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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, ]kdU]}z  
    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, 0u}+n+\g  
    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? 1EMrXnv,  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? o%E-K=a  
    pnpf/T{xpM  
    ]u|5ZCv0  
    Owz>g4l r  
    ?>Bt|[p:s)  
    function z = zernfun(n,m,r,theta,nflag) O:j=L{,d^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. \*mKctpz]6  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Bo4iX,zu  
    %   and angular frequency M, evaluated at positions (R,THETA) on the .A F94OlE/  
    %   unit circle.  N is a vector of positive integers (including 0), and 6m0- he~  
    %   M is a vector with the same number of elements as N.  Each element Wgm{ ]9Q  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) (V>/[Ev  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, nNhb,J  
    %   and THETA is a vector of angles.  R and THETA must have the same G&q@B`I  
    %   length.  The output Z is a matrix with one column for every (N,M) +{\b&q_  
    %   pair, and one row for every (R,THETA) pair. Wb#ON|.2  
    % Mk/ZEyq^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /6FPiASbS  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .*Axr\x3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral P EMuIYm$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +7Qj%x\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized _]M :  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;HmQRiCg  
    % 8Yw V"+Fu/  
    %   The Zernike functions are an orthogonal basis on the unit circle. l(3\ekU!  
    %   They are used in disciplines such as astronomy, optics, and \eCQL(_  
    %   optometry to describe functions on a circular domain. %(~8a  
    % *40Z }1ng  
    %   The following table lists the first 15 Zernike functions. -v~XS-F  
    % R 5Cy%  
    %       n    m    Zernike function           Normalization +pjU4>)  
    %       -------------------------------------------------- R ]y9>5 'U  
    %       0    0    1                                 1 { \5-b:#_  
    %       1    1    r * cos(theta)                    2 <xAlp;8m5  
    %       1   -1    r * sin(theta)                    2 ZIJTGa}B q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) h<FEe~  
    %       2    0    (2*r^2 - 1)                    sqrt(3) D,SL_*r{  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) l?zWi[Zf  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) y:dwx*Q9I  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @v=A)L  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) MKr:a]-'f~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6N'HXL UlQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {-,^3PI\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !' jXN82  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m NApFwZ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Lz;E/a}s  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {y7,n  
    %       -------------------------------------------------- +ywd(Tuzm  
    % !d=Q@oy5  
    %   Example 1: !FR1yO'd>  
    % 7x :j4  
    %       % Display the Zernike function Z(n=5,m=1) da53XEF&  
    %       x = -1:0.01:1; a3oSSkT  
    %       [X,Y] = meshgrid(x,x); dM3V2TT  
    %       [theta,r] = cart2pol(X,Y); !YEU<9  
    %       idx = r<=1; %]@K}!)2  
    %       z = nan(size(X)); i*%2 e)  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Y&_1U/}h  
    %       figure 7&9'=G  
    %       pcolor(x,x,z), shading interp vH^^QI:em  
    %       axis square, colorbar )SYZ*=ezl.  
    %       title('Zernike function Z_5^1(r,\theta)') yD!V;?EnK  
    % 5-pz/%,  
    %   Example 2: ]]Da/^K=Z  
    % 8M;G@ Q80  
    %       % Display the first 10 Zernike functions DqHVc)9  
    %       x = -1:0.01:1;  ]=g |e  
    %       [X,Y] = meshgrid(x,x); kM\O2 ay  
    %       [theta,r] = cart2pol(X,Y); >AW=N  
    %       idx = r<=1; ~f2zMTI|  
    %       z = nan(size(X)); :1wMGk  
    %       n = [0  1  1  2  2  2  3  3  3  3]; +xlxhF  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; a4}2^K  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; :/IcFU~)M  
    %       y = zernfun(n,m,r(idx),theta(idx)); ft[g1  
    %       figure('Units','normalized') +mT}};-TS  
    %       for k = 1:10 VBssn]w  
    %           z(idx) = y(:,k); w%k)J{\  
    %           subplot(4,7,Nplot(k)) Ga%]$4u  
    %           pcolor(x,x,z), shading interp ${ ~UA 6  
    %           set(gca,'XTick',[],'YTick',[]) 05MtQB   
    %           axis square  v|Tg %  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G1~|$X@@  
    %       end v[6BESu  
    % HJ5m5':a  
    %   See also ZERNPOL, ZERNFUN2. =D4EPfQn1  
    38Z"9  
    <' %g $"  
    %   Paul Fricker 11/13/2006 E!VAA=  
    KW&&AuPb}  
    fwAN9zs  
    ),\>'{~5&  
    im+2)9f  
    % Check and prepare the inputs: 6EK+]0  
    % ----------------------------- `CK;,>i   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ai->,<Ig]  
        error('zernfun:NMvectors','N and M must be vectors.') !nj%n  
    end e|Sg?ocR  
    hDlk! #*  
    {u.V8%8  
    if length(n)~=length(m) /"CKVQ  
        error('zernfun:NMlength','N and M must be the same length.') L''0`a. +S  
    end : 6>H\  
    ;wQWt_OtuJ  
    DhZtiqL#_  
    n = n(:); HqXo;`Yy}  
    m = m(:); @IiT8B  
    if any(mod(n-m,2)) Msd!4TrBJ  
        error('zernfun:NMmultiplesof2', ... m]-8?B1`Y  
              'All N and M must differ by multiples of 2 (including 0).') ($S{td;  
    end CY>NU  
    z)VIbEy  
    P/I{q s  
    if any(m>n) AVyZ#`,  
        error('zernfun:MlessthanN', ... K%pmE?%,8  
              'Each M must be less than or equal to its corresponding N.') <,E*,&0W  
    end Q?a"uei[  
    68HX,t  
    ?J~JQe42  
    if any( r>1 | r<0 ) bf74 "  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \4j+pU  
    end D+SpSO7yg  
    sRZ:9de+  
    O h" ^  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,{Ab=xV  
        error('zernfun:RTHvector','R and THETA must be vectors.') \W}EyA  
    end +uLo~GdbE  
    i52R,hz  
    oba*w;  
    r = r(:); 6fV)8,F3  
    theta = theta(:); r/4]b]n  
    length_r = length(r); GBphab|  
    if length_r~=length(theta) Z>,X$ Y6<  
        error('zernfun:RTHlength', ... z;/'OJ[.  
