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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, r0)X]l7  
    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, n`krK"Ii  
    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? wh@;$s"B  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4c*?9r@  
    q}#4bB9  
    gzthM8A  
    L}1|R*b  
    ~P85Or  
    function z = zernfun(n,m,r,theta,nflag) X rVF %  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6WQT,@ ?  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kw>W5tNpf:  
    %   and angular frequency M, evaluated at positions (R,THETA) on the #?Z>o16,u  
    %   unit circle.  N is a vector of positive integers (including 0), and O$ 7R<V  
    %   M is a vector with the same number of elements as N.  Each element YULI y-W  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ?6F\cl0.  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, W0&NX`m  
    %   and THETA is a vector of angles.  R and THETA must have the same 8(e uWS  
    %   length.  The output Z is a matrix with one column for every (N,M) WCc,RI0   
    %   pair, and one row for every (R,THETA) pair. Uv~r]P)  
    % =Vv"\p8  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YzqUOMAt"V  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ao]Dm#HiO  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral m?]X NgT  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, dMw0Aw,2]8  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized .mzy?!w0q  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "|yuP1;L  
    % k[0Gz  
    %   The Zernike functions are an orthogonal basis on the unit circle. [;`B   
    %   They are used in disciplines such as astronomy, optics, and *E0dCY$  
    %   optometry to describe functions on a circular domain. 6px(]QU  
    % ;N4A9/)  
    %   The following table lists the first 15 Zernike functions. 60B6~@]P  
    % 2HNKq<  
    %       n    m    Zernike function           Normalization nCZ&FNi{O~  
    %       -------------------------------------------------- A{Jp>15AVg  
    %       0    0    1                                 1 )a ov]Ns  
    %       1    1    r * cos(theta)                    2 Nr?Z[6O|  
    %       1   -1    r * sin(theta)                    2 ,iKL 68  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rz%8V igb  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 4NaL#3  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) #1-,s.)  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ib(q9!L  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /a}F ;^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8)  uIOnP  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }w{ 6Ua  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) P;7JK=~k  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A}Q6DHh26  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) z']TRjDbT  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I d6H~;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =P}ob eY  
    %       -------------------------------------------------- i^SuVca  
    % iI|mFc|V  
    %   Example 1: [Yr }:B <  
    % kjVUG >e>  
    %       % Display the Zernike function Z(n=5,m=1) EDQKbTaPt  
    %       x = -1:0.01:1; dux.Z9X?  
    %       [X,Y] = meshgrid(x,x); km@V|"ac _  
    %       [theta,r] = cart2pol(X,Y); or~2r8  
    %       idx = r<=1; 1>I4=mj  
    %       z = nan(size(X)); BG>fLp  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); h$p]M^Z7  
    %       figure B 2p/  
    %       pcolor(x,x,z), shading interp :w|ef;  
    %       axis square, colorbar >Q5et1c  
    %       title('Zernike function Z_5^1(r,\theta)') g=)B+SY'  
    % HSXv_  
    %   Example 2: 05o)Q &`  
    % N|JM L  
    %       % Display the first 10 Zernike functions MI^@p`s  
    %       x = -1:0.01:1; -;NGS )RM  
    %       [X,Y] = meshgrid(x,x); /V-uo(n< .  
    %       [theta,r] = cart2pol(X,Y); O+iNR9O  
    %       idx = r<=1; t zn1|  
    %       z = nan(size(X)); b#~K>  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ``X1xiB  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;Gc,-BDFw  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #`Af  
    %       y = zernfun(n,m,r(idx),theta(idx)); J,iS<lV_  
    %       figure('Units','normalized') =VC"X?N  
    %       for k = 1:10 i}u,_ }  
    %           z(idx) = y(:,k); ~Up5+7k@  
    %           subplot(4,7,Nplot(k)) %y96]e1  
    %           pcolor(x,x,z), shading interp / thFs4  
    %           set(gca,'XTick',[],'YTick',[]) ZhqGUb  
    %           axis square `O+}$wP  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #VM+.75o1  
    %       end eELLnU{"  
    % :.DZ~I  
    %   See also ZERNPOL, ZERNFUN2. ~F [V  
    ^(+ X|t  
    -!@]z2uU  
    %   Paul Fricker 11/13/2006 6!39t  
    ^LI\W'K  
    7)RDu,fx  
    =EJ8J;y_f  
