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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, 6f<*1YR F  
    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, d91I  
    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? f'8B[&@L  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Z1 E` I89<  
    HE8'N=0  
    2&3eAJC  
    WlF+unB!9  
    Djg 1Qh  
    function z = zernfun(n,m,r,theta,nflag) ]=O{7#  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. r! cNc  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ";%e~ =  
    %   and angular frequency M, evaluated at positions (R,THETA) on the >^2ZM  
    %   unit circle.  N is a vector of positive integers (including 0), and h'z+8X_t  
    %   M is a vector with the same number of elements as N.  Each element rcD.P?"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5M/%%Ox  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 1_p[*h  
    %   and THETA is a vector of angles.  R and THETA must have the same e)fJd*P  
    %   length.  The output Z is a matrix with one column for every (N,M) {m1t~ S   
    %   pair, and one row for every (R,THETA) pair. UtHmM,*I  
    % $_%2D3-;D  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike eP-R""uPw  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), w yuJSB  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *RUd!]bh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \rB/83[;u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U~JG1#z6  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 81%qM7v9H  
    % %tklup]LF8  
    %   The Zernike functions are an orthogonal basis on the unit circle. *)}Ap4[  
    %   They are used in disciplines such as astronomy, optics, and 8VBkIYgb  
    %   optometry to describe functions on a circular domain. ;@=@N9q K  
    % |TNiKy  
    %   The following table lists the first 15 Zernike functions. QBN=l\m+  
    % *;V2_fWJ@  
    %       n    m    Zernike function           Normalization .j+2x[`l  
    %       -------------------------------------------------- o{ YW  
    %       0    0    1                                 1 {!:|.!-u  
    %       1    1    r * cos(theta)                    2 ?[*@T2Ck  
    %       1   -1    r * sin(theta)                    2 .$}Z:,aB  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Z IGbwL  
    %       2    0    (2*r^2 - 1)                    sqrt(3) X7imUy'.  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) VygXhh^7\  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ePu2t3E  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) |{}d5Z"5;}  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) gn4g 43  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }SJLBy0  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) s`$_  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z!Pdivx  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) }AeE|RNc  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i&SBW0)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) F?+Uar|-a  
    %       -------------------------------------------------- }y -AoG  
    % cE_Xo.:Y,  
    %   Example 1: !@]h@MC$7  
    % t0AqGrn  
    %       % Display the Zernike function Z(n=5,m=1) iX9[Q0g=oQ  
    %       x = -1:0.01:1; =c5 /cpZ^  
    %       [X,Y] = meshgrid(x,x); l,FG:"`Z@  
    %       [theta,r] = cart2pol(X,Y); 9b=^"K  
    %       idx = r<=1; QBBJ1U  
    %       z = nan(size(X)); r)Mx.`d!  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 8zB+%mcF  
    %       figure +'YSpJ  
    %       pcolor(x,x,z), shading interp <}x|@u  
    %       axis square, colorbar ,:Rq  
    %       title('Zernike function Z_5^1(r,\theta)') H?zCIue3  
    % %lqG*dRx0  
    %   Example 2: 4)>\rqF+v  
    % 7=M'n;!Mh  
    %       % Display the first 10 Zernike functions RE*S7[ge  
    %       x = -1:0.01:1; _`Yvfz3  
    %       [X,Y] = meshgrid(x,x); 80l3.z,:  
    %       [theta,r] = cart2pol(X,Y); H&>>]DD  
    %       idx = r<=1; gWU(uBS  
    %       z = nan(size(X)); 3 v,ae7$U&  
    %       n = [0  1  1  2  2  2  3  3  3  3]; -fZShOBY`  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; nH3b<k;S  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; X:} 5L> '  
    %       y = zernfun(n,m,r(idx),theta(idx)); R;,u >P "  
    %       figure('Units','normalized') %onAlf<$:^  
    %       for k = 1:10 6[Pr<4J  
    %           z(idx) = y(:,k); 1wH/#K  
    %           subplot(4,7,Nplot(k)) _tauhwu  
    %           pcolor(x,x,z), shading interp Wn9Mr2r!*,  
    %           set(gca,'XTick',[],'YTick',[]) iRr& 'k  
    %           axis square PTV`=vtj  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]\b1~ki!F  
    %       end gQzJ2LU(  
    % T;pn -  
    %   See also ZERNPOL, ZERNFUN2. G QB^  
    `5 v51TpH  
    ]C:Ifh~  
    %   Paul Fricker 11/13/2006 MAhPO!e5.  
    BKlc{=  
    2W-NCE%K)T  
    J$ih|nP  
    L5N{ie_  
    % Check and prepare the inputs: bJMcI8`  
    % ----------------------------- Q8/0Cb/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }T*xT>p^3  
        error('zernfun:NMvectors','N and M must be vectors.') Y Z2VP  
    end o5G"J"vxe  
    ?~y(--.t;T  
    w!9WCl]9M  
    if length(n)~=length(m) )cmLo0`$  
        error('zernfun:NMlength','N and M must be the same length.') YV!V9   
    end kx#L<   
    aiX;D/t?  
