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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, : ;nvqbd  
    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, 'w8k*@cQ  
    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? QyGTm"9l  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? E26zw9d  
    J\BTrN7  
    0 2lI-xHe  
    9"=1 O  
    6Ch [!=p{  
    function z = zernfun(n,m,r,theta,nflag) .FarKW  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. FC:+[.fi  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R3;,EL{H&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ._uXK[c7P  
    %   unit circle.  N is a vector of positive integers (including 0), and W?n)IBj8  
    %   M is a vector with the same number of elements as N.  Each element b6FC  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 5ir[}I^z  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {*Ag[HS0u  
    %   and THETA is a vector of angles.  R and THETA must have the same e-Xr^@M*Q  
    %   length.  The output Z is a matrix with one column for every (N,M) Lad8C  
    %   pair, and one row for every (R,THETA) pair. &.zG?e.  
    % fq@r6\TI  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,co~@a@9  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), UC!?.  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral #^+C k HX  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, a,GOS:?O5  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized h&t/ L  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R|jt mI?  
    % [O"9OW'2!B  
    %   The Zernike functions are an orthogonal basis on the unit circle. Md4hd#z  
    %   They are used in disciplines such as astronomy, optics, and d-zNvbU"  
    %   optometry to describe functions on a circular domain. :6 , `M,  
    % ; S(KJV  
    %   The following table lists the first 15 Zernike functions. x:7"/H|  
    % jf`QoK  
    %       n    m    Zernike function           Normalization H%LoI)w  
    %       -------------------------------------------------- "~1{|lj|)  
    %       0    0    1                                 1 AG3iKk??T  
    %       1    1    r * cos(theta)                    2 MY8[)<q"  
    %       1   -1    r * sin(theta)                    2 lo1<t<w`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) xppl6v(  
    %       2    0    (2*r^2 - 1)                    sqrt(3) X 5.%e&`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) =RA8^wI  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) * LaL('.>  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) fEdp^oVg  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Lp|7s8?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) X]Aobtz  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) =bx;TV  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #-]!;sY>  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 3 #8bG(  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `b11,lg  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) N;YAG#'9~_  
    %       -------------------------------------------------- SBf8Ipe  
    % #~_ZG% u  
    %   Example 1: GOKca%DT=  
    % `X["Bgk$!T  
    %       % Display the Zernike function Z(n=5,m=1) I"=a:q  
    %       x = -1:0.01:1; XF6ed  
    %       [X,Y] = meshgrid(x,x); wM-I*<L>  
    %       [theta,r] = cart2pol(X,Y); F}f/cG<X  
    %       idx = r<=1; ii3{HJ*C  
    %       z = nan(size(X)); agbG)t0  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1}7Q2Ad w  
    %       figure W~!uSrY  
    %       pcolor(x,x,z), shading interp 0r=KY@D  
    %       axis square, colorbar pie,^-_.g  
    %       title('Zernike function Z_5^1(r,\theta)') CeZ+!-lG  
    % kH.W17D~  
    %   Example 2: !`A]YcQ  
    % 0SHF 8kek  
    %       % Display the first 10 Zernike functions w1Xe9'$Qb  
    %       x = -1:0.01:1; ;kX:k~,]}>  
    %       [X,Y] = meshgrid(x,x); 0b)q,]l]  
    %       [theta,r] = cart2pol(X,Y); wN+3OPM  
    %       idx = r<=1; nlq"OzcH04  
    %       z = nan(size(X)); 5x2m ]u  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ]8m_+:`=  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3axbW f3[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nNEIwlj;  
    %       y = zernfun(n,m,r(idx),theta(idx)); (lz Z=T  
    %       figure('Units','normalized') [ T6MaP?  
    %       for k = 1:10 !4/s|b9K  
    %           z(idx) = y(:,k); o^\L41x3  
    %           subplot(4,7,Nplot(k)) $`wo8A|)  
    %           pcolor(x,x,z), shading interp 4Odf6v,*@  
    %           set(gca,'XTick',[],'YTick',[]) x1O]@Z{d\  
    %           axis square Zv"qA  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .H33C@  
    %       end e8Y;~OAj[  
    % 3G.-JLhs  
    %   See also ZERNPOL, ZERNFUN2. oIJ.Tv@N(  
    Mb1K:U  
    PCcI(b>?l  
    %   Paul Fricker 11/13/2006 J;|i6q q  
    ju8DmC5  
     j1sgvh]D  
    pR,eus;8  
     {ch+G~oS  
    % Check and prepare the inputs: H6vO}pq) r  
    % ----------------------------- 9R1S20O  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?O|CY  
        error('zernfun:NMvectors','N and M must be vectors.')  &$ x1^  
    end S_|VlI  
    )Bb:?!EuEH  
    Q2 S!}A  
    if length(n)~=length(m) ^h5h kIx0  
        error('zernfun:NMlength','N and M must be the same length.') A4mnm6Tf  
    end o6@`aU  
    3m]8>1e1"  
    C}D\^(nLu.  
    n = n(:); T:G8xI1 P  
    m = m(:); )bkJ[ '9  
    if any(mod(n-m,2)) +ak<yV1=  
        error('zernfun:NMmultiplesof2', ... ]T<\d-!CZN  
              'All N and M must differ by multiples of 2 (including 0).') 7A6:*  
    end O~bJ<O=?  
