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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, ,{PN6B  
    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, gw)4P tb!  
    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? <=NnrZOF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? klUV&O+=%  
    \6|y~5Hw{r  
    0`x>p6.)G  
    K,g6y#1"  
    }_nBegv  
    function z = zernfun(n,m,r,theta,nflag) "du(BZw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. z\m$>C|  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cb^IJA9}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the kH eD(Ea  
    %   unit circle.  N is a vector of positive integers (including 0), and ?{ )'O+s  
    %   M is a vector with the same number of elements as N.  Each element n+8YTjd  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) M2Nh3ijr  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, %unn{92)  
    %   and THETA is a vector of angles.  R and THETA must have the same KNeVSZT  
    %   length.  The output Z is a matrix with one column for every (N,M) 8xLQ" l+"  
    %   pair, and one row for every (R,THETA) pair. E{T3Xwg  
    % zIF1A*UH  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Xex7Lr&  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 6]1RxrAV  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~EBaVl ({  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +S~ u,=  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <.ZIhDiEl  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. SD^::bH  
    % k9 r49lb  
    %   The Zernike functions are an orthogonal basis on the unit circle. >V^8<^?G  
    %   They are used in disciplines such as astronomy, optics, and q]="ek&_  
    %   optometry to describe functions on a circular domain. E <yQB39  
    % a?y ucA  
    %   The following table lists the first 15 Zernike functions. w~+*Vd~U  
    % 5$U49j  
    %       n    m    Zernike function           Normalization (csk   
    %       -------------------------------------------------- 1|p\rHGd  
    %       0    0    1                                 1 ;-1KPDIp`  
    %       1    1    r * cos(theta)                    2 aG7Lm2{c"  
    %       1   -1    r * sin(theta)                    2 DNmC   
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) rPB Ju0D"  
    %       2    0    (2*r^2 - 1)                    sqrt(3) I;XM4a  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Kh3i.gm7g  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) s>DFAu!  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z"< S$sDh  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) YMw,C:a4  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) \l=A2i7TQ  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) iYLg[J"  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t 9(,JC0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) bmHj)^v 5]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j/Kul}Ml\*  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) gkK(7=r%  
    %       -------------------------------------------------- qg j;E=7  
    % Oyb9 ql^  
    %   Example 1: Idu'+O4  
    % w8+ phN(-M  
    %       % Display the Zernike function Z(n=5,m=1) r`ftflNh(  
    %       x = -1:0.01:1; 9+(b7L   
    %       [X,Y] = meshgrid(x,x); (Tq)!h35B  
    %       [theta,r] = cart2pol(X,Y); hzAuj0-A  
    %       idx = r<=1; # 9bw'm  
    %       z = nan(size(X)); JXuks`:Q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); */{y%  
    %       figure @[D5{v)S  
    %       pcolor(x,x,z), shading interp ."Pn[$'.  
    %       axis square, colorbar VnN(lJ  
    %       title('Zernike function Z_5^1(r,\theta)') E7$ aT^  
    % <YCjo[(~  
    %   Example 2: ~p+ `pwjY1  
    % fm#7}Y  
    %       % Display the first 10 Zernike functions fhk(<KZvJ  
    %       x = -1:0.01:1; E.C=VfBW  
    %       [X,Y] = meshgrid(x,x); <OiH%:G/1  
    %       [theta,r] = cart2pol(X,Y); )l*3^kwL{U  
    %       idx = r<=1; )[99SM   
    %       z = nan(size(X)); 5bZ0}^FYF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 7yG%E  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 3Q&@l49q  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #x;d+Q@  
    %       y = zernfun(n,m,r(idx),theta(idx)); C^?/9\  
    %       figure('Units','normalized') -Nr*na^H9#  
    %       for k = 1:10 7LaRFL.,kO  
    %           z(idx) = y(:,k); P{RGW.Ci@  
    %           subplot(4,7,Nplot(k)) pw))9~XU  
    %           pcolor(x,x,z), shading interp shLMj)7!  
    %           set(gca,'XTick',[],'YTick',[]) p>U= Jg  
    %           axis square {DVMs|5;^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V%*91t_  
    %       end C\[:{d  
    % asW W@E  
    %   See also ZERNPOL, ZERNFUN2. }w=|"a|,  
    U8aNL sw  
    iQ;lvOja  
    %   Paul Fricker 11/13/2006 RSe av  
    Gp_flGdGQ  
    x[&)\[t  
    9G1ZW=83  
    6R2F,b(_  
    % Check and prepare the inputs: A{3nz DLI  
    % ----------------------------- }L`Z<h*H  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) uC]c`Ue  
        error('zernfun:NMvectors','N and M must be vectors.') *>GRU8_}  
    end 'K23oQwDB  
    ,=pn}\ R  
    B0g?!.#23  
    if length(n)~=length(m) 5rtE/ {A  
        error('zernfun:NMlength','N and M must be the same length.') \^cXmyQ<%  
    end iYW<qgz  
    +Lr`-</VF  
    ( s+}l?  
