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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, J9Ou+6u(  
    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, CI :`<PZ\-  
    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? Q~Hh\Lt  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? .G(llA}  
    )+"'oY$]}  
    Ru>uL@w  
    nJ"YIT1K]p  
    HJ[/|NZU$  
    function z = zernfun(n,m,r,theta,nflag) _uKZMl  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. d,tU#N{Q6  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N !F4@KAv  
    %   and angular frequency M, evaluated at positions (R,THETA) on the n?ctLbg  
    %   unit circle.  N is a vector of positive integers (including 0), and 'vq:D$A  
    %   M is a vector with the same number of elements as N.  Each element RJH,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) <b?!jV7  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, -,aeM~  
    %   and THETA is a vector of angles.  R and THETA must have the same RZ7( J  
    %   length.  The output Z is a matrix with one column for every (N,M) |vMpXiMxxT  
    %   pair, and one row for every (R,THETA) pair. LIVU^Os.  
    % G0{H5_h  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike b}w C|\s  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ?EpSC&S\  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +|{RE.DL  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Q33"u/-v  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ,7)C"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. za9)Q=6FD  
    % Y<b-9ai<w  
    %   The Zernike functions are an orthogonal basis on the unit circle. kR@Yl Yo  
    %   They are used in disciplines such as astronomy, optics, and maY4g&'f  
    %   optometry to describe functions on a circular domain. kctzNGF|  
    % 8W+gl=C~  
    %   The following table lists the first 15 Zernike functions. l|+BC  
    % Rqy0Q8K<  
    %       n    m    Zernike function           Normalization p!V>XY'N^  
    %       -------------------------------------------------- qG/fE'(j&  
    %       0    0    1                                 1 9W>Y#V~|v!  
    %       1    1    r * cos(theta)                    2 f0SAP0M3  
    %       1   -1    r * sin(theta)                    2 m8JR@!t7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) u=NS sTP&  
    %       2    0    (2*r^2 - 1)                    sqrt(3) /.eeO k  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) *tX{MSYW  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =GBI0&U  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) `L5~mb;7*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) f8<o8*`7  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) $ RwB_F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ORWm C!  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $hVYTy~}  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ]:$ O{y  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C#=bW'C  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0Hw-59MK  
    %       -------------------------------------------------- c<BO gNr  
    % y3;q_4.  
    %   Example 1: 5ZPzPUa8~  
    % Gy Qm/I  
    %       % Display the Zernike function Z(n=5,m=1) 3PUAH  
    %       x = -1:0.01:1; qxJQPz  
    %       [X,Y] = meshgrid(x,x); W"xP(7X  
    %       [theta,r] = cart2pol(X,Y); ^D_/=4rz8  
    %       idx = r<=1; +~U=C9[gj  
    %       z = nan(size(X)); O^I[ (8Y8  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "4j:[9vR\  
    %       figure wVA|!>v  
    %       pcolor(x,x,z), shading interp fKa\7{R  
    %       axis square, colorbar 5[9 bWB{  
    %       title('Zernike function Z_5^1(r,\theta)') >(tn"2  
    % ~Z lC '  
    %   Example 2: zN_:nY>  
    % oXt,e   
    %       % Display the first 10 Zernike functions 6`"M  
    %       x = -1:0.01:1; 7C?.L70ZY  
    %       [X,Y] = meshgrid(x,x); l??;3kh1  
    %       [theta,r] = cart2pol(X,Y); kao}(?x%  
    %       idx = r<=1; Y/8K;U|  
    %       z = nan(size(X)); I#FF*@oeM  
    %       n = [0  1  1  2  2  2  3  3  3  3]; $ Cjk  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; bv dR"G  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; i"^<CR@e  
    %       y = zernfun(n,m,r(idx),theta(idx)); D~&Mwsi  
    %       figure('Units','normalized') F[7x*-NO-  
    %       for k = 1:10 2#/p|$;Ec'  
    %           z(idx) = y(:,k); <<|H=![  
    %           subplot(4,7,Nplot(k)) )06iV  
    %           pcolor(x,x,z), shading interp 9<]a!:!^  
    %           set(gca,'XTick',[],'YTick',[]) lZt(&^T  
    %           axis square 6j8 <Q 2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6=PiVwI  
    %       end M\+*P,i  
    % W)SjQp6  
    %   See also ZERNPOL, ZERNFUN2. [ij,RE7,T  
    I(n* _bFq  
    =ziy`#fm,  
    %   Paul Fricker 11/13/2006 gw3NS8 A+  
    qG >DTKIU  
    =O{~Q3z@s  
    WA.\*Nqze  
    /k"hH\Pp  
    % Check and prepare the inputs: %bX0 mN  
    % ----------------------------- 02]xJo  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O'}l lo  
        error('zernfun:NMvectors','N and M must be vectors.')  td(M#a-  
    end JAn1{<Ky  
    $-@$i`Kf/  
    h <[+HsI  
    if length(n)~=length(m) h[ 6hM^n  
        error('zernfun:NMlength','N and M must be the same length.') 1 2]fQkp  
    end iTNqWU-o  
    LnMwx#^*  
    i@<~"~>]7  
    n = n(:); e.6Dl_  
    m = m(:); (@ea|Fd#4  
    if any(mod(n-m,2)) J/4y|8T/y  
        error('zernfun:NMmultiplesof2', ... c%YDt`  
              'All N and M must differ by multiples of 2 (including 0).') It 2UfW  
    end ~2N-k1'-'  
    '=TTa  
    a0zG(7.D  
    if any(m>n) %9c|%#3  
        error('zernfun:MlessthanN', ... bBE^^9G=Z  
              'Each M must be less than or equal to its corresponding N.') 4NVgOr:  
    end ;x>;jS.t  
    ehc<|O9tY  
    &9ki O  
    if any( r>1 | r<0 ) ye r> x  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') NFoZ4R1gy  
    end 5|WOBOh>`&  
    -"Gl 4)  
    @]3*B %t  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) l/V&s<  
        error('zernfun:RTHvector','R and THETA must be vectors.') HRRngk#lV  
    end \3 KfD'L  
    "<dN9l>  
    `03<0L   
    r = r(:); -g2{68 1`r  
    theta = theta(:); Q}uG/HI  
    length_r = length(r); *!u?  
