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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， =nw,*q +  
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， 8x` Kl(  
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？ )|MIWgfWN  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ t4C<#nfo  
P~n8EO1r  
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function z = zernfun(n,m,r,theta,nflag) 6zELe.tq  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xNocGtS  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7=; D0SS  
%   and angular frequency M, evaluated at positions (R,THETA) on the 5/zf x  
%   unit circle.  N is a vector of positive integers (including 0), and QZ6[*_Z6  
%   M is a vector with the same number of elements as N.  Each element 6sO  
%   k of M must be a positive integer, with possible values M(k) = -N(k) $@5%5  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 'nC3:U  
%   and THETA is a vector of angles.  R and THETA must have the same #_?426Wfs  
%   length.  The output Z is a matrix with one column for every (N,M) dx k;@Tz  
%   pair, and one row for every (R,THETA) pair. >Iu]T{QNO  
% s@.`"TF.7  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )w^GP lh  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A%.J%[MVz  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral />2A<{6\=P  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0\8*S3,q  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized uEc0/ a :.  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /8 e2dw: \  
% 6~:W(E}  
%   The Zernike functions are an orthogonal basis on the unit circle. /5L'9e  
%   They are used in disciplines such as astronomy, optics, and tjBh$)  
%   optometry to describe functions on a circular domain. n1fE daa7g  
% *%P>x}6w3  
%   The following table lists the first 15 Zernike functions. !V$6+?2   
% UrD=|-r`  
%       n    m    Zernike function           Normalization vLi/'|7  
%       -------------------------------------------------- &/J.0d-*``  
%       0    0    1                                 1 7.w *+Z>z  
%       1    1    r * cos(theta)                    2 ] P:NnKgK  
%       1   -1    r * sin(theta)                    2 ~0' _K1(H  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ^&f{beU9  
%       2    0    (2*r^2 - 1)                    sqrt(3) P5yJO97  
%       2    2    r^2 * sin(2*theta)             sqrt(6) {[YqGv=fF  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) BLl%D  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) tdMP,0u  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) s0~05{  
%       3    3    r^3 * sin(3*theta)             sqrt(8) I?^Q084  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) q\ \8b{~  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1]D/3!  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) kxr6sO~  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) XwHu:v'=  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ! 0>!tW  
%       -------------------------------------------------- }QX2 :a  
% [[/ }1%  
%   Example 1: Mhu53DT  
% J]kP`  
%       % Display the Zernike function Z(n=5,m=1) !0!P.Q8>&  
%       x = -1:0.01:1; )V9Mcr*Ce6  
%       [X,Y] = meshgrid(x,x); 1)P<cNj  
%       [theta,r] = cart2pol(X,Y); []6ShcqJ[v  
%       idx = r<=1; FcA)RsMI*  
%       z = nan(size(X)); Allt]P>  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); }(f.uN_v  
%       figure V8KTNt%  
%       pcolor(x,x,z), shading interp iECC@g@a  
%       axis square, colorbar zezofW]a  
%       title('Zernike function Z_5^1(r,\theta)') !R] CmK  
% Tv*1q.MB  
%   Example 2: 0,"n-5Im  
% mCC:}n"#  
%       % Display the first 10 Zernike functions G>_42Rp  
%       x = -1:0.01:1; "FLD%3l  
%       [X,Y] = meshgrid(x,x); yPzULO4  
%       [theta,r] = cart2pol(X,Y); X d19GP!  
%       idx = r<=1; [+:mt</HN  
%       z = nan(size(X)); 8zWBXV  
%       n = [0  1  1  2  2  2  3  3  3  3]; OxmlzQ"vM  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %(dV|,|v  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; m"?' hR2  
%       y = zernfun(n,m,r(idx),theta(idx)); Hd=D#u=A4{  
%       figure('Units','normalized') 3u"J4%zg|L  
%       for k = 1:10 fRv S@  
%           z(idx) = y(:,k); NejsI un%  
%           subplot(4,7,Nplot(k)) DS[l,x  
%           pcolor(x,x,z), shading interp YfrTvKX  
%           set(gca,'XTick',[],'YTick',[]) SS45<!i y  
%           axis square $&n240(  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) mNBpb}  
%       end r=P$iG'&  
% _I -0,  
%   See also ZERNPOL, ZERNFUN2. @tjZvRtZ  
%DND&0`  
 B 4*X0x  
%   Paul Fricker 11/13/2006 )l[7;ZIw$  
oRvm*"8B  
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 <'4!G"_EP  
% Check and prepare the inputs: eqUn8<<s  
% ----------------------------- D\_*,Fc  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O+8ApicjTc  
    error('zernfun:NMvectors','N and M must be vectors.') EDa08+Y  
end K9z_=c+  
Ie`SWg*WL  
 Vp{RX8?.  
