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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， !0ly1T 9  
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， TU ]Ed*'&  
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？ a<X8l^Ln  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 49f- u  
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function z = zernfun(n,m,r,theta,nflag) (IX iwu  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. qW]gp7jK4  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N p:M#F:  
%   and angular frequency M, evaluated at positions (R,THETA) on the x_9<&Aj6  
%   unit circle.  N is a vector of positive integers (including 0), and nb(4"|8}  
%   M is a vector with the same number of elements as N.  Each element =! v.VF\;  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ?wE@9 g A  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, =PHl|^  
%   and THETA is a vector of angles.  R and THETA must have the same "8K>Yu17  
%   length.  The output Z is a matrix with one column for every (N,M) VM{`CJ2  
%   pair, and one row for every (R,THETA) pair. vQrce&  
% 1xK'1g72  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike xsK{nM6g  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TJ(PTB;  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral "%~\kJ(G  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V~LZ%NZ8  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized & pwSd  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $iQ>c6  
% t}-[^|)7  
%   The Zernike functions are an orthogonal basis on the unit circle. yu"Ii-9z  
%   They are used in disciplines such as astronomy, optics, and lhg3 }dW  
%   optometry to describe functions on a circular domain. tf64<j6  
% :0o $qz2  
%   The following table lists the first 15 Zernike functions. #1R_* Uh  
% &{QB}r  
%       n    m    Zernike function           Normalization n<MMO=+bg  
%       -------------------------------------------------- ) Kfk\  
%       0    0    1                                 1 ~ `2w ul  
%       1    1    r * cos(theta)                    2 f uojf+i  
%       1   -1    r * sin(theta)                    2 Vzy]N6QT{  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) :QgC Zq  
%       2    0    (2*r^2 - 1)                    sqrt(3) 3{_AzL  
%       2    2    r^2 * sin(2*theta)             sqrt(6) PpMZ-f@  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 8>x.zO_.c>  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 2=ZR}8}9Q:  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Rde_I`Ru  
%       3    3    r^3 * sin(3*theta)             sqrt(8) J*6I@_{/ U  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Ab7hW(/  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T#Pz_ hAu  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }]vj"!?a  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O;M_?^'W  
%       4    4    r^4 * sin(4*theta)             sqrt(10) KsYT3  
%       -------------------------------------------------- IV^LYu  
% FtN1ZZ"<*  
%   Example 1: j)\&#g0u6  
% O<4i)Lx2  
%       % Display the Zernike function Z(n=5,m=1) .jMm-vox}  
%       x = -1:0.01:1; _dqjRhu  
%       [X,Y] = meshgrid(x,x); `XYT:'   
%       [theta,r] = cart2pol(X,Y); #1Mk9sxo  
%       idx = r<=1; cJA0$)JP&  
%       z = nan(size(X)); qx 3.oU  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); :b>Z|7g?  
%       figure /Nq!^=  
%       pcolor(x,x,z), shading interp ?97MW a   
%       axis square, colorbar _yjM_ALjo  
%       title('Zernike function Z_5^1(r,\theta)') (Br$(XJoK}  
% Orh5d 7+S  
%   Example 2: &n<jpMB  
% ]SrKe-*:U  
%       % Display the first 10 Zernike functions IcL3.(!]l  
%       x = -1:0.01:1; Td[w<m+p<P  
%       [X,Y] = meshgrid(x,x); qt]QO1pAd  
%       [theta,r] = cart2pol(X,Y); +C;ZO6%w  
%       idx = r<=1; fEs957$  
%       z = nan(size(X)); 5!#"8|oY  
%       n = [0  1  1  2  2  2  3  3  3  3]; )xQxc.  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; J'9&dt  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; GQqw(2Ub}  
%       y = zernfun(n,m,r(idx),theta(idx)); xy mK|  
%       figure('Units','normalized') |}Mkn4  
%       for k = 1:10 $Br^c< y  
%           z(idx) = y(:,k); x+B~t4A  
%           subplot(4,7,Nplot(k)) N=D Ynz_~  
%           pcolor(x,x,z), shading interp :'y  
%           set(gca,'XTick',[],'YTick',[]) @usQ*k  
%           axis square ]+0-$t7Y  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) y NV$IN%  
%       end PL~k `L  
% Oi BK  
%   See also ZERNPOL, ZERNFUN2. t)XV'J  
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 1@)]+* F*z  
%   Paul Fricker 11/13/2006 ]p'Qk  
CcY.8|HT  
 } Q1$v~  
v{ C]\8  
 uNd;; X  
% Check and prepare the inputs: p5F[( H|9  
% ----------------------------- vCH>Fj"7  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9Z*`{  
    error('zernfun:NMvectors','N and M must be vectors.') H'gPGOd  
end *XH?|SV  
|D]jdd@!a2  
 Jr17pu(t  
if length(n)~=length(m) aS~k.^N  
    error('zernfun:NMlength','N and M must be the same length.') KXw \N!  
