[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， _5SA(0D#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，
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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？ ;sfk@ec  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ iVqa0Gl+}  
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function z = zernfun(n,m,r,theta,nflag) ?j-;;NNf  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. G,JK$j>*l  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N z9IJ%=R  
%   and angular frequency M, evaluated at positions (R,THETA) on the q+}
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%   unit circle.  N is a vector of positive integers (including 0), and QP0[  
%   M is a vector with the same number of elements as N.  Each element G2e0\}q  
%   k of M must be a positive integer, with possible values M(k) = -N(k) eK'ztqQ  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, v#o<. Ig  
%   and THETA is a vector of angles.  R and THETA must have the same d@0&  
%   length.  The output Z is a matrix with one column for every (N,M) Q2 @Ugt$  
%   pair, and one row for every (R,THETA) pair. P1Chmg  
% s2M|ni=  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike SM@RELA'Lb  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), kG70j{gf  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral Q^z&;%q1  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M~#% [?iU  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized CxW-lU3G`  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O]N 8QH  
% ';OZP2  
%   The Zernike functions are an orthogonal basis on the unit circle. ;^Y]nsd  
%   They are used in disciplines such as astronomy, optics, and ! fSM6Vo  
%   optometry to describe functions on a circular domain. a0=5G>G9c  
% T{Rhn V1  
%   The following table lists the first 15 Zernike functions. 2Ed  
% 2h^9lrQcQG  
%       n    m    Zernike function           Normalization -L)b;0%  
%       -------------------------------------------------- Nq=r404  
%       0    0    1                                 1 A-XWG9nL  
%       1    1    r * cos(theta)                    2 qH-':|h7  
%       1   -1    r * sin(theta)                    2 Bk9?
=  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) .<|.nK`6  
%       2    0    (2*r^2 - 1)                    sqrt(3) S|HnmkV66  
%       2    2    r^2 * sin(2*theta)             sqrt(6) mFu0$N6]H  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) u"*Wo'3I|  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) aO]FQ#
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%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) b3RCsIz  
%       3    3    r^3 * sin(3*theta)             sqrt(8) _]~= Kjp  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 4:S?m(ah/  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  }FoO  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (aa}0r5  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qqR8E&Y{  
%       4    4    r^4 * sin(4*theta)             sqrt(10) gkN )`/`*  
%       -------------------------------------------------- _Bhm\|t  
% j/+e5.EX/  
%   Example 1: aur4Ky> :  
% y8QJ=v* B  
%       % Display the Zernike function Z(n=5,m=1) $pOgFA1'  
%       x = -1:0.01:1; d:V6.7>,  
%       [X,Y] = meshgrid(x,x); x!@P|c1nKC  
%       [theta,r] = cart2pol(X,Y); -g;cg7O#(  
%       idx = r<=1; "2~%-;c  
%       z = nan(size(X)); WMw]W&  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); !04zWYHo  
%       figure TUaW'  
%       pcolor(x,x,z), shading interp L[s8`0  
%       axis square, colorbar %oY=.Ok ]  
%       title('Zernike function Z_5^1(r,\theta)') nD8CP[bRo  
% _jr'A-M  
%   Example 2: h72#AN  
% '3MCb  
%       % Display the first 10 Zernike functions D)*OQLHW  
%       x = -1:0.01:1; >TqMb8e_  
%       [X,Y] = meshgrid(x,x);  #mDeA>b  
%       [theta,r] = cart2pol(X,Y); k-uwK-B}v+  
%       idx = r<=1; lIlmXjL0  
%       z = nan(size(X)); (,5,}  
%       n = [0  1  1  2  2  2  3  3  3  3]; KNw{\Pz~w  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; dY'mY~Tv  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; RXF%A5FXh  
%       y = zernfun(n,m,r(idx),theta(idx)); 609_ZW;)  
%       figure('Units','normalized') UD@u hL  
%       for k = 1:10 _CDl9pP36#  
%           z(idx) = y(:,k); v>&sb3I  
%           subplot(4,7,Nplot(k)) !PIpvx{aX  
%           pcolor(x,x,z), shading interp =Q!)xEK  
%           set(gca,'XTick',[],'YTick',[]) ?B!=DC@?H  
%           axis square
#r ;;d(  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,py:e>+^t  
%       end k]<E1 c/  
% RTgR>qI&)  
%   See also ZERNPOL, ZERNFUN2. }>|M6.n "  
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%   Paul Fricker 11/13/2006 RrPo89o  
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% Check and prepare the inputs: W=9Zl(2C  
% ----------------------------- 4R~f  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~baVS-v  
    error('zernfun:NMvectors','N and M must be vectors.') lOc!KZHUp  
end \ M_}V[1+  
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if length(n)~=length(m) aNLkkkJg<;  
    error('zernfun:NMlength','N and M must be the same length.') I,:R~^qJ8v  
end jv C.T]<B  
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n = n(:); EP|OKXRltA  
m = m(:); DeAi'"&  
if any(mod(n-m,2)) (|F}B  
    error('zernfun:NMmultiplesof2', ... FHu -';  
          'All N and M must differ by multiples of 2 (including 0).') Ev R6^n/  
end l|O)B #  
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if any(m>n) ^zMME*G  
    error('zernfun:MlessthanN', ... huu v`$~y  
          'Each M must be less than or equal to its corresponding N.') i09w(k?  
end b~1]}9TJ  
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q Z,7q  
if any( r>1 | r<0 ) U8T"ABvFP  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') KVvzVQ1  
end _msV3JBr  
#DN5S#Ic  
%SwN/rna  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?3{R'Buv]  
    error('zernfun:RTHvector','R and THETA must be vectors.') 4TBK:Vm5  
end wO&edZ]zb^  
j#C1+Us  
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r = r(:); x}`]9XQ  
theta = theta(:); .)7r /1o  
length_r = length(r); fVU9?^0/)9  
if length_r~=length(theta) yC]xYn)  
    error('zernfun:RTHlength', ... R3G+tE/Y  
          'The number of R- and THETA-values must be equal.') uA}w?;  
end ]y4(WG;:  
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],fu#pi=]  
% Check normalization: =?*6lS}gy  
% -------------------- PFqc
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if nargin==5 && ischar(nflag) gbf-3KSp^  
    isnorm = strcmpi(nflag,'norm'); 6O`s&T,t  
    if ~isnorm Y4\BHFq  
        error('zernfun:normalization','Unrecognized normalization flag.') 62R94  
    end eYER"E  
else 8$|<`:~J  
    isnorm = false; Z$0+jpG_s  
end 4>uy+"8PO  
b.`<T"y  
pzo9?/-  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X}Ey6*D:  
% Compute the Zernike Polynomials Y:~A-_  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o)X(;o  
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% Determine the required powers of r: J3]W2m2Zw  
% ----------------------------------- 6I$laHx?  
m_abs = abs(m); 9@Iz:!oqb  
rpowers = []; >q'xW=Y j\  
for j = 1:length(n) YWV"I|Z  
    rpowers = [rpowers m_abs(j):2:n(j)]; $`2rtF  
end +<G |Ru-  
rpowers = unique(rpowers); ;g3z?Uz)  
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m3apeIEi[  
% Pre-compute the values of r raised to the required powers, KjrUTG0oA  
% and compile them in a matrix: pJ` M5pF  
% ----------------------------- 'IorjR@40  
if rpowers(1)==0 O8;`6r  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F:PaVr3q  
    rpowern = cat(2,rpowern{:}); Z~g I)  
    rpowern = [ones(length_r,1) rpowern]; Ub0hISA  
else /Hox]
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    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y:U'3G-
  
