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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 正序阅读 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, Vm"{m/K0  
    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, H\PY\O&cP  
    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? U65a _dakk  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? o8ERU($/  
    n N_Ylw  
    W,D$=Bg  
    c %f'rj  
    l&2pUv=  
    function z = zernfun(n,m,r,theta,nflag) myvn@OsEw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~%D=\iE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GV"X) tGo  
    %   and angular frequency M, evaluated at positions (R,THETA) on the te*|>NRS  
    %   unit circle.  N is a vector of positive integers (including 0), and + lNAog  
    %   M is a vector with the same number of elements as N.  Each element ExW3LM9(  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) -*nd5(lY&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, FSNzBN  
    %   and THETA is a vector of angles.  R and THETA must have the same o-ee3j.  
    %   length.  The output Z is a matrix with one column for every (N,M) dBeZx1Dy  
    %   pair, and one row for every (R,THETA) pair. %"gV>E_u  
    % &2Q0ii#Aa  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kw$*o k  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \Um &  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral wRCv?D`vV  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,UA-Pq3 }  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized uJ:SN;  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (oG-h"^/  
    % 0{k*SCN#  
    %   The Zernike functions are an orthogonal basis on the unit circle. 713)D4y}  
    %   They are used in disciplines such as astronomy, optics, and `*ml/% \  
    %   optometry to describe functions on a circular domain. >>I~v)a>w  
    % m`lxQik  
    %   The following table lists the first 15 Zernike functions. wc~k4B9"  
    % lDf:~  
    %       n    m    Zernike function           Normalization -udKGrT+  
    %       -------------------------------------------------- |WUm;o4E`U  
    %       0    0    1                                 1 ?E|be )  
    %       1    1    r * cos(theta)                    2 Z]\IQDC  
    %       1   -1    r * sin(theta)                    2 Z{p62|+Ck@  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) &`}8Jz=S  
    %       2    0    (2*r^2 - 1)                    sqrt(3) a'prlXr\4  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) J12hjzk6@  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) H vezi>M  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |\# 6?y[o  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,>vI|p,/G*  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) k4!z;Yq  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) +=JJ=F)  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) eI:;l];G9  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) zjlo3=FQX[  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7jtDhsVz  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ><r\ 5`  
    %       --------------------------------------------------  1cvH  
    % ]*\m@lWu  
    %   Example 1: 9i`sSi8   
    % lE 09Y  
    %       % Display the Zernike function Z(n=5,m=1) Ar iW&E  
    %       x = -1:0.01:1; [ KT1.5M[  
    %       [X,Y] = meshgrid(x,x); F =Zc_  
    %       [theta,r] = cart2pol(X,Y); ]fb3>HOTJ  
    %       idx = r<=1; @`S8d%6P  
    %       z = nan(size(X)); m@#@7[6]o  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 'H|=]n0  
    %       figure 1XD|H_JG<j  
    %       pcolor(x,x,z), shading interp u ^Ss8}d  
    %       axis square, colorbar SGA!%=Lp  
    %       title('Zernike function Z_5^1(r,\theta)') "U6:z M  
    % U%zZw)  
    %   Example 2: `a:L%Ex  
    % hn p-x3  
    %       % Display the first 10 Zernike functions  `xm4?6  
    %       x = -1:0.01:1; nApkK1?  
    %       [X,Y] = meshgrid(x,x); 8Z1pQx-P2C  
    %       [theta,r] = cart2pol(X,Y); 48t_?2>  
    %       idx = r<=1; \UR/tlw+/  
    %       z = nan(size(X)); _6/q.  
    %       n = [0  1  1  2  2  2  3  3  3  3]; !ZC0n`  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; O%R*1 P9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; GJB= 5nE  
    %       y = zernfun(n,m,r(idx),theta(idx)); f6O5k8n  
    %       figure('Units','normalized') _=d X01  
    %       for k = 1:10 1~_&XNb&  
    %           z(idx) = y(:,k); HaiaDY)  
    %           subplot(4,7,Nplot(k)) cPL]WI0(  
    %           pcolor(x,x,z), shading interp dX vp-oi  
    %           set(gca,'XTick',[],'YTick',[]) n>["h2  
    %           axis square 1-6[KBQ8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :4'Fq;%C  
    %       end )?qH#>mD6  
    % *M^t@hl  
    %   See also ZERNPOL, ZERNFUN2. 6~b]RZe7  
    ocbNf'W;  
    B6hd*f  
    %   Paul Fricker 11/13/2006 wO&2S-;_K  
    FY(C<fDRo{  
    F>!gwmn~  
    BMlu>,  
    ]Y%U5\$  
    % Check and prepare the inputs: qZ79IX'y  
    % ----------------------------- .)Af&+KT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z.v2 !u  
        error('zernfun:NMvectors','N and M must be vectors.') QTKN6P  
    end z')zV oW,  
    C6P(86?  
