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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, dw v(8  
    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, 9w=GB?/  
    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? ?T(>!m  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]$>O--  
    -K_p? l  
    z|V5/"  
    ~Zc=FP:1  
    y2U^7VrO  
    function z = zernfun(n,m,r,theta,nflag) 2y&m8_s-p  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. KnC;j-j  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N aJC,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the WmRx_d_  
    %   unit circle.  N is a vector of positive integers (including 0), and m"<Sb,"x!  
    %   M is a vector with the same number of elements as N.  Each element b$f@.L  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) hZ0CnY8 '  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 0 7CufoI  
    %   and THETA is a vector of angles.  R and THETA must have the same @k!J}O K  
    %   length.  The output Z is a matrix with one column for every (N,M) fq.ui3lP)  
    %   pair, and one row for every (R,THETA) pair. >h0iq  
    % Z. ))=w6G  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Y?(kE` R  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), c7[<X<yk  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ) /kf  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W -Yv0n3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized (hB&OP5Fne  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. mZ^z%+Ca|  
    % +ou ]|  
    %   The Zernike functions are an orthogonal basis on the unit circle. w(QU'4~  
    %   They are used in disciplines such as astronomy, optics, and >[=fbL@N<@  
    %   optometry to describe functions on a circular domain. Lbka*@  
    % B>3joe}  
    %   The following table lists the first 15 Zernike functions. tSVN}~1\  
    % eC^UL5>%  
    %       n    m    Zernike function           Normalization hE41$9?TJ  
    %       -------------------------------------------------- ze<Lc/;X~  
    %       0    0    1                                 1 JC~L!)f  
    %       1    1    r * cos(theta)                    2 (cX;a/BR  
    %       1   -1    r * sin(theta)                    2 fb7Gy  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) gAA2S5th  
    %       2    0    (2*r^2 - 1)                    sqrt(3) v2e*mNK5  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) {8)Pke  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) X|}yp|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "lcNjyU\O  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Jhclg0q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Fb&Xy{kt1  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) u%J04vG"D  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) la7VeFT  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @5!Mr5;  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G x;U 3iV  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) O,`#h*{N  
    %       -------------------------------------------------- 'u6T^YS  
    % >hkmL](^  
    %   Example 1: b'9\j.By  
    % '?Mt*%J@=$  
    %       % Display the Zernike function Z(n=5,m=1) }Ut*Y*  
    %       x = -1:0.01:1; CdCo+U5z{  
    %       [X,Y] = meshgrid(x,x); Yj/aa0Ka4  
    %       [theta,r] = cart2pol(X,Y); p5|.E  
    %       idx = r<=1; rBd}u+:*  
    %       z = nan(size(X)); :.863_/  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); yrp5\k*{y  
    %       figure AJ_''%$I3:  
    %       pcolor(x,x,z), shading interp k e'aSD  
    %       axis square, colorbar -nVQB146^  
    %       title('Zernike function Z_5^1(r,\theta)') zn| S3c  
    % s}8(__|  
    %   Example 2: qPEtMvL #  
    % J#h2~Hz!  
    %       % Display the first 10 Zernike functions Aofk<O!M  
    %       x = -1:0.01:1; j_::#?o!/  
    %       [X,Y] = meshgrid(x,x); f)`_su U  
    %       [theta,r] = cart2pol(X,Y); toD v~v  
    %       idx = r<=1; {}r#s>  
    %       z = nan(size(X)); 5K_KZL-  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^P4q6BW  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; zX{O"w  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Wpgp YcPS  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0(!j]w"r3  
    %       figure('Units','normalized') b-Q*!U t  
    %       for k = 1:10 Akar@wh  
    %           z(idx) = y(:,k); BE`{? -G  
    %           subplot(4,7,Nplot(k)) ]mDsd*1  
    %           pcolor(x,x,z), shading interp c/:d$o-  
    %           set(gca,'XTick',[],'YTick',[]) C`qo  
    %           axis square :@mBSE/  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ;WydXQ}Q^  
    %       end Lp!4X1/|\  
    % ) qD Ch  
    %   See also ZERNPOL, ZERNFUN2. %sd1`1In  
    (OA-Mgyc  
    m=g\@&N  
    %   Paul Fricker 11/13/2006 up(6/-/.7  
     4RPc&%  
    ?8ZOiY(  
    \<cs:C\h7  
    'CF?pxNQ l  
    % Check and prepare the inputs: R,]J~TfPK  
    % ----------------------------- Y[_{tS#u  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <+7]EwVcn^  
        error('zernfun:NMvectors','N and M must be vectors.') T;7=05k<_  
    end DC9\Sp?  
