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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, 9>zN 27  
    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, {4:En;  
    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?  )?4m}  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]`u{^f  
    BH*vsxe  
    k&^Megcb  
    -3KB:K<  
    q^12Rj;H  
    function z = zernfun(n,m,r,theta,nflag)  .# M 5L  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. R]ppA=1*_l  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N RRq*CLj  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D Zh6/n#q  
    %   unit circle.  N is a vector of positive integers (including 0), and XY%8yII6  
    %   M is a vector with the same number of elements as N.  Each element ((X"D/F]  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) cYGZZC8|K  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ifBJ$x(B.  
    %   and THETA is a vector of angles.  R and THETA must have the same s/A]&! `  
    %   length.  The output Z is a matrix with one column for every (N,M) |y=CmNG,  
    %   pair, and one row for every (R,THETA) pair. UayRT#}]  
    % ;1eu8N8  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rj{'X  /  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), N ~ LR  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral iJsw:Nc  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |,yS>kjp  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized i%\nJs*  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _q8s 7H  
    % /M'b137  
    %   The Zernike functions are an orthogonal basis on the unit circle. 0@xuxm/i  
    %   They are used in disciplines such as astronomy, optics, and t_j.@|/FZ  
    %   optometry to describe functions on a circular domain. 8#oF7eE  
    % gW*ee  
    %   The following table lists the first 15 Zernike functions. W<9G wMU  
    % %X.Q\T  
    %       n    m    Zernike function           Normalization +)7NWR\  
    %       -------------------------------------------------- s&fU|Jk8  
    %       0    0    1                                 1 qi/%&)GZ  
    %       1    1    r * cos(theta)                    2 zV2c `he%z  
    %       1   -1    r * sin(theta)                    2 4CN8>J'-  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ? X:RrZ:/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Q"Bgr&RJ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3K#e]zoI  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) [KjQW/sb'  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) uAJ_`o[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) tKJ) 'v?  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) |E?%Cj^W  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) f0hi70\(X  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :ss9-  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) m\~[^H~g  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "= %-  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =,?@p{g}  
    %       -------------------------------------------------- 50'6l X(v,  
    % tPp }/a%D  
    %   Example 1: mKn[>M1  
    % tL IE^  
    %       % Display the Zernike function Z(n=5,m=1) gjs-j{*  
    %       x = -1:0.01:1; >>!+Ri\@  
    %       [X,Y] = meshgrid(x,x); mybDK'EW  
    %       [theta,r] = cart2pol(X,Y); K}$PIW  
    %       idx = r<=1; {+`ep\.$&  
    %       z = nan(size(X)); w]%r]PwU+  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); g.9MPN  
    %       figure A"P1 B]  
    %       pcolor(x,x,z), shading interp [jLx}\]  
    %       axis square, colorbar |]B]0J#_  
    %       title('Zernike function Z_5^1(r,\theta)') ({i|  
    % d&U;rMEv  
    %   Example 2: "\V:W%23W{  
    % oiR` \uY  
    %       % Display the first 10 Zernike functions _wqFKj  
    %       x = -1:0.01:1; Y< M}'t  
    %       [X,Y] = meshgrid(x,x); V5A7w V3~  
    %       [theta,r] = cart2pol(X,Y); 9GQTe1[t4  
    %       idx = r<=1; ^^?ECnpcU  
    %       z = nan(size(X)); ;N,7#l|wi  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Dic(G[  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; L ~;_R*Th  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 2OZdj  
    %       y = zernfun(n,m,r(idx),theta(idx)); 2 na8G  
    %       figure('Units','normalized') nDPfr\\  
    %       for k = 1:10 fmSA.z  
    %           z(idx) = y(:,k); FEP\5d>  
    %           subplot(4,7,Nplot(k)) e~}+.B0  
    %           pcolor(x,x,z), shading interp CP?\'a"Kt  
    %           set(gca,'XTick',[],'YTick',[]) 0\i&v  
    %           axis square ^^%*2^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Vj:PNt[  
    %       end \[8I5w-  
    % Dbtw>:=  
    %   See also ZERNPOL, ZERNFUN2. !-7(.i-  
    @%jzVF7  
    qI'a|p4fn?  
    %   Paul Fricker 11/13/2006 > h:~*g  
    8>epKFEg  
    T*H4kM  
    f< '~K  
    iI _Fbw8  
    % Check and prepare the inputs: WFh!re%Z  
    % ----------------------------- D #A9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S5uV\Y/A  
        error('zernfun:NMvectors','N and M must be vectors.') c[;I\g  
    end +^"|FtKhE  
    h<QXr'4+  
    jUfc&bi3  
    if length(n)~=length(m) sB"]R%`_  
        error('zernfun:NMlength','N and M must be the same length.') ,v^it+Jc'  
    end 6Es-{u(,  
    I)sCWC:Mq~  
    Oi{jzP  
    n = n(:); )qxL@w.  
