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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, OGcq]ue  
    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, z>lIZ}  
    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? \w#)uYK{i_  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Cg_9V4h.C  
    wAD%1;  
    6pP:Q_U$  
    q]Vxf!0*>  
    ,i)wS1@  
    function z = zernfun(n,m,r,theta,nflag) )[wB:kG  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. fQQj2> 3w  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 717S3knlv  
    %   and angular frequency M, evaluated at positions (R,THETA) on the k~Z;S QyN  
    %   unit circle.  N is a vector of positive integers (including 0), and qBF6LhR  
    %   M is a vector with the same number of elements as N.  Each element &$yxAqdab  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Zz/ z7~{  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *(E]]8o  
    %   and THETA is a vector of angles.  R and THETA must have the same pF/s5z  
    %   length.  The output Z is a matrix with one column for every (N,M) QZ& 4W  
    %   pair, and one row for every (R,THETA) pair. z94#:jPmG  
    % t#d{hEr  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |W*#N8I P  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \r1nMw3&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral r(j:C%?}C  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, AcP d(Pc  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized wU(p_G3  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. u+ b `aB  
    % 18,;2Sr44  
    %   The Zernike functions are an orthogonal basis on the unit circle. fU<_bg  
    %   They are used in disciplines such as astronomy, optics, and Yz)+UF,  
    %   optometry to describe functions on a circular domain. +\-cf,WkI  
    % 7bk`u'0%  
    %   The following table lists the first 15 Zernike functions. E5qt~:C|  
    % =&Z#QD"vl  
    %       n    m    Zernike function           Normalization ;F|8#! (  
    %       -------------------------------------------------- X'{ o/U.  
    %       0    0    1                                 1 nc3u sq  
    %       1    1    r * cos(theta)                    2 "^Vnnb:Z*o  
    %       1   -1    r * sin(theta)                    2 I;Pd}A_}=_  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |@5G\N-  
    %       2    0    (2*r^2 - 1)                    sqrt(3) % oJH 6F  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) u-M Td  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) NY?pvb  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 4s9q Q8?  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) GC`/\~TM  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6<fcG  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) :.= #U  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %mAwK<MY`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) :{,k F  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Qe =8x7oIP  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) c+{ ar^)*  
    %       -------------------------------------------------- j^.|^q<Y  
    % Q[c:A@oW  
    %   Example 1: :}-VLp4b  
    % &o]fBdn  
    %       % Display the Zernike function Z(n=5,m=1) QtA@p  
    %       x = -1:0.01:1; ?)gc;K  
    %       [X,Y] = meshgrid(x,x); RR2M+vQ  
    %       [theta,r] = cart2pol(X,Y); ?$MO!  
    %       idx = r<=1; + B<7]\\M  
    %       z = nan(size(X)); RdB,;Um9f  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); (%'`t(<  
    %       figure NIAji3  
    %       pcolor(x,x,z), shading interp +~EnrrT+W  
    %       axis square, colorbar YJ+l \Wb}  
    %       title('Zernike function Z_5^1(r,\theta)') 0a9[}g1=#  
    % 1 F&}e&}c  
    %   Example 2: h.\p+Qw.  
    % 1,Jy+1G0w  
    %       % Display the first 10 Zernike functions P{HR='2  
    %       x = -1:0.01:1; `#:(F z  
    %       [X,Y] = meshgrid(x,x); )-m/(-  
    %       [theta,r] = cart2pol(X,Y); J| 1!4R~  
    %       idx = r<=1; NtmmPJ|5  
    %       z = nan(size(X)); '|}H ,I{  
    %       n = [0  1  1  2  2  2  3  3  3  3]; MP_/eC ;  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?69E_E  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; E5?$=cL?  
    %       y = zernfun(n,m,r(idx),theta(idx)); cT@H49#uB  
    %       figure('Units','normalized') )Y`ybADd3  
    %       for k = 1:10 eM]>"  
    %           z(idx) = y(:,k); |9Y~k,rF  
    %           subplot(4,7,Nplot(k)) W6RjQ1  
    %           pcolor(x,x,z), shading interp >3,}^`l  
    %           set(gca,'XTick',[],'YTick',[]) &UVqF o  
    %           axis square N/[!$B0H@  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zDBm^ s  
    %       end gH.$B'  
    % mKoDy`s  
    %   See also ZERNPOL, ZERNFUN2. ZENblh8fs  
    s )Xz}QPK.  