    |p4D!M+$7  
    vy:-a G  
    if length(n)~=length(m) ]2:w?+T  
        error('zernfun:NMlength','N and M must be the same length.') ??\1eo2gB  
    end ;Jh=7wx  
    *$%ch=  
    xIOYwVC  
    n = n(:); q mJ#cmN  
    m = m(:); cSbyVC[r  
    if any(mod(n-m,2)) = aO1uC|6C  
        error('zernfun:NMmultiplesof2', ... uPe&i5YR  
              'All N and M must differ by multiples of 2 (including 0).') E#?Bn5-uBs  
    end O4)'78ATp  
    N>zpx U {  
    2p^Jqp`$  
    if any(m>n) @2yoy&IO  
        error('zernfun:MlessthanN', ... )JNUfauyT  
              'Each M must be less than or equal to its corresponding N.') ,@\$PyJ  
    end /$z(BX/  
    =nVEdRU  
    B//2R)HS  
    if any( r>1 | r<0 ) `_MRf[Z}  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') [9<c;&$LU  
    end "b~-`ni  
    VnjhEEM!  
    jDO"?@+  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (h8RthQt  
        error('zernfun:RTHvector','R and THETA must be vectors.') 8QJ^@|7  
    end =&_Y=>rA]0  
    sYfiC`9SO  
    0uZL*4A+C  
    r = r(:); GbXa=* <-<  
    theta = theta(:); a)o-6  
    length_r = length(r); =<BPoGs5  
    if length_r~=length(theta) E;o "^[we  
        error('zernfun:RTHlength', ... zfsGf 'U  
              'The number of R- and THETA-values must be equal.') w\K(kNd(  
    end Qhc>,v)  
    4MFdhJoN  
    |8{c|Qz  
    % Check normalization: =q\Ghqj1  
    % -------------------- 9}*Pb6  
    if nargin==5 && ischar(nflag) \kR:GZ`{UV  
        isnorm = strcmpi(nflag,'norm'); +A;AX.mr  
        if ~isnorm 7hzd.  
            error('zernfun:normalization','Unrecognized normalization flag.') y/.I<5+Bu  
        end dED&-e#  
    else VYo2m  
        isnorm = false; r|ID]}w  
    end .UGbo.e  
    ML!>tCT  
    r%uka5@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _{C =d3  
    % Compute the Zernike Polynomials Tlar@lC|u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2(i@\dZCb<  
    ,hVDGif  
    (i.7\$4  
    % Determine the required powers of r: w(N$$  
    % ----------------------------------- ]aZ3_<b  
    m_abs = abs(m); |?gO@?KDZ  
    rpowers = []; k .#I ;7  
    for j = 1:length(n) Dk^T_7{  
        rpowers = [rpowers m_abs(j):2:n(j)]; l+r3|b  
    end xbNL <3"a  
    rpowers = unique(rpowers); y5/LH~&Ov  
    J=?P`\h  
    s#p\ r  
    % Pre-compute the values of r raised to the required powers, 5OM*NT t  
    % and compile them in a matrix: WbwS!F<au  
    % ----------------------------- TN=!;SvQU  
    if rpowers(1)==0 <hBd #J  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bjr()NM1  
        rpowern = cat(2,rpowern{:}); #zed8I:w  
        rpowern = [ones(length_r,1) rpowern]; &~&oB;uR  
    else x:E:~h[.^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6 =H]p1p~O  
        rpowern = cat(2,rpowern{:}); ..fbRt  
    end hQ80R B  
    ,Zva^5  
    ?m\? #  
    % Compute the values of the polynomials: )qeed-{  
    % -------------------------------------- Yl`)%6'5|  
    y = zeros(length_r,length(n)); 0x2[*pJ|IW  
    for j = 1:length(n) @=6*]:p2.  
        s = 0:(n(j)-m_abs(j))/2; O gtrp)x9  
        pows = n(j):-2:m_abs(j); =`OnFdI  
        for k = length(s):-1:1 hkDew0k  
            p = (1-2*mod(s(k),2))* ... ?BnX<dbi&  
                       prod(2:(n(j)-s(k)))/              ... oC~+K@S  
                       prod(2:s(k))/                     ... m:)s UC0  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v 8B4%1NE  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); aXgngw q  
            idx = (pows(k)==rpowers); Zv5vYe9Ow  
            y(:,j) = y(:,j) + p*rpowern(:,idx);  uWkn}P  
        end {:TOm0eK  
         U.pGp]\Q)G  
        if isnorm q+U&lw|"w  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); :zQNnq:|  
        end X!|K 4Z!k  
    end f/vsf&^O  
    % END: Compute the Zernike Polynomials Y<;KKD5P'j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /wPW2<|"X.  
    6^sH3=#  
    ,FK.8c6g  
    % Compute the Zernike functions: Y(;u)uN_  
    % ------------------------------ 6$&%z Eh  
    idx_pos = m>0; Zq{TY)PI]  
    idx_neg = m<0; 4Cp)!Bq?/  
    FnCMr_  
     NArr2o2  
    z = y; u+m9DNPF  
    if any(idx_pos) jk{m8YP)E  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); P*/ig0_fM  
    end 9cQ;h37J>  
    if any(idx_neg) jGEmf<q&u  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M~ g{}_ 0Z  
    end jP\5bg-}  
    nk"nSXm3SR  
    ]92=PA>75  
    % EOF zernfun 9dFo_a*?  
     
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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)  `{xKU8j^  
    ,=dc-%J  
    DDE还是手动输入的呢? Y, {pG]B$w  
    1B~[L 5p9  
    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