    m = m(:); gmM79^CEF  
    if any(mod(n-m,2)) @ojn< 7W  
        error('zernfun:NMmultiplesof2', ... w.V8-9{  
              'All N and M must differ by multiples of 2 (including 0).') ?^6RFbke+  
    end '7xY ,IY  
    \dCdyl6V  
    )>?K:y8I~  
    if any(m>n) LS \4y&J40  
        error('zernfun:MlessthanN', ... RLIugz{IH  
              'Each M must be less than or equal to its corresponding N.') Cx@,J\rsQ  
    end N8!B2uPQ  
    O c" 2|X  
    gfp#G,/B  
    if any( r>1 | r<0 ) <Siz5qQI4  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') b_6j77  
    end . Bv;Zv  
    &yP9vp="  
    |m?0h.O,  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) !e%#Zb MIo  
        error('zernfun:RTHvector','R and THETA must be vectors.') UZdpKi@  
    end i|2Q}$3t2  
    /FQumqbnt  
    "V^(i%E;  
    r = r(:); 6T)D6;@L  
    theta = theta(:); u9?85  
    length_r = length(r); 0lW}l9}'-  
    if length_r~=length(theta) T}g;kppC  
        error('zernfun:RTHlength', ... Ypp>7J/  
              'The number of R- and THETA-values must be equal.') j:fL_1m  
    end V-)q&cbW]q  
    0chBw~@*s  
    <Z}2A8mjY  
    % Check normalization: ]xFd_OHdb  
    % -------------------- 6r^(VT  
    if nargin==5 && ischar(nflag) :vm*miOF  
        isnorm = strcmpi(nflag,'norm'); xKIm2% U9  
        if ~isnorm G<>`O;i  
            error('zernfun:normalization','Unrecognized normalization flag.') 7Xw #  
        end \N!k)6\  
    else =0O`VSb  
        isnorm = false; Wb^YqqE  
    end 0OlB;  
    aP2  
    I]zCsT.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0Y[mh@(  
    % Compute the Zernike Polynomials ( vgoG5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3F<My+J  
    z}kD:A)a  
    <@JK;qm>S  
    % Determine the required powers of r: ]G&d`DNV  
    % ----------------------------------- [NyR$yD{  
    m_abs = abs(m); X}_kLfP/9  
    rpowers = []; qd(`~a  
    for j = 1:length(n) vJK0>":G  
        rpowers = [rpowers m_abs(j):2:n(j)]; Xul<,U~w6  
    end !m:SRNPg  
    rpowers = unique(rpowers); }Vk#w%EJ  
    <Ms,0YKx  
    mpN|U(n  
    % Pre-compute the values of r raised to the required powers, ]iYjS  
    % and compile them in a matrix: "Bn!<h}mg  
    % ----------------------------- P!1y@R>Ln  
    if rpowers(1)==0 2Fp.m}42i(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Nx,.4CI  
        rpowern = cat(2,rpowern{:}); "1WwSh}Z  
        rpowern = [ones(length_r,1) rpowern]; ..;}EFw5  
    else \M<C6m5  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e=Kf<ZQt  
        rpowern = cat(2,rpowern{:}); ?%#3p[  
    end xyBWV]Y  
    .kyp5CD}4  
    m.Yj{u8zX  
    % Compute the values of the polynomials: W(Xb]t=19  
    % -------------------------------------- "Lw[ $  
    y = zeros(length_r,length(n)); NRgNh5/  
    for j = 1:length(n) sO,,i]a0  
        s = 0:(n(j)-m_abs(j))/2; !S$LRm\ '  
        pows = n(j):-2:m_abs(j); Y{6y.F*Q#  
        for k = length(s):-1:1 `ZC_F! E  
            p = (1-2*mod(s(k),2))* ... +?DP r  
                       prod(2:(n(j)-s(k)))/              ... {p=`"H>  
                       prod(2:s(k))/                     ... Wz;7 |UC  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'QeCJ5p]  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); sWa`-gc  
            idx = (pows(k)==rpowers); { ZrIA+eH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); |eU{cK~e^  
        end f#FAi3  
         ER;?[!  
        if isnorm 6Q"fRXM   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?:H4Xd7  
        end  dtTQY  
    end F-D9nI4{X  
    % END: Compute the Zernike Polynomials A]c'`Nf  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wxS.!9K  
    PFq1Zai}n|  
    /m"O.17N  
    % Compute the Zernike functions: 6QO[!^lY  
    % ------------------------------ .!/w[Z]  
    idx_pos = m>0; ae_Y?g+3  
    idx_neg = m<0; bvzNur_  
    Kg4\:A7Sa.  
    d< j+a1&  
    z = y; -Ri/I4Xj  
    if any(idx_pos) g3B%}!|  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); __LR!F]=i  
    end AWo\u!j  
    if any(idx_neg) R2,Z`I  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); VC~1QPC9  
    end $S(<7[Z  
    ||yx?q6\h  
    ?V&# nA  
    % EOF zernfun ?DJ/Yw>>3  
     
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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)  l0PXU)>C  
    /b]+RXvxj  
    DDE还是手动输入的呢? gI/ SA  
    p4uN+D `.U  
    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