idx) = y(:,k); M7 gM#bv>L  
%           subplot(4,7,Nplot(k)) An=Q`Uxt/  
%           pcolor(x,x,z), shading interp u\@ L|rh  
%           set(gca,'XTick',[],'YTick',[]) A[N>T\  
%           axis square z1LY|8$G  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]*\<k  
%       end :snn-e0l  
% UpqDGd7M  
%   See also ZERNPOL, ZERNFUN2. y0 qq7Dmu  
lPn&,\9@~  
 n$jf($*  
%   Paul Fricker 11/13/2006 )}SiM{g  
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 f/e2td*A  
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 {-,^3PI\  
% Check and prepare the inputs: 3bMUsyJ2  
% ----------------------------- C lzz!v  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -1 _7z{.  
    error('zernfun:NMvectors','N and M must be vectors.') \:-N<[  
end J9`[Qy\  
"6P-0CJ  
 2O)2#N  
if length(n)~=length(m) f n'N^  
    error('zernfun:NMlength','N and M must be the same length.') 2s8(r8AI  
end Y\ G^W8  
TkV$h(#!f&  
 l%9nA.M'  
n = n(:); 8Zvh"Z?  
m = m(:); Y$6W~j  
if any(mod(n-m,2)) da53XEF&  
    error('zernfun:NMmultiplesof2', ... Ab f=b<bu  
          'All N and M must differ by multiples of 2 (including 0).') 9{5 c}bX  
end Ow7I`#P  
rFR2c?j8  
 _\2^s&iJh  
if any(m>n) *oz=k  
    error('zernfun:MlessthanN', ... 7Om)uUjU4  
          'Each M must be less than or equal to its corresponding N.') |A@Gch fd  
end ;t}ux  
05m/iQ  
 blA]z!FU  
if any( r>1 | r<0 ) 7&9'=G  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') r.;(Kx/M  
end IWcYa.=tZ  
`)R@\@jt  
 ~+Da`Wp  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) to*<W,I  
    error('zernfun:RTHvector','R and THETA must be vectors.') rKys:is  
end IRQ3>4hI  
er0ClvB  
 ]]Da/^K=Z  
r = r(:); SAGLLk07G  
theta = theta(:); `g iCytv  
length_r = length(r); o$r]Z1  
if length_r~=length(theta) c{t[iXDG  
    error('zernfun:RTHlength', ... @Q atgYu  
          'The number of R- and THETA-values must be equal.') nNff~u)I  
end W[3)B(Vq<E  
__V6TDehJ$  
 d/OIc){tD  
% Check normalization: ;DKwv}  
% -------------------- =}~h bPJM  
if nargin==5 && ischar(nflag) gaJIc^O  
    isnorm = strcmpi(nflag,'norm'); E$[\Fk}S  
    if ~isnorm 8_tMiIE-pS  
        error('zernfun:normalization','Unrecognized normalization flag.') :22IY> p  
    end ZR'q.y[k)  
else Tl3{)(ezx  
    isnorm = false; :[N[D#/z  
end a".uS4x  
z,oqYU\:  
 >9g`9hB  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e+F5FAMR68  
% Compute the Zernike Polynomials RfB""b8]=  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3Ecm Nwr  
y6o^ Knl  
 tH"SOGfSt  
% Determine the required powers of r: v=|BqG`  
% ----------------------------------- Mf&W<n^j  
m_abs = abs(m); 8E Y< ^:  
rpowers = []; aM9St!i  
for j = 1:length(n) %Ys>PzM  
    rpowers = [rpowers m_abs(j):2:n(j)]; [lA[w Cw  
end 5@Q4[+5&_  
rpowers = unique(rpowers); !DCJ2h%E[_  
bhSpSul  
 <5(8LMF  
% Pre-compute the values of r raised to the required powers, :u{0M&  
% and compile them in a matrix: iEki<e/  
% ----------------------------- |7/B20  
if rpowers(1)==0 .VmI4V?}h  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Rhx7eU#&  
    rpowern = cat(2,rpowern{:}); *ftJ(  
    rpowern = [ones(length_r,1) rpowern]; }FMl4 _}u  
else vd /_`l.D  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ZZE  
    rpowern = cat(2,rpowern{:}); $#@4i4TN-  
end Z=!*7@QY  
_:'m/K3Ee  
 2>O2#53ls0  
% Compute the values of the polynomials: BPrA*u }T  
% -------------------------------------- {7eKv+30  
y = zeros(length_r,length(n)); @\!wW-:A  
for j = 1:length(n) DcbL$9UI  
    s = 0:(n(j)-m_abs(j))/2; ^^?DYC   
    pows = n(j):-2:m_abs(j); MQY1he2M  
    for k = length(s):-1:1 BdO$  
        p = (1-2*mod(s(k),2))* ... &,."=G  
                   prod(2:(n(j)-s(k)))/              ... 2c%}p0<;|?  
                   prod(2:s(k))/                     ... B0z.s+.  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... OV8b~k4=  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); ;*W]]4fy  
        idx = (pows(k)==rpowers); qW7"qw=   
        y(:,j) = y(:,j) + p*rpowern(:,idx); Z{p6Q1u  
    end B@zJ\Ir[  
     f/;\/Q[Z7  
    if isnorm F`I-G~e  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "}SERC7  
    end 4rM77Uw>  
end <YeF?$S}  
% END: Compute the Zernike Polynomials 38q@4U=aiw  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N@MeaO  
pXFNK" jm  
 qfSoF|  
% Compute the Zernike functions: FOk @W&  
% ------------------------------ k)v[/#I  
idx_pos = m>0; &vMH AZd  
idx_neg = m<0; 4(aesZ8h  
o@>c[knJ  
 ($S{td;  
z = y; BRD'5 1]|  
if any(idx_pos) '-i tn  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5&X  
end "]_|c\98  
if any(idx_neg) ImV54h'  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =H,cwSE+%  
end Ar<OP'C  
Ox~'w0c,f  
 ~o/^=:*  
% EOF zernfun Yip9K[  
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)  qE{cCS  
SNU bY6  
DDE还是手动输入的呢？ 68c;Vb  
^ <VE5OM  
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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