[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， Ec]|p6a3  
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， UQT'6* !  
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？ 7m1KR#j  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ |L:Cn J  
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function z = zernfun(n,m,r,theta,nflag) cHD%{xlb  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. u oVNK  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H+ZSPHs  
%   and angular frequency M, evaluated at positions (R,THETA) on the |M5-5)  
%   unit circle.  N is a vector of positive integers (including 0), and UAYd?r  
%   M is a vector with the same number of elements as N.  Each element y,m2(V  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 9dKul,c  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !3}deY8;#  
%   and THETA is a vector of angles.  R and THETA must have the same j9y3hQ+q  
%   length.  The output Z is a matrix with one column for every (N,M) \4bWWy  
%   pair, and one row for every (R,THETA) pair. :tGYs8UK  
% 0 bSA_  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >+9JD%]x]  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &%F@O<:  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral 8cVzFFQP  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, U/w.M_S  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]=&L_(34  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 6yIvaY$KR  
% 3$p#;a:=n  
%   The Zernike functions are an orthogonal basis on the unit circle. Yx)o:#2  
%   They are used in disciplines such as astronomy, optics, and /c-nE3+rn  
%   optometry to describe functions on a circular domain. TEVI'%F
  
% >Pal
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%   The following table lists the first 15 Zernike functions. *\XH+/]+  
% %c/^_.  
%       n    m    Zernike function           Normalization .BZVX=x  
%       -------------------------------------------------- "{BqtU*.  
%       0    0    1                                 1 8X7{vN_3K  
%       1    1    r * cos(theta)                    2 pGWA\}'  
%       1   -1    r * sin(theta)                    2 Rp.W,)i  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) }ot"Sx\.  
%       2    0    (2*r^2 - 1)                    sqrt(3) "Pc$\zJm;  
%       2    2    r^2 * sin(2*theta)             sqrt(6) Q~,YbZ-7  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) oXYMoi  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 2(#Ks's?  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =dH$
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%       3    3    r^3 * sin(3*theta)             sqrt(8) UuJjO^t  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) |X(2Zv^O  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >-cfZ9{!  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 4tc:.  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1~5trsB+5  
%       4    4    r^4 * sin(4*theta)             sqrt(10) >SI<rR[~%  
%       -------------------------------------------------- >1|g5  
% \7 a4uc  
%   Example 1: <+]f`c*Z  
% !,&yyx.  
%       % Display the Zernike function Z(n=5,m=1) JdNF-64ky  
%       x = -1:0.01:1; FLr;`3  
%       [X,Y] = meshgrid(x,x); %5B%KCCN  
%       [theta,r] = cart2pol(X,Y); hU {-a`  
%       idx = r<=1; 8 %Sb+w07  
%       z = nan(size(X)); >)4YP*qIPb  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); +1`t}hO  
%       figure v%91k  
%       pcolor(x,x,z), shading interp }vh Za p^  
%       axis square, colorbar q~Jq/E"f  
%       title('Zernike function Z_5^1(r,\theta)') Px;Cg 6  
% l[Z)@bC1  
%   Example 2: v1.*IV5Y  
% $RO$}!  
%       % Display the first 10 Zernike functions H%i>L?J2/  
%       x = -1:0.01:1; b-<HXn_Fd  
%       [X,Y] = meshgrid(x,x); -T_\f?V88  
%       [theta,r] = cart2pol(X,Y); P%>?[9!Nt  
%       idx = r<=1; ]H[8Z|i""  
%       z = nan(size(X)); *Xr$/N  
%       n = [0  1  1  2  2  2  3  3  3  3]; E`D%PEps+  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; a39hP*  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; oyeG$mpg  
%       y = zernfun(n,m,r(idx),theta(idx)); .5Z_E O  
%       figure('Units','normalized') ?y%t}C\W  
%       for k = 1:10 ;A#~`P  
%           z(idx) = y(:,k); ujzW|HW^v  
%           subplot(4,7,Nplot(k)) 1/iE`Si  
%           pcolor(x,x,z), shading interp bXdY\&fE  
%           set(gca,'XTick',[],'YTick',[]) m4/er539T  
%           axis square Pv){sYUh  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _<Dt z  
%       end ?d-70pm  
% "yh Pm  
%   See also ZERNPOL, ZERNFUN2. FC>d_=V  
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%   Paul Fricker 11/13/2006 @.$MzPQQI  
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% Check and prepare the inputs: C1B'#F9EO  
% ----------------------------- n9oR)&:o  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y1\K;;X  
    error('zernfun:NMvectors','N and M must be vectors.') a6nlt?1?D  
end ycpE=fso'  
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if length(n)~=length(m)
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    error('zernfun:NMlength','N and M must be the same length.') xFb3O|TC  
end [.cq{6-
  
