[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， !;Hi9,<#7g  
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， 4cZig\mE;  
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？ &G63ReW7 @  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ P(iZGOKUs=  
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function z = zernfun(n,m,r,theta,nflag) &qR1fbw"  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. iV+'p->/  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +Smt8O<N  
%   and angular frequency M, evaluated at positions (R,THETA) on the nT7{`aaQl  
%   unit circle.  N is a vector of positive integers (including 0), and ?t;>]Wo;  
%   M is a vector with the same number of elements as N.  Each element "F_o%!l  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 4a'O#;ho  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, si`{>e~`6P  
%   and THETA is a vector of angles.  R and THETA must have the same X`xI~&t_  
%   length.  The output Z is a matrix with one column for every (N,M) 2 uuI_9 "^  
%   pair, and one row for every (R,THETA) pair. oL?[9aww  
% [h"#Gwb=;  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike TTNgnP  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1-z*'Ghys  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral lo}[o0X  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _W@SCV)yH  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized *7L*:g  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 44s K2  
% ,p(4OZz5,  
%   The Zernike functions are an orthogonal basis on the unit circle. w8~J5XS  
%   They are used in disciplines such as astronomy, optics, and $`nKq4Y   
%   optometry to describe functions on a circular domain.
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% sy^k:y?  
%   The following table lists the first 15 Zernike functions. XTIRY4{ d  
% W@S'mxk#*  
%       n    m    Zernike function           Normalization 
84PD`A  
%       -------------------------------------------------- 7Pt*V@DHS  
%       0    0    1                                 1 kBPFk t2  
%       1    1    r * cos(theta)                    2 U3ygFW%  
%       1   -1    r * sin(theta)                    2 pB @l+ n^  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) %9_wDfw~  
%       2    0    (2*r^2 - 1)                    sqrt(3) >.R6\>N%  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 4SG22$7W  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) !U02>X   
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |pIA9/~Z  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ":,HY)z  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ^V^In-[!y:  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) WY@x2bBi  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -25#Vh  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) >40B Fxc  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z_%}pe39B  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ] =Js5  
%       -------------------------------------------------- qA- ya6  
% Q*TxjE7K  
%   Example 1: #vO3*-hs  
% Q9K+k*?{N  
%       % Display the Zernike function Z(n=5,m=1) Z2chv,SqCJ  
%       x = -1:0.01:1; )k&pp^q\  
%       [X,Y] = meshgrid(x,x); 1B3,lYBM  
%       [theta,r] = cart2pol(X,Y);
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%       idx = r<=1; `R@24 )  
%       z = nan(size(X)); Ow\9vf6H  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); F"a^`E&  
%       figure 0w >DU^+  
%       pcolor(x,x,z), shading interp (l22p  
%       axis square, colorbar <$liWAGX\  
%       title('Zernike function Z_5^1(r,\theta)') 6'C!Au  
% S(A0),  
%   Example 2: zIbl[[M&  
% ;{|a~e?Y  
%       % Display the first 10 Zernike functions Q6S[sTKR  
%       x = -1:0.01:1; X7kJ
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%       [X,Y] = meshgrid(x,x); IidZ
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%       [theta,r] = cart2pol(X,Y); \h^bOxh  
%       idx = r<=1; a@@!Eg A  
%       z = nan(size(X)); y? [*qnPj  
%       n = [0  1  1  2  2  2  3  3  3  3]; }\u~He%  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; C!w@Naj  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; bcpH|}[F)  
%       y = zernfun(n,m,r(idx),theta(idx)); tYfhKJzGC  
%       figure('Units','normalized') NrvS/cI!t  
%       for k = 1:10 w8%yX$<  
%           z(idx) = y(:,k); m@JU).NKCS  
%           subplot(4,7,Nplot(k)) 1elx~5v1.=  
%           pcolor(x,x,z), shading interp 
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%           set(gca,'XTick',[],'YTick',[]) >ge-yK 1  
%           axis square Tu_dkif'  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'D(Hqdr;:  
%       end 7kn=j6I  
% \Y9=dE}  
%   See also ZERNPOL, ZERNFUN2. 9[N'HpQ3  
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%   Paul Fricker 11/13/2006 1jmhh!,  
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% Check and prepare the inputs: 9r1pdG_C@  
% ----------------------------- BHj]w*Ov  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~Y)Au?d(a  
    error('zernfun:NMvectors','N and M must be vectors.') pq5)Ug  
end ](_(1  
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if length(n)~=length(m) O8u"Y0$*w  
    error('zernfun:NMlength','N and M must be the same length.') Tf@t.4\  
end @YwaOc_%  

