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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 正序阅读 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, Q&;qFv5-l  
    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, (]dZ+"O{  
    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? f>PU# D@B  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ,5WDYk-  
    r4zS,J;,  
    Kj5f:{Ur  
    ?uv%E*TU  
    }_TdXY #w\  
    function z = zernfun(n,m,r,theta,nflag) )QT+;P.  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3E9j%sYk  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ShxX[k  
    %   and angular frequency M, evaluated at positions (R,THETA) on the V&85<Y%Nl|  
    %   unit circle.  N is a vector of positive integers (including 0), and /y@iaptC  
    %   M is a vector with the same number of elements as N.  Each element 1j(,VW  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Wn5]2D\vkT  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^5F/=TtE G  
    %   and THETA is a vector of angles.  R and THETA must have the same 548BM^^"r  
    %   length.  The output Z is a matrix with one column for every (N,M) @e/dQ:Fb  
    %   pair, and one row for every (R,THETA) pair. $r_gFv  
    % HB:i0m2fJW  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *4E,| IJ  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f{oWd]eAhb  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral qa6up|xUnn  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, GC2<K  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized  X'<xw  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [5-5tipvWp  
    % }+1oD{  
    %   The Zernike functions are an orthogonal basis on the unit circle. &kBs'P8>  
    %   They are used in disciplines such as astronomy, optics, and Sq QB>;/p  
    %   optometry to describe functions on a circular domain. T~E83Jw  
    % ?;Qk!t2U  
    %   The following table lists the first 15 Zernike functions. %{"STbO#>  
    % 6h%(0=^  
    %       n    m    Zernike function           Normalization h'+ swPh  
    %       -------------------------------------------------- Y'9deX+  
    %       0    0    1                                 1 @So"(^  
    %       1    1    r * cos(theta)                    2 Tc :`TE=2  
    %       1   -1    r * sin(theta)                    2 w8Yff[o  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 1<UQJw45  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5**xU+&  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) JZ [&:  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) +l\Dp  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Heu@{t.[!D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !/SFEL@_B  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) HN+z7Q8hH  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) xC(PH?_  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4(]k=c1<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) (XQG"G%U6W  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3 a`-_<  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) YQOGxSi  
    %       -------------------------------------------------- VTU-'q  
    % Wu(GC]lTG  
    %   Example 1: vbp)/I-h  
    % AyDK-8a  
    %       % Display the Zernike function Z(n=5,m=1) #XZ?,neY  
    %       x = -1:0.01:1; U<x3=P  
    %       [X,Y] = meshgrid(x,x); Y9N:%[ :>W  
    %       [theta,r] = cart2pol(X,Y);  "d'@IN  
    %       idx = r<=1; pFh2@O  
    %       z = nan(size(X)); I5mS!m/X  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); =z+zg^wsT  
    %       figure X%sc:V  
    %       pcolor(x,x,z), shading interp ?(z3/ "g]  
    %       axis square, colorbar N*#SY$!y  
    %       title('Zernike function Z_5^1(r,\theta)') i \~4W$4I  
    % 827N?pU$)  
    %   Example 2: _F9 c.BH  
    % : SNp"|  
    %       % Display the first 10 Zernike functions z3,z&Ra  
    %       x = -1:0.01:1; JG `QJ%  
    %       [X,Y] = meshgrid(x,x); R= l/EK  
    %       [theta,r] = cart2pol(X,Y); @({65gJ*  
    %       idx = r<=1; chy7hPxC;  
    %       z = nan(size(X)); 3HD=)k  
    %       n = [0  1  1  2  2  2  3  3  3  3]; >}iYZ[ V  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ZHT.+X:_  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Kf*+Ilq%L  
    %       y = zernfun(n,m,r(idx),theta(idx)); No?pv"  
    %       figure('Units','normalized') pVr,WTr6E  
    %       for k = 1:10 <m!\Ma  
    %           z(idx) = y(:,k); /CP1mn6H  
    %           subplot(4,7,Nplot(k)) .3[YOM7h  
    %           pcolor(x,x,z), shading interp `k+k&t  
    %           set(gca,'XTick',[],'YTick',[]) u}$?r\H'(  
    %           axis square B*{CcQ<5  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &\A$Rj)  
    %       end ^U52 *6  
    % nxG vh4'i8  
    %   See also ZERNPOL, ZERNFUN2. <B)lV'!Bd  
    i<l)To-  
    D_@^XS  
    %   Paul Fricker 11/13/2006 a}yJ$6xi  
    Gc>\L3u  
    83@+X4ptp  
    9T\:ID= h  
    ']V 2V)t  
    % Check and prepare the inputs: T"$"`A"  
