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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, |'U,/  
    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, qa 6=W  
    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? r~Y>+ln.  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? k Zk .]b  
    ER~T'-YMS  
    wUZQB1$F  
    |u^)RB  
    &5kZ{,-eM  
    function z = zernfun(n,m,r,theta,nflag) u;+%Qh  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ee&nU(pK  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N zQL!(2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the y\F`B0#$  
    %   unit circle.  N is a vector of positive integers (including 0), and Po Yr:=S?  
    %   M is a vector with the same number of elements as N.  Each element CDQJ bvx  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) "C:rTIH  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^H5w41  
    %   and THETA is a vector of angles.  R and THETA must have the same _-@ZOhw&  
    %   length.  The output Z is a matrix with one column for every (N,M) C+/Eqq^(  
    %   pair, and one row for every (R,THETA) pair. 9USrgY6_  
    % ,pDp>-vI%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike yD"]{  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qxf+#  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ,3VG.u;U   
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, X!U]`Qh  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized /yx=7<  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2-8YSHlh  
    % a<f;\$h]  
    %   The Zernike functions are an orthogonal basis on the unit circle. gXq!a|eH  
    %   They are used in disciplines such as astronomy, optics, and  Y[f,ia  
    %   optometry to describe functions on a circular domain. m3U+ du  
    % Xy[}Gp  
    %   The following table lists the first 15 Zernike functions. ?D1x;i9<  
    % `[X6#` <  
    %       n    m    Zernike function           Normalization c*.G]nRc  
    %       -------------------------------------------------- sEoZ1E  
    %       0    0    1                                 1 :0nK`$'  
    %       1    1    r * cos(theta)                    2 G+ :bL S#:  
    %       1   -1    r * sin(theta)                    2 NOF?LV  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) |tG05+M  
    %       2    0    (2*r^2 - 1)                    sqrt(3) I") H~  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) B1y<.1k  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) lN);~|IOv7  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) U^B"|lc:[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) '/Cg*o/  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) s0gJ f[  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) / pO{2[  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ov1Wr#s  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NV:>a  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '!pAnsXfO  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =ZG<BG_  
    %       -------------------------------------------------- ah 4kA LO  
    % buRhQ"  
    %   Example 1: A)OdQFet(  
    % u06tDJ[  
    %       % Display the Zernike function Z(n=5,m=1) !)NYW4"  
    %       x = -1:0.01:1; h{\t*U 54'  
    %       [X,Y] = meshgrid(x,x); /CIx$G  
    %       [theta,r] = cart2pol(X,Y); : @s8?eg  
    %       idx = r<=1; )y6QAp  
    %       z = nan(size(X));  NI^{$QMj  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Z#CxQ D%\  
    %       figure {":c@I  
    %       pcolor(x,x,z), shading interp R'Sa?6xS4  
    %       axis square, colorbar >+L7k^[,0  
    %       title('Zernike function Z_5^1(r,\theta)') &xgZF Sq  
    % }(m1ql  
    %   Example 2: Cm^Yl p  
    % Xc{ZN1 4n  
    %       % Display the first 10 Zernike functions 9`&?hi49nK  
    %       x = -1:0.01:1; B i'd5B5  
    %       [X,Y] = meshgrid(x,x); yXkt:O,i  
    %       [theta,r] = cart2pol(X,Y); P%iP:16  
    %       idx = r<=1; 5;}2[3}[  
    %       z = nan(size(X)); hMv2"V-X  
    %       n = [0  1  1  2  2  2  3  3  3  3]; {JXf*IJ  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; `4_c0 q)N4  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; qbH %Hx  
    %       y = zernfun(n,m,r(idx),theta(idx)); SBC~QD>L+  
    %       figure('Units','normalized') Xj%,xm>}!u  
    %       for k = 1:10  +.=1^+a  
    %           z(idx) = y(:,k); XWJ SLN(O  
    %           subplot(4,7,Nplot(k)) s} s|~  
    %           pcolor(x,x,z), shading interp >8%M*-=p  
    %           set(gca,'XTick',[],'YTick',[]) lbd(j{h>4  
    %           axis square \/n+j!  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) JT}.F!q6E  
    %       end uN8/Q2   
    % :Pc(DfkS  
    %   See also ZERNPOL, ZERNFUN2. 36nyu_h:R  
    3\1#eK'TK.  
    -ovoRI^6`}  
    %   Paul Fricker 11/13/2006 B& "RS  
    d)\2U{  
    hzv3F9.x  
    .wP/ai>}  
    ;ed#+$Na  
    % Check and prepare the inputs: w~]T<^fW~  
    % ----------------------------- as(;]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6s5yyy=L%~  
        error('zernfun:NMvectors','N and M must be vectors.') wE?CvL  
    end g@Ld"5$^2  
    #,TELzUVE  
    "w9`cz9a~J  
    if length(n)~=length(m) qIz}$%!A  
        error('zernfun:NMlength','N and M must be the same length.') 7_KXD#  
    end q~j)W$k  
    S"Kq^DN  
    oXdel Ju?  
