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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, l~!fQ$~  
    我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, b\j&!_   
    这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? ;=\5$J9  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? VSpt&19  
    UAXF64w{  
    PeUd  
    Yj7= T%5  
     |iUfM3  
    function z = zernfun(n,m,r,theta,nflag) [^}>AC*im  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Bx : So6:  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N pkN:D+g S  
    %   and angular frequency M, evaluated at positions (R,THETA) on the u$=ogp =0  
    %   unit circle.  N is a vector of positive integers (including 0), and Y!1^@;)^  
    %   M is a vector with the same number of elements as N.  Each element UtBlP+bE?y  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) OG^WZ.YU  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, /\a]S:V-j  
    %   and THETA is a vector of angles.  R and THETA must have the same ENx@Ex  
    %   length.  The output Z is a matrix with one column for every (N,M) % X ,B-h^  
    %   pair, and one row for every (R,THETA) pair. ^&';\O@)  
    % :e<`U~8m  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h$7Fe +#I#  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H"q`k5R  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral hp]ng!I{\u  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {.3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized =Q8H]F  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <\d|=>;  
    % xV>iL(?  
    %   The Zernike functions are an orthogonal basis on the unit circle. j #I:6yA3  
    %   They are used in disciplines such as astronomy, optics, and ?%xhe  
    %   optometry to describe functions on a circular domain. NBqV0>vR  
    % x !:9c<  
    %   The following table lists the first 15 Zernike functions. 0gOrW=  
    % Ng'ZAG;O  
    %       n    m    Zernike function           Normalization lKV\1(`  
    %       -------------------------------------------------- `zzKD2y  
    %       0    0    1                                 1 h/ X5w4  
    %       1    1    r * cos(theta)                    2 U.hERe ~X  
    %       1   -1    r * sin(theta)                    2 =2nn "YVP  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) v :+8U[x  
    %       2    0    (2*r^2 - 1)                    sqrt(3) s@ 2 0#D  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) [UJEU~XC  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) P"bknXL  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) gVnws E  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) A`x -L  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) &vFqe,Z  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (3N"oE.b]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'Uko^R)(  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) O@r.>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) XYb^C s;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 'ybth  
    %       -------------------------------------------------- 77xq/c[)  
    % CP]S-o}yd  
    %   Example 1: z#{ 0;t  
    % 0eqi1;$b]  
    %       % Display the Zernike function Z(n=5,m=1) . Z*j!{@c  
    %       x = -1:0.01:1; f8LrDR  
    %       [X,Y] = meshgrid(x,x); Z&dr0w8  
    %       [theta,r] = cart2pol(X,Y); a/QtJwIV  
    %       idx = r<=1; so!w!O@@  
    %       z = nan(size(X)); 5@+4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {K45~ha9!m  
    %       figure JQ"`9RNb  
    %       pcolor(x,x,z), shading interp ?E+:]j_  
    %       axis square, colorbar .# 6n  
    %       title('Zernike function Z_5^1(r,\theta)') MegE--h  
    % WxVn&c\  
    %   Example 2: .:{h{@a  
    % |*tWF! D6`  
    %       % Display the first 10 Zernike functions j\`EUC  
    %       x = -1:0.01:1; 1p7cv~#95  
    %       [X,Y] = meshgrid(x,x); =My}{n[  
    %       [theta,r] = cart2pol(X,Y); :DdBn.  
    %       idx = r<=1; + mfe*'AU  
    %       z = nan(size(X)); *L%6qxl`V  
    %       n = [0  1  1  2  2  2  3  3  3  3]; L$+d.=]  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; }W:*aU  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; [j)\v^m  
    %       y = zernfun(n,m,r(idx),theta(idx)); {W5ydHXy  
    %       figure('Units','normalized') I 1b  
    %       for k = 1:10 1B)Y;hg6&  
    %           z(idx) = y(:,k); H96BqNoO  
    %           subplot(4,7,Nplot(k)) YgE]d?_h  
    %           pcolor(x,x,z), shading interp M}Nb|V09  
    %           set(gca,'XTick',[],'YTick',[]) <w0NPrS]  
    %           axis square 2;r]gT~  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1Pk mg%+  
    %       end 4S,.R  
    % r]A" Og_U  
    %   See also ZERNPOL, ZERNFUN2. lLuID  
    uY^v"cw/F  
    =n@F$/h  
    %   Paul Fricker 11/13/2006 R K"&l!o  
    $%7I:  
    dB@Wn!Y  
    )W&o?VRfO  
    $[Tt#CJ w  
    % Check and prepare the inputs: r<;l{7lY_  
    % ----------------------------- QS3U)ZO$@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }.0Bl&\UK  
        error('zernfun:NMvectors','N and M must be vectors.') ;mDM5.iF  
    end p"Ot5!F >  
    P^ptsZ%  
    iO!27y  
    if length(n)~=length(m) -O'{:s~  
        error('zernfun:NMlength','N and M must be the same length.') 5]jx5!N  
    end 1 6"#i  
    kTnOmA w  
    $o]r ]#B+  
    n = n(:); Dc08D4   
    m = m(:); i 3m3zXt  
    if any(mod(n-m,2)) P @zz"~f7  
        error('zernfun:NMmultiplesof2', ... QL2Nz@|k  
              'All N and M must differ by multiples of 2 (including 0).') ;W]D ~X&  
    end 4L8z>9D  
    Lp_$?MCD.  
