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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, vL~j6'  
    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, *" ,"u;&  
    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?  @po|07  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? &1ss @-  
    |n\(I$  
    SAGECK[Ix  
    &z%DX   
    Wj\< )cH]  
    function z = zernfun(n,m,r,theta,nflag) *;<>@*  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. xI^nA2g  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N L+TM3*a*  
    %   and angular frequency M, evaluated at positions (R,THETA) on the E]%&)3O[  
    %   unit circle.  N is a vector of positive integers (including 0), and k"J=CDP\  
    %   M is a vector with the same number of elements as N.  Each element 19;F+%no#  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) MI*@^{G  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, @4%x7%+[c  
    %   and THETA is a vector of angles.  R and THETA must have the same F+::UWKA  
    %   length.  The output Z is a matrix with one column for every (N,M) H"%SzU  
    %   pair, and one row for every (R,THETA) pair. If%**o  
    % :y(HOUB  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O - N> X  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Ol1P  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral vm`\0VGSW  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YOY{f:ew  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized _:.'\d(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cS#m\O  
    % MU5#ph  
    %   The Zernike functions are an orthogonal basis on the unit circle. G~`nLC^Y  
    %   They are used in disciplines such as astronomy, optics, and * 2s(TW  
    %   optometry to describe functions on a circular domain. ^%2S,3*0  
    % _,5)  
    %   The following table lists the first 15 Zernike functions. (X}Q'm$n\h  
    % Pqb])-M9p  
    %       n    m    Zernike function           Normalization 50^T \u  
    %       -------------------------------------------------- lO dw H"  
    %       0    0    1                                 1 /d]{ #,k  
    %       1    1    r * cos(theta)                    2 t/0h)mL}  
    %       1   -1    r * sin(theta)                    2 .T }q"  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) <%Afa#  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ~4[4"Pi>|  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) DJ<F8-sb2r  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) CHNIL^B  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) _4MT,kN  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =9'px3:'WR  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) M>"J5yqR  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) T^n0=|  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 34Z$a{ w  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) QX&1BKqWn  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xlU:&=|  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 0I \l_St@  
    %       -------------------------------------------------- /J` ZO$  
    % k4Ub+F  
    %   Example 1: lpHz*NZ0  
    % u[ 2B0a  
    %       % Display the Zernike function Z(n=5,m=1) k&8&D  
    %       x = -1:0.01:1; 3 tIno!|  
    %       [X,Y] = meshgrid(x,x); [d/uy>z,  
    %       [theta,r] = cart2pol(X,Y); C'Z6l^{>  
    %       idx = r<=1; ,zU7UL^I  
    %       z = nan(size(X)); @E@5/N6M  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |FrZ,(\  
    %       figure t+`>zux5(T  
    %       pcolor(x,x,z), shading interp 8ECBi(  
    %       axis square, colorbar !JC!GS"M5  
    %       title('Zernike function Z_5^1(r,\theta)') ]@bu%_s"  
    % g_F-PT>($  
    %   Example 2: "I`g(q#Uo  
    % #K _E/~  
    %       % Display the first 10 Zernike functions 8{ iFxTz  
    %       x = -1:0.01:1; I&Y9  
    %       [X,Y] = meshgrid(x,x); %V3xO%  
    %       [theta,r] = cart2pol(X,Y); 0?d}Oj  
    %       idx = r<=1; `L1lGlt  
    %       z = nan(size(X)); ( [m[<  
    %       n = [0  1  1  2  2  2  3  3  3  3]; M<"H1>q@  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !>Ru= $9  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; /6Vn WrN_  
    %       y = zernfun(n,m,r(idx),theta(idx)); ra*(.<&  
    %       figure('Units','normalized') +`H{  
    %       for k = 1:10 G'qGsKf\  
    %           z(idx) = y(:,k); 6}9`z8  
    %           subplot(4,7,Nplot(k)) tfb_K4h6,  
    %           pcolor(x,x,z), shading interp o(_~ st<  
    %           set(gca,'XTick',[],'YTick',[]) 7y2-8e L  
    %           axis square dN)!B!*aI  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .8K ~ h  
    %       end o#ajBOJ  
    % pJpTOq\h  
    %   See also ZERNPOL, ZERNFUN2. W n43TSs-  
    ?}g#Mc  
    `zZGL&9m`  
    %   Paul Fricker 11/13/2006 t<QSp6n""  
    #(KE9h%  
    :P1/kYg  
    g =)djXW  
    7w]NG`7  
    % Check and prepare the inputs: Oe^oigcM  
    % ----------------------------- | N/Wu9w$  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +zup+=0e  
        error('zernfun:NMvectors','N and M must be vectors.') n :P5m9T  
    end hy?e?^  
    0`L>t  
    Wk"\aoX"E  
    if length(n)~=length(m) hXE_OXZ  
        error('zernfun:NMlength','N and M must be the same length.') D+:}D*_&  
    end 4M4oI .  
