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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, eUvIO+av  
    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, 6zs&DOB  
    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? gwk$|aT@  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? $Z)Dvy|  
    )@.bkzW  
    9`}Wp2  
    U JG)-x  
    w)-@?jN  
    function z = zernfun(n,m,r,theta,nflag) X1U7$/t  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. BQVpp,]  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }OO(uC2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the &T?>Kx  
    %   unit circle.  N is a vector of positive integers (including 0), and a5WVDh, cR  
    %   M is a vector with the same number of elements as N.  Each element >B$ZKE  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~Nf0 1,F  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, \dj&4u3  
    %   and THETA is a vector of angles.  R and THETA must have the same ! *\)7D  
    %   length.  The output Z is a matrix with one column for every (N,M) b u%p,u!  
    %   pair, and one row for every (R,THETA) pair. CBx1.xL  
    % cSCO7L2E18  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike SeAokz>  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5)4*J.  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 0'O;H[nrl  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]xQPSs_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized kvs^*X''Ep  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ";B.^pBv@;  
    % :P`sK&b_  
    %   The Zernike functions are an orthogonal basis on the unit circle. Hno@  
    %   They are used in disciplines such as astronomy, optics, and }xTTz,Oj$  
    %   optometry to describe functions on a circular domain. DG8]FhD^b  
    % /b,+YyWi%  
    %   The following table lists the first 15 Zernike functions. 2|F.JG^  
    % V ~w(^;o@  
    %       n    m    Zernike function           Normalization `+$'bNPn&  
    %       -------------------------------------------------- L>$yslH; b  
    %       0    0    1                                 1 [oOZ6\?HB  
    %       1    1    r * cos(theta)                    2 S!8eY `C.  
    %       1   -1    r * sin(theta)                    2 i_ws*7B<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) zR h1  
    %       2    0    (2*r^2 - 1)                    sqrt(3) [P)'LY6F  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) y %Get  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) bTZ/$7pp9  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) I_.(&hMn  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) # 'G/&&<  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6gwjrGje\  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) BZEY^G  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @PuJre4!;L  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) RL |.y~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )0`;leli  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 6NJ"ty9Bp  
    %       -------------------------------------------------- qC?J`   
    % q a#Fa)g*  
    %   Example 1: 7a0ZI  
    % [CBA Lj5  
    %       % Display the Zernike function Z(n=5,m=1) c#nFm&}dm  
    %       x = -1:0.01:1; `;WiTE)&)  
    %       [X,Y] = meshgrid(x,x); >i~W$; t  
    %       [theta,r] = cart2pol(X,Y); /S1EQ%_  
    %       idx = r<=1; *#e%3N05_  
    %       z = nan(size(X)); Da1BxbDeI  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); o%X_V!B{V  
    %       figure 7CYu"+Ea  
    %       pcolor(x,x,z), shading interp R'qB-v.  
    %       axis square, colorbar %1SA!1>j  
    %       title('Zernike function Z_5^1(r,\theta)') 1i#uKKwE  
    % NUiZ!&  
    %   Example 2: ~\4l*$3(^  
    % LtbL[z>]  
    %       % Display the first 10 Zernike functions ZgF-.(GV  
    %       x = -1:0.01:1; m9ts&b+TE  
    %       [X,Y] = meshgrid(x,x); ,kuJWaUC@  
    %       [theta,r] = cart2pol(X,Y); SSycQ4[{o  
    %       idx = r<=1; D|Wekhm  
    %       z = nan(size(X)); kW\=Z 1\#  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^DXERt&3  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; pl Ii  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; +oBf\!{cW  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2_;.iH 6  
    %       figure('Units','normalized') pSkP8'  ?  
    %       for k = 1:10 (~xFd^W9o  
    %           z(idx) = y(:,k); ^ $Q',  
    %           subplot(4,7,Nplot(k)) [J\5DctX;c  
    %           pcolor(x,x,z), shading interp N}nU\e6 Y  
    %           set(gca,'XTick',[],'YTick',[]) sY7:Lzs.,  
    %           axis square >T;"bc b  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) H`]nY`HYg  
    %       end mm/U9hbp%  
    % >WE3$Q>bi  
    %   See also ZERNPOL, ZERNFUN2. ?|TVz!3  
    Ks@S5:9sp  
    6:>4}WOP  
    %   Paul Fricker 11/13/2006 r!V#@Md  
    l@Ma{*s6=5  
    1c~c_Cc4  
    cf1Ve\(YGI  
    $5yS`Iq S  
    % Check and prepare the inputs: yk!,{Q?<$  
    % ----------------------------- (`GO@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 78#je=MDg  
        error('zernfun:NMvectors','N and M must be vectors.') f<9H#S:  
    end Y oNg3  
    u 9 1;GBY  
    QX ishHk&  
    if length(n)~=length(m) R!W!8rr3  
        error('zernfun:NMlength','N and M must be the same length.') c.m ' %4  
    end ] M "{=z  
    jIK *psaV  
    6hXL`A&},  
    n = n(:); 1lfkb1BM  
    m = m(:); af\>+7x93  
    if any(mod(n-m,2)) X/lLM`  
        error('zernfun:NMmultiplesof2', ... ?(Dkh${@  
              'All N and M must differ by multiples of 2 (including 0).') \ lP c,8)  
    end eHF#ME  
    iOPv % [  
    epCU(d*b  
    if any(m>n) go m< V?$  
        error('zernfun:MlessthanN', ... c 6}d{B[  
              'Each M must be less than or equal to its corresponding N.') JTNQz  
    end @Rj&9/\L  
    _zI9 5  
    mC n,I  
    if any( r>1 | r<0 ) UA>~xJp=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') dc5w_98o  
    end N6cf`xye  
    rK)So#'  
    Ul^/Dh  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _xI'p6C  
        error('zernfun:RTHvector','R and THETA must be vectors.') A`Z!=og=  
    end %'"#X?jk1  
    7]d396%  
    <oI{:KH  
    r = r(:); *I`Sc|A  
    theta = theta(:); ;E(gl$c:  
    length_r = length(r); (u@[}!  
