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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， xaSiG  
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， 1\,wV,  
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？ <_Po/a!c3  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ b WZ X  
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function z = zernfun(n,m,r,theta,nflag) 5adB5)`  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. A832z`  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Uefw  
%   and angular frequency M, evaluated at positions (R,THETA) on the VrRBwvp-K  
%   unit circle.  N is a vector of positive integers (including 0), and k{$Mlt?&-  
%   M is a vector with the same number of elements as N.  Each element Riz!HtyR  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ;6zp,t0  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (V~PYf%  
%   and THETA is a vector of angles.  R and THETA must have the same .We"j_ }  
%   length.  The output Z is a matrix with one column for every (N,M) x~O_v  
%   pair, and one row for every (R,THETA) pair. &5wM`  
% OK9D4 7X  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *(@[E  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), b<rJ@1qtJ  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral v:] AS:  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, = l9H]`T/  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 80ms7 B  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. vV9q5Bj:  
% SA$1rqU=  
%   The Zernike functions are an orthogonal basis on the unit circle. cS1BB#N0  
%   They are used in disciplines such as astronomy, optics, and wq&TU'O  
%   optometry to describe functions on a circular domain. ~v<,6BS<$Z  
% nM)H2'%kL&  
%   The following table lists the first 15 Zernike functions. ~cx/>Hu  
% sh"\ kk9  
%       n    m    Zernike function           Normalization pn~$u  
%       -------------------------------------------------- H0B"?81  
%       0    0    1                                 1 DV/P/1E  
%       1    1    r * cos(theta)                    2 $.@)4Nu!_  
%       1   -1    r * sin(theta)                    2 0Sz iTM  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) N^. !l_  
%       2    0    (2*r^2 - 1)                    sqrt(3) xcYYo'U  
%       2    2    r^2 * sin(2*theta)             sqrt(6) [0e}%!%M  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) L);kwx7{LW  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) f&ym'S  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Gv}h/zu-  
%       3    3    r^3 * sin(3*theta)             sqrt(8) DNaU mz  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) "8 "7AoE  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7MT[fA8^  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) i'%:z]hp9  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yVM 1W"Q  
%       4    4    r^4 * sin(4*theta)             sqrt(10) s],+]<qX  
%       -------------------------------------------------- n300kpv  
% ,Mwj`fgh  
%   Example 1: <3>Ou(F  
% cwxO| .m  
%       % Display the Zernike function Z(n=5,m=1) `?VB)  
%       x = -1:0.01:1; { LJRdV  
%       [X,Y] = meshgrid(x,x); bg)yl iX  
%       [theta,r] = cart2pol(X,Y); 'I_\ELb_  
%       idx = r<=1; ?8X+)nU@  
%       z = nan(size(X)); t$Z#zx X  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); M+ gYKPP  
%       figure Q[y75 [  
%       pcolor(x,x,z), shading interp `1KZ14K  
%       axis square, colorbar ,g$N  
%       title('Zernike function Z_5^1(r,\theta)') KPUc+`cN%  
% :R<n{%~  
%   Example 2: 4PEJ}B W  
% #$Z|)i]w  
%       % Display the first 10 Zernike functions @"H+QVJ@  
%       x = -1:0.01:1; !58-3F%P  
%       [X,Y] = meshgrid(x,x); 16YJQ ue  
%       [theta,r] = cart2pol(X,Y); @fbB3  
%       idx = r<=1; .Tdl'y:..  
%       z = nan(size(X)); 4y+] V~p  
%       n = [0  1  1  2  2  2  3  3  3  3]; E#T-2^nD  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; U&/Jh^Yy  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; o=2y`Eq  
%       y = zernfun(n,m,r(idx),theta(idx)); xgtdmv%  
%       figure('Units','normalized') _~DFZt@T  
%       for k = 1:10 % j7lLSusX  
%           z(idx) = y(:,k); c|Nv^V*2  
%           subplot(4,7,Nplot(k)) R/iw#.Yy  
%           pcolor(x,x,z), shading interp X .g")Bt7  
%           set(gca,'XTick',[],'YTick',[]) ?iUAzM8  
%           axis square J%;TK6  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |_%|  
%       end #oi4!%*M  
% gR(*lXm5w  
%   See also ZERNPOL, ZERNFUN2. 5sx-u!7  
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 >b |l6 #%  
%   Paul Fricker 11/13/2006 5Y)!q?#H  
_#e='~;  
   ]5'  
Srz.-,2PF  
 ]`y4n=L.  