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n = n(:); XQ(`8Jl&^  
m = m(:); Rl5}W\&  
if any(mod(n-m,2)) uy\YJ.WMQ  
    error('zernfun:NMmultiplesof2', ... n]Dq  
          'All N and M must differ by multiples of 2 (including 0).') *7*g! km  
end LO"HwN43h  
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if any(m>n) [RpFC4W  
    error('zernfun:MlessthanN', ... U}A+jJ  
          'Each M must be less than or equal to its corresponding N.') cjN4U [  
end N[pk@M\vX  
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if any( r>1 | r<0 ) BC
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    error('zernfun:Rlessthan1','All R must be between 0 and 1.') q=nMZVVlF(  
end L0&!Qct  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j|aT`UH03  
    error('zernfun:RTHvector','R and THETA must be vectors.')  Mxr#  
end jilO%  "  
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r = r(:); [2\`Wh:%P  
theta = theta(:); T@Q<oNU  
length_r = length(r); G,"$Erx  
if length_r~=length(theta) a|s=d  
    error('zernfun:RTHlength', ... |u}sX5/q  
          'The number of R- and THETA-values must be equal.') *<0g/AL  
end Z#J{tXZc  
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% Check normalization: mmjB1L  
% -------------------- U_8I$v-~  
if nargin==5 && ischar(nflag) 3p4bOT5  
    isnorm = strcmpi(nflag,'norm'); j_H T  
    if ~isnorm }E1Eq
  
        error('zernfun:normalization','Unrecognized normalization flag.') v'@LuF'e8  
    end 9Akwr}  
else =:0(
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    isnorm = false; [c
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end ^X;>?_Bk  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JWdG?[$  
% Compute the Zernike Polynomials 5g5pzww  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AN1bfF:C  
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% Determine the required powers of r: $?GO|.59  
% ----------------------------------- }N|/
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m_abs = abs(m); )I$Mh@F  
rpowers = []; X'F$K!o*,:  
for j = 1:length(n) {vH8X(m  
    rpowers = [rpowers m_abs(j):2:n(j)]; "nefRz%j+  
end )/pPY  
rpowers = unique(rpowers); }wb;ulN)  
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% Pre-compute the values of r raised to the required powers, X5yhS  
% and compile them in a matrix: |S}*
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% ----------------------------- OlJj|?z$  
if rpowers(1)==0 S\rfR N  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 24T
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    rpowern = cat(2,rpowern{:}); E,$uNw']  
    rpowern = [ones(length_r,1) rpowern]; fh3 6  
else W!^=)Qs  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l`]!)j|+  
    rpowern = cat(2,rpowern{:}); qs b4@jt+  
end &ivIv[LV  
n 3]y$
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% Compute the values of the polynomials: [P,nW/H  
% -------------------------------------- cA\W|A)  
y = zeros(length_r,length(n)); Dw[Q,SE  
for j = 1:length(n) 1mV0AE538  
    s = 0:(n(j)-m_abs(j))/2; `ouzeu9}  
    pows = n(j):-2:m_abs(j); &40]sxm  
    for k = length(s):-1:1 NeEV!V8  
        p = (1-2*mod(s(k),2))* ... Ye6O!,R  
                   prod(2:(n(j)-s(k)))/              ... "F}Ip&]hAG  
                   prod(2:s(k))/                     ... FHC7\#p/9Z  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... qQ'@yTVN  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); <i6MbCB  
        idx = (pows(k)==rpowers); eH8
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        y(:,j) = y(:,j) + p*rpowern(:,idx); k}.nH"AQ  
    end u2Obb`p S  
     q}i87a;m  
    if isnorm 4\3t5n  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7KIQ)E'kG|  
    end Uy:.m  
end FM)*>ax{  
% END: Compute the Zernike Polynomials 2cl~Va=  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g-}sVvM  
9R[','x  
;p ('cwU%  
% Compute the Zernike functions: AlxS?f2w  
% ------------------------------ v],DBw9  
idx_pos = m>0; 4Cc
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idx_neg = m<0; ?OyW|jL  
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z = y; +GS=zNw#  
if any(idx_pos) xn8B|axB  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R2`g?5v  
end S/;
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if any(idx_neg) 1n"X?K5;A  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Se8y-AL6x>  
end 6%#'X  
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% EOF zernfun 9(WC#-,  

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)  P^m 6di  

&~A*(+S  
DDE还是手动输入的呢？ ~w9
=Fd6  
xp8f  
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。