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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, _|bIl%W;\'  
    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, _{)e\n  
    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? y5 $h  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ,OsFv}v7  
    l Ot3^`  
    3*"$E_%  
    Gy hoo'<  
    w?d~c*4+  
    function z = zernfun(n,m,r,theta,nflag) >t&Frw/Bl  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. _(&^M[O  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .i>; ?(GH  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 1@6dHFA`o  
    %   unit circle.  N is a vector of positive integers (including 0), and '3O@Nxof4  
    %   M is a vector with the same number of elements as N.  Each element 3,+)3,N  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) qvy~b  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !Low%rP  
    %   and THETA is a vector of angles.  R and THETA must have the same (|I:d!>:U  
    %   length.  The output Z is a matrix with one column for every (N,M) \/g.`Pe  
    %   pair, and one row for every (R,THETA) pair. &u( eu'Q3  
    % Q3vC^}Dmr  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike <[ />M  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), +!6aB|-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [x ?38  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0W<:3+|n4  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3`S|I_$(T"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. K9B_o,  
    % (= } cc  
    %   The Zernike functions are an orthogonal basis on the unit circle. I *YO  
    %   They are used in disciplines such as astronomy, optics, and _]a8lr+_-  
    %   optometry to describe functions on a circular domain. aN?{MA\  
    % [,Q(~Qb  
    %   The following table lists the first 15 Zernike functions. #;sUAR?]  
    % N=^{FZ  
    %       n    m    Zernike function           Normalization Z{s&myd  
    %       -------------------------------------------------- "K n JUXpl  
    %       0    0    1                                 1 ")'o5V  
    %       1    1    r * cos(theta)                    2 @d]I3?`  
    %       1   -1    r * sin(theta)                    2 j}7as&  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) .[%em9u  
    %       2    0    (2*r^2 - 1)                    sqrt(3) rVgz+'rFD[  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) x%ju(B>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 4bLk+EY4A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ~G|un}g=  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 99w;Q 2k  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) LW<DhMV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) S*-n%D0q5  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -Zx hh  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /` 891( f,  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kY*3)KCp  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ]#=43  
    %       -------------------------------------------------- M>W-lp^3  
    % VU3xP2c:  
    %   Example 1: px;5X4U  
    % hfT HP  
    %       % Display the Zernike function Z(n=5,m=1) 35I y\  
    %       x = -1:0.01:1; ivg:`$a[  
    %       [X,Y] = meshgrid(x,x); m99j]w r~c  
    %       [theta,r] = cart2pol(X,Y); 7hwl[knyB  
    %       idx = r<=1; 7OY<*ny  
    %       z = nan(size(X)); >Pne@w!*  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx));  SbQ Ri  
    %       figure  $0>>Z  
    %       pcolor(x,x,z), shading interp "S#4  
    %       axis square, colorbar os6p1"_\f  
    %       title('Zernike function Z_5^1(r,\theta)') R|aA6} /I  
    % :U)>um34e  
    %   Example 2: vFz%#zk>  
    % zK`fX  
    %       % Display the first 10 Zernike functions Gh}k9-L  
    %       x = -1:0.01:1; 0!X;C!v;  
    %       [X,Y] = meshgrid(x,x); 7KIOI,qb6  
    %       [theta,r] = cart2pol(X,Y); td!WgL,m  
    %       idx = r<=1; l9"4"+?j<  
    %       z = nan(size(X)); }% (e`[?1  
    %       n = [0  1  1  2  2  2  3  3  3  3]; dYEF,\Z'  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; .BN~9w  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; fDy Fkhc  
    %       y = zernfun(n,m,r(idx),theta(idx)); &2IrST{d:V  
    %       figure('Units','normalized') P'f0KZL;  
    %       for k = 1:10 b<,Z^Z_  
    %           z(idx) = y(:,k); tYV%izE  
    %           subplot(4,7,Nplot(k)) :Awnj!KNCc  
    %           pcolor(x,x,z), shading interp XQL"D)fw  
    %           set(gca,'XTick',[],'YTick',[]) f>.A^?  
