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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, AF9[2AH=Y  
    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, J1gEjd   
    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? v&[X&Hu[  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? L5-T6CD  
    '[M^f+H|  
    <WQ<<s@#pb  
    q 2_N90u  
    O X5Co <u  
    function z = zernfun(n,m,r,theta,nflag) ex@,F,u>o  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8xD<A|  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8osS OOzM  
    %   and angular frequency M, evaluated at positions (R,THETA) on the U- *8%>Qp  
    %   unit circle.  N is a vector of positive integers (including 0), and "2#-xOCO  
    %   M is a vector with the same number of elements as N.  Each element )JY_eG&2Dx  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) i&}zcGC  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 1Rb XM n  
    %   and THETA is a vector of angles.  R and THETA must have the same ^.Ih,@N6  
    %   length.  The output Z is a matrix with one column for every (N,M) niBjq#bJi  
    %   pair, and one row for every (R,THETA) pair. m p|20`go  
    % He0N  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike OW63^wA`s  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), NSxPN:  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Y?&DEKFbD  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .@8m\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Dh!iY0Lz  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]@ Sc}  
    % Z3abem<Q  
    %   The Zernike functions are an orthogonal basis on the unit circle. Bah.\ZsYQP  
    %   They are used in disciplines such as astronomy, optics, and M0Kh>u  
    %   optometry to describe functions on a circular domain. %0~wtZH_!  
    % U&]p!DV&;  
    %   The following table lists the first 15 Zernike functions. tz0Ttu=xH  
    % dm/\uE'l  
    %       n    m    Zernike function           Normalization |$SvD2^  
    %       -------------------------------------------------- }`<>$2b  
    %       0    0    1                                 1 53,,%Ue  
    %       1    1    r * cos(theta)                    2 4I:JaRT d  
    %       1   -1    r * sin(theta)                    2 ~J. Fl[  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) syC"eH3{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cyHak u+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) IioE<wS)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) qm'C^ X?  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) jL7MmR#y5"  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) bWQORjnd8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) \yX !P1  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ExOB P  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ]\D6;E8P-~  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) AHMV@o`V  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /|u]Y/ *  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) "k6IV&0 3x  
    %       -------------------------------------------------- !OZh fMVd  
    % nnd-pf-  
    %   Example 1: x@ s`;qz  
    % ~0^,L3M  
    %       % Display the Zernike function Z(n=5,m=1) <zDw& s2  
    %       x = -1:0.01:1; |B{$URu  
    %       [X,Y] = meshgrid(x,x); |`(?<m  
    %       [theta,r] = cart2pol(X,Y); Q~w G(0'8  
    %       idx = r<=1; Lx:N!RDw  
    %       z = nan(size(X)); q5\LdI2  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); D 5r   
    %       figure jC Kt;lj  
    %       pcolor(x,x,z), shading interp &zh+:TRm  
    %       axis square, colorbar = C'e1=]  
    %       title('Zernike function Z_5^1(r,\theta)') I_6` Z 0  
    % `Z7ITvF>  
    %   Example 2: aWsKJo>j[#  
    % da?th  
    %       % Display the first 10 Zernike functions Bbt8fJA~  
    %       x = -1:0.01:1; #Hn yE+tD  
    %       [X,Y] = meshgrid(x,x); \2<yZCn  
    %       [theta,r] = cart2pol(X,Y); HsgTHe  
    %       idx = r<=1; b%!`fn-;  
    %       z = nan(size(X)); N;ecT@U g  
    %       n = [0  1  1  2  2  2  3  3  3  3]; QV H'06 "{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; mQA<t)1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ^n45N&916  
    %       y = zernfun(n,m,r(idx),theta(idx)); r4NT`&`g?  
    %       figure('Units','normalized') 3JE;:2O~P  
    %       for k = 1:10 ='bmjXu  
    %           z(idx) = y(:,k); *ckrn>E{h  
    %           subplot(4,7,Nplot(k)) FTYLMQ i  
    %           pcolor(x,x,z), shading interp  wpdEI(  
    %           set(gca,'XTick',[],'YTick',[]) ? -F'0-t4%  
    %           axis square 33KPo0g7  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) UH^wyK bM  
    %       end 8(_g]u#B;  
    % iBiA0 W  
    %   See also ZERNPOL, ZERNFUN2. j_WF38o  
    e$^!~+J7  
    oJ@PJvmR&a  
    %   Paul Fricker 11/13/2006 *T2&$W|_a  
    F+$@3[Q`N  
    WmVw>.]@~  
    +$= Wms-z  
    D_ZBx+/_?  