    rpowern = cat(2,rpowern{:}); (,5oqU9s@  
end r/X4Hy0!lT  
Ywj=6 +;  
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% Compute the values of the polynomials: K[wOK  
% -------------------------------------- DCJmk6p%0  
y = zeros(length_r,length(n)); z (N3oBW  
for j = 1:length(n) E8TJ*ZU  
    s = 0:(n(j)-m_abs(j))/2; hSxlj7Eo^T  
    pows = n(j):-2:m_abs(j); !]%M  
    for k = length(s):-1:1 IETdL{`~  
        p = (1-2*mod(s(k),2))* ... o/EN3J  
                   prod(2:(n(j)-s(k)))/              ... i+/:^tc;  
                   prod(2:s(k))/                     ... Kn9O=?Xh;  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zW`Zmt\T2  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); W\(u1>lj  
        idx = (pows(k)==rpowers); jlmP1b9  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ;j#$d@VG"  
    end BrW1:2w >\  
     ~s}0z&v^te  
    if isnorm 5ryzAB O\2  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '}3m('u  
    end 'Zq$W]i  
end l!n<.tQW  
% END: Compute the Zernike Polynomials sU {'  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f@ &?K<  
'%W'HqVcG1  
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% Compute the Zernike functions: Wvhg:vup  
% ------------------------------ T&?0hSYt  
idx_pos = m>0; >28.^\?H4  
idx_neg = m<0; G1;.\i  
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?
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z = y; < q6z$c)K  
if any(idx_pos) <Tq&Va_w  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); OD,"8JF  
end ?/mkFDN  
if any(idx_neg) ryz [A:^G  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); O"otzla  
end %5X}4k!p  
~R`Rj*Q2Y  
dg%Orvuz  
% EOF zernfun &&iZ?JteZ  

慢慢研究，这个专业性很强的。用的人又少。

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

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)  O=}4?Xv  

wZVY h  
DDE还是手动输入的呢？ l%*KBME  
@cPb*  
zygo和zemax的zernike系数，类型对应好就没问题了吧

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支持一下，慢慢研究