    %           axis square '}\{4Qst  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k8fvg4  
    %       end )9'eckt  
    % 6&/H XqP  
    %   See also ZERNPOL, ZERNFUN2. n';"c;Ye)  
    Z#7T!/28  
    W+k`^A|@  
    %   Paul Fricker 11/13/2006 {!5"Y(>X  
    'Gwa[ |6i  
    )zw}+z3st  
    XrR@cDNx{  
    Eq$Q%'5*ua  
    % Check and prepare the inputs: ly`p)6#R=  
    % ----------------------------- tY$ .(2Ua  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) p&1IK8i"  
        error('zernfun:NMvectors','N and M must be vectors.') 'Okitq+O  
    end n{vp&  
    i! G^=N  
    vqh@)B+)  
    if length(n)~=length(m) Ju5<wjQR\  
        error('zernfun:NMlength','N and M must be the same length.') *o]Q<S>lH  
    end 9L3#aE]C  
    !(\OT  
    Abr:UEG  
    n = n(:); `P1jg$(eA  
    m = m(:); _r!''@B  
    if any(mod(n-m,2)) zrfE'C8O  
        error('zernfun:NMmultiplesof2', ... v4]7"7GuW  
              'All N and M must differ by multiples of 2 (including 0).') Ao%E]M  
    end :x e/7-  
    pT Yq#9  
    #6g-{OBv  
    if any(m>n) #PYTFB%  
        error('zernfun:MlessthanN', ... GRpS^%8i@  
              'Each M must be less than or equal to its corresponding N.') CD#:*  
    end J/(3: a>  
    :UjHP}s  
    c-nBB  
    if any( r>1 | r<0 ) pQ0yZpN%;  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3md yY\+&  
    end K{[ySB  
    >L$g ;(g  
    (Bs0 /C  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) uS :3Yo  
        error('zernfun:RTHvector','R and THETA must be vectors.') ahgm*Cpc  
    end }>:v  
    >T3HkOT  
    X$,#OR  
    r = r(:); /7Z0|Zw]  
    theta = theta(:); [~$Ji&Dd  
    length_r = length(r); M ,.++W\  
    if length_r~=length(theta) z,XM|-"#<K  
        error('zernfun:RTHlength', ... 9TGjcZ1S'  
              'The number of R- and THETA-values must be equal.') _%Q\G,a;  
    end -L7Q,"a$  
    fd >t9.  
    @D K,ka(  
    % Check normalization: w?kdM1T  
    % -------------------- <Q)6N!Tp^  
    if nargin==5 && ischar(nflag) =!3G,qV  
        isnorm = strcmpi(nflag,'norm'); e#`wshtN:  
        if ~isnorm oD_'8G}  
            error('zernfun:normalization','Unrecognized normalization flag.') "El$Sat`  
        end <~# ZtD$G  
    else #p~tkQ:'1  
        isnorm = false; W&`_cGoP  
    end l= 5kd.{  
    ?}^e,.M0?s  
    ]dk44,EL  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =g' 7 xA  
    % Compute the Zernike Polynomials V/i&8UMw  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s$xctIbm?,  
    $oK,&_  
    }8 A]  
    % Determine the required powers of r: o?uTL>Zin  
    % ----------------------------------- '3=[xVnv  
    m_abs = abs(m); (PU0\bGA  
    rpowers = []; z<_{m 4I;  
    for j = 1:length(n) qNER 6  
        rpowers = [rpowers m_abs(j):2:n(j)]; h &9Ld:p  
    end H@2JL.(k  
    rpowers = unique(rpowers); >L#&L ?#  
    <x DD*u  
    @TC_XU)&  
    % Pre-compute the values of r raised to the required powers, Sj{z  
    % and compile them in a matrix: %,%s09tO  
    % ----------------------------- g':mM*j&  
    if rpowers(1)==0 hX\XNiCiK8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3EB8ls2  
        rpowern = cat(2,rpowern{:}); k!O#6Z  
        rpowern = [ones(length_r,1) rpowern]; |0n h  
    else  ~d_Z?Z  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); uy{mSx?td  
        rpowern = cat(2,rpowern{:}); %*]3j^b Q+  
    end 2;.7c+r0  
    =3lUr<Ze  
    Hx2.2 A^  
    % Compute the values of the polynomials: oH!sJ&"#_  
    % -------------------------------------- A:[La#h|p  
    y = zeros(length_r,length(n)); x-) D@dw<  
    for j = 1:length(n) 3nf+ imAF  
        s = 0:(n(j)-m_abs(j))/2; G\tTwX4  
        pows = n(j):-2:m_abs(j); vV.'&."g  
        for k = length(s):-1:1 ftF?T.dx  
            p = (1-2*mod(s(k),2))* ... vjaIFyj  
                       prod(2:(n(j)-s(k)))/              ... i%>]$*  
                       prod(2:s(k))/                     ... orf21N+[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... & PrV+Lv  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); w(n&(5FzB<  
            idx = (pows(k)==rpowers); 5gZ0a4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 1t=Y+|vA9  
        end sLzcTGa2:z  
         wlBdA  
        if isnorm 2fTkHBhn&  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); P (jlWr$$  
        end hf0G-r_ow  
    end [ivz/r(Rj  
    % END: Compute the Zernike Polynomials ^CI.F.#X|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zMt"ST.  
    <M=U @  
    3G8BYP  
    % Compute the Zernike functions: v>.nL(VLjP  
    % ------------------------------ fG;)wQJ  
    idx_pos = m>0; d /&aC#'B  
    idx_neg = m<0; ui`xgR\6Rh  
    75wQH*  
    Q bfm*JP~  
    z = y; au}rS0) +  
    if any(idx_pos) Q[scmP^$^  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Tz+2g&+  
    end I>bLgt]u3  
    if any(idx_neg) tc\LK_@$/F  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %~J90a  
    end FyJI@PZdI-  
    uDK`;o'F  
    I:u xj%  
    % EOF zernfun $iDatQ[  
     
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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)  REW *6:  
    9TO  
    DDE还是手动输入的呢? LZ|G"5X[  
    .Ig`v  
    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