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    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

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    离线jssylttc
     
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    只看楼主 倒序阅读 楼主  发表于: 2012-04-23
    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, WTK )SKa,.  
    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, 7XrXx:*a5  
    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? kbu.KU+  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 6_}& WjU'  
    <u`m4w  
    f_'#wc6  
    _!_%Afz  
     vf}.)  
    function z = zernfun(n,m,r,theta,nflag) `,~8(rIM  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. x`9IQQ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H+lBb$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the rW),xfo0  
    %   unit circle.  N is a vector of positive integers (including 0), and 1!/WC.0  
    %   M is a vector with the same number of elements as N.  Each element n;QMiz:yY  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $1KvL8  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -aSj-  
    %   and THETA is a vector of angles.  R and THETA must have the same ol#| .a2O  
    %   length.  The output Z is a matrix with one column for every (N,M) /N=;3yWF  
    %   pair, and one row for every (R,THETA) pair. 3FetyW l'  
    % ;fi H=_{us  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *UxN~?N|  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {zhajY7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral :9?y-X  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, "Zr+>a  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Lfr>y_i;F  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. s\/$`fuhx  
    % )b\89 F  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4rDa Jd>,  
    %   They are used in disciplines such as astronomy, optics, and >tGl7Ov  
    %   optometry to describe functions on a circular domain. KdN+$fe*g  
    % RZ +SOZs7H  
    %   The following table lists the first 15 Zernike functions. _4^#VD#f  
    % ^p7g[E&  
    %       n    m    Zernike function           Normalization VelR8tjP  
    %       -------------------------------------------------- V;@kWE>3  
    %       0    0    1                                 1 xQU$E|I  
    %       1    1    r * cos(theta)                    2 lD+f{GR  
    %       1   -1    r * sin(theta)                    2 lJ>OuSd  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) <36z,[,kZ@  
    %       2    0    (2*r^2 - 1)                    sqrt(3)  iup "P  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) #]SiS2lM#  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) LWX,u  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) mto=_|gn  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <4Ev3z*;Z  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) t?l0L1;  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Lkf}+aY  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o W<Z8s;p  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )y#~eYn  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zLt7jxx  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) xQKRUHDc  
    %       -------------------------------------------------- `2I<V7SF$  
    % v$JhC'  
    %   Example 1: {BI5lvx:  
    % 1ZZ}ojq  
    %       % Display the Zernike function Z(n=5,m=1) + $Yld{i  
    %       x = -1:0.01:1; ]:g;S,{  
    %       [X,Y] = meshgrid(x,x); 'O:QS)  
    %       [theta,r] = cart2pol(X,Y); ~[*\YN);  
    %       idx = r<=1; P;' xa^Y  
    %       z = nan(size(X)); Bk44 wz2 X  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .ey=gI!x0  
    %       figure KB@F^&L {  
    %       pcolor(x,x,z), shading interp u&-Zh@;Q7  
    %       axis square, colorbar RX\l4H5;  
    %       title('Zernike function Z_5^1(r,\theta)') +CaA%u  
    % XLq%nVBM8\  
    %   Example 2: t^')ST  
    % 31-:xUIX  
    %       % Display the first 10 Zernike functions D-KQRe2@  
    %       x = -1:0.01:1; _$vAitUe4S  
    %       [X,Y] = meshgrid(x,x); 'n$TJp|s  
    %       [theta,r] = cart2pol(X,Y); Tm) (?y  
    %       idx = r<=1; Ex`!C]sQ  
    %       z = nan(size(X)); bf*VY&S- T  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Ho!dtEs  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |%HTBF  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; _*1{fvv0{  
    %       y = zernfun(n,m,r(idx),theta(idx)); )}|b6{{<  
    %       figure('Units','normalized') r@)_>(  
    %       for k = 1:10 V/,@hv`+  
    %           z(idx) = y(:,k); + r<d z  
    %           subplot(4,7,Nplot(k)) @w[2 BaDt  
    %           pcolor(x,x,z), shading interp 9]]isE8r  
    %           set(gca,'XTick',[],'YTick',[]) kKlcK_b;  
    %           axis square u|eV'-R)s  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [OU[i(,{  
    %       end <n|ayxA)  
    % W3~xjS"h  
    %   See also ZERNPOL, ZERNFUN2. A@81wv  
    g q|]t<'  
    jwQ(E  
    %   Paul Fricker 11/13/2006 (fUpj^E)p  
    =F 9!)r  
    !M*$p Qi}  
    sngM4ikhs  
    .W*"C  
    % Check and prepare the inputs: y(92Th$  
    % ----------------------------- 8x/]H(J  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) nP5T*-~  
