[讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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， oQV3  
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， islHtX V
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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？ I3r")}P  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 0<uLQVoR2n  
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function z = zernfun(n,m,r,theta,nflag) @6+_0^  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. N#RC;  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X[$|I9  
%   and angular frequency M, evaluated at positions (R,THETA) on the 0QPY+6  
%   unit circle.  N is a vector of positive integers (including 0), and -bdWG]w"  
%   M is a vector with the same number of elements as N.  Each element &nVekE:!  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ntPj9#lf  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, /=AFle2(  
%   and THETA is a vector of angles.  R and THETA must have the same -:'%YHxX  
%   length.  The output Z is a matrix with one column for every (N,M) 6)ZaK  
%   pair, and one row for every (R,THETA) pair. :*[mvF  
% ._A4:  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h@1/  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), fJ _MuAv  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral cmU0=js.  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T95FoA  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized \h s7>5O^K  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *>zOWocxD  
% :6)!#q'g  
%   The Zernike functions are an orthogonal basis on the unit circle. iR{@~JN=)  
%   They are used in disciplines such as astronomy, optics, and |T"j7  
%   optometry to describe functions on a circular domain. mC\<fo-u  
% ;w]1H&mc*A  
%   The following table lists the first 15 Zernike functions. EXH,+3fQp  
% 33eOM(`D[  
%       n    m    Zernike function           Normalization JgP%4)]LV  
%       -------------------------------------------------- sm G?y
~  
%       0    0    1                                 1 {&D$U'ye  
%       1    1    r * cos(theta)                    2 gN(kRhp  
%       1   -1    r * sin(theta)                    2 N.]~%)K:{  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) lWJYT<kt  
%       2    0    (2*r^2 - 1)                    sqrt(3) jgXr2JQ<  
%       2    2    r^2 * sin(2*theta)             sqrt(6) "d~<{(:N^  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) l& sEdEA  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \SwqBw  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @V1FBw9S!@  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ouoIbA9X  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Yh"9,Z&wiR  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \}=W*xxB  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }z\t}lven  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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%       4    4    r^4 * sin(4*theta)             sqrt(10) a5pM~.]  
%       -------------------------------------------------- ;"9Ks.  
% ~=HPqe8  
%   Example 1: Zg4wd/y?  
% J &=5h.G$  
%       % Display the Zernike function Z(n=5,m=1) /J!hKK^k  
%       x = -1:0.01:1; ^:f)XZ  
%       [X,Y] = meshgrid(x,x); Iw#[K  
%       [theta,r] = cart2pol(X,Y); PV=sqLM~  
%       idx = r<=1; !&@t  
%       z = nan(size(X)); gr.G']9lNq  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); UaQW<6
+  
%       figure 3y:),;|5  
%       pcolor(x,x,z), shading interp Qch'C0u  
%       axis square, colorbar X ) =-a  
%       title('Zernike function Z_5^1(r,\theta)') 4{6,Sx  
% ?=kH}'igq  
%   Example 2: |Df`Aq(eYJ  
% Y1qbu~!  
%       % Display the first 10 Zernike functions E J6|y'  
%       x = -1:0.01:1; k4:=y9`R}$  
%       [X,Y] = meshgrid(x,x); a~-k} G5  
%       [theta,r] = cart2pol(X,Y); }>YEtA  
%       idx = r<=1; OH 88d:  
%       z = nan(size(X)); wb]Z4/j#  
%       n = [0  1  1  2  2  2  3  3  3  3];
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%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; C'sA0O@O  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; ;1TQr3w  
%       y = zernfun(n,m,r(idx),theta(idx)); YsTF10  
%       figure('Units','normalized') ~"R;p}5"  
%       for k = 1:10 #.OCoc  
%           z(idx) = y(:,k); `<R^ZL,  
%           subplot(4,7,Nplot(k)) ?i~mt'O  
%           pcolor(x,x,z), shading interp =}SC .E\  
%           set(gca,'XTick',[],'YTick',[]) x.(Sv]+[  
%           axis square }~<9*M-P  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \9:IL9~F  
%       end D;DI8.4`N  
% ;,B $lgF  
%   See also ZERNPOL, ZERNFUN2. f+QDjJ?z  
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%   Paul Fricker 11/13/2006 :C#(yp  
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% Check and prepare the inputs: =l}XKl->  
% ----------------------------- )TkXdA?.  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <\\,L@  
    error('zernfun:NMvectors','N and M must be vectors.') 1jj.oa]  
end p+{*&Hm5  
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if length(n)~=length(m) zCpXF<_C  
    error('zernfun:NMlength','N and M must be the same length.') zT _[pa)O`  
end 2?9gf,U  
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n = n(:); P&8QKX3 j^  
m = m(:); 83gp'W{|  
if any(mod(n-m,2)) 3*[YM7y  
    error('zernfun:NMmultiplesof2', ... 9]*hP](  
          'All N and M must differ by multiples of 2 (including 0).') 4.i< `'  
end A>Oi9%OY:  
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if any(m>n) dfA4OZ&  
    error('zernfun:MlessthanN', ... VA@t8H,  
          'Each M must be less than or equal to its corresponding N.') -JB~yO?0  
end ;kiL`K  
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D= LLm$y  
if any( r>1 | r<0 ) y6 _,U/9
 