    if length_r~=length(theta) vI{JBWE,S  
        error('zernfun:RTHlength', ... #w*1 !  
              'The number of R- and THETA-values must be equal.') a)MjX<y  
    end x6* {@J&5*  
    <>A:Oi3^  
    1}tZ,w>  
    % Check normalization: :7D&=n)  
    % -------------------- 9b@L^]Kg  
    if nargin==5 && ischar(nflag) /YR*KxIx  
        isnorm = strcmpi(nflag,'norm'); [^A.$,  
        if ~isnorm {0q;:7Bt  
            error('zernfun:normalization','Unrecognized normalization flag.') =2s 5>Oz+  
        end 1B+MCt4  
    else /[VafR!  
        isnorm = false; ?xGxr|+a  
    end w8wF;:>  
    ~&CaC  
    J<p<5):R;  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }el. qZ  
    % Compute the Zernike Polynomials ve.iyr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p \; * :  
    ^uW](2  
    u ?7(A %  
    % Determine the required powers of r: zawU  
    % ----------------------------------- 3uwu}aw  
    m_abs = abs(m); miCt)Qd  
    rpowers = []; WiH%URFB  
    for j = 1:length(n) -TU7GCb=  
        rpowers = [rpowers m_abs(j):2:n(j)]; r[vMiVb  
    end 0L $v7, 5  
    rpowers = unique(rpowers); `~WxMY0M  
    [3nhf<O  
    _J 6|ju\  
    % Pre-compute the values of r raised to the required powers, d;|e7$F'  
    % and compile them in a matrix: ZwAX+0  
    % ----------------------------- &0K; Vr~D  
    if rpowers(1)==0 30*^ERO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); },2-\-1  
        rpowern = cat(2,rpowern{:}); TYw0#ZXo  
        rpowern = [ones(length_r,1) rpowern]; $lOx 6rL  
    else _U1~^ucV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); DifRpj I-0  
        rpowern = cat(2,rpowern{:}); 7ks09Cy  
    end @CP"AYB #  
    rFM`ne<zh  
    Fm3-Sn|Po  
    % Compute the values of the polynomials: 8 2&JYx  
    % -------------------------------------- p)f OAr  
    y = zeros(length_r,length(n)); V`TXn[7  
    for j = 1:length(n) X"(!\{ySI;  
        s = 0:(n(j)-m_abs(j))/2; i)1E[jc{p!  
        pows = n(j):-2:m_abs(j); U> (5J,G  
        for k = length(s):-1:1 gd_w;{WP  
            p = (1-2*mod(s(k),2))* ... mq[(yR  
                       prod(2:(n(j)-s(k)))/              ... &O#a==F!(  
                       prod(2:s(k))/                     ... U; ?%rM6  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |H2{%!  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); n4 KiC!*i0  
            idx = (pows(k)==rpowers); Bg-C:Ok 2'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); - DlKFN  
        end k)'hNk"x  
         $G"PZ7  
        if isnorm doBfpQ2  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); yrdJX  
        end D2[wv+#)  
    end H:`W\CP7_  
    % END: Compute the Zernike Polynomials RI:x`do  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <.HHV91  
    X9/V;!  
    T73oW/.0X?  
    % Compute the Zernike functions: {v+i!a'+  
    % ------------------------------ X@b$C~+  
    idx_pos = m>0; NO +j    
    idx_neg = m<0;  uw LT$  
    .hg<\-:_  
    4adCMfP7.  
    z = y; m1gJ"k6 `j  
    if any(idx_pos) ;QR|v  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -vGyEd7  
    end ;J2U5Y NO  
    if any(idx_neg) doFp53NhV  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Qf414 oW  
    end o2 W pi  
    bFVz ;  
    s C e7ni  
    % EOF zernfun fm;1Iu#  
     
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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)  8f8+3  
    5wGyM10  
    DDE还是手动输入的呢? Plm3vk=  
    %}'sFu m`  
    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