    |f.R]+cH  
    XK 09x1r  
    n = n(:); y$n`+%_  
    m = m(:); xf?6_=  
    if any(mod(n-m,2)) ^s=p'&6  
        error('zernfun:NMmultiplesof2', ... (2vf <x  
              'All N and M must differ by multiples of 2 (including 0).') ,9:0T LLR  
    end (i*;V0  
    yj+HU5L4  
    z]J pvw`p  
    if any(m>n) EN!Q]O|  
        error('zernfun:MlessthanN', ... (VxWa#P  
              'Each M must be less than or equal to its corresponding N.') EFYyr f@  
    end MCcWRbE5#  
    kroO~(\  
    =p lG9  
    if any( r>1 | r<0 ) 9J!@,Zsh  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0f<$S$~h  
    end lN-[2vT<  
    ?H.7 WtTC  
    &74*CO9B9  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) uWSfr(loX  
        error('zernfun:RTHvector','R and THETA must be vectors.') F/qx2E$*wo  
    end u9 yXHf  
    34$qV{Y%y  
    X!w&ib-  
    r = r(:); z^q ~|7  
    theta = theta(:); 8+irul{H_  
    length_r = length(r); k^ZcgHHgb  
    if length_r~=length(theta) F9SkEf]99  
        error('zernfun:RTHlength', ... dgIEc]#pH  
              'The number of R- and THETA-values must be equal.') h 'F\9t  
    end @]EJbiGv  
    3]iBX`Ni  
    y_=},a  
    % Check normalization: _Zq2 <:  
    % -------------------- }gr6naz  
    if nargin==5 && ischar(nflag) YlY3C  
        isnorm = strcmpi(nflag,'norm'); *:*Kdt`'G  
        if ~isnorm 'z=QV{ni  
            error('zernfun:normalization','Unrecognized normalization flag.') N>'T"^S/  
        end 1gZW~6a}  
    else m'Thm{Y,?n  
        isnorm = false; ^nS'3g^"  
    end O'G,   
    $g?`yE(K  
    Yzr|Z7r q}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Zu$30&U  
    % Compute the Zernike Polynomials 'WA]DlO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q0}Sju+HX  
    wd/"! A4(  
    +])St3h  
    % Determine the required powers of r: }h6 N.vz  
    % ----------------------------------- ]y3'6!  
    m_abs = abs(m); ;LBq!  
    rpowers = []; Q+O3Wgjy  
    for j = 1:length(n) Dm"@59x  
        rpowers = [rpowers m_abs(j):2:n(j)]; m 8Q[+_:$H  
    end j>5D4}*]f  
    rpowers = unique(rpowers); fFHT`"bD:  
    tWNz:V  
    M] +FTz  
    % Pre-compute the values of r raised to the required powers, t/=xY'7  
    % and compile them in a matrix: %Q}T9%Mtj  
    % ----------------------------- O%(E 6 n  
    if rpowers(1)==0 Wa8?o~0"L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); azj:Hru&t#  
        rpowern = cat(2,rpowern{:}); 1w)#BYc=L  
        rpowern = [ones(length_r,1) rpowern]; p.gaw16}>  
    else QzwA*\G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z.Sq5\d  
        rpowern = cat(2,rpowern{:}); s4$Z.xwr  
    end bUW`MH7yJ  
    {~"&$DY2  
    2VNMz[W'  
    % Compute the values of the polynomials: ?0Qm  
    % -------------------------------------- RaS7IL:e  
    y = zeros(length_r,length(n)); Zz\e:/  
    for j = 1:length(n) =)B@`"  
        s = 0:(n(j)-m_abs(j))/2; `XwFH#_  
        pows = n(j):-2:m_abs(j); @bN`+DC!<  
        for k = length(s):-1:1 PTu~PVbp4  
            p = (1-2*mod(s(k),2))* ... +$eEZ;4  
                       prod(2:(n(j)-s(k)))/              ... #'qEm=%  
                       prod(2:s(k))/                     ... hJ+;N  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... SWX;sM  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !,#42TY*X  
            idx = (pows(k)==rpowers); '$]u?m  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ![wV}. }  
        end 6wzTX8  
         +%$'( t s  
        if isnorm ?#8s=t  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u0;FQr2  
        end nyZUf{:  
    end 26Y Y1T\B)  
    % END: Compute the Zernike Polynomials %|l^oC+E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% / M(A kNy  
    < c[+60p"  
    7/ "g} F}Q  
    % Compute the Zernike functions: ,cR=W|6cQm  
    % ------------------------------ MCOz-8@|Y  
    idx_pos = m>0; I/ pv0  
    idx_neg = m<0; 3[RbVT  
    %)7HBj(*J  
    ;:nO5VFOg  
    z = y; N798("  
    if any(idx_pos) `TM[7'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); N7=L^]  
    end T@[(FVA N  
    if any(idx_neg) niZ/yW{w  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4s?x 8oAy  
    end "y_A xOH  
    MtYi8"+<e.  
    QGtKu:c.81  
    % EOF zernfun C3Mr)  
     
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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)  6bCC6G  
    #LEK?]y  
    DDE还是手动输入的呢? d V#h~  
    %\-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