    @\*`rl]  
    % Check and prepare the inputs: wH?]kV8Q  
    % ----------------------------- .-Z=Aa>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8SZZ_tS3r  
        error('zernfun:NMvectors','N and M must be vectors.') 'zJBp 9a%  
    end %I^schE*  
    /1y\EEc  
    "]SA4Ud^  
    if length(n)~=length(m) "xI70c{  
        error('zernfun:NMlength','N and M must be the same length.') q1^bH 6*fl  
    end tZXq<k9  
    YD9|2S!G  
    q!10 G  
    n = n(:); c9ye[81  
    m = m(:); dz6&TdEl  
    if any(mod(n-m,2))  *KV^ X(/  
        error('zernfun:NMmultiplesof2', ... xcQD]"   
              'All N and M must differ by multiples of 2 (including 0).') a S;z YD  
    end S4S}go*G[  
    qdPmTaak  
    %!\iII  
    if any(m>n) \? n<UsI  
        error('zernfun:MlessthanN', ... A3Xfu$[u  
              'Each M must be less than or equal to its corresponding N.') %zKTrsMZ  
    end :Z[|B(U  
    t5aX9WIW  
    ]\1H=g%Ou  
    if any( r>1 | r<0 ) YB+My~fw{l  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *ZkOZ  
    end 6vfut$)[{  
    /B 53Z[yL  
    Pk3b#$+E  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) wzj :PS  
        error('zernfun:RTHvector','R and THETA must be vectors.') Q<Q?#v7NX  
    end 'WNq/z"X  
    \zJb}NbnT  
    {zI>"%$u  
    r = r(:); N0pA ,&  
    theta = theta(:); %oOSmt  
    length_r = length(r); 84_Y+_9  
    if length_r~=length(theta) W5uC5C*,l  
        error('zernfun:RTHlength', ... _<6E>"*m  
              'The number of R- and THETA-values must be equal.') |;(>q  
    end }U^iVq*  
    Bdcs}Ga  
    \;+TZ1i_  
    % Check normalization: ?>1wZ  
    % -------------------- Y1;jRIOA  
    if nargin==5 && ischar(nflag) P\y ZcL  
        isnorm = strcmpi(nflag,'norm'); v'Pbx  
        if ~isnorm q:1n=i Ei  
            error('zernfun:normalization','Unrecognized normalization flag.') 65vsQ|Zw  
        end ,`8:@<e  
    else U UhlKV|5  
        isnorm = false; ?X+PNw|pf  
    end @8Cja.H  
    98maQQWD  
    cpm *m"Nk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q8)w Al  
    % Compute the Zernike Polynomials Jsa;pG=3&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O YfRtfE  
    K}DrJ/s  
    b%h.>ij?  
    % Determine the required powers of r: ;*{Ls#  
    % ----------------------------------- OD~yIV  
    m_abs = abs(m); 9aYVbq""  
    rpowers = []; F;MACu;x  
    for j = 1:length(n) 3U! l8N2  
        rpowers = [rpowers m_abs(j):2:n(j)]; BxiR0snf0q  
    end YB_fy8Tfx  
    rpowers = unique(rpowers); O<J<)_W)  
    XaaR>HljJ  
    v=daafO  
    % Pre-compute the values of r raised to the required powers, wkY$J\J  
    % and compile them in a matrix: DB0?H+8t  
    % ----------------------------- s)+] pxV0-  
    if rpowers(1)==0 tlYB'8bJY  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bduHYs+rq  
        rpowern = cat(2,rpowern{:}); ";upu  
        rpowern = [ones(length_r,1) rpowern]; |+Xh ^E  
    else y"iK)SH  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D|2lBU  
        rpowern = cat(2,rpowern{:}); 7HJH9@8V  
    end s~A:*2\  
    x=N0H  
    EvT"+;9/p  
    % Compute the values of the polynomials: \1eWI  
    % -------------------------------------- QS@eqN  
    y = zeros(length_r,length(n)); 0S\HO<~k  
    for j = 1:length(n) \okvL2:!  
        s = 0:(n(j)-m_abs(j))/2; Z^.qX\<M  
        pows = n(j):-2:m_abs(j); /PpZ6ne~ [  
        for k = length(s):-1:1 EiS2-Uh*TT  
            p = (1-2*mod(s(k),2))* ... H{uR+&<  
                       prod(2:(n(j)-s(k)))/              ... bR J]avR  
                       prod(2:s(k))/                     ... wS [k}  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .PCbGPbk  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); lr[&*v?h  
            idx = (pows(k)==rpowers); A{wk$`vH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?{~. }Vn  
        end qxHsmGV  
         C9j5Pd5q1L  
        if isnorm \,G19o}`Es  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Pu}PE-b  
        end }7i}dyQv}  
    end ^AT#A<{1(  
    % END: Compute the Zernike Polynomials @9g!5dcT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0C717  
    7 .xejz  
    T'7x,8&2|  
    % Compute the Zernike functions: CWkAc5  
    % ------------------------------ qX]ej 2  
    idx_pos = m>0; S/6I9zOP  
    idx_neg = m<0; ^3nB2G.ax  
    {/XU[rn  
    %sS7o3RW\  
    z = y; % %QAC4  
    if any(idx_pos) o2^?D`Jr  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); t`0(5v  
    end aIE\B4w  
    if any(idx_neg) {ZgycMS  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); NmV][0(BS  
    end `(L<Q%  
    L/:u  
    cWa> rUsF  
    % EOF zernfun (z'!'?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)
    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)  /*bS~7f1  
    _azg 0.)  
    DDE还是手动输入的呢? y$At$i>u  
    >]k'3|vV  
    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