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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, Pqw<nyC.  
    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, IGT9}24  
    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? =6O*AJ  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? {:#nrD"  
    <<E 9MIn_  
    QxGcRlpLK  
    ?FjnG_Uz`D  
    y22DBB8  
    function z = zernfun(n,m,r,theta,nflag) bk;uKV+<  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5V\",PA W  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ?@;)2B|q  
    %   and angular frequency M, evaluated at positions (R,THETA) on the l'aCpzf  
    %   unit circle.  N is a vector of positive integers (including 0), and P9f`<o  
    %   M is a vector with the same number of elements as N.  Each element ^G(Ee+PN@  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) OG$v"Yf~  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, u%+k\/Scp.  
    %   and THETA is a vector of angles.  R and THETA must have the same )7.DF|A  
    %   length.  The output Z is a matrix with one column for every (N,M) %D8.uGsh  
    %   pair, and one row for every (R,THETA) pair. Ox&G  [  
    % i%i />;DF  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .|5$yGEF_+  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ed}#S~4q  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *B}O  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .RJMtmp  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %lWOW2~R  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ..+#~3es#y  
    % _oCNrjt9  
    %   The Zernike functions are an orthogonal basis on the unit circle. Qni`k)4  
    %   They are used in disciplines such as astronomy, optics, and Up'#OkTx  
    %   optometry to describe functions on a circular domain.  k4dC  
    % S\< i`q  
    %   The following table lists the first 15 Zernike functions. dt,Z^z+" E  
    % ^]D1':  
    %       n    m    Zernike function           Normalization QDV+(  
    %       -------------------------------------------------- "t(_r@qU/  
    %       0    0    1                                 1 Iia.`"S  
    %       1    1    r * cos(theta)                    2 rzn,N FI  
    %       1   -1    r * sin(theta)                    2 i!e8-gVMP&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6)  0.0-rd>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) >h#w~@e::  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {vCtp   
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) p^k0Rad  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) X(MS!RV  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) y32$b,%Xi,  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 0]iaNR %  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) @v2ko5  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ktx| c19  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <?5|(Q"@:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HOFxOBV  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) }UB@FRPF  
    %       -------------------------------------------------- z|D*ymz*EY  
    % =urGs`\  
    %   Example 1: wN4#j}C  
    % X_hDU~5{wC  
    %       % Display the Zernike function Z(n=5,m=1) (BeJ,K7  
    %       x = -1:0.01:1; `(0B09~7  
    %       [X,Y] = meshgrid(x,x); ?zm]KxIC  
    %       [theta,r] = cart2pol(X,Y); 2a48(~<_  
    %       idx = r<=1; @;P ;iI  
    %       z = nan(size(X)); l[ $bn!_ e  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); -|uoxj>  
    %       figure tMX$8W0 c  
    %       pcolor(x,x,z), shading interp /}m*|cG/  
    %       axis square, colorbar jd-]q2fQ|  
    %       title('Zernike function Z_5^1(r,\theta)') M\5|  
    % 8Ejb/W_  
    %   Example 2: [%N?D#;  
    % iP"sw0V8  
    %       % Display the first 10 Zernike functions dM^Z,; u  
    %       x = -1:0.01:1; DJ:'<"zH7  
    %       [X,Y] = meshgrid(x,x); DI{*E  
    %       [theta,r] = cart2pol(X,Y); Q'jw=w!|g  
    %       idx = r<=1; t'Wv? ,  
    %       z = nan(size(X)); 3>@VPMi  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /z*Z+OT2  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %NxQb'  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 5P-t{<]tx  
    %       y = zernfun(n,m,r(idx),theta(idx)); kt978qfk  
    %       figure('Units','normalized') 3^+D,)#D^  
    %       for k = 1:10 V&s|IoTR  
    %           z(idx) = y(:,k); <4q H0<  
    %           subplot(4,7,Nplot(k)) src+z#  
    %           pcolor(x,x,z), shading interp Fds 11 /c7  
    %           set(gca,'XTick',[],'YTick',[]) 6~x'~T  
    %           axis square % ERcFI]G  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \xCCJWek  
    %       end ~E7IU<B  
    % XH$r(@Z\7  
    %   See also ZERNPOL, ZERNFUN2. $3g{9)}  
    \}?X5X>  
    ]2wxqglh)  
    %   Paul Fricker 11/13/2006 Q7$o&N{  
    {4G/HW28  
    5?^L))  
    _V-KyK  
    1^}I?PbqV  
