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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, *wZV*)}  
    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, GQAg ex)D  
    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? {_N(S]Z  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ZjbG&oc  
    P*=3$-`  
    zSufU2  
    r\- k/0  
     Jy[8,X  
    function z = zernfun(n,m,r,theta,nflag) RpXGgw  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. lSv;wwEg  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @9P9U`ZP  
    %   and angular frequency M, evaluated at positions (R,THETA) on the (dnc7KrM  
    %   unit circle.  N is a vector of positive integers (including 0), and 'Bn_'w~j{  
    %   M is a vector with the same number of elements as N.  Each element ED_5V@  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) /faP]J)  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, MBrVh6z>  
    %   and THETA is a vector of angles.  R and THETA must have the same \B +SzW  
    %   length.  The output Z is a matrix with one column for every (N,M) ?PtRb:RHt  
    %   pair, and one row for every (R,THETA) pair. exU=!3Ji  
    % (w  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike tl#s:  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), [4yQbqe;  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Yzx0[_'u  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hf5SpwxLiH  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized PS;*N 8  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k"-#ox!  
    % } ZGpd9D  
    %   The Zernike functions are an orthogonal basis on the unit circle. A{T@O5ucj  
    %   They are used in disciplines such as astronomy, optics, and &!fcLJd  
    %   optometry to describe functions on a circular domain. Gl:T  
    % ;XuE Mq,Di  
    %   The following table lists the first 15 Zernike functions. ITPp T  
    % <T[ui  
    %       n    m    Zernike function           Normalization |W];v@b\y  
    %       -------------------------------------------------- ``CADiM:S  
    %       0    0    1                                 1 >5W"a?(  
    %       1    1    r * cos(theta)                    2 N2Hb19/k  
    %       1   -1    r * sin(theta)                    2 RIx6& 7$  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 2{: J1'pC  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ?2>v5p  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) hvZR4|k>  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) OEi9 )I  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) zhL,BTH  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) =x]dP.  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ="E V@H?U  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) YIqfGXu8  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m(]IxI  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) > PA,72e   
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?LM'5  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) L#b Q`t  
    %       -------------------------------------------------- e:occT  
    % "b7C0NE  
    %   Example 1: bUL9*{>G  
    % )C6 7qY  
    %       % Display the Zernike function Z(n=5,m=1) _3>zi.J/  
    %       x = -1:0.01:1; ^Z+D7Q  
    %       [X,Y] = meshgrid(x,x); :N:8O^D^<  
    %       [theta,r] = cart2pol(X,Y); 3&:fS|L~c  
    %       idx = r<=1; EOC"a}Cq-  
    %       z = nan(size(X)); 6[7k}9`alz  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); >*CK@"o  
    %       figure #C}(7{Vt  
    %       pcolor(x,x,z), shading interp =1Jo-!{{  
    %       axis square, colorbar l]&)an  
    %       title('Zernike function Z_5^1(r,\theta)') 4+bsG6i  
    % L<`g}iw  
    %   Example 2: Dw,f~D$+ic  
    % O,#[m:Ejb  
    %       % Display the first 10 Zernike functions 4/_|Qy  
    %       x = -1:0.01:1; v21?  
    %       [X,Y] = meshgrid(x,x); _gh7_P^H=d  
    %       [theta,r] = cart2pol(X,Y); PCjY,O  
    %       idx = r<=1; @kymL8"2w  
    %       z = nan(size(X)); j]SkBZgik  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 7C^ nk z  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; px@\b]/  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; B[50{;X  
    %       y = zernfun(n,m,r(idx),theta(idx)); PD4E& k  
    %       figure('Units','normalized') 49GCj`As  
    %       for k = 1:10 :LG%8Z{R  
    %           z(idx) = y(:,k); W -&5 v  
    %           subplot(4,7,Nplot(k)) 4pv :u:Z  
    %           pcolor(x,x,z), shading interp pXa? Q@ 6  
    %           set(gca,'XTick',[],'YTick',[]) p60D{UzU  
    %           axis square 7 i/Cax  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) l[k$O$jo  
    %       end RGmpkQEp  
    % O!tD1^O!1}  
    %   See also ZERNPOL, ZERNFUN2. :DJ@HY  
    3R {y68-S  
    C"<@EMU9  
    %   Paul Fricker 11/13/2006 wt;aO_l  
    Ea?.H Rxl  
    EM}z-@A>  
