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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, KJUH(]>F  
    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, oN~&_*FE  
    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? 'T;P;:!\  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ];$L &5^  
    ^rR1ZVY  
    f1RWP@iar  
    {GT*ZU*  
    & bm 1Fz  
    function z = zernfun(n,m,r,theta,nflag) 23eX;gL  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. tyDU @M  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ' ,wFTV&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the fSj5ZsO  
    %   unit circle.  N is a vector of positive integers (including 0), and Pl06:g2I  
    %   M is a vector with the same number of elements as N.  Each element wc@X.Q[  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) WF+99?75  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, |-67 \p]  
    %   and THETA is a vector of angles.  R and THETA must have the same #powub  
    %   length.  The output Z is a matrix with one column for every (N,M) A0s ZOCky  
    %   pair, and one row for every (R,THETA) pair. wo{gG?B  
    % &{n.]]%O.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +4~_Ei[i  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Igt#V;kK"2  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *!t/"b  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, nsC3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized U[-o> W#  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. =%K;X\NB  
    % epe)a  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3BUSv#w{i  
    %   They are used in disciplines such as astronomy, optics, and Ms#M+[a  
    %   optometry to describe functions on a circular domain. N7zft  
    % yjX9oxhtL  
    %   The following table lists the first 15 Zernike functions. ZgcMv,=  
    % h 0Q5-EA  
    %       n    m    Zernike function           Normalization '3tCH)s  
    %       -------------------------------------------------- ibk6|pp  
    %       0    0    1                                 1 7hcYD!DS  
    %       1    1    r * cos(theta)                    2 f|c{5$N!  
    %       1   -1    r * sin(theta)                    2 9ULQrq$?  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ,AFu C <  
    %       2    0    (2*r^2 - 1)                    sqrt(3) s?}e^/"v  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) (k.[GfCbD  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) hBUn \~z  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]y '>=a|T  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ql{ OETn#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) n0 {i&[I~+  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 3z?> j]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Do7Tj  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) I;|B.j  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) }B+C~@j  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) lvz7#f L~  
    %       -------------------------------------------------- 8qTys8  
    % BC.87Fji/  
    %   Example 1: \ :sUL!  
    % *Kg ks4  
    %       % Display the Zernike function Z(n=5,m=1) t\,PB{P:J  
    %       x = -1:0.01:1; =s2*H8]  
    %       [X,Y] = meshgrid(x,x); ,!y$qVg'\f  
    %       [theta,r] = cart2pol(X,Y); Y"aJur=`  
    %       idx = r<=1; S`0(*A[W*  
    %       z = nan(size(X)); & l&:`nsJ  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q,|j]+9q  
    %       figure 9}<ile7^  
    %       pcolor(x,x,z), shading interp +gtbcF@rx  
    %       axis square, colorbar JIOR4'9  
    %       title('Zernike function Z_5^1(r,\theta)') pJ"qu,w  
    % ]Ie 0S~  
    %   Example 2: Be2DN5)  
    % Ckuh:bs  
    %       % Display the first 10 Zernike functions 6j]0R*B7`Q  
    %       x = -1:0.01:1; ucW-I;"  
    %       [X,Y] = meshgrid(x,x); [!#L6&:a8  
    %       [theta,r] = cart2pol(X,Y); 6iE<T&$3P  
    %       idx = r<=1; Hk.TM2{w  
    %       z = nan(size(X)); /]Md~=yNp  
    %       n = [0  1  1  2  2  2  3  3  3  3]; &.Qrs :U  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Yu^4VXp~M%  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; MaQqs=  
    %       y = zernfun(n,m,r(idx),theta(idx)); P* BmHz4KL  
    %       figure('Units','normalized') %RRNJf}z  
    %       for k = 1:10 BG]#o| KW  
    %           z(idx) = y(:,k); YfKdR"i+.  
    %           subplot(4,7,Nplot(k)) E]n&=\  
    %           pcolor(x,x,z), shading interp Hd ={CFip  
    %           set(gca,'XTick',[],'YTick',[]) s$`0yGmQ  
    %           axis square u^I|T.w<r6  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZG8DIV\D7  
    %       end =K[yT:  
    % EUX\^c]n  
    %   See also ZERNPOL, ZERNFUN2. )g%d:xI  
    Flm%T-Dl  
    @:vwb\azVD  
    %   Paul Fricker 11/13/2006 |3"KK  
    KU(&%|;g  
    %XQ(fj>  
    #r\4sVg  
    #f]SK[nR  
    % Check and prepare the inputs: 16(QR-  
    % ----------------------------- >@_^fw)  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2-EIE4ds  
        error('zernfun:NMvectors','N and M must be vectors.') E4/Dr}4  
    end Ioa$51&  
    3,qr-g|;jM  
    ~HsJUro  
    if length(n)~=length(m) nMUw_7Y6  
        error('zernfun:NMlength','N and M must be the same length.') iz PDd{[  
    end d^ 8ZeC#  
    j6 z^Tt12  
    ?NsW|w_  
    n = n(:); _Q4)X)F  
    m = m(:); ndMA-`Ny,  
    if any(mod(n-m,2)) 7[XRd9a5(  
        error('zernfun:NMmultiplesof2', ...  d{3QP5  
              'All N and M must differ by multiples of 2 (including 0).') &B1WtW  
    end 9qzHS~l  
    Feq]U?  