    if length_r~=length(theta) mahi7eU P  
        error('zernfun:RTHlength', ... Fi{mr*}  
              'The number of R- and THETA-values must be equal.') x\;GoGsez  
    end U~g@TfU;  
    z`9l<Q/  
    :Ba-u  
    % Check normalization: /2:Q6J  
    % --------------------  ,(hY%M&\  
    if nargin==5 && ischar(nflag) u-/3(dKt  
        isnorm = strcmpi(nflag,'norm'); :+pPr Gj"  
        if ~isnorm xhD$e= g  
            error('zernfun:normalization','Unrecognized normalization flag.') `1p?*9Ssn  
        end #8qyg<F  
    else s#Q _Gu  
        isnorm = false; WA$ p_% r=  
    end "w1(g=n  
    %~(~W>^A  
    Cs;<'[_?YO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <d<RK@2-  
    % Compute the Zernike Polynomials $pBr &,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I_L;T  
    j) <[j&OWw  
    ]J~g'">  
    % Determine the required powers of r: 7#/|VQX<A  
    % ----------------------------------- |=OpzCs  
    m_abs = abs(m); @>9A$w$H|a  
    rpowers = []; Q~CpP9%  
    for j = 1:length(n) $@4e(Zrmo  
        rpowers = [rpowers m_abs(j):2:n(j)]; Kk56/(_S  
    end 6NKF'zh  
    rpowers = unique(rpowers); ~)!VV)  
    9/Q S0  
    c; d"XiA  
    % Pre-compute the values of r raised to the required powers, Suj}MEiv  
    % and compile them in a matrix: V'$oTZ`  
    % ----------------------------- yL4 -4  
    if rpowers(1)==0 @%keTTZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 92NC]_jw  
        rpowern = cat(2,rpowern{:}); /T4VJ{D  
        rpowern = [ones(length_r,1) rpowern]; k4* ! Q_A  
    else T7X!#j" \  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jS}'cm-  
        rpowern = cat(2,rpowern{:}); zZw@c?  
    end Hm<M@M$aG  
    _fe0,  
    l+'`BBh*]  
    % Compute the values of the polynomials: 4jPwL|#  
    % -------------------------------------- 4I+.^7d  
    y = zeros(length_r,length(n)); jFS 'I*1+  
    for j = 1:length(n) L)=8mF.  
        s = 0:(n(j)-m_abs(j))/2; oO}>i0ax*  
        pows = n(j):-2:m_abs(j); Y~}QJ+`?  
        for k = length(s):-1:1 QDl)92z  
            p = (1-2*mod(s(k),2))* ... CJtr0M<U+  
                       prod(2:(n(j)-s(k)))/              ... xg4T` ])  
                       prod(2:s(k))/                     ... "&s9cO.H  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... MH2OqiCI  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); .Lp Nm'=R  
            idx = (pows(k)==rpowers); U5 -zB)V  
            y(:,j) = y(:,j) + p*rpowern(:,idx); v^57j:sD  
        end L)j]~^P$-  
         `mWQWx$V!  
        if isnorm vC s6#PR$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); oa?!50d  
        end DPR;$yV  
    end ktdz@f  
    % END: Compute the Zernike Polynomials 9 #.<E5:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f45;fT>   
    lsN /$ M|}  
    {: Am9B  
    % Compute the Zernike functions: D6"~fjHh  
    % ------------------------------ Qj{$dqmDN  
    idx_pos = m>0; p,!fIx  
    idx_neg = m<0; `,hW;p>-  
    7Q<Kha  
    wGZ>iLe:  
    z = y; &T5f H!?4  
    if any(idx_pos) e@6RC bj  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 7/[TE  
    end _m) gO/02A  
    if any(idx_neg) ~Tpe,juG_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); w+URCj  
    end 3W%f#d$`  
    |SwZi'p  
    !- Cs?  
    % EOF zernfun $ l0eI  
     
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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)  +ke42Jwt  
    M$5%QM}  
    DDE还是手动输入的呢? +h\W~muR  
    <=GzK:4L  
    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