    RUKSGj_NJ  
    >Z% `&D~u  
    % Check and prepare the inputs: = @o}  
    % ----------------------------- Q2Rj0E`  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) b??1Up  
        error('zernfun:NMvectors','N and M must be vectors.') I "4B1g  
    end d .A0(*k,  
    -f=hL7NW  
    Lw`\J|%p  
    if length(n)~=length(m) 5>Q)8` @E  
        error('zernfun:NMlength','N and M must be the same length.') X$f%Ss  
    end iXFaQ  
    E12k1gC`  
    $'q(Z@  
    n = n(:); "Cb<~Dy  
    m = m(:); >.|gmo>b  
    if any(mod(n-m,2)) *b EsWeP  
        error('zernfun:NMmultiplesof2', ... :F&WlU$L  
              'All N and M must differ by multiples of 2 (including 0).') "f_Z.6WMY  
    end o*_D  
    43XuQg4  
    CggEAi~  
    if any(m>n) #eYVZ=E  
        error('zernfun:MlessthanN', ... }^muAr  
              'Each M must be less than or equal to its corresponding N.') Sls> OIc  
    end Pp2 )P7  
    Npqbxb  
    VM[8w`  
    if any( r>1 | r<0 ) *rLs!/[Z_  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') pC6_ jIZ  
    end /7^~*  
    s><co]  
    &^.'g{\Y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) zlfm})+G  
        error('zernfun:RTHvector','R and THETA must be vectors.') 3>+;G4  
    end c'b,=SM  
    \!r^6'A   
    }wV rmDh \  
    r = r(:); -MjRFa  
    theta = theta(:); ArY'NE\Htt  
    length_r = length(r); %[J( ,rm  
    if length_r~=length(theta) y.zQ `  
        error('zernfun:RTHlength', ... Ty=}A MMyE  
              'The number of R- and THETA-values must be equal.') S4w/ kml3  
    end =R05H2hs  
    amRtFrc|  
    &+v&Dd&  
    % Check normalization: x+pFu5,  
    % -------------------- {Fj`'0Xu;  
    if nargin==5 && ischar(nflag) k{~5pxd-t  
        isnorm = strcmpi(nflag,'norm'); O%r<I*T^r  
        if ~isnorm cnR>)9sX  
            error('zernfun:normalization','Unrecognized normalization flag.') -Q; w4@  
        end >qE$:V "_5  
    else }49?Z3  
        isnorm = false; pfT7  
    end E O5Vg  
    )l=j,4nn  
    DcOLK\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b}fH$.V@  
    % Compute the Zernike Polynomials X\;y;pmRH  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xInWcQ  
    Oe$C5KA>LW  
    R`c5-0A  
    % Determine the required powers of r: }^H_|;e1p  
    % ----------------------------------- M-NR!?9  
    m_abs = abs(m); f =Nm2(e  
    rpowers = []; 2,+H;Ypi!  
    for j = 1:length(n) (~jOtUyT  
        rpowers = [rpowers m_abs(j):2:n(j)]; Z1Wra-g  
    end 1n^xVk-G  
    rpowers = unique(rpowers); V|7 c dX#H  
    FW2} 9#R  
    y3x_B@}BY  
    % Pre-compute the values of r raised to the required powers, q45n.A6a  
    % and compile them in a matrix: -8]$a6`{_  
    % ----------------------------- | !Knd ^}  
    if rpowers(1)==0 %\A~w3E  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i[B%:q:&  
        rpowern = cat(2,rpowern{:}); M-n +3E9  
        rpowern = [ones(length_r,1) rpowern]; D3]_AS&\  
    else 'G&w[8mqY  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); d$!ibL#o  
        rpowern = cat(2,rpowern{:}); YJ6Xq||_  
    end Cd4G&(=  
    (j(6%U  
    ]]+"`t,-  
    % Compute the values of the polynomials: 2'D2>^os  
    % -------------------------------------- >">-4L17m  
    y = zeros(length_r,length(n)); .L}ar7  
    for j = 1:length(n) C`fQ` RL\  
        s = 0:(n(j)-m_abs(j))/2; /wQDcz  
        pows = n(j):-2:m_abs(j); q N>j2~  
        for k = length(s):-1:1 dwRJ0D]&  
            p = (1-2*mod(s(k),2))* ... ~!I \{(  
                       prod(2:(n(j)-s(k)))/              ... i9d.Ls  
                       prod(2:s(k))/                     ... 0VPa=AW  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7z}NI,R}1  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 8"+Kz  
            idx = (pows(k)==rpowers); \QVL%,.%M  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :>|[ o&L  
        end a$ Z06j  
         Gd!y,n&s  
        if isnorm j sm{|'  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /0A}N$?>:  
        end OmsNo0OA  
    end  0y?bwxkc  
    % END: Compute the Zernike Polynomials YQ]W<0(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \j4TDCs_[  
    &U:;jlST9  
    /)j:Y:5  
    % Compute the Zernike functions: LKhUqW  
    % ------------------------------ T{Av[>M  
    idx_pos = m>0; W_%Dg]l   
    idx_neg = m<0; gkDB8,C<j  
    _k&vW(O=:  
    WmeV[iI  
    z = y; +5voAx!  
    if any(idx_pos) HUZI7rC[=)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $%ps:ui~X  
    end )KG.:BO<  
    if any(idx_neg) vLq_l4l  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @PutUYz  
    end s~3"*,3@  
    QN":Qk(,q  
    dW6sA65<Y  
    % EOF zernfun  Hi#hf"V  
     
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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)  ! c`&L_ "!  
    Q3Pu<j}Y  
    DDE还是手动输入的呢? vJxE F&X  
    O}>@G  
    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