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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， FuZ7xM,  
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， ldJ:A*/M6  
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？ K#=)]qIk  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ k-LB %\p  
:eK;:pN  
 Kf XE=v{t  
<uugT9By  
 #-9;Hn4x  
function z = zernfun(n,m,r,theta,nflag) 2EubMG  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. "RG.27  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hi>sDU< x  
%   and angular frequency M, evaluated at positions (R,THETA) on the W*q[f!@  
%   unit circle.  N is a vector of positive integers (including 0), and zS*X9|p  
%   M is a vector with the same number of elements as N.  Each element \ORNOX:  
%   k of M must be a positive integer, with possible values M(k) = -N(k) nj*B-M\p  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MorR&K  
%   and THETA is a vector of angles.  R and THETA must have the same > Xq:?}-m2  
%   length.  The output Z is a matrix with one column for every (N,M) )fz)Rrr  
%   pair, and one row for every (R,THETA) pair. 6#+&_ #9  
% &)Fp  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike V~+{douq  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), a.a5qwG  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral llbj-9OZL  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, C:n55BE9  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized a*d>WN.;U  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UW+|1Bj_:  
% eKlh }v  
%   The Zernike functions are an orthogonal basis on the unit circle. #c5 NFU}9  
%   They are used in disciplines such as astronomy, optics, and TxYxB1C)  
%   optometry to describe functions on a circular domain. 3S-nsMs.  
% nT0FonK>  
%   The following table lists the first 15 Zernike functions. }LNpr  
% L Ty [)  
%       n    m    Zernike function           Normalization gqaENU>  
%       -------------------------------------------------- OLc/Vij;  
%       0    0    1                                 1 n&=3Knbd@d  
%       1    1    r * cos(theta)                    2 8CxC`*L(  
%       1   -1    r * sin(theta)                    2 &eQF[8 ,  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ^tIi;7k  
%       2    0    (2*r^2 - 1)                    sqrt(3) 00'R1q4  
%       2    2    r^2 * sin(2*theta)             sqrt(6) e,qc7BJzK  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 2G8f4vsC[  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) %|[+\py$Q  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) B:=*lU.n  
%       3    3    r^3 * sin(3*theta)             sqrt(8) u>j:8lhtV  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) D +/27#  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Pe w-6u"  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) d-g&TSGd  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4r!8_$fN?G  
%       4    4    r^4 * sin(4*theta)             sqrt(10) BlQu9{=n  
%       -------------------------------------------------- N3Ub|$}q  
% ajuwP1I  
%   Example 1: q9w6 6R  
% 9u/"bj  
%       % Display the Zernike function Z(n=5,m=1) )/h~csy:~  
%       x = -1:0.01:1; xtyzy@)QL  
%       [X,Y] = meshgrid(x,x); K oPTY^  
%       [theta,r] = cart2pol(X,Y); ]R/VE"-  
%       idx = r<=1; 89:Ys=  
%       z = nan(size(X)); m M!H}|  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2E^zQ>;01  
%       figure "gXz{$q  
%       pcolor(x,x,z), shading interp `#hdb=3  
%       axis square, colorbar |HXI4 MU"  
%       title('Zernike function Z_5^1(r,\theta)') \3(d$_:b  
% ;Y#~2eYCz  
%   Example 2: T_O\L[]p*  
% 2~+_T  
%       % Display the first 10 Zernike functions r#wMd9])  
%       x = -1:0.01:1; FA ?xp1E  
%       [X,Y] = meshgrid(x,x); ]Kb  
%       [theta,r] = cart2pol(X,Y); }!b9L]  
%       idx = r<=1; _B)LRD+Hj  
%       z = nan(size(X)); 2xH9O{  
%       n = [0  1  1  2  2  2  3  3  3  3]; ZKyK#\v<  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; mmm025.   
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; dL'hC#!h  
%       y = zernfun(n,m,r(idx),theta(idx)); l?v-9l M  
%       figure('Units','normalized') QA\eXnR  
%       for k = 1:10 iIu  
%           z(idx) = y(:,k); CXGq>cQ=d  
%           subplot(4,7,Nplot(k)) VZ{aET!  
