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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, >sm<$'vZ/  
    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, h][$1b&B  
    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? ctu`FQ  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Xi81?F?[  
    y6N }R  
    KVZ-T1K  
    5.zv0tJku  
    $ {5|{`  
    function z = zernfun(n,m,r,theta,nflag) S_dM{.!Z(,  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. M Qlx&.>  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vC>8:3Z aq  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]U)Yg  
    %   unit circle.  N is a vector of positive integers (including 0), and bz\-%$^k  
    %   M is a vector with the same number of elements as N.  Each element {<y.G1<.  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) _"688u'88  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (bo-JOOdY(  
    %   and THETA is a vector of angles.  R and THETA must have the same BoHpfx1C  
    %   length.  The output Z is a matrix with one column for every (N,M) |++\"g  
    %   pair, and one row for every (R,THETA) pair. \O(~:KN  
    % Ue2%w/Yo  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fH*1.0f]6  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #/LU@+  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral  Va3/#is'  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Y]])Tq;h5  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized {bD:OF  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #f-pkeaeq  
    % UTR`jXCg  
    %   The Zernike functions are an orthogonal basis on the unit circle. P1IL ]  
    %   They are used in disciplines such as astronomy, optics, and \ICc?8oL  
    %   optometry to describe functions on a circular domain. $Z[W}7{pt#  
    % 'jj|bN  
    %   The following table lists the first 15 Zernike functions. e]q(fPK  
    % vj hh4$k  
    %       n    m    Zernike function           Normalization r0(*]K:.  
    %       -------------------------------------------------- %8$ldNhV  
    %       0    0    1                                 1 gjDxgNpa  
    %       1    1    r * cos(theta)                    2 8c^Hfjr0  
    %       1   -1    r * sin(theta)                    2 VL2+"<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6)  G%5ZG$as  
    %       2    0    (2*r^2 - 1)                    sqrt(3) bTbF  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) E2u9>m4_J  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) }(/\vTn*1  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) bK#SxV  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ()o[(Hx+ph  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $i]G'fj  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) "'v^X!"  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uN+]q qCf  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) L5 Q^cY]p  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) + [~)a 4#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ~Y3X*  
    %       -------------------------------------------------- ckdXla  
    % 8Ai\T_l  
    %   Example 1: $~)YI/b  
    % WO!'("  
    %       % Display the Zernike function Z(n=5,m=1) B&>z&!}  
    %       x = -1:0.01:1; gi #dSd1\&  
    %       [X,Y] = meshgrid(x,x);  KGJ *h  
    %       [theta,r] = cart2pol(X,Y); Ci_Qra 6  
    %       idx = r<=1; i)th] 1K%  
    %       z = nan(size(X)); H7dT6`<~Y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7ro&Q%  
    %       figure 14z ?X%  
    %       pcolor(x,x,z), shading interp Pmdf:?B  
    %       axis square, colorbar E4, J"T|@  
    %       title('Zernike function Z_5^1(r,\theta)') XJ e}^k  
    % Z]08gH  
    %   Example 2: ;LqpX!Pi f  
    % YDYN#Ob(;  
    %       % Display the first 10 Zernike functions i !;9A6D  
    %       x = -1:0.01:1; bYBEh n  
    %       [X,Y] = meshgrid(x,x); $0XR<D  
    %       [theta,r] = cart2pol(X,Y); ;(&S1Rv9  
    %       idx = r<=1;  L30$  
    %       z = nan(size(X)); t-Uo  
    %       n = [0  1  1  2  2  2  3  3  3  3]; z)Lw\H^/  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; vEw8<<cgg  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; |8?e4yVd  
    %       y = zernfun(n,m,r(idx),theta(idx)); 53WCF[  
    %       figure('Units','normalized') X^Fc^U8  
    %       for k = 1:10 $:RR1.Tv  
    %           z(idx) = y(:,k); 6/6{69tnr  
    %           subplot(4,7,Nplot(k)) Z rv:uEl  
    %           pcolor(x,x,z), shading interp OJiwI)a9  
    %           set(gca,'XTick',[],'YTick',[]) QJ+Ml  
    %           axis square mgMa)yc!dp  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A DVUx}  
    %       end h43py8v  
    % |y pX O3  
    %   See also ZERNPOL, ZERNFUN2. "x3x$JQZy  
    jN-!1O._G  
    4W#DLip9  
    %   Paul Fricker 11/13/2006 XAZPbvG|$  
    #I1q,fm  
    " v<O)1QT  
    n8tw8o%&[  
    R@){=8%z  
    % Check and prepare the inputs: %{-r'Yi%  
    % ----------------------------- C5g9Gg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Vh?RlIUA  
        error('zernfun:NMvectors','N and M must be vectors.') -Fq`#"  
    end cn: L]%<  
    ZUkM8M$c  
    .N7<bt@~)  
    if length(n)~=length(m) hn~btu 9h  
        error('zernfun:NMlength','N and M must be the same length.') Q5lt[2Zyzd  
    end 3CH> !QOA  
    OG9 '[o`8  
    U\(71 =  
    n = n(:); /_qHF-  
    m = m(:); pHXs+Ysw+  
    if any(mod(n-m,2)) D?=4'"@v  
        error('zernfun:NMmultiplesof2', ... W-*HAS  
              'All N and M must differ by multiples of 2 (including 0).') 6_:I~TTX  
    end 5'(T*"  
    `~z[Hj=2  
    f `D( V-4  
    if any(m>n) k*v${1&  
        error('zernfun:MlessthanN', ... bB>.dC  
              'Each M must be less than or equal to its corresponding N.') aIDv~#l  
    end mfG m>U  
    S*gm[ZLQ  
    iL2__TO  
    if any( r>1 | r<0 ) AOJ[/YpM  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') e{9~m  
    end /EG'I{oC  
    Y'5(exW  
    YUHiD *  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :dpwr9)  
        error('zernfun:RTHvector','R and THETA must be vectors.') KK6fRtKv>q  
    end 9g\;L:'  
    )E}@h%d  
    }LeS3\+UHl  
    r = r(:); d14n>  
    theta = theta(:); ALV(fv$cD  
    length_r = length(r); 4$WR8  
    if length_r~=length(theta) %`QgG   
        error('zernfun:RTHlength', ... I)yF!E &  
              'The number of R- and THETA-values must be equal.') <&#MX  
    end f%i%QZP  
    PXqG;o*Q*?  
