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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, `<<9A\Y-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, AOcUr)  
    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? z+wegF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? a+k3wzJ  
    Y|hd!C-x  
    T7/DH  
    B|9XqQ EI  
    Da6l =M  
    function z = zernfun(n,m,r,theta,nflag) \k=%G_W  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0 .T5% _ /  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LqJV  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 0Db=/sJ>  
    %   unit circle.  N is a vector of positive integers (including 0), and wEI? 9  
    %   M is a vector with the same number of elements as N.  Each element FdEUZ[IT`{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) XA.1Y)  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 3bo [34  
    %   and THETA is a vector of angles.  R and THETA must have the same awQGu,<N  
    %   length.  The output Z is a matrix with one column for every (N,M) HP<a'|r  
    %   pair, and one row for every (R,THETA) pair. |{ZdAr.;  
    % FBouXu#  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike lm&^`Bn)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), r #w7qEtD  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 0xCe6{86  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x=x%F;  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized +tg${3ti_  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mO]dP;,  
    % K~3Y8ca  
    %   The Zernike functions are an orthogonal basis on the unit circle. > MRuoJ  
    %   They are used in disciplines such as astronomy, optics, and ? }`mQ<~  
    %   optometry to describe functions on a circular domain. r6aIW8  
    % L 9cXgd  
    %   The following table lists the first 15 Zernike functions. 6jm/y@|F!  
    % Z}>;@c  
    %       n    m    Zernike function           Normalization *a{WJbau]  
    %       -------------------------------------------------- @PQd6%@  
    %       0    0    1                                 1 7,alZ"%W  
    %       1    1    r * cos(theta)                    2 :1gpbfW  
    %       1   -1    r * sin(theta)                    2 #(+V&< K  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) b^}U^2S%  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ; }ThBb3  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) -3b_}by  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) o^owv(  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) wHx_lsY;   
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) dShGIH?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^4<&"aoo  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Q~' \oWz  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A>FWvlLw'm  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) oY; C[X  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `P:[.hRu  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %CgV:.,K  
    %       -------------------------------------------------- 3%Q9521  
    % fuF{8-ua  
    %   Example 1: ]TcQGW@'  
    % U. $Th_  
    %       % Display the Zernike function Z(n=5,m=1) ^/x\HGrw  
    %       x = -1:0.01:1; x1E;dbOZ  
    %       [X,Y] = meshgrid(x,x); m] -cRf)9  
    %       [theta,r] = cart2pol(X,Y); Vu E$-)&)  
    %       idx = r<=1; uAoZ&8D6  
    %       z = nan(size(X)); !3DY#  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 2vsV :LS.  
    %       figure ;, \!&o6  
    %       pcolor(x,x,z), shading interp AA=eWg  
    %       axis square, colorbar $ye>;Ek  
    %       title('Zernike function Z_5^1(r,\theta)') 88?O4)c  
    % zE/\2F$  
    %   Example 2: =2} kiLKO  
    % 3Z#WAhfS:  
    %       % Display the first 10 Zernike functions t&EY$'c  
    %       x = -1:0.01:1; wg\ p&avvb  
    %       [X,Y] = meshgrid(x,x); fd>&RbUp  
    %       [theta,r] = cart2pol(X,Y); +#<Z/  
    %       idx = r<=1; Ve)BF1YG  
    %       z = nan(size(X)); [/n@BK  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ja&m-CFK  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; )1 HWD]>4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; L*vKIP<EMM  
    %       y = zernfun(n,m,r(idx),theta(idx)); _F|}=^Z`  
    %       figure('Units','normalized') T"gk^.  
    %       for k = 1:10 r=54@`O!  
    %           z(idx) = y(:,k); U)aftH *Pk  
    %           subplot(4,7,Nplot(k)) B_b5&M@  
    %           pcolor(x,x,z), shading interp &CN(PZv  
    %           set(gca,'XTick',[],'YTick',[]) s2 :Vm\  
    %           axis square K1]3zLnS  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) B##X94aTT  
    %       end #V#!@@c;?  