    % Check and prepare the inputs: muX4Y1M_  
    % ----------------------------- E)_!Hi0<s  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qCkg\)Ks5I  
        error('zernfun:NMvectors','N and M must be vectors.') 4p.{G%h  
    end cf!k 9x9Z  
    3Q~&xNf  
    Nt^&YE7d:  
    if length(n)~=length(m) K<w5[E9V.  
        error('zernfun:NMlength','N and M must be the same length.') k`~br249  
    end e/Oj T  
    S 2 h  
    'sQO0611S  
    n = n(:); PRlo"kN  
    m = m(:); P_g0G#`4  
    if any(mod(n-m,2)) ,0~ {nQj]  
        error('zernfun:NMmultiplesof2', ... iY'hkrw  
              'All N and M must differ by multiples of 2 (including 0).') XXwhs-:o  
    end Mh.eAM8_  
    U1|4vd9  
    gwz _b  
    if any(m>n) xAz4ZXj=q  
        error('zernfun:MlessthanN', ... FC(cXPX}  
              'Each M must be less than or equal to its corresponding N.') =+=|{l?F  
    end kGqf@ I+  
    >(ww6vk2  
    +;iesULXn  
    if any( r>1 | r<0 ) (l_de)N7  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8=o(nFJw  
    end %1 ^jd\  
    o4f9EJY   
    EF=D}"E6pO  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,k!f`  
        error('zernfun:RTHvector','R and THETA must be vectors.') > ,Bu^] C  
    end KJC9^BAr  
    &2]D+aL|h  
    e CUcE(  
    r = r(:); [=1?CD  
    theta = theta(:); q<uLBaL_]r  
    length_r = length(r); 7CMgvH)O  
    if length_r~=length(theta) oNsx Fi:  
        error('zernfun:RTHlength', ... t8N9/DZ}Q  
              'The number of R- and THETA-values must be equal.') p2vUt  
    end (a!,)  
    mT~>4xi0  
    4H? Ma|,  
    % Check normalization: _NnO mwK7  
    % -------------------- }t-|^mY>  
    if nargin==5 && ischar(nflag) +i!M[  
        isnorm = strcmpi(nflag,'norm'); 0_pwY=P  
        if ~isnorm W1`ZS*12D  
            error('zernfun:normalization','Unrecognized normalization flag.') qm5pEort  
        end 3D dG$@  
    else [ =2In;  
        isnorm = false; Df3v"iCq}  
    end 2U+p@}cQUA  
    r3vj o(  
    $rYu4^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7 ~8Fs@  
    % Compute the Zernike Polynomials SZD2'UaG  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M%^laf  
    [te7 uZv-  
    :uDB3jN[  
    % Determine the required powers of r:  /?xn  
    % ----------------------------------- aKtTx~$@  
    m_abs = abs(m); Ud*[2Oi|R  
    rpowers = []; W3rvKqdw5  
    for j = 1:length(n) K3D $ hb  
        rpowers = [rpowers m_abs(j):2:n(j)]; S$On$]~\"  
    end IfCqezd  
    rpowers = unique(rpowers); o9\m? ~g!E  
    J vsB^F.4  
    to=##&ld<  
    % Pre-compute the values of r raised to the required powers, +[[gU;U"v  
    % and compile them in a matrix: 5c7a\J9>  
    % ----------------------------- n7uD(cL  
    if rpowers(1)==0 GTNTx5H  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E_rC"_Zte  
        rpowern = cat(2,rpowern{:}); /n:fxdhe  
        rpowern = [ones(length_r,1) rpowern]; hI{Yg$H1  
    else L"/ato  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  m:Abq`C  
        rpowern = cat(2,rpowern{:}); (Z +C  
    end iUBni&B  
    e'&{KD,-T  
    W%cPX0  
    % Compute the values of the polynomials: hDMp^^$  
    % -------------------------------------- j=S"KVp9NF  
    y = zeros(length_r,length(n)); 0pOha(,~  
    for j = 1:length(n) n #/m7  
        s = 0:(n(j)-m_abs(j))/2; \ y",Qq?  
        pows = n(j):-2:m_abs(j); _Z2)e*(  
        for k = length(s):-1:1 ,[#f}|s_  
            p = (1-2*mod(s(k),2))* ... iNSJOS  
                       prod(2:(n(j)-s(k)))/              ... Mv =;+?z!  
                       prod(2:s(k))/                     ... jQ}| ]pj+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... c'R|Wyf  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); xII!2.  
            idx = (pows(k)==rpowers); tH(#nx8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); '~J6 mojE  
        end Su#1yw>  
         rzLl M  
        if isnorm \_bX2Lg  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >.4Sx~VH2  
        end +8I0.,'  
    end r |/9Dn%  
    % END: Compute the Zernike Polynomials h+(s/o?\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "O "@HVF@  
    _P1-d`b0 a  
    |D:0BATRP  
    % Compute the Zernike functions: w2[R&hJ  
    % ------------------------------ xpwzzO*U  
    idx_pos = m>0; kw'D2692  
    idx_neg = m<0; ^XVa!s,d  
    @oNrR$7  
    oZtz"B  
    z = y; DmsloPB?_  
    if any(idx_pos) lUd,-  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |\t_I~de  
    end pE N`&'4  
    if any(idx_neg) 7F\g3^ z9`  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %BKTN@;7  
    end H'.eqZM  
    YeJdkt  
    Ip x:k+J  
    % EOF zernfun f *vziC<m  
     
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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)  C}!|K0t?  
    V;m3=k0U  
    DDE还是手动输入的呢? p7*\]HyE)  
    L{42?d  
    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