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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, #+3I$ k  
    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, k1WyV_3  
    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? ,Hsu ;I~  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? w?p8)Q6m  
    odq3@ ziO  
    (WS<6j[q  
    jM(!!A jpC  
    h1?.x  
    function z = zernfun(n,m,r,theta,nflag) '8Lc}-M4  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. pvd9wKz  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N IRDD   
    %   and angular frequency M, evaluated at positions (R,THETA) on the nHF  
    %   unit circle.  N is a vector of positive integers (including 0), and V P4ToYc  
    %   M is a vector with the same number of elements as N.  Each element [k6,!e[/uG  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) s!BZrVM%I`  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l' Z `%}R  
    %   and THETA is a vector of angles.  R and THETA must have the same X<$8'/p r  
    %   length.  The output Z is a matrix with one column for every (N,M) ~f%AbDye  
    %   pair, and one row for every (R,THETA) pair. E;x~[MA  
    % \U'TL_Ql  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike kPZ1OSX  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), F30jr6F\  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral i(>v~T,(  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _N`pwxpsb  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized =R\-mov$  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. qxW 2q8QHo  
    % 6MR S0{  
    %   The Zernike functions are an orthogonal basis on the unit circle. GB_ m&t  
    %   They are used in disciplines such as astronomy, optics, and s97L/iH  
    %   optometry to describe functions on a circular domain. ed)!Snz   
    % KbV%8nx!!  
    %   The following table lists the first 15 Zernike functions. C(*)7| m  
    % IN9o$CZ:  
    %       n    m    Zernike function           Normalization S$I:rbc  
    %       -------------------------------------------------- >taZw '  
    %       0    0    1                                 1 XUT\nN-N  
    %       1    1    r * cos(theta)                    2 )Z 3fytY  
    %       1   -1    r * sin(theta)                    2 ,/>~J]:\;  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 57b;{kl  
    %       2    0    (2*r^2 - 1)                    sqrt(3) t`mLZ <X  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) $bKa"T*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) |"Oazll  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ZvO:!u0+"  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Kn9 ,N@bU_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) a[8_ O-   
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Fk,3th  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ptuW}"F  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @*O(dw  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }a_: oR  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =kLg)a |  
    %       -------------------------------------------------- L3~E*\cV  
    % jr:LLn#}  
    %   Example 1: 0\U28zbMJw  
    % QrPWS-3~!  
    %       % Display the Zernike function Z(n=5,m=1) 7_/.a9$G  
    %       x = -1:0.01:1; (Qq$ql27  
    %       [X,Y] = meshgrid(x,x); #UJ@P Dwil  
    %       [theta,r] = cart2pol(X,Y); 3-8Vw$u  
    %       idx = r<=1; Yazpfw 7'd  
    %       z = nan(size(X));  . yu  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); e"|9%AW@<  
    %       figure %]Nz54!  
    %       pcolor(x,x,z), shading interp Ahq^dx#o  
    %       axis square, colorbar O'-lBf+<  
    %       title('Zernike function Z_5^1(r,\theta)') 5J&n<M0G1  
    % X>|.BvY|  
    %   Example 2: 1^Zx-p3J  
    % M=N`&m\  
    %       % Display the first 10 Zernike functions >8tE`2[i*  
    %       x = -1:0.01:1; nw5#/5xw  
    %       [X,Y] = meshgrid(x,x); %NS]z;G  
    %       [theta,r] = cart2pol(X,Y); e:V,>RbC0s  
    %       idx = r<=1; 28BiuxVW  
    %       z = nan(size(X)); |#2<4sd  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Qw5M\   
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; SqTm/ t  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28];   6^: l  
    %       y = zernfun(n,m,r(idx),theta(idx)); v @N8v  
    %       figure('Units','normalized') ypd  
    %       for k = 1:10 SKfv.9  
    %           z(idx) = y(:,k); f (n{7  
    %           subplot(4,7,Nplot(k)) {2:H`|x  
    %           pcolor(x,x,z), shading interp #t(?8!F  
    %           set(gca,'XTick',[],'YTick',[]) LbYI{|_Js  
    %           axis square BkqIfV%O  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7\/O"Ot  
    %       end dadMwe_l0  
    % Iwn@%?7  
    %   See also ZERNPOL, ZERNFUN2. 0`ib_&yI  
    aQ~x$T|  
    b]g.>$[nX  
    %   Paul Fricker 11/13/2006 v}Aw!Dv/  
    Zz3#Kt5t3  
    t=e0z^2i+  
    dna f>G3  
    7T2W% JT-,  
    % Check and prepare the inputs: gCm?nb)  
    % -----------------------------  +NXj/  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [ $"iO#oO  
        error('zernfun:NMvectors','N and M must be vectors.') Nm081ic2<  
    end 1VZ>*Tl  
    \qTNWA #'  
    1)PR]s:-m@  
    if length(n)~=length(m) z~Pmh%b  
        error('zernfun:NMlength','N and M must be the same length.') B]qh22Yib  
    end 7kwG_0QO  
    =4%C?(\  
    a "1$z`ln  
    n = n(:); q-3KF  
    m = m(:); 4 ?c1c  
    if any(mod(n-m,2)) q9!#S  
        error('zernfun:NMmultiplesof2', ... UQdQtj1'  
              'All N and M must differ by multiples of 2 (including 0).') s,29_z7  
    end OLR1/t`V  
    ( gFA? aD<  
    V_1#7  
    if any(m>n) =Fs LF  
        error('zernfun:MlessthanN', ... GSFT(XX  
              'Each M must be less than or equal to its corresponding N.') D8#q.OR]  
    end ]`kvq0Gyb  
    7cy~qg  
    RQ'c~D)X  
    if any( r>1 | r<0 ) <Ztda !  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') lKD@2  
    end (hV"z;rI  
    ii2Z }qe  
    jlP7'xt1%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &e)p6Egl  
        error('zernfun:RTHvector','R and THETA must be vectors.') 3"hR:'ts  
    end >V$#Um?AXj  
    Rp|&1nS  
    ykeUS zz2  
    r = r(:); S- \lN|  
    theta = theta(:); 3# (5Kco  
    length_r = length(r); yfW^wyDd2o  
    if length_r~=length(theta) M^f1D&A  
        error('zernfun:RTHlength', ... DnFl*T>  
              'The number of R- and THETA-values must be equal.') Os!22 O  
    end [Z+,)-ke  
    n6Zx0ad?  
