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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, ?zsRs?rc0  
    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, kN<;*jHV  
    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? }3 ~*/30V  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? WM$Z?CN%KB  
    Vd.XZ*}r*  
    5' t9/8i  
    9nO&d(r g  
    wuCZz{c7  
    function z = zernfun(n,m,r,theta,nflag) OPq6)(Q  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. dEf5x_TGm  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~ c~j  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Eos;7$u[  
    %   unit circle.  N is a vector of positive integers (including 0), and k|]l2zlT  
    %   M is a vector with the same number of elements as N.  Each element .d#Hh&jj  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) o2YHT \P n  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, $y<`Jy]+)~  
    %   and THETA is a vector of angles.  R and THETA must have the same ZS3T1 <z  
    %   length.  The output Z is a matrix with one column for every (N,M) ept:<!4  
    %   pair, and one row for every (R,THETA) pair. S._h->5f  
    % %0815 5M  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]=|iO~WN  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `"~X1;  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral `yhc,5M  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f~jd N~  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized v+C D{Tc  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BlqfST#6  
    % >9g^-~X;v  
    %   The Zernike functions are an orthogonal basis on the unit circle. 4Im}!q5;:<  
    %   They are used in disciplines such as astronomy, optics, and E}36  
    %   optometry to describe functions on a circular domain. ;%>X+/.y0  
    % 0icB2Jm:D}  
    %   The following table lists the first 15 Zernike functions. DAN"&&  
    % FNl^ lj`Y  
    %       n    m    Zernike function           Normalization "tK3h3/Xv  
    %       -------------------------------------------------- u7p:6W  
    %       0    0    1                                 1 bx".<q(  
    %       1    1    r * cos(theta)                    2 Jju?v2y`  
    %       1   -1    r * sin(theta)                    2 X5tV Xd  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) zb9vUxN [  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Gv(n2r  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ~F~hgVS5  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ,=%c e  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) I= z+`o8  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9FT==>  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;ov}%t>UD  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) x||b :2  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QX-M'ur99  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ,.gI'YPQC  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sg]g;U  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) "bjbJC&T  
    %       -------------------------------------------------- )4+uM'2%  
    % .e:+Ek+  
    %   Example 1: O#U_mgfzJ  
    % f K4M:_u  
    %       % Display the Zernike function Z(n=5,m=1) LGCeYXic  
    %       x = -1:0.01:1; y2C/DyuAY|  
    %       [X,Y] = meshgrid(x,x); 0`WZ  
    %       [theta,r] = cart2pol(X,Y); AkqGk5e ^  
    %       idx = r<=1; tkix@Q!;\  
    %       z = nan(size(X)); A<g5:\3  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); eR8>5:V_  
    %       figure 6Qm .k$[  
    %       pcolor(x,x,z), shading interp o\y qf:V8  
    %       axis square, colorbar jmnrpXaAx  
    %       title('Zernike function Z_5^1(r,\theta)') X`daaG_l  
    % [  **F  
    %   Example 2: yj`xOncE}  
    % VzFzVeJ  
    %       % Display the first 10 Zernike functions (pQ$<c  
    %       x = -1:0.01:1; ~_SVQ7P  
    %       [X,Y] = meshgrid(x,x); n~&e>_;(.  
    %       [theta,r] = cart2pol(X,Y); *WXqN!:  
    %       idx = r<=1; Yf^/YLLS  
    %       z = nan(size(X)); =~QC)y_  
    %       n = [0  1  1  2  2  2  3  3  3  3]; i>rsq[l  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 4> [tjz.?k  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; > qIZ  
    %       y = zernfun(n,m,r(idx),theta(idx)); X1h*.reFAL  
    %       figure('Units','normalized') fm,:8%  
    %       for k = 1:10 AqP\g k  
    %           z(idx) = y(:,k); `?Xt ,  
    %           subplot(4,7,Nplot(k)) 4=n%<U`Z/  
    %           pcolor(x,x,z), shading interp |a[ :L  
    %           set(gca,'XTick',[],'YTick',[]) o)6udRzBv  
    %           axis square - e"XEot~  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <b;Oap3  
    %       end 7llEB*dSA  
    % W.U|mNJ$  
    %   See also ZERNPOL, ZERNFUN2. WN?meZ/N/  
    'Xzi$}E D  
    Sst`*PX:  
    %   Paul Fricker 11/13/2006 i0~L[v9l<  
    +^.Q%b0Xx  
    ('px X+  
    gbRdng7(}  
    t]%! vXo  
    % Check and prepare the inputs: =Hs~fHa)  
    % ----------------------------- > 'KQL?!F  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) UwxrYouv~@  
        error('zernfun:NMvectors','N and M must be vectors.') V 5ihplAk  
    end 3/hAxd  
    pV))g e\  
    0CO6-&F9n  
    if length(n)~=length(m) |tS~\_O/  
        error('zernfun:NMlength','N and M must be the same length.') x5Ee'G(  
    end YPq`su7m9  
    ;42D+q=s  
    ~d?\rj3=  
    n = n(:); Mky8qVQ2  
    m = m(:); /C}fE]n{X  
    if any(mod(n-m,2)) 5Gsj;   
        error('zernfun:NMmultiplesof2', ... rJm%qSZz  
              'All N and M must differ by multiples of 2 (including 0).') jNNl5.  