    -Lu&bVt<>  
    % Check normalization: [uK{``"  
    % -------------------- iPkCuLQ}  
    if nargin==5 && ischar(nflag) #lg R"%  
        isnorm = strcmpi(nflag,'norm'); lZuH:AH  
        if ~isnorm TQmrL  
            error('zernfun:normalization','Unrecognized normalization flag.') m[KmXPFht1  
        end PZ`11#bbm  
    else Q4hY\\Hi  
        isnorm = false; H%XF~tF:  
    end Dk>6PBl  
    "l9aBBiu  
    +wJ!zab`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #\ `kg#&  
    % Compute the Zernike Polynomials ;-XfbqZ\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @"MQ6u G>  
    _Z!@#y@j  
    ^!d0a bA  
    % Determine the required powers of r: aWlIq(dU  
    % ----------------------------------- w]yVNB  
    m_abs = abs(m); ,oh;(|=  
    rpowers = []; C l,vBjl h  
    for j = 1:length(n) 8*@{}O##  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z.u 1Dz  
    end #CaPj:>[  
    rpowers = unique(rpowers); QF\nf_X  
    [!yA#{xl,  
    ~mARgv  
    % Pre-compute the values of r raised to the required powers, B ~N3k  
    % and compile them in a matrix: \0d'y#Gp*  
    % ----------------------------- Hcwfe=K&/  
    if rpowers(1)==0 5.oIyC^Ik  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); : S3+UT  
        rpowern = cat(2,rpowern{:}); pITF%J@_]  
        rpowern = [ones(length_r,1) rpowern]; ~bx ev/$d  
    else [#q]B=JB  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I](a 5i  
        rpowern = cat(2,rpowern{:}); 4$[o;t>  
    end Wz s=BNm9  
    uT4|43< G  
    C_kuW+H  
    % Compute the values of the polynomials: [GI~ &  
    % -------------------------------------- Xs2 jR14`  
    y = zeros(length_r,length(n)); 0Zi+x#&d  
    for j = 1:length(n) 3g;,  
        s = 0:(n(j)-m_abs(j))/2; {V2"Pym?  
        pows = n(j):-2:m_abs(j); ~(ke'`gJ0-  
        for k = length(s):-1:1 xNf}f 9 l  
            p = (1-2*mod(s(k),2))* ... a @2fJ}  
                       prod(2:(n(j)-s(k)))/              ... fDf[:A,8  
                       prod(2:s(k))/                     ... gK`w|kh`  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X<}}DZSu a  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); PnA{@n\  
            idx = (pows(k)==rpowers); ]|.ked  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 9+^)?JUYll  
        end .{h"0<x  
         <[cpaZT,  
        if isnorm n jWe^  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5b7(^T^K  
        end $TXxhd 6  
    end #BUq;5  
    % END: Compute the Zernike Polynomials *uhQP47B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1xW!j!A;  
    M% \ T5  
    &,k!,<IF  
    % Compute the Zernike functions: 3- Kgz  
    % ------------------------------ );7 d_#  
    idx_pos = m>0; 6M*z`B{hV  
    idx_neg = m<0; # dWz,e3   
    tF`L]1r>  
    \Y)HSJR;e  
    z = y; v'@gUgC  
    if any(idx_pos) T+}|$/Tv  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); '" "v7  
    end (BVqmi{  
    if any(idx_neg) Ayw_LCUD  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 't5ufAT  
    end p|-MwCeH  
    SH{@yS[c!  
    G;G*!nlWf  
    % EOF zernfun x|0C0a\"A  
     
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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)  ^ij0<*ca9  
    h(|T.  
    DDE还是手动输入的呢? \L Q+ n+  
    U=n7RPw  
    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