    \Uq(Zga4)  
    if any(m>n) &}B|"s[  
        error('zernfun:MlessthanN', ... [waIi3Dv\  
              'Each M must be less than or equal to its corresponding N.') Iit; F  
    end ENs&RZ;  
    @lrztM  
    )Y{L&A  
    if any( r>1 | r<0 ) V {ddr:]4  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') FWgpnI\X|{  
    end S;#'M![8  
    h MD|#A-<  
    -k e's  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) >_T-u<E  
        error('zernfun:RTHvector','R and THETA must be vectors.') )1`0PJoHE  
    end T+H!_ky`A  
    >!1-lfa8  
    tFOhL9T  
    r = r(:); Btn]}8K  
    theta = theta(:); Z,Dl` w  
    length_r = length(r); j{+.tIzpq[  
    if length_r~=length(theta) ` 7V]y -  
        error('zernfun:RTHlength', ... .Vvx,>>D  
              'The number of R- and THETA-values must be equal.') #?- wm  
    end ,(^*+G.i  
    ^o&. fQ*  
    q#9RW(o  
    % Check normalization: v;D~Pa  
    % -------------------- H8}oIA"b  
    if nargin==5 && ischar(nflag) M@v.c; Lt  
        isnorm = strcmpi(nflag,'norm'); ')<hON44EX  
        if ~isnorm {q^[a-h>  
            error('zernfun:normalization','Unrecognized normalization flag.') i5@ z< \  
        end *#+An<iT ;  
    else *_\_'@1|J)  
        isnorm = false; Q K<"2p?  
    end -;WGS o  
    Y\g3h M  
    uiR8,H9*M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w@w(-F!%l  
    % Compute the Zernike Polynomials >7DhTM-A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fd9 [pU  
    z1X`o  
    A;?|& `f  
    % Determine the required powers of r: ,/|T-Ka  
    % ----------------------------------- suDQ~\ n  
    m_abs = abs(m); (V2fRv  
    rpowers = []; ml }{|Yz  
    for j = 1:length(n) Y9XEP7  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1\I}2;  
    end AFE~ v\Gz  
    rpowers = unique(rpowers); LyFN.2qw  
    6?c7$Y  
    8&b,qQ~  
    % Pre-compute the values of r raised to the required powers, 8[{ Vu0R  
    % and compile them in a matrix: &\*(Q*2N  
    % ----------------------------- OYn}5RN  
    if rpowers(1)==0 v0.#Sl-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *VxgARIL  
        rpowern = cat(2,rpowern{:}); }b.%Im<3R  
        rpowern = [ones(length_r,1) rpowern]; XUuN )i  
    else X$W~mQma6  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^.QzQ1=D  
        rpowern = cat(2,rpowern{:}); :,6\"y-  
    end L) T (<  
    {&1/V  
    V)N%WX G  
    % Compute the values of the polynomials: svH !1 b  
    % -------------------------------------- 1o{Mck  
    y = zeros(length_r,length(n)); ,r\o}E2  
    for j = 1:length(n) Wg]Qlw`\|  
        s = 0:(n(j)-m_abs(j))/2; ;>7De8v@@  
        pows = n(j):-2:m_abs(j); WNrk}LFof  
        for k = length(s):-1:1 r3UUlR/Do  
            p = (1-2*mod(s(k),2))* ... 9~[Y-cpoi  
                       prod(2:(n(j)-s(k)))/              ... XU(eEnmo m  
                       prod(2:s(k))/                     ... #?:lb1  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... k@W1-D?  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Oxd]y1  
            idx = (pows(k)==rpowers); X45%e!  
            y(:,j) = y(:,j) + p*rpowern(:,idx); aAUvlb  
        end DEZve Qr=  
         6qnzBA7  
        if isnorm Z/+#pWBI!  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zIAD9mQex  
        end E hMNap}5"  
    end $*fMR,~t&  
    % END: Compute the Zernike Polynomials \ }G> 8^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wz%Nb Ly-  
    sd|).;s}  
    wI/iuc  
    % Compute the Zernike functions: ?gGHj-HYJ  
    % ------------------------------ 5$C-9  
    idx_pos = m>0; \bw2u!  
    idx_neg = m<0; R8'RA%O9J  
    -nV9:opD  
    h~zT ydnH  
    z = y; YUk\Q%  
    if any(idx_pos) ZPYS$Ydy  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vx5Zl&6r  
    end [d ]9Oa4  
    if any(idx_neg) /mzlH  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z4ImV~m  
    end [/8%3  
    CzrC%xy  
    {"KMs[M  
    % EOF zernfun I-l_TpM)  
     
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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)  |;{6& S  
    cso8xq|b7  
    DDE还是手动输入的呢? :[d9tm  
    bW+:C5'  
    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