    % Check and prepare the inputs: Tn#Co$<  
    % ----------------------------- *(F`NJ 3  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wwB3m&  
        error('zernfun:NMvectors','N and M must be vectors.') dWvVK("Wj  
    end gVOAB-nw  
    Nhjq.&  
    W8^m-B&  
    if length(n)~=length(m) "^n,(l*4x  
        error('zernfun:NMlength','N and M must be the same length.') E=p+z"Ui  
    end GBbnR:hM  
    a4__1N^Qj  
    PC#^L$cg}  
    n = n(:); IT_I.5*A2  
    m = m(:); Go)$LC0Mi  
    if any(mod(n-m,2)) 9qB0F_xl  
        error('zernfun:NMmultiplesof2', ... I4X9RYB6c  
              'All N and M must differ by multiples of 2 (including 0).') T$xB H  
    end l4oyF|oJTH  
    J, 9NVw$  
    No'?8+i  
    if any(m>n) 6:7[>|okQ  
        error('zernfun:MlessthanN', ... Cku"vVw,  
              'Each M must be less than or equal to its corresponding N.') "d_wu#fO)  
    end _qxI9Q}<"  
    L=4+rshl!_  
    F~mIV;BP  
    if any( r>1 | r<0 ) W&?Qs=@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') lT^su'+bk  
    end R-13DVK  
    *|fF;-#v  
    br[iRda@  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;Y?MbD  
        error('zernfun:RTHvector','R and THETA must be vectors.') }X&rJV  
    end U#` e~d t<  
    `t~jHe4!Y  
    ;.A}c)b  
    r = r(:); s<9g3Gh  
    theta = theta(:); m+TAaK  
    length_r = length(r); 9r!8BjA  
    if length_r~=length(theta) k {*QU(  
        error('zernfun:RTHlength', ... $ F2Uv\7=  
              'The number of R- and THETA-values must be equal.') =:- fK-d  
    end ci~#G[_$S  
    o|kykxcq  
    ,@`?I6nKy  
    % Check normalization: }e?H(nZS7h  
    % -------------------- ;o_F<68QP  
    if nargin==5 && ischar(nflag) )/T[Cnx.Nc  
        isnorm = strcmpi(nflag,'norm'); : uncOd.  
        if ~isnorm *GT=U(d  
            error('zernfun:normalization','Unrecognized normalization flag.') 513,k$7  
        end g4IF~\QRVi  
    else Zse&{  
        isnorm = false; `\kihNkJn3  
    end s^wm2/Yw  
    WAa45G  
     \i%'M%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% va6Fp2n<1*  
    % Compute the Zernike Polynomials ! _S#8"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pHV^K v#  
    YV O$`W^N  
    v/+ <YU  
    % Determine the required powers of r: 2{-29bq  
    % ----------------------------------- ?b (iWq  
    m_abs = abs(m); KGz Nj%  
    rpowers = []; u_(~zs.N]  
    for j = 1:length(n) =&}@GsXdo  
        rpowers = [rpowers m_abs(j):2:n(j)]; DX s an  
    end cb}"giXQTB  
    rpowers = unique(rpowers); "rv~I_zl  
    Eb8pM>'qM  
    |&; ^?M  
    % Pre-compute the values of r raised to the required powers, !}(B=-  
    % and compile them in a matrix: Ipp_}tl_  
    % ----------------------------- BI1M(d#1L"  
    if rpowers(1)==0 k^J8 p#`6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IPQRdBQ  
        rpowern = cat(2,rpowern{:}); *WwM"NFHDd  
        rpowern = [ones(length_r,1) rpowern]; mMAN* }`O  
    else ?:(y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <LHhs <M'  
        rpowern = cat(2,rpowern{:}); x/*lNG/  
    end )l3Uf&v^f  
    ;J%:DD  
    3:)z+#Uk6  
    % Compute the values of the polynomials: )GD7 rsC`<  
    % -------------------------------------- %~u]|q<{  
    y = zeros(length_r,length(n)); hFrMOc&  
    for j = 1:length(n) LP2~UVq  
        s = 0:(n(j)-m_abs(j))/2; #@R0$x  
        pows = n(j):-2:m_abs(j);  F B]Y~;(  
        for k = length(s):-1:1 _D '(R  
            p = (1-2*mod(s(k),2))* ... Rs%`6et}\  
                       prod(2:(n(j)-s(k)))/              ... YvR bM  
                       prod(2:s(k))/                     ... ARH~dN*C  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... V=O52?8  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); A;oHji#*  
            idx = (pows(k)==rpowers); >B BV/C'9  
            y(:,j) = y(:,j) + p*rpowern(:,idx); AGlBvRX7e  
        end F.9}jd{  
         SE'Im  
        if isnorm broLC5hbQU  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 8q2a8I9g  
        end x~5uc$  
    end As:O|!F  
    % END: Compute the Zernike Polynomials iq#{*:1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D6"=2XR4n  
    e4z`:%vy  
    >)>f~>  
    % Compute the Zernike functions: ;Afz`Se1@  
    % ------------------------------ honh 'j  
    idx_pos = m>0; +|A`~\@N  
    idx_neg = m<0; b1&tk~D  
    $7x2TiAL  
    ':*H#}Br-#  
    z = y; R\j~X@vI  
    if any(idx_pos) ohx[_}xN  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 77>oQ~q  
    end ]aX@(3G1s  
    if any(idx_neg) VkQ@c;C  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); }EK{UM9y  
    end { bj!]j  
    55Ss%$k@  
    9YzV48su#  
    % EOF zernfun eqx }]#  
     
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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)  (*S<2HN5  
    +?J  N_aR  
    DDE还是手动输入的呢? f,G*e367:  
    M}8P _<,  
    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