[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， )m'_>-`^:  
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， c:6w >:  
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？ ?hmb"^vlG  
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ ,9pi9\S  
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function z = zernfun(n,m,r,theta,nflag) ~ZNhU;%YW  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 4bBxZY  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N EsWszpRqb  
%   and angular frequency M, evaluated at positions (R,THETA) on the CS{9|FNz  
%   unit circle.  N is a vector of positive integers (including 0), and \l2 s^7G_  
%   M is a vector with the same number of elements as N.  Each element [>:gwl _\  
%   k of M must be a positive integer, with possible values M(k) = -N(k) bn`zI~WS  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 4eEs_R  
%   and THETA is a vector of angles.  R and THETA must have the same =_H39)|
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%   length.  The output Z is a matrix with one column for every (N,M) :mrGB3x{  
%   pair, and one row for every (R,THETA) pair. kwL)&@  
% &wuV}S7  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )q^vitkjup  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), q"Md)?5N  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral y|dXxd9  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, d<Os TA  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8<kme"%s  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '=H^m D+gl  
% ?>y-5B[K/(  
%   The Zernike functions are an orthogonal basis on the unit circle. u
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%   They are used in disciplines such as astronomy, optics, and )1vojp 4Za  
%   optometry to describe functions on a circular domain. ~ YK<T+  
% #{M -3  
%   The following table lists the first 15 Zernike functions. vVW=1(QWI#  
% @5y(>>C}8%  
%       n    m    Zernike function           Normalization z~H Gc"~  
%       -------------------------------------------------- Wj:QC<5 v  
%       0    0    1                                 1 )^\='(s  
%       1    1    r * cos(theta)                    2 tTt3D]h(  
%       1   -1    r * sin(theta)                    2 3+-(;>>\  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) %5G BMMn  
%       2    0    (2*r^2 - 1)                    sqrt(3) lnDDFsA  
%       2    2    r^2 * sin(2*theta)             sqrt(6) [&CM-` N  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) Q 8rtZ  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Oi0;.<kX  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) +V@=G &Ou0  
%       3    3    r^3 * sin(3*theta)             sqrt(8) kYB <FwwB  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) /;rN/ot2o  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )DmiN^:  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) r?!xL\C\  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m-89nOls  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ,}tdfkZFYl  
%       -------------------------------------------------- tA-B3 ]  
% 9oP{Al  
%   Example 1: skz]@{38  
% f~FehN7  
%       % Display the Zernike function Z(n=5,m=1) (U_Q7hja?  
%       x = -1:0.01:1; 'pY;]^M  
%       [X,Y] = meshgrid(x,x); ~ ZL`E  
%       [theta,r] = cart2pol(X,Y); Z0eB
x  
%       idx = r<=1; mi<D bnou  
%       z = nan(size(X)); 9|?Lz  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); !Blk=L+p  
%       figure wYA/<0'yH  
%       pcolor(x,x,z), shading interp |{)xC=  
%       axis square, colorbar el?V2v[  
%       title('Zernike function Z_5^1(r,\theta)') 5j,qAay9  
% ,-3(^d\1F  
%   Example 2: ;q; C^l  
% `4H9f&8(  
%       % Display the first 10 Zernike functions A+*oT(`  
%       x = -1:0.01:1; \83A|+k  
%       [X,Y] = meshgrid(x,x); 8 
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%       [theta,r] = cart2pol(X,Y); 50ew/fZj|  
%       idx = r<=1; %^=!s  
%       z = nan(size(X)); -1  
%       n = [0  1  1  2  2  2  3  3  3  3]; <cv1$ x ~P  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; '$XHRS/q]  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Bh,)5E^m  
%       y = zernfun(n,m,r(idx),theta(idx)); +MZO%4  
%       figure('Units','normalized') /iy*3P,`  
%       for k = 1:10 5^K#Tj ;2  
%           z(idx) = y(:,k); ~H|LWCU)K8  
%           subplot(4,7,Nplot(k)) 
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%           pcolor(x,x,z), shading interp SP*JleQN  
%           set(gca,'XTick',[],'YTick',[]) d?M!
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%           axis square bmVgTm&  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) qH Ga  
%       end ;Uqx&5P}  
% 'e>sHL  
%   See also ZERNPOL, ZERNFUN2. DRW.NL o  
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%   Paul Fricker 11/13/2006 !mxh]x<e  
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% Check and prepare the inputs: bepYe
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% ----------------------------- QHzX 5$IM  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k,R~oSA'n  
    error('zernfun:NMvectors','N and M must be vectors.') '<D`:srV  
end to!W={S<ol  
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if length(n)~=length(m) $',GkK{NX  
    error('zernfun:NMlength','N and M must be the same length.') \+Rwm:lI  
end Kt90mA  
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n = n(:); (L0hS'  
m = m(:); JXY!c\,  
if any(mod(n-m,2)) a^XTW7]r  
    error('zernfun:NMmultiplesof2', ... ;WS7.  