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') j?cE0 hz  
end bpWEF b'f  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2K&5Kt/  
    error('zernfun:RTHvector','R and THETA must be vectors.') n{v[mqm^  
end $bT<8:g  
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r = r(:); W|aFEY  
theta = theta(:); ?3{:[*  
length_r = length(r); BK16~Wl  
if length_r~=length(theta) Qt+;b  
    error('zernfun:RTHlength', ... yw9)^JU8"  
          'The number of R- and THETA-values must be equal.') uzpW0(_i3a  
end F7~T=X)1
 
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% Check normalization: "TboIABp:H  
% --------------------  UX& ?^]  
if nargin==5 && ischar(nflag) >SR!*3$5  
    isnorm = strcmpi(nflag,'norm'); OL
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    if ~isnorm O0VbKW0h3  
        error('zernfun:normalization','Unrecognized normalization flag.') =6N%;2`84  
    end N-O"y3W}  
else /T_@rm  
    isnorm = false; +jPs0?}s  
end  A/zZ%h  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a*4l!-7  
% Compute the Zernike Polynomials  *Fe  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  @{|vW  
Z+*t=?L,,G  
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% Determine the required powers of r: Pq@%MF]5  
% ----------------------------------- )ZcwG(o0  
m_abs = abs(m); 3{wmKo|_X  
rpowers = []; cR0OJ'w  
for j = 1:length(n) LR5X=&k  
    rpowers = [rpowers m_abs(j):2:n(j)]; >8*J ;(:W  
end v7ShXX:  
rpowers = unique(rpowers); ?IWLH-fkP  
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% Pre-compute the values of r raised to the required powers, VX!UT=;  
% and compile them in a matrix: |NsrO8H   
% ----------------------------- EO<{Bj=2  
if rpowers(1)==0 0pYCh$TL1  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &ZmHR^Flz  
    rpowern = cat(2,rpowern{:}); Q>,EYb>wI  
    rpowern = [ones(length_r,1) rpowern]; @w H+,]xE  
else COF_a%  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m3cO{ 1I  
    rpowern = cat(2,rpowern{:}); ?-y!FD}m&  
end LZ97nvK  
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% Compute the values of the polynomials: mr`Lxy9e  
% -------------------------------------- f^tCD'Vmi  
y = zeros(length_r,length(n)); w4S0aR:yL  
for j = 1:length(n) 'g4t !__  
    s = 0:(n(j)-m_abs(j))/2; !zd]6YL$  
    pows = n(j):-2:m_abs(j); KNkVI K  
    for k = length(s):-1:1 4z_>Ci
A  
        p = (1-2*mod(s(k),2))* ... cZ!%#Az  
                   prod(2:(n(j)-s(k)))/              ... yEaim~  
                   prod(2:s(k))/                     ... t3~ZGOn  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *bl*R';  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); S690Y]:h$v  
        idx = (pows(k)==rpowers); hU:M]O0uw  
        y(:,j) = y(:,j) + p*rpowern(:,idx); n^svRM]eQ  
    end [a\
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     y%y F34  
    if isnorm bJ[{[|yEd  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J?yNZK$WqN  
    end 4`x.d  
end _1s\ztDpw  
% END: Compute the Zernike Polynomials kl0!*j  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q'07  
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?r~|B/]  
% Compute the Zernike functions: }b//oe
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% ------------------------------ FC'v= *  
idx_pos = m>0; W[5a'}OV  
idx_neg = m<0; /M:R|91:_  
uxGY/Zf  
X+'z@xpj  
z = y; !>~W5c^  
if any(idx_pos) 2wHvHH!  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); VtNY~  
end bNUb  
if any(idx_neg) *rY@(|  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); g4NxNjM;  
end P/BWFN1  
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X[Lwx.Ly8  
% EOF zernfun _\HMF  

支持一下，慢慢研究

顶顶·········

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)  4>x$I9^Y!  

y4\(ynk  
DDE还是手动输入的呢？ ;mT|0&o>#  
.uh>S!X, ]  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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

慢慢研究，这个专业性很强的。用的人又少。