    % ----------------------------- `O#y%*E  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7w9) ^  
        error('zernfun:NMvectors','N and M must be vectors.') ^'}Td~(  
    end :)+cI?\#  
    ]5^u^  
    ZEB1()GB  
    if length(n)~=length(m) 7%X$6N-X  
        error('zernfun:NMlength','N and M must be the same length.') t{$t3>p-t  
    end T =:^k+  
    9 eP @}C6  
    18Ty )7r'  
    n = n(:); T^"d%au  
    m = m(:); kWWb<WRW:  
    if any(mod(n-m,2)) Ih.o;8PpK  
        error('zernfun:NMmultiplesof2', ... }hGbF"clqg  
              'All N and M must differ by multiples of 2 (including 0).') )%*uMuF  
    end -IPc;`<  
    KNV$9&Z  
    uvT]MgT  
    if any(m>n) 6,k}v:  
        error('zernfun:MlessthanN', ... >J4_/p>Qs  
              'Each M must be less than or equal to its corresponding N.') =!7yX ;|  
    end Zcc6E2  
    `74A'(u_  
    K2&pTA~OR  
    if any( r>1 | r<0 ) ,#<"VU2bC  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') yHCBf)N7\  
    end \i{=%[c  
    tvP"t{C6,  
    &0M^UvO  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @L`t/OD  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2+0'vIw}  
    end =\tg$  
    QQqWJq~  
    "}EydG"=  
    r = r(:); c" yf>0  
    theta = theta(:); &}rh+z  
    length_r = length(r); ^G15]Pyw  
    if length_r~=length(theta) P\SE_*&  
        error('zernfun:RTHlength', ... ,rQznE1e  
              'The number of R- and THETA-values must be equal.') /+%1Kq.hP  
    end fY\QI =  
    R7+k=DI  
    --y .q~d  
    % Check normalization: o <sX6a9e  
    % -------------------- UA}k"uM  
    if nargin==5 && ischar(nflag) $BCqz! 4K  
        isnorm = strcmpi(nflag,'norm'); Dg \fjuK9  
        if ~isnorm |Zz3X  
            error('zernfun:normalization','Unrecognized normalization flag.') QO0T<V  
        end }56"4/  Z  
    else H=EvT'g  
        isnorm = false; !DD|dVA{  
    end Ju+r@/y%  
    #AE'arT<  
    \#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r'-)@|  
    % Compute the Zernike Polynomials t[%9z6t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3. fIp5g  
    Z3=t"  
    +Nyx2(g<m  
    % Determine the required powers of r: e%#9|/uP  
    % ----------------------------------- _<&IpT{w+  
    m_abs = abs(m); (V}D PA  
    rpowers = []; |>Kf_b Y#  
    for j = 1:length(n) BHqJ~2&FDW  
        rpowers = [rpowers m_abs(j):2:n(j)]; gQh;4v  
    end 3%>"|Ye}A  
    rpowers = unique(rpowers); 76(&O  
    yin"+&<T  
    (yn!~El3  
    % Pre-compute the values of r raised to the required powers,  ]Ocf %(  
    % and compile them in a matrix: CZt)Q4  
    % ----------------------------- =]E;wWC  
    if rpowers(1)==0 mbU[fHyV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DO(FG-R  
        rpowern = cat(2,rpowern{:}); (WX,&`a<$  
        rpowern = [ones(length_r,1) rpowern]; USfOc  
    else E:L =>}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t :sKvJ  
        rpowern = cat(2,rpowern{:}); Xb5n;=)  
    end >?'cZTNk]  
    UeX3cD  
    /&Khk #  
    % Compute the values of the polynomials: R@u6mMX{N,  
    % -------------------------------------- x4Y+?2  
    y = zeros(length_r,length(n)); y;3vr1?  
    for j = 1:length(n) gs7H9%j{U  
        s = 0:(n(j)-m_abs(j))/2; 6uOR0L  
        pows = n(j):-2:m_abs(j); JO1KkIV  
        for k = length(s):-1:1 Rq<T2}K  
            p = (1-2*mod(s(k),2))* ... T[*=7jnJQ  
                       prod(2:(n(j)-s(k)))/              ... L00,{g6wqb  
                       prod(2:s(k))/                     ... JY~s-jxa  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... *4dA(N\k"  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); T1LtO O  
            idx = (pows(k)==rpowers); ;a[56W  
            y(:,j) = y(:,j) + p*rpowern(:,idx); (Rve<n6{A  
        end Gmf.lHr$%  
         &Dgho  
        if isnorm "n=`{~F  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Da0E)  
        end ]+{Cy\*kR  
    end H_3S#.  
    % END: Compute the Zernike Polynomials 1BmevE a)  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O sbY}*S  
    .yd{7Te  
    YO|Kc {j2e  
    % Compute the Zernike functions: Ot`jjZ&  
    % ------------------------------ VX2 KE@  
    idx_pos = m>0; u yzc"d i  
    idx_neg = m<0; 5M;fh)fT  
    ck){N?y  
    4t|ril``]  
    z = y; C7[_#1Oz  
    if any(idx_pos) K, WNM S  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); XTUxMdN  
    end *1$rg?yGf  
    if any(idx_neg)  S`)KC-  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); O$V 6QJ  
    end W7c(] tg.  
    ICN>8|O`&  
    7%c9 nY  
    % EOF zernfun By]XD~gcP  
     
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    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
    离线jssylttc
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    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    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)  X&nkc/erx  
    7r 07N'  
    DDE还是手动输入的呢? hV#+joT8i  
    QFU;\H/  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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。
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