          'All N and M must differ by multiples of 2 (including 0).') \ ~LU 'j  
end 5'kTe=  
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if any(m>n) I4<{
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    error('zernfun:MlessthanN', ... HcBH!0  
          'Each M must be less than or equal to its corresponding N.') {{]=zt|69  
end @x=BJuUuX  
PF'5z#] NP  
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if any( r>1 | r<0 ) f1Yv hvWL  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') siRnH(^J  
end h#@4@x{  
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I^M#[xA  
    error('zernfun:RTHvector','R and THETA must be vectors.') 11B{gUv.]  
end {wpMg  
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r = r(:); !k:zLjtp  
theta = theta(:); T^'*_*m  
length_r = length(r); %89"A'g  
if length_r~=length(theta) {V%%^Zhwy  
    error('zernfun:RTHlength', ... 8L7Y A)u  
          'The number of R- and THETA-values must be equal.') EFRZ% Y  
end 0r0\b*r  
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% Check normalization: hB P$9GR  
% -------------------- )4!CR/ao  
if nargin==5 && ischar(nflag) rysP)e  
    isnorm = strcmpi(nflag,'norm'); + 9\:$wMN  
    if ~isnorm NoJnchiU  
        error('zernfun:normalization','Unrecognized normalization flag.') +H[}T ]  
    end Ok}{jwJ%W;  
else FI?gT  
    isnorm = false; >J^7}J  
end NIGB[2V(  
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yN9$gfJC^  
% Compute the Zernike Polynomials -uv 9(r\P  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M.xhVgFf)  
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% Determine the required powers of r: L,C? gd@"  
% ----------------------------------- Tn4W\?R  
m_abs = abs(m); !paN`Fz\a  
rpowers = []; m4Phn~>Gg  
for j = 1:length(n) 6\,DnO   
    rpowers = [rpowers m_abs(j):2:n(j)]; ,zAK3d&h
j  
end }zkL[qu;  
rpowers = unique(rpowers); >W`S(a Mn  
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% Pre-compute the values of r raised to the required powers, mP9cBLz  
% and compile them in a matrix: 22)0zY%\  
% ----------------------------- J
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if rpowers(1)==0 b$dJ?%W  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E fP>O  
    rpowern = cat(2,rpowern{:}); 3)6+1Yc  
    rpowern = [ones(length_r,1) rpowern]; uSABh^  
else B;xZ%M]  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0V<Aub[${  
    rpowern = cat(2,rpowern{:}); $Cz1C  
end c=b+g+*xd  
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% Compute the values of the polynomials: |. J,8~x  
% -------------------------------------- - *:p.(c  
y = zeros(length_r,length(n)); SX1X<9  
for j = 1:length(n) t2=a(N-/,  
    s = 0:(n(j)-m_abs(j))/2; 8O~0RYk  
    pows = n(j):-2:m_abs(j); gW%pM{PW  
    for k = length(s):-1:1 TA Ftcs:  
        p = (1-2*mod(s(k),2))* ... {V}t'x`4c  
                   prod(2:(n(j)-s(k)))/              ... &?SX4c~?u  
                   prod(2:s(k))/                     ... KKLR'w,A>  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /Jh1rck  
                   prod(2:((n(j)+m_abs(j))/2-s(k)));  7]p>XAb  
        idx = (pows(k)==rpowers); <e;jWK  
        y(:,j) = y(:,j) + p*rpowern(:,idx); EfFz7j&X  
    end Gx.P
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     {I4%  
    if isnorm v2Dt3$@H6  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4cott^K.  
    end +J"'  'cZ  
end [dl+:P:zc  
% END: Compute the Zernike Polynomials Xl#Dw bx  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M0RRmW@f.a  
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% Compute the Zernike functions: 3Bejp+xX  
% ------------------------------ fZ[kh{|  
idx_pos = m>0; Vd,' s  
idx_neg = m<0; py]KTRzy  
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z = y; -M6L.gi)oJ  
if any(idx_pos) wAw42{M  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8s<^]sFP  
end A'GlCp  
if any(idx_neg) 92ZWU2"  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); w'A tf  
end :d.1;st  
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% EOF zernfun O]t\B*%}  

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)   hw=GR_,  

dv!r.  
DDE还是手动输入的呢？ ~3CVxbB^<  
G1d(,4Xp  
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