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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 正序阅读 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, kem(U{m  
    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, l_8ibLyo  
    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? OT$++cj^  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? :."6g)T  
    Gd'_X D  
    Dd,i^,4Gj  
    | +uc;[`  
    UR S=1+  
    function z = zernfun(n,m,r,theta,nflag) Pp_? z0M  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9]lyV  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aT+w6{%Z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the f! )yE`4-  
    %   unit circle.  N is a vector of positive integers (including 0), and ]m7x&N2  
    %   M is a vector with the same number of elements as N.  Each element ie>mOsz  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) f"NWv!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, hy@b/Y![M  
    %   and THETA is a vector of angles.  R and THETA must have the same .<xD'54  
    %   length.  The output Z is a matrix with one column for every (N,M)  p: eaZ  
    %   pair, and one row for every (R,THETA) pair. Y"^.6  
    % B52dZb  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike L@_o*"&j  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 94]i|2qj*  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral U.b|3E/^  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, *UXa.kT@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized %o0H#7'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ${}9/(x/^  
    % 1'iQlnMO@  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3+ 2&9mm  
    %   They are used in disciplines such as astronomy, optics, and k,; (`L  
    %   optometry to describe functions on a circular domain. # JY>  
    % F1L[C4'  
    %   The following table lists the first 15 Zernike functions. <b\8<mTr  
    % .7:ecFKk  
    %       n    m    Zernike function           Normalization q_L. Sy|)  
    %       -------------------------------------------------- 1mR@Bh  
    %       0    0    1                                 1 -V[!qI  
    %       1    1    r * cos(theta)                    2 p,uM)LD  
    %       1   -1    r * sin(theta)                    2 Uz[#ye  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 'A\0^EvVv  
    %       2    0    (2*r^2 - 1)                    sqrt(3) l<ZHS'-;8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) (:%t  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) x9 n(3Oa  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) rY1jC\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) x{GFCy7  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +_gA"I  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ~?)y'?  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0>e]i[P.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) $2blF)uYE  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yS[HYq  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) vq-;wdq?2  
    %       -------------------------------------------------- qK~]au:C  
    % hx/A215L  
    %   Example 1: L `=*Pwcj  
    % UlKg2p  
    %       % Display the Zernike function Z(n=5,m=1) L'"c;FF02i  
    %       x = -1:0.01:1; ">S1,rhgS  
    %       [X,Y] = meshgrid(x,x); [a}Idi` K  
    %       [theta,r] = cart2pol(X,Y); E @Rb+8},"  
    %       idx = r<=1; "gDk?w  
    %       z = nan(size(X)); ; TwqZw[.  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); TIaiJvo  
    %       figure olXfR-2>1  
    %       pcolor(x,x,z), shading interp oYJ<.Yxeb  
    %       axis square, colorbar MBU4Awj  
    %       title('Zernike function Z_5^1(r,\theta)') EU'rdG*t/R  
    % $?VYHkX  
    %   Example 2: U2~|AkL  
    % [ :Sl~  
    %       % Display the first 10 Zernike functions ]gF=I5jn]  
    %       x = -1:0.01:1; -~H "zu`  
    %       [X,Y] = meshgrid(x,x); / T_v8 {D  
    %       [theta,r] = cart2pol(X,Y); 9#~jlq(  
    %       idx = r<=1; BGOS(  
    %       z = nan(size(X)); 1]A\@(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Zw%:mZN  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; i~M-V=Zg  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; %zDi|WZ  
    %       y = zernfun(n,m,r(idx),theta(idx)); fjuPGg~  
    %       figure('Units','normalized') vkM_a}%<  
    %       for k = 1:10 \8vZZt  
    %           z(idx) = y(:,k); <;jg/  
    %           subplot(4,7,Nplot(k)) U^DR'X=  
    %           pcolor(x,x,z), shading interp A8AeM `  
    %           set(gca,'XTick',[],'YTick',[]) KF!d?  
