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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, kW&TJP+5*  
    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, ]?[fsdAQW  
    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? Jg| XH L)  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ~R92cH>L  
    RrQJ/ts7}  
    !Q0w\j h  
    6,{$J  
    BR yl4  
    function z = zernfun(n,m,r,theta,nflag) `+Q%oj#FF  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. N//K Ph  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N '1s0D]  
    %   and angular frequency M, evaluated at positions (R,THETA) on the  ZExlGC  
    %   unit circle.  N is a vector of positive integers (including 0), and B_m8{44zM  
    %   M is a vector with the same number of elements as N.  Each element OpYY{f  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) s,&Z=zt0R  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, pcWPH.  
    %   and THETA is a vector of angles.  R and THETA must have the same `RL"AH:+  
    %   length.  The output Z is a matrix with one column for every (N,M) WEi2=3dV  
    %   pair, and one row for every (R,THETA) pair. }Kbb4]t|"  
    % q5)O%l!  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G*P#]eO  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7%eK37@u  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral x+@rg];m  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,1o FPa{?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 5C5sgR C  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]-/VHh  
    % ckE-",G  
    %   The Zernike functions are an orthogonal basis on the unit circle. u5f9Jw}  
    %   They are used in disciplines such as astronomy, optics, and b B3powy9  
    %   optometry to describe functions on a circular domain. <B6H. P =  
    % Qjv}$`M  
    %   The following table lists the first 15 Zernike functions. l]l'4@1   
    % QE`bSI  
    %       n    m    Zernike function           Normalization .jWC$SVR  
    %       -------------------------------------------------- n]o<S+z  
    %       0    0    1                                 1 L>4"(  
    %       1    1    r * cos(theta)                    2 68WO~*  
    %       1   -1    r * sin(theta)                    2 lp%pbx43s  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) m`^q <sj  
    %       2    0    (2*r^2 - 1)                    sqrt(3) H%Q7D-  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) t=W}SH  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) D7Q$R:6|  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) -fW*vE:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Q:d]imw!O  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ?QdWrE_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) R|87%&6']  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =R$u[~Xl2X  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) )W _v:?A9  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Tqn@P  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Ig0VW)@  
    %       -------------------------------------------------- z/2//mM  
    % q ,]L$  
    %   Example 1: ra gXn  
    % mLLDE;7|}  
    %       % Display the Zernike function Z(n=5,m=1) j/c&xv 7=  
    %       x = -1:0.01:1; eF-."1  
    %       [X,Y] = meshgrid(x,x); O:{~urV  
    %       [theta,r] = cart2pol(X,Y); " C Qa.%  
    %       idx = r<=1; YHygo#4=8  
    %       z = nan(size(X)); 4*cEag   
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); a![{M<Y~  
    %       figure j[J-f@F \Y  
    %       pcolor(x,x,z), shading interp #r~# I}U  
    %       axis square, colorbar (m(JK^  
    %       title('Zernike function Z_5^1(r,\theta)') A&Usddcp  
    % kxIF#/8  
    %   Example 2: yEoF4bt  
    % >rmqBDKaQ  
    %       % Display the first 10 Zernike functions q01wbO3-"  
    %       x = -1:0.01:1; w4{<n /"  
    %       [X,Y] = meshgrid(x,x); x}I+Iggi  
    %       [theta,r] = cart2pol(X,Y); ~1AgD-:Jz  
    %       idx = r<=1; \aUC(K~o\;  
    %       z = nan(size(X)); z3m85F%dR  
    %       n = [0  1  1  2  2  2  3  3  3  3]; $AjHbU.I{  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :g=qz~2Xk  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 6@F9G 4<Z  
    %       y = zernfun(n,m,r(idx),theta(idx)); cO+qs[ BQ  
    %       figure('Units','normalized') Y0dEH^I  
    %       for k = 1:10 ' ;FnIZ  
    %           z(idx) = y(:,k); W ]?G}Q;  
    %           subplot(4,7,Nplot(k)) Eib5  
    %           pcolor(x,x,z), shading interp a;qryUyG  
    %           set(gca,'XTick',[],'YTick',[]) ~#[yJNYQ  
    %           axis square i0kak`x0  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Q}K"24`=  
    %       end m{cGK`/\  
    % CMG&7(MR  
    %   See also ZERNPOL, ZERNFUN2. G+"t/?/  
    5o'FS{6U  
    :tB1D@Cb6  
    %   Paul Fricker 11/13/2006 w3obIJm  
    qJa H ,  
    kY|utoAP  
    mt+Oi70  
    m_?~OL S  
    % Check and prepare the inputs: %G/ hD  
    % ----------------------------- e L^ |v  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) oAJM]%g{  
        error('zernfun:NMvectors','N and M must be vectors.') s_OF(o  
    end BB!THj69a6  
    ,,&* :<Q  
    .B]MpmpK  
    if length(n)~=length(m) vkx7paY_  
        error('zernfun:NMlength','N and M must be the same length.') $=8  NED5  
    end *K6g\f]b#  
    ]7F=u!/`<C  
    HJLG=mU  
    n = n(:); ll<Xz((o  
    m = m(:); $%CF8\0  
    if any(mod(n-m,2)) $m%f wB  
        error('zernfun:NMmultiplesof2', ... r6MMCJ|G  
              'All N and M must differ by multiples of 2 (including 0).') P}y +G|  
    end 9~5uaP$S  
    RXpw!  