end tB(Q-c  
tHoFnPd\|  
 m.K"IXD  
n = n(:); Rp`}"x9  
m = m(:); @Jvw"=  
if any(mod(n-m,2)) @TgCI`E   
    error('zernfun:NMmultiplesof2', ... 9q* sR1  
          'All N and M must differ by multiples of 2 (including 0).') }QJE9;<e  
end Y2<#%@%4  
aF'Ik XG d  
 J=zZGd%  
if any(m>n) nWXI*%m5  
    error('zernfun:MlessthanN', ... K:'pK1zy  
          'Each M must be less than or equal to its corresponding N.') |lJXI:G G  
end =Rb,`%  
00;=6q]TA  
 $6y1';A  
if any( r>1 | r<0 ) ;uoH+`pf  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') ][G<CO`k  
end ybS7uo  
 ~-M7  
 c"O\fX  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |{ 9"n<JW  
    error('zernfun:RTHvector','R and THETA must be vectors.') O)9T|, U  
end @Wx_4LOhf  
a S<JsB  
 Z]SCIU @+  
r = r(:); HwU \[f  
theta = theta(:); QyHUuG|g  
length_r = length(r); +!_^MBkk  
if length_r~=length(theta) /o|@]SAe.  
    error('zernfun:RTHlength', ... XLmbpEh  
          'The number of R- and THETA-values must be equal.') j,1,;  
end v11mu2  
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 a[(n91J0  
% Check normalization: '|FM|0~-J  
% -------------------- 3[V|C=u0  
if nargin==5 && ischar(nflag) @F,HyCSN  
    isnorm = strcmpi(nflag,'norm'); 'kh%^_FH7  
    if ~isnorm L\-T[w),z7  
        error('zernfun:normalization','Unrecognized normalization flag.') Qy6Avw/$  
    end #Jm_~k  
else CS"p[-0  
    isnorm = false; ^2mXXAQf7^  
end QGLm4 Wl9  
Z&.FJZUP  
 "\>3mVOb  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x&9 I2"  
% Compute the Zernike Polynomials &rNXn?>b  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $iP#8La:Y  
*g=*}2  
 Q]|+Y0y}X  
% Determine the required powers of r: y!v$5wi  
% ----------------------------------- mB1)!  
m_abs = abs(m); MiSFT5$v6  
rpowers = []; 9s7B1Pf  
for j = 1:length(n) \DQ;v  
    rpowers = [rpowers m_abs(j):2:n(j)]; tXp)o >"  
end KX9ZwsC0  
rpowers = unique(rpowers); |/<iydP  
Wc]Fg9E  
 \\/X+4|o'  
% Pre-compute the values of r raised to the required powers, gf3/kll9  
% and compile them in a matrix: 1i>)@{P&BN  
% ----------------------------- S((8DSt*  
if rpowers(1)==0 ?&X6VNbU  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pixI&iQ  
    rpowern = cat(2,rpowern{:}); "^trHh8=  
    rpowern = [ones(length_r,1) rpowern]; HFDg@@  
else nB:Bw8U"Q  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b#y}VY)?  
    rpowern = cat(2,rpowern{:}); qh&K{r*T  
end ~cZ1=,P  
[]Fy[G.)H  
 R -h7c!ko  
% Compute the values of the polynomials: 3$kZu  
% -------------------------------------- 'rF TtT  
y = zeros(length_r,length(n)); L`Ic0}|lzy  
for j = 1:length(n) A5/h*`Q\\  
    s = 0:(n(j)-m_abs(j))/2; -!}1{   
    pows = n(j):-2:m_abs(j); <y'ttxeS  
    for k = length(s):-1:1 |aVv Lz  
        p = (1-2*mod(s(k),2))* ... u(~s$ENl  
                   prod(2:(n(j)-s(k)))/              ... Ec[:6}  
                   prod(2:s(k))/                     ... xp &I~YPH  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xj~6,;83xR  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); qMUqd}=P  
        idx = (pows(k)==rpowers); w%ip"GT,  
        y(:,j) = y(:,j) + p*rpowern(:,idx); B#gmT2L  
    end "*T)L<G  
     },"g*  
    if isnorm 1rKR=To  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); asJYGqdF  
    end e-s@@k  
end NKGCz|- 9  
% END: Compute the Zernike Polynomials Hv=coS>g:  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vd ;wQ  
81n%2G  
 %sq=lW5R{b  
% Compute the Zernike functions: K)14v;@  
% ------------------------------ 4-"wFp  
idx_pos = m>0; [L\w] 6  
idx_neg = m<0; O]Hg4">f  
'|cuVxcE55  
 BNByaC  
z = y; ^g0 Ig2'  
if any(idx_pos) ysa"f+/  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); u)V*o  
end }E=kfMu  
if any(idx_neg) J\%:jg( m  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e,x@?L*  
end 0N}5sF  
sDF J  
 h}oQr0"c  
% EOF zernfun ::R^ w"  
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)  Mz^s^aJEE  
KFhn}C3 i  
DDE还是手动输入的呢？  LJ))  
? 1{S_  
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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