    n = n(:); W+K.r?G<j  
    m = m(:); 07FT)QTE  
    if any(mod(n-m,2)) v$;@0t:;#  
        error('zernfun:NMmultiplesof2', ... h D.)M  
              'All N and M must differ by multiples of 2 (including 0).') Ch t%uzb,  
    end Y([d;_#P  
    i-]U+m*  
    F_v-}bbcFQ  
    if any(m>n) &atyDFJ'  
        error('zernfun:MlessthanN', ... m<3w^mww  
              'Each M must be less than or equal to its corresponding N.') Kr]z]4.d@  
    end eVx~n(m!}  
    \sITwPA[z  
    e8-ehs>  
    if any( r>1 | r<0 ) lov%V*tL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') SB/3jH  
    end z0 \N{rP&  
    8ljuc5,J  
    <aXoB*Y  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W~yLl%  
        error('zernfun:RTHvector','R and THETA must be vectors.') zqf[Z3  
    end !b63ik15O~  
    $ser+Jt=  
    L#2ZMy  
    r = r(:); !D;c,{Oz  
    theta = theta(:); VX!hv`E  
    length_r = length(r); GyK(Vb"h6  
    if length_r~=length(theta) bcn7,ht  
        error('zernfun:RTHlength', ... @$c!/  
              'The number of R- and THETA-values must be equal.') K{2h9 ]VF  
    end 3ev -Iqz  
    = ^s$ <  
    #w|5 jN?  
    % Check normalization: MMd.0JuaO  
    % -------------------- )~dOmfw%|  
    if nargin==5 && ischar(nflag) |IN[uQ  
        isnorm = strcmpi(nflag,'norm'); 8k H<$9  
        if ~isnorm `[Sl1saZ$S  
            error('zernfun:normalization','Unrecognized normalization flag.') TF2KZL#A|  
        end I .P6l*$  
    else ISBF\ wQY  
        isnorm = false; *)D1!R<\,R  
    end >f@ G>H)+  
    ]2$x| #Gg}  
    fEwifSp.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Sc_5FX\Yx  
    % Compute the Zernike Polynomials `tVy_/3(9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QNpu TZn#Q  
    l!@ 1u^v2  
    #U"1 9@|}  
    % Determine the required powers of r: I_>`hTiR  
    % ----------------------------------- gr+Pl>C{  
    m_abs = abs(m); ]r959+\$  
    rpowers = []; x.UaQ |F  
    for j = 1:length(n) F0.zi>5  
        rpowers = [rpowers m_abs(j):2:n(j)]; oY.\)eJ~>  
    end H=<LutnZ  
    rpowers = unique(rpowers); ) rpq+~b  
    b#='^W3  
    T 1zi0fa'  
    % Pre-compute the values of r raised to the required powers, MI*Sq\-i  
    % and compile them in a matrix: taDQ65  
    % ----------------------------- .iT4-  
    if rpowers(1)==0 Hi8Y6|y$D  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); C:j]43`  
        rpowern = cat(2,rpowern{:}); &*gbK6JB  
        rpowern = [ones(length_r,1) rpowern]; !_x*m@/  
    else J\A8qh8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); HeozJ^u\?  
        rpowern = cat(2,rpowern{:}); mb{q(WEPP  
    end @GeHWv  
    <5IQc[3]aP  
    p6[ (81  
    % Compute the values of the polynomials: ri=+(NKo-  
    % -------------------------------------- {y-`QS  
    y = zeros(length_r,length(n)); h<NRE0-  
    for j = 1:length(n) ,YB1 y)x  
        s = 0:(n(j)-m_abs(j))/2; A3q*$.[  
        pows = n(j):-2:m_abs(j); (B}+h   
        for k = length(s):-1:1 j^ EbO3  
            p = (1-2*mod(s(k),2))* ... 28UVDG1?  
                       prod(2:(n(j)-s(k)))/              ... s MZ[d\  
                       prod(2:s(k))/                     ... ^y Vl"/  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zP nC=h|g  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ChE_unw  
            idx = (pows(k)==rpowers); ?,XC =}  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ti9}*8  
        end P {H{UKs#  
         <L&eh&4c  
        if isnorm hW' HT  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i0ybJOa4  
        end a.,_4;'UE1  
    end % rcFT_  
    % END: Compute the Zernike Polynomials {ERjeuDm]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m =k%,J_  
    r/PKrw sC  
    .@k*p>K  
    % Compute the Zernike functions: "&-C$J5 Id  
    % ------------------------------ 7>,rvW:]  
    idx_pos = m>0; TB#N k5  
    idx_neg = m<0; D^$OCj\  
    oD0EOT/E  
    _]o5R7[MQ  
    z = y; X4Xf2aXI  
    if any(idx_pos) o5 WW{)Q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Funj!x'uE  
    end 3_zSp.E\l  
    if any(idx_neg) 2 ~-( A  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ' ^a!`"Bc  
    end 8*Zvr&B,G  
    @q)E=G1<o0  
    9L"?wv  
    % EOF zernfun [Vp\$;\nT  
     
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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)  "u5KbJW  
    T+(M8 qb  
    DDE还是手动输入的呢? n{$}#NdV  
    lJlhl7  
    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