    U~l.%mui  
    C U 8s*  
    if any(m>n) ebTwU]Nb  
        error('zernfun:MlessthanN', ... !=B=1th4  
              'Each M must be less than or equal to its corresponding N.') 7FYq6wi  
    end ZR |n\.  
    G1?m}{D)  
    -n#fj;.2_  
    if any( r>1 | r<0 ) KM&bu='L^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') hVLV Mqd  
    end Pg%k>~i  
    } >z l  
    |3K]>Lio  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &XF@Dvv  
        error('zernfun:RTHvector','R and THETA must be vectors.') ])vWvNx  
    end T{B\1|2w  
    Y3M"a8e'  
    e{k)]]J  
    r = r(:); ;]A:(HSZj  
    theta = theta(:); 7c>{og6  
    length_r = length(r); .cCB,re  
    if length_r~=length(theta) )ipTm{  
        error('zernfun:RTHlength', ... I;rh(FMV  
              'The number of R- and THETA-values must be equal.') hG!|ts  
    end (! "+\KY  
    u7G9 eN  
    FXO{i:Zo  
    % Check normalization: nR#'BBlI  
    % -------------------- >DkRl  
    if nargin==5 && ischar(nflag) &l;wb.%ijW  
        isnorm = strcmpi(nflag,'norm'); b 8~7C4  
        if ~isnorm <HMmsw  
            error('zernfun:normalization','Unrecognized normalization flag.') M?I^Od'8  
        end I>n2# -8  
    else &O;' ?/4 S  
        isnorm = false; cK _:?G  
    end ov%.+5P  
    US2Tdmy@05  
    =c Krp'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2 i:tPe&  
    % Compute the Zernike Polynomials biy[h3b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1Uf8ef1,  
    o)SA^5  
    ?I}0[+)V  
    % Determine the required powers of r: { cMf_qQ  
    % ----------------------------------- =!P  
    m_abs = abs(m); ZB5u\NpcW  
    rpowers = []; z>9gt  
    for j = 1:length(n) l>{+X )  
        rpowers = [rpowers m_abs(j):2:n(j)]; OJTEvb6nPg  
    end >X0c:p Pu  
    rpowers = unique(rpowers); Wt $q{g{C  
    ^/_1y[j  
    _^Yav.A=  
    % Pre-compute the values of r raised to the required powers, >#8J@=iuqv  
    % and compile them in a matrix: ly)L%hG  
    % ----------------------------- NUb:5tL  
    if rpowers(1)==0 n^:Wc[[m  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g)UYpi?p-}  
        rpowern = cat(2,rpowern{:}); e_z"<yq  
        rpowern = [ones(length_r,1) rpowern]; :j4i(qcF  
    else >^(Q4eU7!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); O}cg1Q8p  
        rpowern = cat(2,rpowern{:}); g4CdzN~  
    end Yt#e[CYnu  
    y+K21(z.  
    /Q*cyLv  
    % Compute the values of the polynomials: wML5T+  
    % -------------------------------------- ^Z~'>J  
    y = zeros(length_r,length(n)); T*i rCe  
    for j = 1:length(n) ]id5jVY  
        s = 0:(n(j)-m_abs(j))/2; }Pf7YuUZZ  
        pows = n(j):-2:m_abs(j); hY^-kdQ>M  
        for k = length(s):-1:1 Ey**j  
            p = (1-2*mod(s(k),2))* ... Ii4lwZnz  
                       prod(2:(n(j)-s(k)))/              ... dt=5 Pnf[y  
                       prod(2:s(k))/                     ... xfqW~&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -4!i(^w[m/  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); e Zb8x  
            idx = (pows(k)==rpowers); :.%Hu9=GL  
            y(:,j) = y(:,j) + p*rpowern(:,idx); q"%;),@  
        end "J(7fL$!  
         ?iQA>P9B  
        if isnorm UB&)U\hn  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2 bQC 2  
        end p/GYfa dU  
    end Ls~F4ar$/  
    % END: Compute the Zernike Polynomials Gkq<?q({t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]&kzIxh  
    M.QXwIT  
    $Y Cy,Ew   
    % Compute the Zernike functions: c7$U0JO  
    % ------------------------------ zZ\2fKrpg  
    idx_pos = m>0; a|\ZC\(xI  
    idx_neg = m<0; KN"V(<!)~  
    <^,5z!z }  
    (HZzA7eph  
    z = y; #Kp/A N5YC  
    if any(idx_pos) ,0=@cJ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lY_&P.B  
    end >kJEa8  
    if any(idx_neg) u  teI[Q  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); QX$i ]y%S  
    end _a3,Zuv  
    z9#iU>@  
    TXlxnB  
    % EOF zernfun E?/Bf@a28=  
     
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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)S[\  
    kh /n|2  
    DDE还是手动输入的呢? _-$"F>  
    t@[&8j2B>  
    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