%           pcolor(x,x,z), shading interp 09`5<9/  
%           set(gca,'XTick',[],'YTick',[]) a9qB8/Gg[  
%           axis square t0p^0   
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #q40  >)]  
%       end S P)$K=  
% /|Za[  
%   See also ZERNPOL, ZERNFUN2. &yv%"BPV  
,/{mRw%  
 "| V{@)!t  
%   Paul Fricker 11/13/2006 g4 _DEBh  
N7k<q=r-  
 w~QUG^0Fx  
O`U&0lKi'  
 @47MJzC  
% Check and prepare the inputs: o0^'x Vv  
% ----------------------------- \[oU7r}?/V  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !EuU @ +  
    error('zernfun:NMvectors','N and M must be vectors.') 'WkDp a  
end EAp6IhW{  
 FqAW><  
 \2)a.2mAz  
if length(n)~=length(m) gUzCDB^.:  
    error('zernfun:NMlength','N and M must be the same length.') iD#HB o  
end )h&s.k  
1$e z}k,  
 [TvH7ott'1  
n = n(:); {;]:}nA  
m = m(:); 8=OK8UaU  
if any(mod(n-m,2)) fw,ruROqD  
    error('zernfun:NMmultiplesof2', ... 'm9f:iTr  
          'All N and M must differ by multiples of 2 (including 0).') 7 N+;K0  
end n}PK0  
)vO;=% GQ  
 ~` v 7  
if any(m>n) V*xT5TljS-  
    error('zernfun:MlessthanN', ... z|[#6X6tT  
          'Each M must be less than or equal to its corresponding N.') aW]!$  
end ,A9pj k'  
qN}kDT  
 2 |w;4  
if any( r>1 | r<0 ) I <`9ANe  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') G=a.Wff  
end Z{RRhJ  
$Z(fPKRN/  
 ]YYjXg}%  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )[Bwr bn  
    error('zernfun:RTHvector','R and THETA must be vectors.') [RG&1~  
end /-JBz U$  
XbdoTriE  
 Yf >SV #  
r = r(:); t_5b  
theta = theta(:); q 1a}o%  
length_r = length(r); YUd*\_  
if length_r~=length(theta) Vc|r(lM  
    error('zernfun:RTHlength', ... J;4x-R$W  
          'The number of R- and THETA-values must be equal.') "H\'4'hg  
end }yCJ#}  
9.ZhkvR4A  
 "f\2/4EIl  
% Check normalization: 'gd3 w~  
% -------------------- [?$ZB),L8  
if nargin==5 && ischar(nflag) 2)]C'  
    isnorm = strcmpi(nflag,'norm'); 6r"uDV #0  
    if ~isnorm B MU@J  
        error('zernfun:normalization','Unrecognized normalization flag.') TtEc~m  
    end YgiwtZ5FY  
else rBLkowDP*  
    isnorm = false; =ZM#_uW  
end Y, K): ~T  
gv$6\1  
 o8\@R  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V!G&Aen  
% Compute the Zernike Polynomials <y1V2Np  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q/r0p>  
X||Z>w}v  
 6J0HaL  
% Determine the required powers of r: -IhFPjQ  
% ----------------------------------- .QOQqU*2I  
m_abs = abs(m); .b>1u3  
rpowers = []; %J4]T35^2  
for j = 1:length(n) _KiaeVE  
    rpowers = [rpowers m_abs(j):2:n(j)];  ,!_  
end "8 |y  
rpowers = unique(rpowers); 'SF+P)Kmz  
(.\GI D+i  
 B;t U+36nM  
% Pre-compute the values of r raised to the required powers, |,M&ks  
% and compile them in a matrix: 3;=nQ{0b  
% ----------------------------- h+F@apUS  
if rpowers(1)==0 ?6.vd]oNO  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8>a/x,  
    rpowern = cat(2,rpowern{:}); n's3!HQY[  
    rpowern = [ones(length_r,1) rpowern]; W Da;wt  
else 3U=q3{%1  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cC w,b]  
    rpowern = cat(2,rpowern{:}); YAnt}]u!"  
end L(Q v78F  
Q(h,P+  
 EJY[M  
% Compute the values of the polynomials: p#~' xq  
% -------------------------------------- Im%|9g;P  
y = zeros(length_r,length(n)); [^t"Hf  
for j = 1:length(n) ,}F2l|x_  
    s = 0:(n(j)-m_abs(j))/2; j{N;2#.u  
    pows = n(j):-2:m_abs(j); tVQfR*=  
    for k = length(s):-1:1 t]{qizfOB  
        p = (1-2*mod(s(k),2))* ... \V`O-wcJ]S  
                   prod(2:(n(j)-s(k)))/              ... hKjvD.6]%  
                   prod(2:s(k))/                     ... _`Ey),c_  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... eU_|.2  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Z y@35;r  
        idx = (pows(k)==rpowers); 5} |O  
        y(:,j) = y(:,j) + p*rpowern(:,idx); {;^boo q  
    end k9UmTvX  
     2#&9qGR  
    if isnorm D4'"GaCv  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (WiA  
    end GyJp! xFB  
end ^T"9ZBkb  
% END: Compute the Zernike Polynomials V[,/Hw~d%  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8yax.N j  
=:`1!W0I  
 pVn 6>\xa  
% Compute the Zernike functions: U,)Ngnd  
% ------------------------------ A@*P4E`xp  
idx_pos = m>0; VpMpZ9oM<  
idx_neg = m<0; mH*42XC*  
b_ Sh#d&  
 >JS\H6  
z = y; Pgf$GXE  
if any(idx_pos) *{tn/ro6a  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); FOpOS?Cr'  
end !Jb?r SJ.h  
if any(idx_neg) ? Ldw\  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V S2p"0$3D  
end @]tFRV  
VA4vAF  
 K @"m0  
% EOF zernfun KrVF>bq+  
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)  wED~^[]f  
R_uA!MoLs  
DDE还是手动输入的呢？ !OPK?7   
=NAL*4c+  
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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