    n = n(:); ),,0T/69+9  
    m = m(:); Dz$dJF1 8  
    if any(mod(n-m,2)) G[d]t$f=  
        error('zernfun:NMmultiplesof2', ... Cpn!}!Gnf  
              'All N and M must differ by multiples of 2 (including 0).') 1Fsa}UK  
    end 1yS: `  
    i:&$I=  
    g/!tp;e  
    if any(m>n) do.XMdit  
        error('zernfun:MlessthanN', ... Q{ g{  
              'Each M must be less than or equal to its corresponding N.') F$s:\ N  
    end 5o^\jTEl^  
    ##" Hui  
    ;b|=osyT\  
    if any( r>1 | r<0 ) 2/))Y\~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') r0<zy_d'  
    end xjYH[PgfX  
    M3fTU CR  
    =QwT)KRB%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) WQ{^+C9g'1  
        error('zernfun:RTHvector','R and THETA must be vectors.') msq2/sS~  
    end Lu71Qdu09  
    nx!+: P ,  
    LmKY$~5P  
    r = r(:); ACEVd! q  
    theta = theta(:); U]M5&R=?  
    length_r = length(r); wO]H+t  
    if length_r~=length(theta) HSACaTVK  
        error('zernfun:RTHlength', ... [t?:CgI)E  
              'The number of R- and THETA-values must be equal.') 'kJyE9*xU.  
    end ~'Korxa  
    OP``+z>  
    c&g*nDuDj  
    % Check normalization: C)cuy7<  
    % -------------------- rj29$d?Y9  
    if nargin==5 && ischar(nflag) 2\"T&  
        isnorm = strcmpi(nflag,'norm'); ] `;Fc8$  
        if ~isnorm YCG $GD  
            error('zernfun:normalization','Unrecognized normalization flag.') G1SOvdq  
        end 5hDm[*83  
    else `nd$6i^#W  
        isnorm = false; Nm#[A4  
    end .sZ"|j9m  
    m&- -$sr  
    q^}iXE~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5_rx$avm  
    % Compute the Zernike Polynomials !3ji]q;uF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h\|T(597.  
    2t3)$\ylQp  
    Dyj>dh-  
    % Determine the required powers of r: <,t6A?YoMP  
    % ----------------------------------- ,/eAns`ZU  
    m_abs = abs(m); F{eI[A  
    rpowers = []; %/r:iD  
    for j = 1:length(n) b}ODc]3  
        rpowers = [rpowers m_abs(j):2:n(j)]; &3 x [0DV  
    end :>3?|Z"Aj  
    rpowers = unique(rpowers); }n k [WW  
    > q8)~  
    }4q1"iMlO  
    % Pre-compute the values of r raised to the required powers, l(Uwci  
    % and compile them in a matrix: 3oPyh $*  
    % ----------------------------- nR,QqIFFw  
    if rpowers(1)==0 fy>~ GFk(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); N LSJ D  
        rpowern = cat(2,rpowern{:}); j^mpkv<P  
        rpowern = [ones(length_r,1) rpowern]; nx5I  
    else 5>fAO =u!Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #?DoP]1Y  
        rpowern = cat(2,rpowern{:}); B# o6UO\  
    end Lr*\LP6jx3  
    ugs9>`fF&  
    4mm>6w8NT  
    % Compute the values of the polynomials: iE^=Vf;  
    % -------------------------------------- $v1_M1  
    y = zeros(length_r,length(n)); ,HK-mAH   
    for j = 1:length(n) [`!%u3  
        s = 0:(n(j)-m_abs(j))/2; xC 4L`\  
        pows = n(j):-2:m_abs(j); |+Tq[5&R  
        for k = length(s):-1:1 V=H:`n3k  
            p = (1-2*mod(s(k),2))* ... 5wC,:c[H7  
                       prod(2:(n(j)-s(k)))/              ...  =tc!"{  
                       prod(2:s(k))/                     ... Fzy5k?R  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... yg82a7D  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); z6 .^a-sU5  
            idx = (pows(k)==rpowers); M AL;XcRR  
            y(:,j) = y(:,j) + p*rpowern(:,idx); HnKXO  
        end /1b7f'  
         yKC1h`2  
        if isnorm G BM8:IG \  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bv}e[yH  
        end vU9:` @beu  
    end "-Wb[*U;  
    % END: Compute the Zernike Polynomials C40o_1g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pz]! T'  
    6K^O.VoV^J  
    /z: mi  
    % Compute the Zernike functions: YRU95K [  
    % ------------------------------ aAgQ^LY  
    idx_pos = m>0; _P*QX  
    idx_neg = m<0; yV*4|EkvW  
    KY\=D 2m  
    N t\ZM  
    z = y; e s<  
    if any(idx_pos) b8glZb*$  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 9A *gW j  
    end @]Lu"h#u=  
    if any(idx_neg) xL"O~jTS  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 6!wk5#  
    end >+):eB L  
    ]AX3ov6z9;  
    5t-, 5  
    % EOF zernfun |mcc?*%t8  
     
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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)  U3Z=X TB  
    DA.k8M  
    DDE还是手动输入的呢? 28`s+sH  
    c!/ +0[  
    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