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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, L0%hnA@  
    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, XI Jlc~2  
    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? L5IbExjV  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? \uG`|D n  
    qpJ{2Q  
    ]ALc;lb-}  
    /?/#B `  
    : t$l.+B  
    function z = zernfun(n,m,r,theta,nflag) fWGOP~0  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. S>q>K"j^!  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1\L[i];L8  
    %   and angular frequency M, evaluated at positions (R,THETA) on the pWE`x|J  
    %   unit circle.  N is a vector of positive integers (including 0), and |DF9cd^  
    %   M is a vector with the same number of elements as N.  Each element -V % gVI[  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 'z=:[#b  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 'j%F]CK  
    %   and THETA is a vector of angles.  R and THETA must have the same V2|3i}V"  
    %   length.  The output Z is a matrix with one column for every (N,M) M!M!Ni  
    %   pair, and one row for every (R,THETA) pair. BsZ{|,oQnZ  
    % qJR!$?  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~9Cz6yF  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1on'^8]0  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +~sd"v6  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, p3^jGj@  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized '[P}&<ie,  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )<4_:  
    % ']o od!  
    %   The Zernike functions are an orthogonal basis on the unit circle. qu6DQ@ ~YC  
    %   They are used in disciplines such as astronomy, optics, and 7yI @"c#O  
    %   optometry to describe functions on a circular domain. ! o, 5h|\  
    % pL1s@KR  
    %   The following table lists the first 15 Zernike functions. tZWrz e^  
    % ;%q39U}  
    %       n    m    Zernike function           Normalization FdOFE.l  
    %       -------------------------------------------------- (3,.3)%`  
    %       0    0    1                                 1 j%Y\A~DV  
    %       1    1    r * cos(theta)                    2 )wzV $(~  
    %       1   -1    r * sin(theta)                    2 g+zJ?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) $<)Yyi>6E  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }b["Jk\2  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 5iFV;W  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Y\/gU8w/  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ?T: jk4+  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) oholt/gb+0  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) q$ghLGz  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) QYE7p\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j;P+_Hfe/E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) j,%EW+j$  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) eQX`,9:5  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) YwT-T,oD  
    %       -------------------------------------------------- W,hWOO  
    % P]<= ! F  
    %   Example 1: wod/&!)]A  
    % M7y|EB))  
    %       % Display the Zernike function Z(n=5,m=1) hY \{|  
    %       x = -1:0.01:1; yDd[e]zS`  
    %       [X,Y] = meshgrid(x,x); Db03Nk>#  
    %       [theta,r] = cart2pol(X,Y); u"VS* hSH  
    %       idx = r<=1; - HOnB=  
    %       z = nan(size(X)); Bmr<O !  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +GNWF% zN  
    %       figure )q?$p9  
    %       pcolor(x,x,z), shading interp \%_ZV9cKF  
    %       axis square, colorbar jD< pIHau  
    %       title('Zernike function Z_5^1(r,\theta)') ~5#)N{GbY  
    % 9fVj 8G  
    %   Example 2: }~enEZ  
    % oFg'wAO.  
    %       % Display the first 10 Zernike functions #+sF`qR,  
    %       x = -1:0.01:1; jqoPLbxT  
    %       [X,Y] = meshgrid(x,x); mA{#]Yvf1  
    %       [theta,r] = cart2pol(X,Y); iK}v`xq  
    %       idx = r<=1; 0o/B{|rv  
    %       z = nan(size(X)); 2*[Un(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,Q2N[Jwd$  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; CI^|k/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,?b78_,2  
    %       y = zernfun(n,m,r(idx),theta(idx)); -Ds|qzrN%  
    %       figure('Units','normalized') ;~tsF.=  
    %       for k = 1:10 IKm&xzV-  
    %           z(idx) = y(:,k); Yw"P)Zp  
    %           subplot(4,7,Nplot(k)) ckwF|:e 7*  
    %           pcolor(x,x,z), shading interp ?n*fy  
    %           set(gca,'XTick',[],'YTick',[]) hLA;Bl  
    %           axis square !UNNjBBP7  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .ewZV9P)t  
    %       end VO9f~>`(  
    % R7aXR\ R  
    %   See also ZERNPOL, ZERNFUN2. x0x $  9  
    0$Ff#8  
    wu^q`!ml  
    %   Paul Fricker 11/13/2006 F.KrZ3%4iB  
    0 BC`iql5  
    mU]s7` %<>  
    z>:U{!5k  
    c^-YcGwa  
    % Check and prepare the inputs: i_Ar<9a~  
    % ----------------------------- =J.EH|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f'_ S1\  
        error('zernfun:NMvectors','N and M must be vectors.') 8eww7k^R  
    end ,P{ HE8.  
