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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, %p0b{P j_p  
    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, 0iYP  
    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? Ovv ny$  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? j]pohxn$5  
    /61by$E  
    i"L }!5  
    LeY+p]n~  
    RcgRaQ2^  
    function z = zernfun(n,m,r,theta,nflag) XwcMt r*  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. |*:tyP%m^  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @l'G[jN5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the "H>.':c"+3  
    %   unit circle.  N is a vector of positive integers (including 0), and {3hqp*xl  
    %   M is a vector with the same number of elements as N.  Each element N~;*bvW{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) eGLO!DdxZ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, vQUZVq5M  
    %   and THETA is a vector of angles.  R and THETA must have the same <eY %sFq,  
    %   length.  The output Z is a matrix with one column for every (N,M)  ]nUR;8  
    %   pair, and one row for every (R,THETA) pair. ##H;Yb  
    % wW-Ab  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]/Vh{d|I&  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [|4}~UV  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *zq.C  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tM)Iir*U#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~n WsP}`n  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ! .AhzU1%Y  
    % GuT6K}~|D  
    %   The Zernike functions are an orthogonal basis on the unit circle. LfEvc2 v=g  
    %   They are used in disciplines such as astronomy, optics, and z!$gVWG  
    %   optometry to describe functions on a circular domain. 3:l DL2  
    % AH^e]<2-  
    %   The following table lists the first 15 Zernike functions. ~\$=w10  
    % /}Yqf`CZy  
    %       n    m    Zernike function           Normalization F;u7A]H^  
    %       --------------------------------------------------  1Ao6y.S  
    %       0    0    1                                 1 ,9mgYp2  
    %       1    1    r * cos(theta)                    2 `mzb(b E  
    %       1   -1    r * sin(theta)                    2 ~Rs#|JWB2V  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ;hwzYXWF  
    %       2    0    (2*r^2 - 1)                    sqrt(3) bni) Qw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) <FUon  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) F.<L> G7{1  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) f ba&`  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) &|b4\uj9  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) I5qM.@%zB  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) .s2$al  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ca(U!T68  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1AF%-<`?s  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;1 |x  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) x? N.WABr;  
    %       -------------------------------------------------- Lz@$3(2  
    % HY;9?KJ'  
    %   Example 1: 9_ZBV{   
    % U&P{?>{u  
    %       % Display the Zernike function Z(n=5,m=1) 8Atq,GcG  
    %       x = -1:0.01:1; x vmt.>f  
    %       [X,Y] = meshgrid(x,x); Q(Gyq:L=>  
    %       [theta,r] = cart2pol(X,Y); qbiK^g R  
    %       idx = r<=1; WULAty  
    %       z = nan(size(X)); ZjD)? 4  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ;F5"}x  
    %       figure s\gp5MT  
    %       pcolor(x,x,z), shading interp R4{-Qv#8 q  
    %       axis square, colorbar @vRwzc\   
    %       title('Zernike function Z_5^1(r,\theta)') 7?J3ci\  
    % >;4!O%F  
    %   Example 2: <sX VW  
    % j13DJ.xu  
    %       % Display the first 10 Zernike functions 0C$8g Y*  
    %       x = -1:0.01:1;  l{$[}<  
    %       [X,Y] = meshgrid(x,x); $.rzc]s  
    %       [theta,r] = cart2pol(X,Y); #DFp[\)1  
    %       idx = r<=1; ~$<UE}qp  
    %       z = nan(size(X)); |sIr?RL{C  
    %       n = [0  1  1  2  2  2  3  3  3  3]; M:|8]y@  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; $6h*l T<  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; `G!M>h@  
    %       y = zernfun(n,m,r(idx),theta(idx)); A]o4Mf0>I  
    %       figure('Units','normalized') (ChD]PWQ  
    %       for k = 1:10 SV.z>p  
    %           z(idx) = y(:,k); WOX}Sw"  
    %           subplot(4,7,Nplot(k)) m#e*c [*G  
    %           pcolor(x,x,z), shading interp < Ek/8x  
    %           set(gca,'XTick',[],'YTick',[]) :Q_3hK  
    %           axis square %}3qR~;  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 6w| J -{2  
    %       end lnh+a7a)  
    % NHm]`R,  
    %   See also ZERNPOL, ZERNFUN2. };,/0Fu  
    l_{8+\`!  
    YoKs:e2/:  
    %   Paul Fricker 11/13/2006 }Fa%%}  
    ,Na^%A@TJ  
    8wK ~ i  
    S6xgiem  
    ?o*I9[Z)  
    % Check and prepare the inputs: DM{ 4@*]  
    % ----------------------------- e6E?t[hEeS  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;_O)p,p  
        error('zernfun:NMvectors','N and M must be vectors.') s?rBE.g@}  
    end 0w=R_C)s  
    b2C`g]ibQ  
    By)u-)g9  
    if length(n)~=length(m) % 0T+t.  
        error('zernfun:NMlength','N and M must be the same length.') F$V/K&&W  
    end ^oM|<";!?D  
    !w39FfU{  
    YA:nOvd@O  
    n = n(:); ~" i0x  
    m = m(:); k*mt4~KLT8  
    if any(mod(n-m,2)) !RW `3  
        error('zernfun:NMmultiplesof2', ... pkgjTXR2b  
              'All N and M must differ by multiples of 2 (including 0).') ?jx1R^  
    end QDx$==Fo  
    ZcJ\ZbE|  
    ntVS:F  
    if any(m>n) P{Lf5V9# <  
        error('zernfun:MlessthanN', ... Ztr Cv?  
