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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, uF9C -H@:  
    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, 0;AA/  
    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? '":lB]hS  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 4'a=pnE$  
    7|$:=4  
    w1OI4C)~  
    oPXkYW  
    uj R_"r|l  
    function z = zernfun(n,m,r,theta,nflag) i*Sqda $  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. LE9(fe) fe  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @,cowar*  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7!EBH(,z  
    %   unit circle.  N is a vector of positive integers (including 0), and #t: S.A@  
    %   M is a vector with the same number of elements as N.  Each element &:dH,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 3L_\`Ia9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, }%'?p<^M  
    %   and THETA is a vector of angles.  R and THETA must have the same x7jC)M<k0  
    %   length.  The output Z is a matrix with one column for every (N,M) iS WU'K  
    %   pair, and one row for every (R,THETA) pair. s'E2P[:  
    % h?BFvbAt  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike i+S) K  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]yx$(6_U  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Ay5i+)MD  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V )x$|!(  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized t\{'F7  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ktFhc3);!  
    % #Ssx!+q?  
    %   The Zernike functions are an orthogonal basis on the unit circle. T|7}EAR=b  
    %   They are used in disciplines such as astronomy, optics, and %_RQx2  
    %   optometry to describe functions on a circular domain. Lvq>v0|  
    % s;S?;(QI  
    %   The following table lists the first 15 Zernike functions. T arIPp  
    % 723bkJw V  
    %       n    m    Zernike function           Normalization T92UeG  
    %       -------------------------------------------------- #h8Sq~0  
    %       0    0    1                                 1 v9w'!C)b  
    %       1    1    r * cos(theta)                    2 s:#V(<J  
    %       1   -1    r * sin(theta)                    2 h_:C+)13`x  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) wsIW |@  
    %       2    0    (2*r^2 - 1)                    sqrt(3) aT)BR?OYSJ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4'`{H@]tb  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) vY  }A  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) bx{$Y_L+p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) p?7v$ev_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Y^8C)p9r  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) VY;{/.Sa  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =BSzsH7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) -@yh> 8v  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Pe3@d|-,MU  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) x(etb<!jd  
    %       -------------------------------------------------- wkA+j9.  
    % %Y<|;0v  
    %   Example 1: uxtWybv  
    % tyXuG<  
    %       % Display the Zernike function Z(n=5,m=1) ?Z Rs\+{vG  
    %       x = -1:0.01:1; \Xm,OE_v"  
    %       [X,Y] = meshgrid(x,x); .S(TxksCz  
    %       [theta,r] = cart2pol(X,Y); m?pstuUK(  
    %       idx = r<=1; ,SynnE68  
    %       z = nan(size(X)); 5][Ztx  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); -+ SF  
    %       figure Cjqklb/  
    %       pcolor(x,x,z), shading interp DoJ\ q+  
    %       axis square, colorbar F(k.,0Nc  
    %       title('Zernike function Z_5^1(r,\theta)') U3T#6Rptl  
    % z=rT%lz6  
    %   Example 2: Ir`eL  
    % ,&j hlZ i  
    %       % Display the first 10 Zernike functions ;1`fC@rI  
    %       x = -1:0.01:1; @R/07&lBR  
    %       [X,Y] = meshgrid(x,x); 8oUpQcim  
    %       [theta,r] = cart2pol(X,Y); 4]G?G]lS>  
    %       idx = r<=1;  tBq nf v  
    %       z = nan(size(X)); r=5{o 1"  
    %       n = [0  1  1  2  2  2  3  3  3  3]; z.$4!$q  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; o<<xY<  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; BSYzC9h`  
    %       y = zernfun(n,m,r(idx),theta(idx)); d F9!G;V  
    %       figure('Units','normalized') 4 Y ;Nm1 @  
    %       for k = 1:10 U=DmsnD,  
    %           z(idx) = y(:,k); C8[&S&<_<  
    %           subplot(4,7,Nplot(k)) 9o;^[Ql-  
    %           pcolor(x,x,z), shading interp 9xO#tu]  
    %           set(gca,'XTick',[],'YTick',[]) i@P)a'W_  
    %           axis square ]+|~cRQ9I  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |uqf:V`z:  
    %       end GNXHM*~  
    % @ zs'Y8  
    %   See also ZERNPOL, ZERNFUN2. /2UH=Q!x4E  
    [s"O mAy4  
    }4Tc  
    %   Paul Fricker 11/13/2006 xIxn"^'  
    FME3sa$  
    LQtj~c>X-|  
    v1BDP<qU2  