    I@PJl  
    qc-C>Ra  
    if length(n)~=length(m) Y\8+}g;KR  
        error('zernfun:NMlength','N and M must be the same length.') C"No5r'K3  
    end Y(z }[`2  
    zlMlMyG4  
    MgnE-6_c  
    n = n(:); M71R -B`-  
    m = m(:); *f*f&l%  
    if any(mod(n-m,2)) LhKY}R  
        error('zernfun:NMmultiplesof2', ... Kw*~W i  
              'All N and M must differ by multiples of 2 (including 0).') Vj7Hgc-,  
    end 0Q3YN(  
    (,TH~("{  
    >nNl^ yqW  
    if any(m>n) ?d,M.o{0]  
        error('zernfun:MlessthanN', ... Qi|?d7k0  
              'Each M must be less than or equal to its corresponding N.') `t9.xB#Z  
    end GiqBzV3"  
    @YNGxg~*g  
    + o{*r#  
    if any( r>1 | r<0 ) 4g'}h`kh  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ] j1 vbk  
    end TPqvp|~2  
    D?J#u;h~f  
    !3?~#e{_  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) p  .aE  
        error('zernfun:RTHvector','R and THETA must be vectors.') Wa}"SqYr h  
    end >gGil|I  
    cS 4T\{B;  
    Nc"NObe  
    r = r(:); 1!s!wQgS  
    theta = theta(:); }(cY|  
    length_r = length(r); w?/f Zx  
    if length_r~=length(theta) jRwa0Px(  
        error('zernfun:RTHlength', ... ytob/tc  
              'The number of R- and THETA-values must be equal.') W%H]Uyt  
    end 1::LN(`<  
    \@:j  
    i)8gCDc  
    % Check normalization: GM77Z.Y  
    % -------------------- .CvFE~  
    if nargin==5 && ischar(nflag) +qZc} 7rJF  
        isnorm = strcmpi(nflag,'norm'); =lm nzu<  
        if ~isnorm h/{8bC@bi  
            error('zernfun:normalization','Unrecognized normalization flag.') Yim#Pq&_  
        end 8}9Ob~on  
    else [Q=4P*G}X  
        isnorm = false; `L;OY 4  
    end uh1S 7!^  
    e-jw^   
    rF'<r~Lw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cL ae=N  
    % Compute the Zernike Polynomials @,GjeF]!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =_uol8v  
    "TUPYFK9  
    4xpj<  
    % Determine the required powers of r: H[Cj7{V  
    % ----------------------------------- 8Y7 @D$=w  
    m_abs = abs(m); #*\Ry/9Q  
    rpowers = []; a&8l[xe1  
    for j = 1:length(n) h k/+  
        rpowers = [rpowers m_abs(j):2:n(j)]; y3Y2 QC(  
    end # UjEY9"M  
    rpowers = unique(rpowers); \y@ eBW  
    {GAsFnZk  
    H a90  
    % Pre-compute the values of r raised to the required powers, |E? ,xWN  
    % and compile them in a matrix: -S`TEX  
    % ----------------------------- aQxe)  
    if rpowers(1)==0 <Ak:8&$O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &bn*p.=G  
        rpowern = cat(2,rpowern{:}); zv`zsqDJ  
        rpowern = [ones(length_r,1) rpowern]; 9A(n _Rs7?  
    else FF8WTuzB+  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?-4OfGN  
        rpowern = cat(2,rpowern{:}); }WA<=9e  
    end /(y4V  
    L,O>6~9:^1  
    Ia=&.,xub  
    % Compute the values of the polynomials: *>G ^!e.u  
    % -------------------------------------- @Ap@m6K?q  
    y = zeros(length_r,length(n)); m9%yR"g9  
    for j = 1:length(n) N&x@_t""   
        s = 0:(n(j)-m_abs(j))/2; Zp^)_ 0  
        pows = n(j):-2:m_abs(j); |&9tU  
        for k = length(s):-1:1 `CPZPp,l6`  
            p = (1-2*mod(s(k),2))* ... t;h+Cf4  
                       prod(2:(n(j)-s(k)))/              ... ]aREQ?ma&z  
                       prod(2:s(k))/                     ... vM5k4%D  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... XSD"/_xD  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 58qaA\iw  
            idx = (pows(k)==rpowers); >3<&V{<K  
            y(:,j) = y(:,j) + p*rpowern(:,idx); IvPA|8(  
        end M4R%Gr,La  
         qxRT1B]{Wx  
        if isnorm MoZU(j  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w2.qT+; v  
        end ,oC= {^l{  
    end TXA. 6e  
    % END: Compute the Zernike Polynomials .WxFm@]/\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Iz 1*4@  
    [3Wsc`Q  
    ['Hp?Q|k  
    % Compute the Zernike functions: 8h55$j  
    % ------------------------------ mvUVy1-c  
    idx_pos = m>0; }w;Q^EU  
    idx_neg = m<0; U/}AiCdj@  
    r0rJ.}!  
    I|Vk.,  
    z = y; B7NmET4  
    if any(idx_pos) iuvtj]/  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); XHU<4l:kl  
    end l|4xKBCV]  
    if any(idx_neg) z:0-aDe M  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); T2c_vY   
    end v e6N  
    ;r0|_mnf  
    URmAI8fq*M  
    % EOF zernfun VR5e CJ:i  
     
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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)  n\"LN3  
    $H:h(ia:  
    DDE还是手动输入的呢? ! 7*_Z=  
    JTh =JHJ  
    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