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n = n(:); !#WqA9<  
m = m(:); <r\I"z$  
if any(mod(n-m,2)) \<
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    error('zernfun:NMmultiplesof2', ... !v>ew9  
          'All N and M must differ by multiples of 2 (including 0).') #A<"4#}  
end JAP(|  
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if any(m>n) "tR}j,=S:D  
    error('zernfun:MlessthanN', ... 9g@NcJ]  
          'Each M must be less than or equal to its corresponding N.') =AkX4k  
end kk_zVrQ<  
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if any( r>1 | r<0 ) ]|LgVXEpx  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') W~dE  
end e'~ Q@_D  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vLn> 4SK  
    error('zernfun:RTHvector','R and THETA must be vectors.') ScJu_Af  
end v5 Y)al@  
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r = r(:); h48 bb.p2  
theta = theta(:); ;=p;v .l  
length_r = length(r); {B^pnLc  
if length_r~=length(theta) n\>.T[$"  
    error('zernfun:RTHlength', ... }x0Z( `  
          'The number of R- and THETA-values must be equal.') `|,tCM&-  
end 'j#a%j@{  
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% Check normalization: /v$]X4 S`  
% -------------------- P^4'|#~2T  
if nargin==5 && ischar(nflag) 9::YR;NY  
    isnorm = strcmpi(nflag,'norm'); .tp=T  
    if ~isnorm +Ag#B*   
        error('zernfun:normalization','Unrecognized normalization flag.') cYD1~JX.  
    end i tW~
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else 9|a)sb7/  
    isnorm = false; 8A_TIyh?  
end uXNp!tY  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]+,nA R  
% Compute the Zernike Polynomials ?>TbTfmR  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P^;WB*V  
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u]#8$M2  
% Determine the required powers of r: ZjavD^ky  
% ----------------------------------- p[gAZ9  
m_abs = abs(m); Iq@IUFpc7~  
rpowers = []; p1?J  
for j = 1:length(n) ,s8&#1rJ-  
    rpowers = [rpowers m_abs(j):2:n(j)]; .lG
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end b)y<.pS\  
rpowers = unique(rpowers); 0k7"H]J  
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% Pre-compute the values of r raised to the required powers, eo]a'J9(  
% and compile them in a matrix: '%$-]~  
% ----------------------------- 6d(b'S^  
if rpowers(1)==0 98ayA$  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); KZ @l/s  
    rpowern = cat(2,rpowern{:}); s9kTuhoK  
    rpowern = [ones(length_r,1) rpowern]; *fOIq88  
else EyPy*_A  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); X)Ocn`
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    rpowern = cat(2,rpowern{:}); Qvs(Rt3?y  
end +E `06
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% Compute the values of the polynomials: B=^2g}mgK  
% -------------------------------------- ,Zr YJ<  
y = zeros(length_r,length(n)); R+x%r&L5F  
for j = 1:length(n) &a~L_`\'  
    s = 0:(n(j)-m_abs(j))/2; wfWS-pQ  
    pows = n(j):-2:m_abs(j); l.yJA>\24I  
    for k = length(s):-1:1 F^[M  
        p = (1-2*mod(s(k),2))* ... P'gT6*an,"  
                   prod(2:(n(j)-s(k)))/              ... b^;N>zx  
                   prod(2:s(k))/                     ... s2wwmtUCN  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >DkN+S  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 8UlB~fVg  
        idx = (pows(k)==rpowers); 7Im}~3NJG  
        y(:,j) = y(:,j) + p*rpowern(:,idx); FC~|&  
    end WJBW:2=;  
     b^@`uDb6  
    if isnorm M_|M&lR>  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); / *Ou$  
    end m]  EDuW  
end t>m8iS>  
% END: Compute the Zernike Polynomials `W D*Q-&n  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [%7IQ4`{  
Z={UM/6w  
$Cut  
% Compute the Zernike functions: .[]{
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% ------------------------------ |~Htj4K/  
idx_pos = m>0; X*43!\  
idx_neg = m<0; R4[. n@  
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tP/GDC;  
z = y; FA<Z37:  
if any(idx_pos) Cj`pw2.  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I67k M{V  
end WXRHG)nvL  
if any(idx_neg) Z^jGT+
2  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Xd90n>4S  
end hCSRsk3  
4'd;'SvF  
?tW%"S^D  
% EOF zernfun 1gf/#+$\  

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)  hdNZ":1s  

\T#(rt\j  
DDE还是手动输入的呢？ c)P%O  
}E\+e!'!2  
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。