    4~Pto f@  
    % Check normalization: 11T\2&Q  
    % -------------------- @(?4g-*E  
    if nargin==5 && ischar(nflag) pdHb  
        isnorm = strcmpi(nflag,'norm'); bx^EaXj(r  
        if ~isnorm T!A}ipqb  
            error('zernfun:normalization','Unrecognized normalization flag.') B3t>M) 9  
        end ?t42=nvf  
    else c):*R ]=  
        isnorm = false; @/(7kh +  
    end jq)|7_N  
    EXcjF  
    LD~'^+W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &FVlTo1  
    % Compute the Zernike Polynomials Hu7zmh5FF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4Z<l>!  
    @<=#i  
    tQaCNS$=  
    % Determine the required powers of r: {:X];A$  
    % ----------------------------------- L,pSdeq  
    m_abs = abs(m); JJ0 CM:xe  
    rpowers = []; F\rSYjMyk  
    for j = 1:length(n) $)]FCuv  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z^6(&Rh  
    end %pkq ?9  
    rpowers = unique(rpowers); MEnHC'nI  
    mVAm^JK  
    I<K/d  
    % Pre-compute the values of r raised to the required powers, %u0;.3Gw  
    % and compile them in a matrix: 'm5(MC,  
    % ----------------------------- O9X:1>a@i  
    if rpowers(1)==0 gA1in  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5a!e%jj  
        rpowern = cat(2,rpowern{:}); I 47GQho  
        rpowern = [ones(length_r,1) rpowern]; 7U`S9DDwq  
    else }a1Sfl@`3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @h$0S+?:  
        rpowern = cat(2,rpowern{:}); z\8yB`8b^  
    end )V/lRR&  
    4`P2FnJ?  
    67\Ojl~(1  
    % Compute the values of the polynomials: I*S`I|{J  
    % -------------------------------------- 9PqgBq   
    y = zeros(length_r,length(n)); 9Pp|d"6]y  
    for j = 1:length(n) 7XWBI\SW  
        s = 0:(n(j)-m_abs(j))/2; @H%=%ZwpO  
        pows = n(j):-2:m_abs(j); sVdK^|j  
        for k = length(s):-1:1 H!.D2J   
            p = (1-2*mod(s(k),2))* ... LA`V qJ  
                       prod(2:(n(j)-s(k)))/              ... tq:tY}:4  
                       prod(2:s(k))/                     ... ?b7g9 G4  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9Y%?)t.2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); UdW(\%  
            idx = (pows(k)==rpowers); za]p,bMX  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ,Zf!KQw  
        end #"8[8jyV  
         jJ"EGFa8  
        if isnorm k-pEBh OH  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +aw>p_\  
        end "I}Z2  
    end m_"p$m;  
    % END: Compute the Zernike Polynomials 2ZG1n#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :-\ yy  
    C{hcK 1-K  
    j"IM,=  
    % Compute the Zernike functions: Dp*$GQ  
    % ------------------------------ l \^nC2  
    idx_pos = m>0; )ozcr^  
    idx_neg = m<0; zYG,x*IH  
    IbJ[Og^Qyu  
    3[=`uO0\7  
    z = y; n1>,#|#  
    if any(idx_pos) K>cz63}S  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x:?a;muf  
    end #xP!!.DF(  
    if any(idx_neg) MFH"$t+  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i ,IM?+4  
    end OOBcJC  
    3Q$'qZw p  
    ~` \9Q  
    % EOF zernfun Rz zFhU#r  
     
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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)  `SV"ElRV  
    bGv* -;*  
    DDE还是手动输入的呢? CI`N8 f=v  
    5Go0}'*%  
    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