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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, s"L&y <?)  
    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, !E70e$Th  
    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? $C16}^  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <J%qzt}  
    1=VyD<dNG6  
    QE]@xLz   
    LUbhTc  
    3 ML][|TR  
    function z = zernfun(n,m,r,theta,nflag) eSPS3|YYn  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. vrn4yHoZ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SA, ~q&  
    %   and angular frequency M, evaluated at positions (R,THETA) on the '2,~'Zk  
    %   unit circle.  N is a vector of positive integers (including 0), and /4{WT?j  
    %   M is a vector with the same number of elements as N.  Each element ]&'!0'3`  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :@w~*eK~  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, f5}afPk  
    %   and THETA is a vector of angles.  R and THETA must have the same zzG=!JR  
    %   length.  The output Z is a matrix with one column for every (N,M) YSjc=  
    %   pair, and one row for every (R,THETA) pair. &9jJ\+:7  
    % wGHft`Z  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o)Q4+njT@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2"0VXtv6  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 2OG/0cP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3=S |U,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized tpI/I bq  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]dycesc'  
    % N2h5@*1Y  
    %   The Zernike functions are an orthogonal basis on the unit circle. qxRsq&_  
    %   They are used in disciplines such as astronomy, optics, and hV3]1E21"  
    %   optometry to describe functions on a circular domain. a )O"PA}2  
    % ] 0i[=  
    %   The following table lists the first 15 Zernike functions. +V=<vT  
    % ui]iO p  
    %       n    m    Zernike function           Normalization 5nPvEN/  
    %       -------------------------------------------------- >N3X/8KL%  
    %       0    0    1                                 1 L5hF-Ek! 3  
    %       1    1    r * cos(theta)                    2 /%YW[oY{V  
    %       1   -1    r * sin(theta)                    2 l&& i`  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ^Ks1[xc*`  
    %       2    0    (2*r^2 - 1)                    sqrt(3) A-x^JC=  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) at>_EiS  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) UG vIHm  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) r*HSi.'21  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,~L*N*ML  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) /fQcrd7h  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ~|u;z,\  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wXNng(M7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) a$W O} g?  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o*T?f)_[p  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 6 `6 I<OJ\  
    %       -------------------------------------------------- PpRS4*nR  
    % :GvC#2 p  
    %   Example 1: '[ c-$X2Ak  
    % 2d[tcn$;h]  
    %       % Display the Zernike function Z(n=5,m=1) ~XUUrg;  
    %       x = -1:0.01:1; EXdX%T\  
    %       [X,Y] = meshgrid(x,x); 1@Ba7>%'  
    %       [theta,r] = cart2pol(X,Y); {[uhIJD3g6  
    %       idx = r<=1; +kI}O*s  
    %       z = nan(size(X)); su0K#*P&I  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); .)*&NY!nsl  
    %       figure nS#F*)  
    %       pcolor(x,x,z), shading interp CW`^fI9H  
    %       axis square, colorbar `=Mk6$%Cs  
    %       title('Zernike function Z_5^1(r,\theta)') cg )(L;  
    % Eu |/pH=:  
    %   Example 2: HOD?i_  
    % ~'*23]j  
    %       % Display the first 10 Zernike functions ,]wab6sY  
    %       x = -1:0.01:1; Vc\g"1 x  
    %       [X,Y] = meshgrid(x,x); CfOyHhhKX  
    %       [theta,r] = cart2pol(X,Y); d 6Y9D=O  
    %       idx = r<=1; \]Y<d  
    %       z = nan(size(X)); o. $ 48h(  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \m`IgP*  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; TT/=0^"  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; #h.N#{9  
    %       y = zernfun(n,m,r(idx),theta(idx)); `&I6=,YLp  
    %       figure('Units','normalized') 2NFk#_9e~  
    %       for k = 1:10 b$w66q8  
    %           z(idx) = y(:,k); 28JVW3&)  
    %           subplot(4,7,Nplot(k)) *wAX&+);  
    %           pcolor(x,x,z), shading interp +sJ{9#6  
    %           set(gca,'XTick',[],'YTick',[]) tE>FL  
    %           axis square  -raK  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) oD%n}  
    %       end NO/$} vw  
    % hzbvR~rn  
    %   See also ZERNPOL, ZERNFUN2. BTsvL>Wy  
    H28-;>'`  
    !/`AM<`o  
    %   Paul Fricker 11/13/2006 VK4UhN2  
    i<&z'A6&]*  
    f$</BND  
    2SU'lh\E  
    n4?;!p<F  
    % Check and prepare the inputs: 5I* 1CIO  
    % ----------------------------- ko.% @Y(=  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qV=O;  
        error('zernfun:NMvectors','N and M must be vectors.') e_Zs4\^ef  
    end y**L^uvr  
    DN8I[5O  
    [/ CB1//Y  
    if length(n)~=length(m) 2C0j.Ib  
        error('zernfun:NMlength','N and M must be the same length.') \>T1&JT  
    end r<]^.]3zj  
    SdufI_'B  
    9[t-W:3c7  
    n = n(:); Z$0mKw   
    m = m(:); .yzXw8~S  
    if any(mod(n-m,2)) ( *26aMp  
        error('zernfun:NMmultiplesof2', ... I9TNUZq('  
              'All N and M must differ by multiples of 2 (including 0).') 7ey|~u2  
    end "% i1zQo&  
    qoan<z7  
    ILNghtm-  
    if any(m>n) KW0KXO06a  
        error('zernfun:MlessthanN', ... WbFCj0  
              'Each M must be less than or equal to its corresponding N.') v&sp;%I6=  
    end 4&]NC2I  
    )`=N+k]  
    >iJxq6!  