    (:^YfG~e  
    %   Paul Fricker 11/13/2006 Y5y7ONcn  
    l GJN;G7  
    HIUP =/x  
    :QF`Orb!^  
    2Sk hBb=d  
    % Check and prepare the inputs: vs>Pd |p;  
    % ----------------------------- s`pdy$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i6S["\h>  
        error('zernfun:NMvectors','N and M must be vectors.')  N!Xn)J  
    end F$'po#  
    l3y}nh+ 8  
    -c{Y+M`  
    if length(n)~=length(m) r*ziO#[  
        error('zernfun:NMlength','N and M must be the same length.') t*fH&8(  
    end )(rr1^Xer  
    : rudo[L  
    %TO&  
    n = n(:); (=j/"Mb  
    m = m(:); %L$ ?Mey  
    if any(mod(n-m,2)) .J=QWfqt  
        error('zernfun:NMmultiplesof2', ... Bc`L ]<  
              'All N and M must differ by multiples of 2 (including 0).') Urol)_3X  
    end n<F3&2w  
    HG)$ W  
    'e6J&X  
    if any(m>n) 4sfq,shRq  
        error('zernfun:MlessthanN', ... qxcTY|&  
              'Each M must be less than or equal to its corresponding N.') flz7{W  
    end .krEfY&  
    F=PBEaX  
    (4IP&^j:\  
    if any( r>1 | r<0 ) ulk/I-y  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') `-Tb=o}.  
    end oTr,zRL  
    `=Rxnl,<U  
    I,"q:QS+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o5YL_=7m  
        error('zernfun:RTHvector','R and THETA must be vectors.') I]42R;Sc  
    end ^W`RBrJay  
    fhha-J  
    9yu#G7  
    r = r(:); 3Vk\iJ  
    theta = theta(:); 4QYStDFe  
    length_r = length(r); ZkdSgc')  
    if length_r~=length(theta) mR|']^!SE  
        error('zernfun:RTHlength', ... &x4*YM h  
              'The number of R- and THETA-values must be equal.') '}OAl  
    end ) r"7"i  
    h\5~&}Hp  
    [.$/o}  
    % Check normalization: ezhfKt]j  
    % -------------------- dp2FC   
    if nargin==5 && ischar(nflag) I]cZcx,<q  
        isnorm = strcmpi(nflag,'norm'); IR&b2FTcU  
        if ~isnorm Ef3=" }AI;  
            error('zernfun:normalization','Unrecognized normalization flag.') k4!p))ql  
        end P,#l~\  
    else 7.+vp@+  
        isnorm = false; @PK 1  
    end iAeq%N1(0  
    {$7vd  
    {cjp8W8hS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #WE lL2&  
    % Compute the Zernike Polynomials 'b6qEU#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K.}jyhKIKi  
    dZddo z_  
    ) bd`U  
    % Determine the required powers of r: d" a\`#  
    % ----------------------------------- !u/c'ZLZ>  
    m_abs = abs(m); -vh\XO  
    rpowers = []; %fXgV\xY  
    for j = 1:length(n) IK8" 3+(  
        rpowers = [rpowers m_abs(j):2:n(j)]; j9}.U \  
    end h?fp(  
    rpowers = unique(rpowers); .$+,Y4q~(  
    DweF8c  
    kk OjAp{<t  
    % Pre-compute the values of r raised to the required powers, '*`1uomeo  
    % and compile them in a matrix: 5!57<n  
    % ----------------------------- cet|k!   
    if rpowers(1)==0 fF5\\_,  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hn$jI5*`  
        rpowern = cat(2,rpowern{:}); )/z+W[t  
        rpowern = [ones(length_r,1) rpowern]; #8%~u+"N  
    else :#UA!| nV  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L9l]0C37e  
        rpowern = cat(2,rpowern{:}); Wi*HLP!lNC  
    end 2Y;iqR  
    rT;_"y}  
    Smd83W&  
    % Compute the values of the polynomials: Bi0&F1ZC!  
    % -------------------------------------- b8 6c[2  
    y = zeros(length_r,length(n)); cA{,2CYc  
    for j = 1:length(n) =7S\-{  
        s = 0:(n(j)-m_abs(j))/2; @[5]?8\o  
        pows = n(j):-2:m_abs(j); ?9~|K/`l  
        for k = length(s):-1:1 y#nyH0U  
            p = (1-2*mod(s(k),2))* ... T+:GYab/  
                       prod(2:(n(j)-s(k)))/              ... _1jeaV9@  
                       prod(2:s(k))/                     ... !1<>][F  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... A8ClkLC;I  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); l HZ4N{n  
            idx = (pows(k)==rpowers); o%h[o9i  
            y(:,j) = y(:,j) + p*rpowern(:,idx); +1eb@b X  
        end Xx^v%[!`+  
         +@do<2l]  
        if isnorm 4EhWK;ra  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {y<E_y x1  
        end ~-A"M_n ?  
    end T1RICIf 1F  
    % END: Compute the Zernike Polynomials l i%8X.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j$k/oQ  
    tU-jtJ  
    W)`H(J  
    % Compute the Zernike functions: pQ`S%]k.<  
    % ------------------------------ zKf0 :X  
    idx_pos = m>0; ZRUI';5x  
    idx_neg = m<0; OuB [[L  
    raZ0B,;eFu  
    De49!{\a  
    z = y; n&E/{o(  
    if any(idx_pos) ,(kaC.Em  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); l YjPrA]TC  
    end UJ&gm_M+kL  
    if any(idx_neg) fBP J8VY  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); VS+5{w:t  
    end okBaQH2lUl  
    @Z@S;RWSU  
    o H]FT{  
    % EOF zernfun px^brzLQo  
     
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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)  -2Azpeh  
    %epK-q9[  
    DDE还是手动输入的呢? {:9P4<%H  
    jj 9eFB  
    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