    Pg0x/X{t  
    if any(m>n) 9N%We|L,c  
        error('zernfun:MlessthanN', ... D9 CaFu  
              'Each M must be less than or equal to its corresponding N.') Vod\a 5c  
    end Pw7]r<Q  
    nQX:T;WL@  
    * 8yAG]z  
    if any( r>1 | r<0 ) F3v !AvA|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') [#<-ZC#T*  
    end 8>2.UrC  
    b8`)y<7  
    M=.n7RY-  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7a =gH2]&  
        error('zernfun:RTHvector','R and THETA must be vectors.') VgG0VM  
    end * J7DY f  
    @;zl  
    /yDz/>ID\  
    r = r(:); !N\@'F!  
    theta = theta(:); zUkgG61  
    length_r = length(r); ^pAAzr"hv  
    if length_r~=length(theta) nX6u(U  
        error('zernfun:RTHlength', ... xjuN-  
              'The number of R- and THETA-values must be equal.') 8`q:Gz=M\  
    end t9kzw*U9  
    c@!_ /0  
    25nt14Y 0u  
    % Check normalization: G6q }o)[m)  
    % -------------------- Zw 26  
    if nargin==5 && ischar(nflag) P_dCR  
        isnorm = strcmpi(nflag,'norm'); VuhGx:Xl  
        if ~isnorm knu,"<  
            error('zernfun:normalization','Unrecognized normalization flag.') 6"L cJ%o  
        end =1FRFZI!j  
    else I+%[d^,  
        isnorm = false; {NmWQyEv  
    end U8s2|G;K  
     acajHs  
    Ex Y]Sdx  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GfxZ'VIn  
    % Compute the Zernike Polynomials 9|^2",V  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~WeM TXF>y  
    Z, zWuE3  
    Q 04al=  
    % Determine the required powers of r: #px+;k 5  
    % ----------------------------------- /wQy17g  
    m_abs = abs(m); -/wtI   
    rpowers = []; [N-Di"  
    for j = 1:length(n) @ wGPqg  
        rpowers = [rpowers m_abs(j):2:n(j)]; LiC*@W  
    end |IeTqEu9  
    rpowers = unique(rpowers); Avge eJi  
    0#7>o^2  
    BA@lk+aW  
    % Pre-compute the values of r raised to the required powers, .wEd"A&j  
    % and compile them in a matrix: "(3[+W{|  
    % ----------------------------- gDQ^)1k  
    if rpowers(1)==0 B?eCe}*f;B  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); j2t7'bO_  
        rpowern = cat(2,rpowern{:}); JK7G/]j+Ez  
        rpowern = [ones(length_r,1) rpowern]; 9@SC}AF.  
    else !{+,B5 Hc  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ?(@ 7r_j  
        rpowern = cat(2,rpowern{:}); G*?8MTP8![  
    end s$zLiQF;  
    lF<]8m%F  
    v^sv<4*%  
    % Compute the values of the polynomials: !4ocZmj\  
    % -------------------------------------- aj-Km`5r}  
    y = zeros(length_r,length(n)); Hc;[Cs0  
    for j = 1:length(n) +r�  
        s = 0:(n(j)-m_abs(j))/2; prUN)r@U   
        pows = n(j):-2:m_abs(j); Fx]WCQo  
        for k = length(s):-1:1 k90YV(  
            p = (1-2*mod(s(k),2))* ... I {SjlN}d  
                       prod(2:(n(j)-s(k)))/              ... <.%4 ! }f8  
                       prod(2:s(k))/                     ... 3p$?,0ELH  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Oz75V|D  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); =1@u  
            idx = (pows(k)==rpowers); ,5P0S0*{  
            y(:,j) = y(:,j) + p*rpowern(:,idx); s-NX o  
        end k`cfG\;r  
         ^7`BP%6  
        if isnorm v1#otrf  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); CmWeY$Jb  
        end ]]HNd7Vh  
    end "-E\[@/  
    % END: Compute the Zernike Polynomials =?5]()'*n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K,tQ!kk  
    <q)#  
    p . %]Q*8  
    % Compute the Zernike functions: 3RUy, s  
    % ------------------------------ b3P+H r  
    idx_pos = m>0; Q*GN`07@?d  
    idx_neg = m<0; 2/U.| *mH  
    ; t)3F  
    3h]g}&k  
    z = y; k<z )WNBf  
    if any(idx_pos) t"sBPLU\  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); wC"FDr+  
    end M^A48u{,"  
    if any(idx_neg)  X hR4ru`  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); TbMW|0 #w  
    end 9FF0%*tGo  
    {aZ0;  
    xKbXt;l2  
    % EOF zernfun v<k?Vu  
     
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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)  ;V_e>TyG  
    K g*Q  
    DDE还是手动输入的呢? f)rq%N &  
    b6M[q_   
    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