    if any( r>1 | r<0 ) 7DW-brd   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9<Zm}PE32  
    end M/[9ZgDc  
    nP=/XiCj  
    PC=s:`Y}R  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) s5b<KQ.  
        error('zernfun:RTHvector','R and THETA must be vectors.') acpc[ ^'  
    end B_r:daCS:  
    ^[HX#JJ~  
    _qit$#wK;  
    r = r(:); X7aj/:fXe  
    theta = theta(:); Yk4ah$}%-^  
    length_r = length(r); gi A(VUwI>  
    if length_r~=length(theta) Xp^>SSt:4  
        error('zernfun:RTHlength', ... )sEAP Ika  
              'The number of R- and THETA-values must be equal.') (ds*$]  
    end XF4NRs  
    7")&njQ/x  
    i;)r|L `V?  
    % Check normalization: Qe<c@i"  
    % -------------------- F fzY3r+   
    if nargin==5 && ischar(nflag) {-IRX)m*  
        isnorm = strcmpi(nflag,'norm'); R[lA@q:  
        if ~isnorm m<9W#  
            error('zernfun:normalization','Unrecognized normalization flag.') z Hj_q%A  
        end $L"-JNS  
    else v2#qs*sW8  
        isnorm = false; Z*5]qh2r8  
    end (i'wa6[E8  
    4p&SlJ  
    RG_)<U/B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K)oN^  
    % Compute the Zernike Polynomials H% c{ }F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0xutG/-&N  
    5al44[  
    xeHqC9Ou  
    % Determine the required powers of r: VM ny>g&3  
    % ----------------------------------- `=foB-(zt  
    m_abs = abs(m); "_&HM4%!  
    rpowers = []; Sytx9`G 5  
    for j = 1:length(n) j@s,5:;[  
        rpowers = [rpowers m_abs(j):2:n(j)]; T\HP5&  
    end Xp3cYS*u  
    rpowers = unique(rpowers); #^/&fdK~A  
    [26([H  
    KWV{wW=-  
    % Pre-compute the values of r raised to the required powers, q} R"  
    % and compile them in a matrix: 65A>p:OO  
    % ----------------------------- [+y/qx79  
    if rpowers(1)==0 u"n ~ 9!G  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3?(||h{  
        rpowern = cat(2,rpowern{:}); D&)gcO`\  
        rpowern = [ones(length_r,1) rpowern]; Ol@ YSkd  
    else ]+S.#x`#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hE/y"SP3  
        rpowern = cat(2,rpowern{:}); I1(, J  
    end Ts:pk  
    g'b)]Q  
    \oGZM0j  
    % Compute the values of the polynomials: 'W j Q  
    % -------------------------------------- ,Gd8 <  
    y = zeros(length_r,length(n)); p>p=nLK  
    for j = 1:length(n) f&>Q 6 {*]  
        s = 0:(n(j)-m_abs(j))/2; = %7:[#n  
        pows = n(j):-2:m_abs(j); 3'6>zp  
        for k = length(s):-1:1 ',* 6vbII  
            p = (1-2*mod(s(k),2))* ... {4{ACp  
                       prod(2:(n(j)-s(k)))/              ... \*w*Q(&3  
                       prod(2:s(k))/                     ... |3g:q  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... N1!|nS3w  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Hw/1~O$T  
            idx = (pows(k)==rpowers); Hca)5$yL  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /T*]RO4%>]  
        end j:,*Liz  
         m5LP~Gb  
        if isnorm _hLM\L  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ni]gS0/  
        end .Isg1qrC  
    end ZA ii"F  
    % END: Compute the Zernike Polynomials L+QEFQ:r5  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Da8qR+*x  
    )5X7|*LP  
    ? B^*YCo7(  
    % Compute the Zernike functions: aX1|&erI  
    % ------------------------------ 4S.%y7d\  
    idx_pos = m>0; 4//Ww6W:  
    idx_neg = m<0; 0:nQGX!N  
    M ~!*PCd5  
    c66Iy"  
    z = y; /^0Hi4+\  
    if any(idx_pos) D1lHq/  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); e}}xZ%$4|  
    end w>rglm&  
    if any(idx_neg) 8c3 X9;a  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); zYj8\iER  
    end P*(lc:  
    M>_S%V4a  
     c:~o e  
    % EOF zernfun ScfW;  
     
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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 B=H1+  
    oAA%pZ@  
    DDE还是手动输入的呢? RAR"9 N .  
    S;MS,R  
    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