              'The number of R- and THETA-values must be equal.') .u*].As=  
    end zl:D|h77  
    $1?X%8V  
    <=inogf  
    % Check normalization: o(``7A@7a  
    % -------------------- @}?D<O8#"#  
    if nargin==5 && ischar(nflag) V^{!d}  
        isnorm = strcmpi(nflag,'norm'); {6n \532@  
        if ~isnorm `e9uSF:9C  
            error('zernfun:normalization','Unrecognized normalization flag.') *h4m<\^U  
        end dI !/:x  
    else Qwa"AY 5pW  
        isnorm = false; [;=ky<K0E  
    end -*lP1Nbp  
    }}1/Ede{5  
    v 2GhR*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yy)a,clZ*$  
    % Compute the Zernike Polynomials "?{yVu~9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PbPP1G')  
    668bJ.M\O  
    9Bk}g50$#  
    % Determine the required powers of r: )A0&16<  
    % ----------------------------------- [~<',,tA0|  
    m_abs = abs(m); D%idlL2%J  
    rpowers = []; 9-Qtj49  
    for j = 1:length(n) _;q-+"6L;  
        rpowers = [rpowers m_abs(j):2:n(j)]; O|RO j  
    end lDU:EJ&DHE  
    rpowers = unique(rpowers); 8-5 jr_*  
    #Q@6:bBzv  
    a1`cI5n  
    % Pre-compute the values of r raised to the required powers, nh=Us^xD  
    % and compile them in a matrix: 'q'Y:A?,  
    % ----------------------------- ptv 4v[gQ  
    if rpowers(1)==0 'Xl>,\'6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &{/>Sv!6#  
        rpowern = cat(2,rpowern{:}); H27Oq8  
        rpowern = [ones(length_r,1) rpowern]; OZ;E&IL  
    else Zax]i,Bx  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =+h!JgY/L  
        rpowern = cat(2,rpowern{:}); S.)7u6/_!  
    end NoAb}1uae  
    (1,#=e+  
    gwaC?tf[  
    % Compute the values of the polynomials: ePRMv  
    % -------------------------------------- ba9<(0`  
    y = zeros(length_r,length(n)); &E &iaw!  
    for j = 1:length(n) U9o*6`"o  
        s = 0:(n(j)-m_abs(j))/2; m90R8  V  
        pows = n(j):-2:m_abs(j); eH!|MHe  
        for k = length(s):-1:1 6&QTVdK'O  
            p = (1-2*mod(s(k),2))* ... m=.7f9  
                       prod(2:(n(j)-s(k)))/              ... q7 oR9  
                       prod(2:s(k))/                     ... .&x?`pER  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :ZfUjqRE  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); cNr][AzU@  
            idx = (pows(k)==rpowers); pt cLJ]+)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :/[YY?pg-  
        end q uGPk)c  
         Z)O>h^0  
        if isnorm /-YlC (kL  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <SM&VOiaOz  
        end uP=_-ZUW  
    end 9;Pu9s[q2  
    % END: Compute the Zernike Polynomials HjK<)q8b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3:8nwt  
    Vc52s+7=8  
    KO]?>>5S6  
    % Compute the Zernike functions: khN:+V|  
    % ------------------------------ ]6%%X+$7  
    idx_pos = m>0; `{|}LFS>  
    idx_neg = m<0; @oqi@&L'C  
    h NOYFH  
    x\bRj>%(  
    z = y; YTjuSV  
    if any(idx_pos) 9poEUjBI  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v8vh~^X%P  
    end k *;{n8o?)  
    if any(idx_neg) h,'mN\6t  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); mf;^b.mKh  
    end ilRm}lU|x  
    c7l!G~yx'  
    rI^~9Rz  
    % EOF zernfun N]s7/s  
     
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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)  'GEBxNH:  
    9">zdFC'  
    DDE还是手动输入的呢? "rme~w Di  
    82j'MgGP  
    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