    %           axis square Q7UQwAN'  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) AP4s_X+=  
    %       end W3^^aD-  
    % <KStl fX  
    %   See also ZERNPOL, ZERNFUN2. 8vfC  
    U9 Q[K`  
    5>=4$!`  
    %   Paul Fricker 11/13/2006 04}c_XFFE  
    RmO kb~  
    [[Nn~7  
    _6]CT0  
    {W%XS E  
    % Check and prepare the inputs: ^?A>)?Sq  
    % ----------------------------- [ p(0g;bx  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W*n|T{n  
        error('zernfun:NMvectors','N and M must be vectors.') vAOThj)  
    end 3#\C!T0y  
    Z]5xy_La  
    &0d5".|s  
    if length(n)~=length(m) )]\-Uy$x  
        error('zernfun:NMlength','N and M must be the same length.') Y 7?q `  
    end }We-sZ/w7r  
    Q#&6J=}  
    w7\ \m9  
    n = n(:); R[m+s=+  
    m = m(:); Kv#Q$$)r  
    if any(mod(n-m,2)) F+W{R+6  
        error('zernfun:NMmultiplesof2', ... >rYMOC~  
              'All N and M must differ by multiples of 2 (including 0).') 6\y?+H1  
    end xsvJjs;=  
    A-M6MW  
    @f,/K1k  
    if any(m>n) :01B)~^  
        error('zernfun:MlessthanN', ... 3b`#)y^y?%  
              'Each M must be less than or equal to its corresponding N.') IL?"g{w  
    end bcAk$tA2  
    ,WAJ& '^  
    (tiE%nF+  
    if any( r>1 | r<0 ) 3 [O+wVv  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') R#rfnP >  
    end %"|W qxv  
    \(zUI  
    PM QlJ&  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) H5CL0#I  
        error('zernfun:RTHvector','R and THETA must be vectors.') iWkC: fQz  
    end oTTE<Ct [  
    Ac}5,  
    BJp~/H`vd  
    r = r(:); dkQP.Tj$i  
    theta = theta(:); `@So6%3Y|  
    length_r = length(r); ]v+yeGIKS  
    if length_r~=length(theta) /38XaKc{6  
        error('zernfun:RTHlength', ... QQ %W3D @  
              'The number of R- and THETA-values must be equal.') .B!  Z0  
    end -"x@V7X  
    A yOy&]g  
    8}Q 2!,9Q  
    % Check normalization: meGL T/   
    % -------------------- :8]y*j  
    if nargin==5 && ischar(nflag) R\x3'([A5  
        isnorm = strcmpi(nflag,'norm'); 7IrH(~Fo  
        if ~isnorm I`x[1%y2 F  
            error('zernfun:normalization','Unrecognized normalization flag.') IUD@Kf]S  
        end `1lGAKv  
    else sdN1BV2  
        isnorm = false; n-OQCz9Xl  
    end ,Z8)DC=  
    ROO@EQ#`Z  
    TrQUhmS/!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %kP=VUXj  
    % Compute the Zernike Polynomials [7,q@>:CS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NFqGbA|  
    L08lkq,  
    7s Gf_`Z  
    % Determine the required powers of r: N_l_^yD  
    % ----------------------------------- F4IU2_CnPD  
    m_abs = abs(m); C>QWV[F  
    rpowers = [];  k =O  
    for j = 1:length(n) v z&88jt  
        rpowers = [rpowers m_abs(j):2:n(j)]; 4v9d& m!<  
    end Y<_;8%S  
    rpowers = unique(rpowers); :4r*Jju<V  
    )G*xI`(@  
    q w @g7  
    % Pre-compute the values of r raised to the required powers, fT YlIT9  
    % and compile them in a matrix: bKEiS8x  
    % ----------------------------- gSe3S-Lt  
    if rpowers(1)==0 WYIv&h<h"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !1Ht{cA0  
        rpowern = cat(2,rpowern{:}); \p^'[B(O77  
        rpowern = [ones(length_r,1) rpowern]; ZzxWKIE'c  
    else FbXur-et^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s(r4m/  
        rpowern = cat(2,rpowern{:}); {HFx+<JG  
    end 'LR|DS[Ne  
    >Sb3]$$  
    pm[+xM9PB  
    % Compute the values of the polynomials: \m=k~Cf:f  
    % -------------------------------------- vhDtjf/*  
    y = zeros(length_r,length(n)); }]=@Y/p  
    for j = 1:length(n) N*)O_Ki  
        s = 0:(n(j)-m_abs(j))/2; OP\L  
        pows = n(j):-2:m_abs(j); wVX2.D'n<  
        for k = length(s):-1:1 b.RFvq5Z  
            p = (1-2*mod(s(k),2))* ... 2rb@Md]dx  
                       prod(2:(n(j)-s(k)))/              ... g8@F/$HY  
                       prod(2:s(k))/                     ... FrE#l.)?!  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Mh {>#Gs  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); l(\F2_,2W  
            idx = (pows(k)==rpowers); `$q0fTz  
            y(:,j) = y(:,j) + p*rpowern(:,idx); tq51;L  
        end I+31:#d  
         T1'\!6_5  
        if isnorm ncTMcu  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Y~?Z'uR  
        end '%YE#1*gH  
    end )JJF}m=  
    % END: Compute the Zernike Polynomials <(H<*Xf9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^F&j;8U  
    ~YByyJG   
    - FJLM  
    % Compute the Zernike functions: n~0MhE0H  
    % ------------------------------ 7k00lKA\w  
    idx_pos = m>0; 3[8p,wx  
    idx_neg = m<0; B:Awy/XMi  
    lQy-&d|=#^  
    M27H{} v  
    z = y; 2\;/mQI2A  
    if any(idx_pos) /y6I I$AvM  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Sh?eb  
    end T|0d2aa  
    if any(idx_neg) Ijk hV  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H!>>|6OPF  
    end ~Yc!~Rz  
    [ako8  
    c _!!DEe7  
    % EOF zernfun c2?VjuB0  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  p<#aXs jy  
    !{+.)%d'g  
    DDE还是手动输入的呢? R=<::2_Y96  
    0"T/a1S7bl  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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