% Check and prepare the inputs: ~-6Kl3Y  
% ----------------------------- 6pQ#Zg()vp  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Tfgx>2  
    error('zernfun:NMvectors','N and M must be vectors.') I, .`w/I+  
end >GgX-SZ%  
%"DEgI P  
 u /cL[_Q  
if length(n)~=length(m) 6A/Nlk.  
    error('zernfun:NMlength','N and M must be the same length.') ID5?x8o#k  
end S0g5Ym ia  
Lqbu]  
 3`k 1  
n = n(:); 7##nY3",^  
m = m(:); t[F tIj6  
if any(mod(n-m,2)) GOa](oD}  
    error('zernfun:NMmultiplesof2', ... f 7et  
          'All N and M must differ by multiples of 2 (including 0).') sN 7I~  
end .7Ys@;>B  
Y 1Bj++?2  
 Uz\B^"i|  
if any(m>n) )AqM?FE4R  
    error('zernfun:MlessthanN', ... ,ibI@8;#~'  
          'Each M must be less than or equal to its corresponding N.') g^ ^%4Y  
end EUe2<G  
`t:7&$>T  
 <2ffcBv  
if any( r>1 | r<0 ) 1?)<*[  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') -^$CGRE6A  
end }!& w<wR  
r[ ' T.yo  
 u sR19_E-  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) B8|=P&L7N  
    error('zernfun:RTHvector','R and THETA must be vectors.') Fkz+Qz  
end =q^o6{d0"  
C 1|e1  
 X`:(-3T  
r = r(:); l?a(=  
theta = theta(:); A`=ESz  
length_r = length(r); g;\zD_":l  
if length_r~=length(theta) R/b)hP ~  
    error('zernfun:RTHlength', ... ).N}x^  
          'The number of R- and THETA-values must be equal.') Z,,Da|edH  
end iyu%o9_0  
@*q\$Eg}2  
 ?9v!UT&#  
% Check normalization: c"H4/,F  
% -------------------- cIja^xD  
if nargin==5 && ischar(nflag) L`x:Y>C(  
    isnorm = strcmpi(nflag,'norm'); WaN0$66[:  
    if ~isnorm mv SNKS  
        error('zernfun:normalization','Unrecognized normalization flag.') X+P& up06  
    end (bH"x  
else .-`7Av+7  
    isnorm = false; b\][ x6zJp  
end .+ai dWd  
w^p 'D{{  
 i{T0[\4  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kdQ=%  
% Compute the Zernike Polynomials =NF},j"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6O$OM  
}N2T/U  
 Kdx?s;i  
% Determine the required powers of r: ECg/ge2  
% ----------------------------------- 4'Vuhqk  
m_abs = abs(m); _9#4  
rpowers = []; z:RwCd1\  
for j = 1:length(n) 2y ~]Uo  
    rpowers = [rpowers m_abs(j):2:n(j)]; rA8neO)  
end ~Rk6@&ZS}  
rpowers = unique(rpowers); =o{zw+|% %  
vay_QxB5  
 @w,-T@nAW  
% Pre-compute the values of r raised to the required powers, 9j:?s;B  
% and compile them in a matrix: ` B : Ydf  
% ----------------------------- exTpy  
if rpowers(1)==0 O#Xq0o  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); UG&/0{j5XV  
    rpowern = cat(2,rpowern{:}); Z\(+awv  
    rpowern = [ones(length_r,1) rpowern]; G:c)e ,pD  
else 2ztP'  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !(uyqplTk  
    rpowern = cat(2,rpowern{:}); h+,zfVJu  
end ?%;7k'0"  
yFl@ z  
 Rc0OEs%7P  
% Compute the values of the polynomials: 1f~unb\Gg  
% -------------------------------------- ud'r ?QDM  
y = zeros(length_r,length(n)); p!|Wp  
for j = 1:length(n) vs7Hg )F  
    s = 0:(n(j)-m_abs(j))/2; 9N5 &N3  
    pows = n(j):-2:m_abs(j); 3)atqM)i  
    for k = length(s):-1:1 k/j]*~"  
        p = (1-2*mod(s(k),2))* ... mAk)9`f/  
                   prod(2:(n(j)-s(k)))/              ... t$]lK6  
                   prod(2:s(k))/                     ... &=<x&4H+  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... p8%x@%k  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); E2LpQNvN%g  
        idx = (pows(k)==rpowers); p r(:99~3  
        y(:,j) = y(:,j) + p*rpowern(:,idx); G9N6iKP!  
    end 3"6lPUS  
     r*&gd|sn  
    if isnorm LUHj3H  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w%S\)wjS  
    end 80![aj}z4G  
end BV9B}IV  
% END: Compute the Zernike Polynomials O<Ht-TN&  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9Fv1D  
)f*&}SV  
 2%. A{!  
% Compute the Zernike functions: 2?z3s|+[  
% ------------------------------ x: `oqbd  
idx_pos = m>0; 9=ns.r  
idx_neg = m<0; 7xO =:*  
 pzg|?U  
 -3_-n*k!  
z = y; (Z,v)TOXjV  
if any(idx_pos) rt_%_f>qd  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #[qmhU{s  
end 4~P{H/]  
if any(idx_neg) W[AX?  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); pBL,kqYNA>  
end qTj7mUk  
Xg^`fRg =T  
 ;8'hvc3i$  
% EOF zernfun !0zbWB9  
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)  pM],-7UM  
UbNA|`H  
DDE还是手动输入的呢？ ROlef;/A  
 J -tOO  
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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