    ap&?r`Tu  
    % Check and prepare the inputs: 0'V5/W  
    % ----------------------------- RIb4!!',c  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zo+nq%=  
        error('zernfun:NMvectors','N and M must be vectors.') q}~3C1  
    end JRSSn]pw  
    gZLP\_CL  
    xl6,s>ob  
    if length(n)~=length(m) Xe<sJ. &Wf  
        error('zernfun:NMlength','N and M must be the same length.') lV1G<qP  
    end \@8+U;d  
    &j4xgh9  
    E=e*VEjy  
    n = n(:); [z9 `)VIe  
    m = m(:); c0%"&a1]]V  
    if any(mod(n-m,2)) 1QLbf*zeIW  
        error('zernfun:NMmultiplesof2', ... FN\E*@>X=  
              'All N and M must differ by multiples of 2 (including 0).') A6:es_  
    end BFL`!^  
    t?}zdI(4  
    ]z l [H7  
    if any(m>n) B$b +Ymu  
        error('zernfun:MlessthanN', ... AtdlZ  
              'Each M must be less than or equal to its corresponding N.') k p<OJy  
    end 7w'wjX-  
    x|mqL-Q f  
    Zb1<:[  
    if any( r>1 | r<0 ) o$w_Es]Ma  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') H*[ M\gN$  
    end R{ a"Y$  
    2Ou[u#H  
    _9=Yvc=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Ezr:1 GJ  
        error('zernfun:RTHvector','R and THETA must be vectors.') H-~6Z",1  
    end ^:#D0[  
    i%/Jp[e\W>  
    tS|(K=$  
    r = r(:); ,=Mt`aN  
    theta = theta(:); oW<5|FaN  
    length_r = length(r); xn5l0'2  
    if length_r~=length(theta) %kdE un  
        error('zernfun:RTHlength', ... "br,/Dk>MX  
              'The number of R- and THETA-values must be equal.') So0f)`A  
    end BsEF'h'Owh  
    }UWL-TkEjF  
    8>0e*jC  
    % Check normalization: DpUbzr41+k  
    % -------------------- z"0I>gl  
    if nargin==5 && ischar(nflag) 1UE6 4Kl:S  
        isnorm = strcmpi(nflag,'norm'); .ox8*OO<  
        if ~isnorm D'J 0wT#  
            error('zernfun:normalization','Unrecognized normalization flag.') <$X3Hye  
        end P %#<I}0C  
    else O+]Ifm[  
        isnorm = false; }[ 4r4 1[  
    end 8=gjY\Dp  
    ',:*f8Jk  
    /(iFcMT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]=>F.GE  
    % Compute the Zernike Polynomials 1IZ3=6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1`a5C.v  
    ~zJ?H<>  
    ;i<|9{;  
    % Determine the required powers of r: qYlhlHD  
    % ----------------------------------- go'-5in(  
    m_abs = abs(m); MM (xk  
    rpowers = []; %`&2+\`  
    for j = 1:length(n) =5kY6%E7c  
        rpowers = [rpowers m_abs(j):2:n(j)]; A1.7 O  
    end w-Da~[J  
    rpowers = unique(rpowers); Q0&H#xgt  
    YgUvOyaQXf  
    g7O qX \  
    % Pre-compute the values of r raised to the required powers, TrLu~4  
    % and compile them in a matrix: OH">b6>\  
    % ----------------------------- ][?G/*k  
    if rpowers(1)==0 oxz OA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \lZf<f  
        rpowern = cat(2,rpowern{:}); *a#rM"6P  
        rpowern = [ones(length_r,1) rpowern]; $`)/0{qY-  
    else rvXWcu-"  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m^GJuP LW  
        rpowern = cat(2,rpowern{:}); F.w 5S!5Q  
    end E0MGRI"me  
    a2 Y;xe  
    ] :BX!<  
    % Compute the values of the polynomials: /.Ww6a~  
    % -------------------------------------- w*oQ["SL  
    y = zeros(length_r,length(n)); UrYZ` J  
    for j = 1:length(n) :=wT vz  
        s = 0:(n(j)-m_abs(j))/2; b\-&sM(W"  
        pows = n(j):-2:m_abs(j); wnM9('\  
        for k = length(s):-1:1 DDPxmuNG  
            p = (1-2*mod(s(k),2))* ... rdJ d#S  
                       prod(2:(n(j)-s(k)))/              ... 5[* qi?w=  
                       prod(2:s(k))/                     ... ,PWgH$+  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l zYnw)Pv  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); IHJ=i-  
            idx = (pows(k)==rpowers); %we u 1f  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /4` 0?/V  
        end P DrZY.-  
         ^OstR`U3  
        if isnorm V- Cv,8   
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;uw Ryd  
        end ]y,==1To  
    end >i5acuth  
    % END: Compute the Zernike Polynomials &:, dJ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?sMP~RHQ  
    rz@=pR :  
    b+f'[;  
    % Compute the Zernike functions: lJE93rXU  
    % ------------------------------ LAd\Tvms  
    idx_pos = m>0; ++V=s\d7  
    idx_neg = m<0; U2ZD]q  
    3>R#zJf  
    '+$EhFwD  
    z = y; BI $   
    if any(idx_pos) TW[_Ko86  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $ep.-I>  
    end l&?}hq^'Dn  
    if any(idx_neg) jz5qQt]^  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); t=-SH^$SR  
    end ^sVX)%  
    ?84 s4BpV1  
    L (khAmm  
    % EOF zernfun s;64N'HH  
     
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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)  R3!@?mcr  
    bI y sl  
    DDE还是手动输入的呢? ;_(PVo  
    ad_`x  
    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