if length(n)~=length(m) Lct+cKKU  
    error('zernfun:NMlength','N and M must be the same length.') >{LJ#Dc6  
end 8HH.P`Vk#  
GD6'R"tJ  
 /kviO@jm4(  
n = n(:); tWX+\ |  
m = m(:); M;Mdz[Q  
if any(mod(n-m,2)) wHN` - 5%  
    error('zernfun:NMmultiplesof2', ... UNZVu~WnF  
          'All N and M must differ by multiples of 2 (including 0).')  h?pGw1Q  
end sVm'9k  
I!0$% ]F  
 r^o}Y  
if any(m>n) H @&"M%  
    error('zernfun:MlessthanN', ... k}Clq;G  
          'Each M must be less than or equal to its corresponding N.') +IOKE\,Y  
end qU x7S(a  
Lw2YP[CR  
 "*>QxA%c4  
if any( r>1 | r<0 ) dE9aE#o  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') uwS'*5tU  
end fY+ .#V  
J<P/w%i2  
 :#!F 7u  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) MgK(gL/&[  
    error('zernfun:RTHvector','R and THETA must be vectors.') 8KdcLN@  
end =B{$U~}  
7[?}kG   
 Z`1o#yZ  
r = r(:); CPCB!8-5  
theta = theta(:); 4#Nd;gM2  
length_r = length(r); qI%9MI;BV  
if length_r~=length(theta) Y8CYkJTAD-  
    error('zernfun:RTHlength', ... }y=n#%|i.  
          'The number of R- and THETA-values must be equal.') []fj~hj  
end XuAc3~HAd  
W,oV$ s^  
 1p5q}">z  
% Check normalization: ah@GSu;7  
% -------------------- >8HRnCyp/  
if nargin==5 && ischar(nflag) FU v)<rK  
    isnorm = strcmpi(nflag,'norm'); w7ABnX  
    if ~isnorm _q!ck0_  
        error('zernfun:normalization','Unrecognized normalization flag.') ojs/yjvx  
    end H-y-7PW*~  
else F9G$$%Q-Z  
    isnorm = false; YwTtI ID%  
end _@ 3O`  
"kuBjj2  
 ewlc ^`  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BO cEL%+  
% Compute the Zernike Polynomials 2!& ;ZcT,  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `fE:5y  
M(f*hOG{Y  
 ;0}"2aGY  
% Determine the required powers of r: .;sPG  
% ----------------------------------- a{YVz\?d}  
m_abs = abs(m); B!C32~[  
rpowers = []; Qz90 mb  
for j = 1:length(n) oM7-1O  
    rpowers = [rpowers m_abs(j):2:n(j)]; YO4ppL~xe  
end KE1@z]  
rpowers = unique(rpowers); [~H`9Ab=  
;iI2K/ 3  
 FQ&VM6_  
% Pre-compute the values of r raised to the required powers, ^\t">NJ^  
% and compile them in a matrix: GnHf9 JrR  
% ----------------------------- ll^O+>1dO  
if rpowers(1)==0 4>eg@sN  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); JZ0+VB-3U  
    rpowern = cat(2,rpowern{:}); `)_FO]m}jS  
    rpowern = [ones(length_r,1) rpowern]; :<G+)hIK  
else T{2//$T?  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,%^0 4sl  
    rpowern = cat(2,rpowern{:}); ^~od*:  
end }#M|3h;q9+  
UYUd IIoL  
 }Q_i#e(S  
% Compute the values of the polynomials: cPSpPx  
% -------------------------------------- G_@H:4$3  
y = zeros(length_r,length(n)); N3)EG6vE*  
for j = 1:length(n) Os;\\~e5  
    s = 0:(n(j)-m_abs(j))/2; ,`b9c=6;  
    pows = n(j):-2:m_abs(j); D1a4+AyI  
    for k = length(s):-1:1 #e[5O| V~  
        p = (1-2*mod(s(k),2))* ... V7<} ;Lzm  
                   prod(2:(n(j)-s(k)))/              ... Nt?B(.G  
                   prod(2:s(k))/                     ... gB#t"s)  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ty ?y&~axk  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); jC=_>\<|X*  
        idx = (pows(k)==rpowers);  LvaF4Y2v  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Qpc>5p![3  
    end $I%]jAh6  
     xe' *%3-v)  
    if isnorm %!RQ:?=  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (nm&\b~j  
    end ;pJ7k23(  
end tFCeE=4%  
% END: Compute the Zernike Polynomials S_zE+f+ 2  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3$TpI5A  
$= gv  
 IN"qJ3<k  
% Compute the Zernike functions: "cZ.86gG`:  
% ------------------------------ y!j1xnzki  
idx_pos = m>0; *hba>LZ  
idx_neg = m<0; ]4PG[9J@  
'"6VfF)*  
 r BaK$Ut  
z = y; :hr%iu  
if any(idx_pos) vhKD_}}aP  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 3JwmLGj}  
end aE+E'iL  
if any(idx_neg) >i '3\  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }0I! n@  
end z 'j%.Dd8  
sy~mcH:%+  
 ry:tL0;;e#  
% EOF zernfun <5 okwcJ^  
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)  Dzb@H$BQ7  
zvE]4}VL?  
DDE还是手动输入的呢？ 8g$pfHt|e  
l]GLkE  
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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