[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， 2D3mTpw  
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， :"+3Uk2  
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？ Imo?)dYK  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ %XXjQ5p  
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function z = zernfun(n,m,r,theta,nflag) M%Q_;\?]  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Jd33QL}Hj  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $^#q0Yx  
%   and angular frequency M, evaluated at positions (R,THETA) on the +^4HCyW  
%   unit circle.  N is a vector of positive integers (including 0), and ]:4\rBR3  
%   M is a vector with the same number of elements as N.  Each element P;ZVv{mT  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 8%b-.O:_$  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, O%Qz6R  
%   and THETA is a vector of angles.  R and THETA must have the same +#@2,  
%   length.  The output Z is a matrix with one column for every (N,M) mWVq>~  
%   pair, and one row for every (R,THETA) pair. ~jC$C2A0  
% k{^iv:  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike  w4UJXc  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tYUo;V  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral ]TsmWob  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^3Z~RK\}  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ?A 5;"  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4&B|rf  
% 2j[;M-3  
%   The Zernike functions are an orthogonal basis on the unit circle. @^b>S6d"  
%   They are used in disciplines such as astronomy, optics, and o~VZ%B  
%   optometry to describe functions on a circular domain. <!?ZH"F0  
% }y%mG&KSz  
%   The following table lists the first 15 Zernike functions. $oi8<8Y  
% nA+gqY6 6|  
%       n    m    Zernike function           Normalization byIP]7Ld  
%       -------------------------------------------------- Dh9C9<Ta:  
%       0    0    1                                 1 NcIr; }  
%       1    1    r * cos(theta)                    2 W!a'KI'  
%       1   -1    r * sin(theta)                    2 Pm|S>r  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) Ntpw(E<$f  
%       2    0    (2*r^2 - 1)                    sqrt(3) vVbS 4_  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 4/&.N]  
%       3   -3    r^3 * cos(3*theta)             sqrt(8)
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%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Kr%O}<"  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) -qB{TA-.\  
%       3    3    r^3 * sin(3*theta)             sqrt(8) F'njtrO3  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) F\+!\b*lP  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WAGU|t#."  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) .[vYT.LE  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (gUxS.zU  
%       4    4    r^4 * sin(4*theta)             sqrt(10) D(Q=EdlO  
%       -------------------------------------------------- Odbm"Y  
% D(">bR)1  
%   Example 1: }<H0CcG  
% -qDL':  
%       % Display the Zernike function Z(n=5,m=1) xGK"`\V  
%       x = -1:0.01:1;  h x hl  
%       [X,Y] = meshgrid(x,x); h.aXW]]}(P  
%       [theta,r] = cart2pol(X,Y); hKN/&P^  
%       idx = r<=1; uBo~PiJ2"  
%       z = nan(size(X)); Pb/[945  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); Y9nyKL  
%       figure TiSV`V
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%       pcolor(x,x,z), shading interp UphZRgT!N  
%       axis square, colorbar [vcSt5R=  
%       title('Zernike function Z_5^1(r,\theta)') iiV'-!3w  
% bU_P@GKB  
%   Example 2: x7c#kU2A&Z  
% Dmn{ppfyb  
%       % Display the first 10 Zernike functions Qy|6A@  
%       x = -1:0.01:1; r-c1_ [Q#  
%       [X,Y] = meshgrid(x,x); 8>ODtKI*  
%       [theta,r] = cart2pol(X,Y); 1tFx Z#(G  
%       idx = r<=1; jGOE CKP  
%       z = nan(size(X)); ~|=G3(I[  
%       n = [0  1  1  2  2  2  3  3  3  3]; M[Mx g  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; gqACIXR  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; !FbW3p f  
%       y = zernfun(n,m,r(idx),theta(idx)); 3qrjb]E%}  
%       figure('Units','normalized') 2<^eVpNJR  
%       for k = 1:10 -!:h]
 