        error('zernfun:NMvectors','N and M must be vectors.') I/vQP+w O  
    end c7R<5f  
    |08'd5  
    duT'$}2@>  
    if length(n)~=length(m) tX'2 $}  
        error('zernfun:NMlength','N and M must be the same length.') ='z4bU  
    end 0*{ 2^\  
    [5T{`&  
    )Bo]+\2  
    n = n(:); uJ@C-/BD!M  
    m = m(:); 8H7=vk+  
    if any(mod(n-m,2)) ~A-Y%P  
        error('zernfun:NMmultiplesof2', ... 6aq=h`Y  
              'All N and M must differ by multiples of 2 (including 0).') N:% }KAc  
    end *k^'xL  
    _GF{Duxh  
    cy{ ado2  
    if any(m>n) P+2@,?9#  
        error('zernfun:MlessthanN', ... tsf)+`vt  
              'Each M must be less than or equal to its corresponding N.') tH^]`6"QUa  
    end 15dbM/Gj  
    k[<Uxh%  
    JC#M,j2  
    if any( r>1 | r<0 ) P g.j]  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 4Uzx2   
    end 3-6Lbe9H  
    viXt]0  
    vp2s)W8W  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Uz$.sa  
        error('zernfun:RTHvector','R and THETA must be vectors.') /OtLIM+7~{  
    end efUa[XO  
    [#mRlL0yk  
    'fS&WVR?  
    r = r(:); + rN&@}Jt.  
    theta = theta(:); n~Qo@%Jr  
    length_r = length(r); {$P')> /  
    if length_r~=length(theta) /O {iL:`  
        error('zernfun:RTHlength', ... 2Sb68hJIE  
              'The number of R- and THETA-values must be equal.') /kH 7I  
    end 1ww#]p`1  
    J2avt  
    5!jU i9  
    % Check normalization: 0hv}*NYd  
    % -------------------- ,.,spoV  
    if nargin==5 && ischar(nflag) zkb[u"  
        isnorm = strcmpi(nflag,'norm'); Mv_-JE9#>o  
        if ~isnorm kT12  
            error('zernfun:normalization','Unrecognized normalization flag.') eFXQ~~gOj  
        end YQN@;  
    else ,qu7XFYrY  
        isnorm = false; e754g(|>b  
    end >j6"\1E+Dz  
    C.N#y`g  
    a%XF"*^v  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N;mJHr3[F  
    % Compute the Zernike Polynomials G:4'')T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9YEE.=]T  
    hUP?r/B  
    3](At%ss  
    % Determine the required powers of r: ?)V|L~/  
    % ----------------------------------- ./i5VBP5  
    m_abs = abs(m); tYUg%2G  
    rpowers = []; q5#6PYIq  
    for j = 1:length(n) @Pb%dS  
        rpowers = [rpowers m_abs(j):2:n(j)]; opv<r* !  
    end hn[lhC  
    rpowers = unique(rpowers); 6R#.AD\  
    *|({(aZ  
    . ytxe!O  
    % Pre-compute the values of r raised to the required powers, T o$D [-  
    % and compile them in a matrix: JsK_q9]$e  
    % ----------------------------- kHz?vVE/l  
    if rpowers(1)==0 5H }d\=z  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,R[<+!RS  
        rpowern = cat(2,rpowern{:}); E>isl"  
        rpowern = [ones(length_r,1) rpowern]; ]Wg&r Y0  
    else #7Jvk_r9Y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .g>0FP  
        rpowern = cat(2,rpowern{:}); ,Fzuo:{uy  
    end 4L<;z'   
    7Sl"q=>  
    MHQM'  
    % Compute the values of the polynomials: h pKrP  
    % -------------------------------------- &6&$vF65c  
    y = zeros(length_r,length(n)); e !N%   
    for j = 1:length(n) ZKF  #(G  
        s = 0:(n(j)-m_abs(j))/2; 63HtZ=hO7  
        pows = n(j):-2:m_abs(j); BT|n+Y[  
        for k = length(s):-1:1 on.m '-s  
            p = (1-2*mod(s(k),2))* ... 3eN(Sw@p  
                       prod(2:(n(j)-s(k)))/              ... auHP^O> 4L  
                       prod(2:s(k))/                     ... }iRRf_   
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... \2[sUY<W  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); S N ;1F  
            idx = (pows(k)==rpowers); Nn{/_QG  
            y(:,j) = y(:,j) + p*rpowern(:,idx); q85 4k+C  
        end yC\!6pg  
         2Q)pT$  
        if isnorm UL0n>Wa5  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1xjw=  
        end 1EQLsg`d^  
    end p+}eP|N  
    % END: Compute the Zernike Polynomials 6yK"g7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i?n#ge  
    ZN}U^9m=  
    {nH*Wu*^  
    % Compute the Zernike functions: jwO7r0?\`G  
    % ------------------------------ Lm7fz9F%  
    idx_pos = m>0; UUEbtZH;  
    idx_neg = m<0; qJK-HF:#  
    hx hs>eY  
    C^ZD Uj`  
    z = y; CGs5`a  
    if any(idx_pos) ;Swj`'7  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [m6%_3zV  
    end 7-MyiCt  
    if any(idx_neg) VWW(=j  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); V PI_pK  
    end "#]V^Rzxh  
    (|sqN8SbA  
    J<-2dvq  
    % EOF zernfun q],/%W  
     
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    离线phoenixzqy
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    只看该作者 1楼 发表于: 2012-04-23
    慢慢研究,这个专业性很强的。用的人又少。
    2024年6月28-30日于上海组织线下成像光学设计培训,欢迎报名参加。请关注子在川上光学公众号。详细内容请咨询13661915143(同微信号)
    离线sansummer
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    只看该作者 2楼 发表于: 2012-04-27
    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
    离线jssylttc
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    只看该作者 3楼 发表于: 2012-05-14
    回 sansummer 的帖子
    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)   ~ A4_  
    +BkmI\  
    DDE还是手动输入的呢? o 9{~F`{p  
    ~"i4"Op&  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
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    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究