    O?J:+L(  
    n = n(:); ,ce^"yG  
    m = m(:); s/&]gj "  
    if any(mod(n-m,2)) xwp?2,<  
        error('zernfun:NMmultiplesof2', ... YbBH6R Zr  
              'All N and M must differ by multiples of 2 (including 0).') EYD{8Fw-  
    end 1kw4'#J8  
    U\GZ  
    %[CM;|?B4  
    if any(m>n) *t*&Q /W  
        error('zernfun:MlessthanN', ... < 3+&DV-<N  
              'Each M must be less than or equal to its corresponding N.') DT]p14@t9  
    end A =#-u&l  
    h9smviU7u  
    Lj1 @yokB  
    if any( r>1 | r<0 ) 1E_Ui1[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Qi]Z)v{^  
    end L;t~rW!1  
    A|OC?NZY  
    SpiC0  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /<pQ!'/G  
        error('zernfun:RTHvector','R and THETA must be vectors.') zi[M{bm  
    end [)0k}  
    ffd 3QQ  
    s3!LR2qiF  
    r = r(:); mnaD KeA  
    theta = theta(:); D)Rf  
    length_r = length(r); myX0<j3G5  
    if length_r~=length(theta) G")EE#W$}  
        error('zernfun:RTHlength', ... U+M?<4J) "  
              'The number of R- and THETA-values must be equal.') V/%;:u l.  
    end g'7hc~=  
    ov>L-  
    1Sk6[h'CL  
    % Check normalization: xTJ5VgG  
    % -------------------- L umD.3<  
    if nargin==5 && ischar(nflag) {S(T1ua  
        isnorm = strcmpi(nflag,'norm'); <s3(   
        if ~isnorm Dx)XC?'xO  
            error('zernfun:normalization','Unrecognized normalization flag.') ,]qX_`qF  
        end {# _C  
    else *uM*)6O 3  
        isnorm = false; VjM uU"++@  
    end &JM;jS z  
    ebK wCZwK*  
    ;CBdp-BUj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =nZd"t'p|  
    % Compute the Zernike Polynomials =)5a=^ 6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6u;(R0n  
    J :(\o=5 5  
    shZ<j7gqI  
    % Determine the required powers of r: @!O{>`  
    % ----------------------------------- N)Kr4GC  
    m_abs = abs(m); aC 0Jfo  
    rpowers = []; 2MeavTr  
    for j = 1:length(n) U# B  
        rpowers = [rpowers m_abs(j):2:n(j)]; VbR.tz  
    end :!h H`l}p  
    rpowers = unique(rpowers); y@JYkp>I  
    EBLoRW=8ld  
    C;>Ll~f_  
    % Pre-compute the values of r raised to the required powers, 7?] p\`  
    % and compile them in a matrix: ~C x2Q4E  
    % ----------------------------- qNL~m'  
    if rpowers(1)==0 !,"G/}'^;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5 Vqvb|  
        rpowern = cat(2,rpowern{:}); s$6#3%h  
        rpowern = [ones(length_r,1) rpowern]; zy;w07-)  
    else v}D!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); J< M;vB)  
        rpowern = cat(2,rpowern{:}); [G/X  
    end 0n=E.qZ9c  
    &N;6G`3  
    #9Z-Hd<  
    % Compute the values of the polynomials: %L.+r!.  
    % -------------------------------------- *[n^6)  
    y = zeros(length_r,length(n)); i[#Tn52D  
    for j = 1:length(n) V|7CYkB8  
        s = 0:(n(j)-m_abs(j))/2; TKX#/  
        pows = n(j):-2:m_abs(j); Q2=~  
        for k = length(s):-1:1 lh5d6VUA  
            p = (1-2*mod(s(k),2))* ... c qp#1oM4M  
                       prod(2:(n(j)-s(k)))/              ... >m!.l{*j>N  
                       prod(2:s(k))/                     ... v g]&T  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +dv@N3GV  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); )I4tl/  
            idx = (pows(k)==rpowers); A?zW!'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }Jfo(j  
        end )`^:G3w  
         rg~CF<  
        if isnorm jFfki.H  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); *93 N0m4Rl  
        end 8 Hn{CJ~'  
    end Ui&$/%Z|  
    % END: Compute the Zernike Polynomials "Wp<^ssMo  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  HV(Kz  
    #v-!GK_<  
    ,z3b2$ &A  
    % Compute the Zernike functions: gP@ni$n  
    % ------------------------------ kZNZ?A<D  
    idx_pos = m>0; =Wa\yBj_;m  
    idx_neg = m<0; L?fv5 S3  
    s-B\8&^C  
    Xk$lQMwZ  
    z = y; 9@06]EI_  
    if any(idx_pos) G w[&P%  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i_"I"5pBF  
    end nC^'2z  
    if any(idx_neg) xo$ZPnf(zv  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); "6i9f$N  
    end TfPx   
    %`'VXR?`h=  
    &bRH(yF  
    % EOF zernfun %}[??R0  
     
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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)  $]Q_x?  
    `S!`=26Z!  
    DDE还是手动输入的呢? $Ec;w~e  
    S&VN</p  
    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