    Ls&+XlrX8  
    if any(m>n) G+0><,S  
        error('zernfun:MlessthanN', ... ,eR8 ~(`=  
              'Each M must be less than or equal to its corresponding N.') b9!.-^<8y  
    end 94\t1fE  
    &~RR&MdZ2  
    BR+nL6sU  
    if any( r>1 | r<0 ) z9[[C^C  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') U4Z[!s$  
    end pD"YNlB^  
    X*i/A<Y`=  
    W+_RhJ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WzjL-a(  
        error('zernfun:RTHvector','R and THETA must be vectors.') >*IN  
    end ~ |6dH  
    W4(v6>5l  
     >1A*MP4  
    r = r(:); 7KU~(?|:h  
    theta = theta(:); P''X_1oMC  
    length_r = length(r); 'l~6ErBSg  
    if length_r~=length(theta) r!7Y'|  
        error('zernfun:RTHlength', ... cB#nsu>  
              'The number of R- and THETA-values must be equal.') \#CM <%  
    end -T7%dLHY  
    ;6ky5}z  
    J{`eLmTu  
    % Check normalization: 98fu>>*G{  
    % -------------------- ` @8`qXg  
    if nargin==5 && ischar(nflag) 'n0 .#E_  
        isnorm = strcmpi(nflag,'norm'); 1"}cdq.  
        if ~isnorm 'B_\TU0 O  
            error('zernfun:normalization','Unrecognized normalization flag.') 7 {f_fkbs  
        end  B$^7h!  
    else .-0%6] cFD  
        isnorm = false; k@V#HC{t  
    end } VEq:^o.  
    ZsZcQj6G,  
    r [s!F=^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V >Hf9sZ  
    % Compute the Zernike Polynomials NBjeH tT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AVG>_$<  
    k6!4Zz_8  
    *:_P8G;  
    % Determine the required powers of r: B<7/,d'  
    % ----------------------------------- EATu KLP\  
    m_abs = abs(m); y:d{jG^  
    rpowers = []; @m~RtC-Q  
    for j = 1:length(n) B6] <G-  
        rpowers = [rpowers m_abs(j):2:n(j)]; o%[U  
    end |%1?3Mpn  
    rpowers = unique(rpowers); /RT%0!  
    1f#mHt:(  
    [Il~K  
    % Pre-compute the values of r raised to the required powers, WZZ4]cC  
    % and compile them in a matrix: wvMW|  
    % ----------------------------- ]JE TeZ^/  
    if rpowers(1)==0 at|g%$%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); S[,8TErz  
        rpowern = cat(2,rpowern{:}); {f/]5x(_  
        rpowern = [ones(length_r,1) rpowern]; LZ U$  
    else W0XF~  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); YE}s  
        rpowern = cat(2,rpowern{:}); 9 [jTs3l:  
    end PXzT6)  
    T[?6[,.  
    _q?<at}y  
    % Compute the values of the polynomials: 0)!Ll*L!p  
    % -------------------------------------- 1mH%H*#  
    y = zeros(length_r,length(n)); ^YvB9XN  
    for j = 1:length(n) X"q!Y#)  
        s = 0:(n(j)-m_abs(j))/2; [zkikZy  
        pows = n(j):-2:m_abs(j); FP^{=0  
        for k = length(s):-1:1 Nt:9MG>1  
            p = (1-2*mod(s(k),2))* ... nkDy!"K  
                       prod(2:(n(j)-s(k)))/              ... AoaN22  
                       prod(2:s(k))/                     ... xJZ@DR,#  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2; `=P5V  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); %7hB&[ 5  
            idx = (pows(k)==rpowers); 2Y!S_Hw8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Bi3+)k>u7  
        end LN2D  
         Oco YV J  
        if isnorm =Gk/k}1  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J#2!ZQE 3  
        end C'A]i5  
    end ,`A?!.K$  
    % END: Compute the Zernike Polynomials KvPX=/&Zu  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a`(a)9i  
    p4K.NdUH  
    h*B|fy4K9U  
    % Compute the Zernike functions: ULH0'@BJ  
    % ------------------------------ C0*@0~8$9  
    idx_pos = m>0; mTNVU@TY=  
    idx_neg = m<0; (Y% Q|u  
    Q&'}BeUbm  
    p&-'|'![l  
    z = y; A@*:<Hs%  
    if any(idx_pos) U-k VNBs  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5kNzv~4B,;  
    end LPYbHo3fq  
    if any(idx_neg) )~6zYJ2  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');  Ez~'^s@  
    end 6$fYt&1  
    4 1a. #o  
    gb=/#G0R  
    % EOF zernfun `(6r3f~XJ  
     
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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)  _/wV;h~R  
    5uxBK"q  
    DDE还是手动输入的呢? e9Nk3Sj]  
    gn3jy^5  
    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