              'Each M must be less than or equal to its corresponding N.') vy9 w$ls  
    end 9$ qm>,o  
    Az;t"  
    f+_h !j  
    if any( r>1 | r<0 ) #OO>rm$  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') g%[c<l9  
    end ` Ag{)  
    7!WA)@6  
    42J {aJVH  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [* ?Awf`   
        error('zernfun:RTHvector','R and THETA must be vectors.') Uu 8,@W+  
    end `-h8vj5uG  
    0'YG6(h  
    c2t=_aAIPQ  
    r = r(:); pi<TFe@eG  
    theta = theta(:); Gs+3e8  
    length_r = length(r); Zwz&rIQpT  
    if length_r~=length(theta) UcOk3{(z$q  
        error('zernfun:RTHlength', ... ^. dsW0"0  
              'The number of R- and THETA-values must be equal.') aI^/X {d  
    end {8)zg<rL+M  
    }XO K,Hw  
    Ez|oN,  
    % Check normalization: Ms~{9?  
    % -------------------- 2EZb )&Q  
    if nargin==5 && ischar(nflag) - K9c@?  
        isnorm = strcmpi(nflag,'norm'); Oy U  
        if ~isnorm F *FwRj  
            error('zernfun:normalization','Unrecognized normalization flag.') n8*;lK8  
        end u/cg|]x&T  
    else .ZvM^GJb  
        isnorm = false; $hivlI-7Ko  
    end QUU;g2k  
    35E_W>n  
    M`rl!Ci#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %?e& WLS  
    % Compute the Zernike Polynomials \b%kf99  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F a'k0/_j  
    8"sb;  
    0"CG7Vg,zh  
    % Determine the required powers of r: +qh[N@F  
    % ----------------------------------- Ecd;<$tk  
    m_abs = abs(m); Hemq +]6^  
    rpowers = []; Jn{OWw2  
    for j = 1:length(n) ='`/BY(m[  
        rpowers = [rpowers m_abs(j):2:n(j)]; {&jb5-*f  
    end  IiY/(N+J  
    rpowers = unique(rpowers); #Q2Y&2`yGT  
    T:5fc2Ngv  
    (M*FIX  
    % Pre-compute the values of r raised to the required powers, cWoPB _  
    % and compile them in a matrix: UK<Nj<-'t  
    % ----------------------------- ZosP(Tdq  
    if rpowers(1)==0 G6T_O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c-B cA  
        rpowern = cat(2,rpowern{:}); b )B? F  
        rpowern = [ones(length_r,1) rpowern]; eeyHy"@  
    else G1vNt7  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {phNds%  
        rpowern = cat(2,rpowern{:}); Ney/[3 A  
    end :A/d to  
    Y;?{|  
    S:h{2{  
    % Compute the values of the polynomials: ILGMMA_2  
    % -------------------------------------- ogyTO|V=  
    y = zeros(length_r,length(n)); ;M)QwF1  
    for j = 1:length(n) 9I}-[|`u  
        s = 0:(n(j)-m_abs(j))/2; M7pOLP_1jB  
        pows = n(j):-2:m_abs(j); ;lHr =e7  
        for k = length(s):-1:1  #"@|f  
            p = (1-2*mod(s(k),2))* ... ~_/(t'9  
                       prod(2:(n(j)-s(k)))/              ... `{dm;j5/y  
                       prod(2:s(k))/                     ... 03q 5e  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... R%?9z 8-  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Xu%'Z".>:  
            idx = (pows(k)==rpowers); wOU_*uY@6'  
            y(:,j) = y(:,j) + p*rpowern(:,idx); C{U?0!^  
        end }H^+A77v  
         E=nIRG|g  
        if isnorm %5(I/zB  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '@_d(N1jTw  
        end 4 o Fel.o  
    end ynthDE o  
    % END: Compute the Zernike Polynomials |?,A]|j  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sB7# ~p A  
    .+$ Q<L  
    8WXQ Oo8  
    % Compute the Zernike functions: :tV*7S=)  
    % ------------------------------ a<^v(r  
    idx_pos = m>0; t'n pG}`tE  
    idx_neg = m<0; JRB9rSN^  
    KVclhT<F  
    hgPa6Kd  
    z = y; k>;`FFQU>  
    if any(idx_pos) ].-1v5  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); d3\qKL!~  
    end c2l@6<Ww  
    if any(idx_neg) F3On?x)  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); l9{hq/V  
    end CsGx@\jN  
    Hj^1or3R]  
    H#,W5EJzM  
    % EOF zernfun >qnko9V  
     
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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)  ,V7nzhA2  
    B`EJb71^Xy  
    DDE还是手动输入的呢? ?al'F  q  
    ko!)s  
    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