    v@KP~kp  
    if length(n)~=length(m) SF#Rc>v  
        error('zernfun:NMlength','N and M must be the same length.') {6uhUb  
    end XnCrxj  
    :lGH31GG  
    8fI&-uP{g  
    n = n(:); HGJfj*JH  
    m = m(:); qV`JZ\n  
    if any(mod(n-m,2)) MaX:o GF,  
        error('zernfun:NMmultiplesof2', ... sHwn,4|iY  
              'All N and M must differ by multiples of 2 (including 0).') {#Vck\&  
    end o"5[~$O  
    Q[U_ 0O,A9  
    ['l.]k-b}  
    if any(m>n) *[MK{m  
        error('zernfun:MlessthanN', ... /Wqx@#  
              'Each M must be less than or equal to its corresponding N.') qp6*v&  
    end Bt\z0*t=s  
    NcVsQV  
    yavoGk  
    if any( r>1 | r<0 ) O*d4zBT  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'z}Hg *  
    end [=xJh?*P  
    rW&# Xw/a  
    -< 0PBl  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) M zbs#v0  
        error('zernfun:RTHvector','R and THETA must be vectors.') |0jmOcZF  
    end w_sA8B  
    %^C.e*  
    .}V&*-ep  
    r = r(:); =:;KY uTr  
    theta = theta(:); 8% ;K#,>  
    length_r = length(r); 5v uB87`  
    if length_r~=length(theta) ~il{6Z+#n  
        error('zernfun:RTHlength', ... lv* fK  
              'The number of R- and THETA-values must be equal.') @/ m|T]'8  
    end +z2+z  
    O!c b-  
    nvdo|5  
    % Check normalization: [v!TQwMU  
    % -------------------- sMikTwR/^  
    if nargin==5 && ischar(nflag) >(t_  
        isnorm = strcmpi(nflag,'norm'); {MaFv  
        if ~isnorm ZPISclSA+  
            error('zernfun:normalization','Unrecognized normalization flag.') TBzOz:k  
        end (Wm4JmX%  
    else DG&[.dR+  
        isnorm = false; J f,)Y>EI  
    end 'xC83}!k  
    gtBnP~zT\B  
    8?S)>-mwv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P2+Z^J`Y>  
    % Compute the Zernike Polynomials 8jnz;;|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,;2x.We  
    )/hb9+S  
    "1U:qr2-H  
    % Determine the required powers of r: 2eOde(K+  
    % ----------------------------------- 'Si 1r%'m#  
    m_abs = abs(m); -[I}"Glz:  
    rpowers = []; v=~=Q*\l  
    for j = 1:length(n) #jja#PF]7  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1f"LAs`%  
    end Z L3aO,G2  
    rpowers = unique(rpowers); :V%XEN)  
    F_Q?0 Do0'  
    c==` r C  
    % Pre-compute the values of r raised to the required powers, l&z)Q/>?pZ  
    % and compile them in a matrix: Nz ,8NM]  
    % ----------------------------- `+!GoXI  
    if rpowers(1)==0 z'G~b[kG4n  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); I#]$H#}Av  
        rpowern = cat(2,rpowern{:}); ,AC+s"VS  
        rpowern = [ones(length_r,1) rpowern]; tsFwFB*  
    else Ng6(2Wt0e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GYD`  
        rpowern = cat(2,rpowern{:}); 88dq8T4  
    end \gh`P S-B  
    {&'u1yR  
    v;9VX   
    % Compute the values of the polynomials: NC*h7  
    % -------------------------------------- =Of!1TR(  
    y = zeros(length_r,length(n)); cNW [i"  
    for j = 1:length(n) 0aMw  
        s = 0:(n(j)-m_abs(j))/2; Ba$Ibq,r/  
        pows = n(j):-2:m_abs(j); GHMoT  
        for k = length(s):-1:1 g2=5IU<  
            p = (1-2*mod(s(k),2))* ... fR>(b?C  
                       prod(2:(n(j)-s(k)))/              ... y s5b34JN  
                       prod(2:s(k))/                     ... K#=)]qIk  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... He4sP` &I  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ;P-xKRU!Xx  
            idx = (pows(k)==rpowers); f!`,!dZgkd  
            y(:,j) = y(:,j) + p*rpowern(:,idx); b@OL !?JP  
        end }ST9&w i~  
         (9N75uCa  
        if isnorm  H4HWr6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); e ,_b  
        end hi>sDU< x  
    end Z=sCYLm  
    % END: Compute the Zernike Polynomials xud  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z#wmEc.}C  
    $vS`w4Y  
    Bf Lh%XC  
    % Compute the Zernike functions: =o5ZcC  
    % ------------------------------ .)W'{2J-  
    idx_pos = m>0; "+js7U-  
    idx_neg = m<0; H)$-T1Wx4  
    @`N)`u85[  
    V~+{douq  
    z = y; 8J:6uO c|  
    if any(idx_pos) m8Q6ESg<*u  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dkuB{C,  
    end vj I>TIy  
    if any(idx_neg) ellj/u61bj  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); nn@"68]g  
    end T!uK _  
    l>RW&C&T  
    8X`Gm!)  
    % EOF zernfun Kc=&jCn  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  #P$=P2o  
    E^_P  
    DDE还是手动输入的呢? <#JJS}TLk  
    ?"\`u;  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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)