%           z(idx) = y(:,k); )F%zT[Auph  
%           subplot(4,7,Nplot(k)) r$;u4FR  
%           pcolor(x,x,z), shading interp n,%/cUl  
%           set(gca,'XTick',[],'YTick',[]) Ve\P,.  
%           axis square X6EnC57  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) IFF3gh42.  
%       end ci{WyIh  
% Ct9*T`Gl  
%   See also ZERNPOL, ZERNFUN2. 
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%   Paul Fricker 11/13/2006 p!2t/X
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% Check and prepare the inputs: h>&t
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% ----------------------------- iLJBiZ+  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +We=- e7  
    error('zernfun:NMvectors','N and M must be vectors.') 3+WostOx  
end ]gB:ht  
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if length(n)~=length(m) yI_MYL[  
    error('zernfun:NMlength','N and M must be the same length.') >7nOR  
end jMQ7^(9-  
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n = n(:); G0pqiU6  
m = m(:); >Gxh=**F  
if any(mod(n-m,2)) 1F94e)M)"  
    error('zernfun:NMmultiplesof2', ... ;&]oV`Ib  
          'All N and M must differ by multiples of 2 (including 0).') "!_,N@\t  
end #F6!x3Z  
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if any(m>n) s f<NC>
-  
    error('zernfun:MlessthanN', ... Z6_E/S  
          'Each M must be less than or equal to its corresponding N.') L QA6iZBP  
end yVbyw(gS  
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if any( r>1 | r<0 ) h&m4"HBL_  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') n3JSEu;J  
end H(F9&6}  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fIm=^}?fwK  
    error('zernfun:RTHvector','R and THETA must be vectors.') P_%kYcX'  
end 5{O9<~,  
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r = r(:); $BWA=2$  
theta = theta(:); QeipfK+me  
length_r = length(r); UWg+7RL  
if length_r~=length(theta) ({kOgOeC  
    error('zernfun:RTHlength', ... V.Qy4u7m  
          'The number of R- and THETA-values must be equal.') EGJrnz8  
end xzOM\Nq?O  
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% Check normalization: :`20i*  
% -------------------- Ur2)];WZ  
if nargin==5 && ischar(nflag) ,NoWAmv  
    isnorm = strcmpi(nflag,'norm'); ck K9@RQ  
    if ~isnorm gtw?u b  
        error('zernfun:normalization','Unrecognized normalization flag.') (ixlFGvEq  
    end P~7p~ke  
else QsH?qI&2jp  
    isnorm = false; g,d'&r"JWt  
end /F7X"_(H  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rwvCp_pN.  
% Compute the Zernike Polynomials f`"@7-N  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /[#5<;  
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% Determine the required powers of r: BewJ!,A!  
% ----------------------------------- 2;&!]2vo$  
m_abs = abs(m); t6a$ZN;  
rpowers = []; E.+BqWZ!  
for j = 1:length(n) '?dT<w=Y&  
    rpowers = [rpowers m_abs(j):2:n(j)]; T~b6Zu6  
end +DA,|~k_  
rpowers = unique(rpowers); b 3i34,  
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&1$|KbmV4  
% Pre-compute the values of r raised to the required powers, tA]Y=U+Q  
% and compile them in a matrix: g.d~`R@v  
% ----------------------------- ?N(opggiD  
if rpowers(1)==0 )w'GnUqWz  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h;S?  
    rpowern = cat(2,rpowern{:}); BhCOT+i;c  
    rpowern = [ones(length_r,1) rpowern]; 2L|)uCb  
else 5;Q9Z1 `  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^P}c0}^  
    rpowern = cat(2,rpowern{:}); j]bNOC2.L  
end 4oA9|}<FR  
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% Compute the values of the polynomials: z`KP }-  
% -------------------------------------- A~%h*nZc%I  
y = zeros(length_r,length(n)); APM!xX=N  
for j = 1:length(n) ?QG?F9?  
    s = 0:(n(j)-m_abs(j))/2; q_[V9  
    pows = n(j):-2:m_abs(j); S^*ME*DDz  
    for k = length(s):-1:1 [ %:%C]4  
        p = (1-2*mod(s(k),2))* ... DZ5QC aA  
                   prod(2:(n(j)-s(k)))/              ... d<+@cf_9  
                   prod(2:s(k))/                     ... HlC[Nu^6U  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !@wG22iC4d  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); fs;pX/:FR  
        idx = (pows(k)==rpowers); r"\g6<RP  
        y(:,j) = y(:,j) + p*rpowern(:,idx); p{S#>JT
r  
    end P2>Y0"bY  
     B[B(=4EzMP  
    if isnorm do&0m[x%  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %"g;K  
    end fNaboNj[  
end f5dctDHP  
% END: Compute the Zernike Polynomials j^qI~|#  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !j-JMa?  
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% Compute the Zernike functions: S1G3xY$0  
% ------------------------------ 6*tbil_G+  
idx_pos = m>0; ]#t5e>o|  
idx_neg = m<0; ST7Xgma-  
v7@O ,%  
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z = y; 1G62Qu$O  
if any(idx_pos) }j6<S-s~  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6Z7J<0  
end vQhi2J'  
if any(idx_neg) TB(!*t  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \bzT=^Z;2  
end >C
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SlojB^%  
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% EOF zernfun 1H]E:Bq  

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)  j7M[]/|  

G$"$k=[  
DDE还是手动输入的呢？ /\_wDi+#  
@Ja8~5:  
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。