    % 've[Mx  
    %   See also ZERNPOL, ZERNFUN2. #reW)P>  
    ?N!kYTR%}  
    b`=g#B|  
    %   Paul Fricker 11/13/2006 ~(GN Y5  
    DZ`m{l3H  
    pv-c>8Wb6  
    e+{lf*"3  
    ~{vB2  
    % Check and prepare the inputs: N>]J$[j  
    % ----------------------------- lmL$0{Yr  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~<s =yjTu+  
        error('zernfun:NMvectors','N and M must be vectors.') O7uCTB+  
    end ,w BfGpVb  
    Dh?I   
    K4YD}[  
    if length(n)~=length(m) <yBa5m@/  
        error('zernfun:NMlength','N and M must be the same length.') u |.7w 2  
    end D>HbJCG4^  
    8Gnf_lkI  
    *kYGXT,f]  
    n = n(:); J.M&Vj:  
    m = m(:); woBx609Aak  
    if any(mod(n-m,2)) X ,^([$  
        error('zernfun:NMmultiplesof2', ... 1<_/Qu>V  
              'All N and M must differ by multiples of 2 (including 0).') +{I" e,Nk  
    end [;sTl~gC  
    IAq o(Qm  
    M6Np!0G  
    if any(m>n) p3{Ff5FZ  
        error('zernfun:MlessthanN', ... 8"ZS|^#  
              'Each M must be less than or equal to its corresponding N.') \hBzP^*"n  
    end ; D/6e6  
    N2duhI6  
    Vp|?R65S*  
    if any( r>1 | r<0 ) %9Z0\ a)[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') K5 BL4N  
    end Q9xb7)G  
    "d0=uHd5\  
    '=#fELMW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =y)K er  
        error('zernfun:RTHvector','R and THETA must be vectors.') N)R5#JX  
    end }f?[m&<  
    QKlsBq  
    NX.5 u8Pf  
    r = r(:); BK6 X)1R  
    theta = theta(:); q^Oj/ws  
    length_r = length(r); OHsA]7S  
    if length_r~=length(theta) pq&[cA_w  
        error('zernfun:RTHlength', ... c"Vp5lo0  
              'The number of R- and THETA-values must be equal.') xz.Jmv  
    end t H.L_< N  
     HG?+b  
    NlKVl~_ C  
    % Check normalization: PM#3N2?|E  
    % -------------------- kROIVO1|`  
    if nargin==5 && ischar(nflag) Z${eDl6i  
        isnorm = strcmpi(nflag,'norm'); uW=G1 *n-  
        if ~isnorm ]77f`<q<}!  
            error('zernfun:normalization','Unrecognized normalization flag.') \U>&W  
        end 2Ki_d  
    else S)j( %g  
        isnorm = false; L/C~l3  
    end Mb 4"bDBsl  
    CW?Z\  
    83t/ \x,Q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P~=yTW  
    % Compute the Zernike Polynomials aK@ Y) Ju'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z_m<x!  
    x;z=[eE  
    'o#oRK{#  
    % Determine the required powers of r: p'2IlQ\  
    % ----------------------------------- F=1 #qo<?  
    m_abs = abs(m); 'g,h  
    rpowers = []; ;<m`mb4x[  
    for j = 1:length(n) d!0rq4v7  
        rpowers = [rpowers m_abs(j):2:n(j)]; % _E?3  
    end prz COw  
    rpowers = unique(rpowers); -8Mb~Hfl0  
    3c3;8h$k  
    n{sk  
    % Pre-compute the values of r raised to the required powers, 4Zwbu  
    % and compile them in a matrix: e7xBi!I)~  
    % ----------------------------- |`#fX(=  
    if rpowers(1)==0 $KGMAg/H  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); j_N<aX  
        rpowern = cat(2,rpowern{:}); &TQ~!ZMOR"  
        rpowern = [ones(length_r,1) rpowern]; 0h*Le  
    else Jl`^`Yv  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R2sG'<0B0  
        rpowern = cat(2,rpowern{:}); "}*D,[C5e  
    end b2UDPW  
    In96H`  
    \\KjiT'  
    % Compute the values of the polynomials: NOXP}M  
    % -------------------------------------- PD0&ep1h7G  
    y = zeros(length_r,length(n)); CMW4Zqau*  
    for j = 1:length(n) _Ik?WA_;  
        s = 0:(n(j)-m_abs(j))/2; tSJ#  
        pows = n(j):-2:m_abs(j); uo]xC+^  
        for k = length(s):-1:1 %(/E `  
            p = (1-2*mod(s(k),2))* ... ^ WO3,  
                       prod(2:(n(j)-s(k)))/              ... e>Z&0lV:  
                       prod(2:s(k))/                     ... T3{~f  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $5JeN{B  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); i3N{Dt  
            idx = (pows(k)==rpowers); y&,|+h  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Gd%i?(U,R  
        end m.m6.  
         qs ep9z.  
        if isnorm '@.6Rd 8  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); #:gl+  
        end & mOn]  
    end ,X^3.ILz  
    % END: Compute the Zernike Polynomials 1#,4P1"  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s;OGb{H7  
    rC^ 5Z  
    M0fN[!*z  
    % Compute the Zernike functions: qS/}aDk&  
    % ------------------------------ ))|d~m  
    idx_pos = m>0; SZ9Oz-?  
    idx_neg = m<0; .h=n [`RB  
    T(?w}i  
    ]|CcQ1#|H  
    z = y; m1pA]}Y/5o  
    if any(idx_pos) A[+)PkR  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mufGv%U2  
    end qhxMO[f  
    if any(idx_neg) w{*kbGB8s7  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); FE!jN-#  
    end 8j#S+=l>  
    Ra|P5  
    ):G%o  
    % EOF zernfun Ow/,pC >V  
     
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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)
    在线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)  ,<L4tp+y0  
     )58O9b  
    DDE还是手动输入的呢? O6Py  
    *:H,-@  
    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