    end s2ys>2k  
    YB}_zuZ4&  
    cBA2;5E  
    if any(m>n) way-Q7  
        error('zernfun:MlessthanN', ... 1P\_3.V{  
              'Each M must be less than or equal to its corresponding N.') DD hc^(  
    end \:?H_^^ d  
    ] H[FZY  
    ARu^hz=  
    if any( r>1 | r<0 ) ;3_Q7;y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') , 0rC_)&B  
    end l9.wMs*`X  
    '17u Wq  
    b\\?aR |  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {:enoV"  
        error('zernfun:RTHvector','R and THETA must be vectors.') y!^RL,HIL  
    end ':w6 {b  
    /qL&)24  
    <`9:hPp0  
    r = r(:); &,&oTd.  
    theta = theta(:); m%E7V{t  
    length_r = length(r); u;:N 4d=f'  
    if length_r~=length(theta) 6C/D&+4  
        error('zernfun:RTHlength', ... ()>\D  
              'The number of R- and THETA-values must be equal.') |R*fw(=W  
    end rd 1&?X  
    9H3#8T] ;  
    Aq|LeH  
    % Check normalization: 5J&n<M0G1  
    % -------------------- ]@ [=FK^  
    if nargin==5 && ischar(nflag) ^J~}KOH  
        isnorm = strcmpi(nflag,'norm'); Qzh:*O  
        if ~isnorm 6<t\KMd  
            error('zernfun:normalization','Unrecognized normalization flag.') 1 )j%]zd2  
        end j`'=K_+nU  
    else W#y)ukRv  
        isnorm = false; D0k7)\puQ  
    end +TAm9eDNV  
    ]@?3,N  
    ($W9 ?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ak;Z;  
    % Compute the Zernike Polynomials xHr  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]-fZeyY$  
    xG}eiUbM`  
    cdIy[ 1  
    % Determine the required powers of r: !P92e1  
    % ----------------------------------- u%[*;@;9+  
    m_abs = abs(m); $~~=SOd0  
    rpowers = []; \K?./*  
    for j = 1:length(n) {Ue6DK %  
        rpowers = [rpowers m_abs(j):2:n(j)]; cW GU?cv}  
    end a 5)[?ol  
    rpowers = unique(rpowers); >PGm}s_  
    S5Px9&N8(  
    @xkM|N?  
    % Pre-compute the values of r raised to the required powers, Ol%*3To  
    % and compile them in a matrix: n`:l`n>N$  
    % ----------------------------- uN\9c Q  
    if rpowers(1)==0 *,n7&  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &gEu%s^wR  
        rpowern = cat(2,rpowern{:}); CWN=6(y  
        rpowern = [ones(length_r,1) rpowern]; *<A;jP  
    else =k/n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Xs`:XATb/  
        rpowern = cat(2,rpowern{:}); f@/qW!o  
    end bL[PNUG  
    R&alq  
    <s7{6n')  
    % Compute the values of the polynomials: 25 ~$qY_  
    % -------------------------------------- .N+xpxdG,  
    y = zeros(length_r,length(n)); /Bwea];^Q  
    for j = 1:length(n) F'~/  
        s = 0:(n(j)-m_abs(j))/2; ' R@<4Ib|  
        pows = n(j):-2:m_abs(j); p4AXQuOP  
        for k = length(s):-1:1 RU >vnDaC  
            p = (1-2*mod(s(k),2))* ... 0F~9t !  
                       prod(2:(n(j)-s(k)))/              ... EqmJXDm  
                       prod(2:s(k))/                     ... k?8W2fC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8|tm`r`*Az  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 88*RlxU  
            idx = (pows(k)==rpowers); +8Px` v1L  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }-Ma ~/  
        end aw4+1.xy  
         .>nd@oU  
        if isnorm Ni)#tz_9  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1*jL2P]D  
        end %7@H7^s}9  
    end i|O7nB@  
    % END: Compute the Zernike Polynomials B*AMo5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [uY 2N h  
    W|Tew-H{h_  
    v}Nx*%  
    % Compute the Zernike functions: JS:lysu  
    % ------------------------------ H'`(|$:|  
    idx_pos = m>0; 3"hR:'ts  
    idx_neg = m<0; A-u5  
    0X-2).n u  
    )/AvWDKvO  
    z = y; U;xWW9  
    if any(idx_pos) Tz-cN  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); prs<ZxbQb  
    end dtBV0$  
    if any(idx_neg) (R}X( u  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c>!J@[,  
    end oQXkMKZ  
    c+{4C3z  
    htRZ}e  
    % EOF zernfun *!/#39  
     
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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)  8{<[fZyC  
    a950M7  
    DDE还是手动输入的呢? DGZY~(]  
    /3[ 9{r  
    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