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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, hXb%;GL  
    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, _DrJVC~6@  
    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? ,jC3Fcly  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? (YY~{W$w(  
    `;YU.*  
    uZZU{U9h  
    4Q IE8f Y  
    >Bs#Xb_B]  
    function z = zernfun(n,m,r,theta,nflag) \o\nr!=k  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V97,1`  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N CiR%Ujf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the h?-#9<A  
    %   unit circle.  N is a vector of positive integers (including 0), and A<\JQ  
    %   M is a vector with the same number of elements as N.  Each element Hg9CZM ko  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) JT9N!CGZ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ?=VOD#)  
    %   and THETA is a vector of angles.  R and THETA must have the same EwS!]h?  
    %   length.  The output Z is a matrix with one column for every (N,M) x+]!m/  
    %   pair, and one row for every (R,THETA) pair. or k=`};  
    % |ou b!fG4  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c*`>9mv  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), []0mX70N  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Fb/XC:AD  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ZhNdB  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 7~ztwL  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~2d:Q6  
    % ?:|-Dq,  
    %   The Zernike functions are an orthogonal basis on the unit circle. }n7t h  
    %   They are used in disciplines such as astronomy, optics, and m%"uPv\  
    %   optometry to describe functions on a circular domain. p'sc0@}_O  
    % }pa9%BQI  
    %   The following table lists the first 15 Zernike functions. -dv %H{  
    % w'X]M#Q><  
    %       n    m    Zernike function           Normalization _5MNMV LwW  
    %       -------------------------------------------------- w#N?l!5  
    %       0    0    1                                 1 $ n,Z  
    %       1    1    r * cos(theta)                    2 ~^ ^ NHq  
    %       1   -1    r * sin(theta)                    2 c9j*n;Q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) uY< H#k  
    %       2    0    (2*r^2 - 1)                    sqrt(3) jKZt~I  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) !GW ,\y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) >xA),^ YT  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Z?J:$of*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {B*W\[ns  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^v9|%^ug  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ]O{u tm  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zq1mmFIO  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e4I^!5)N  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r}u%#G+K,  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) qn"D#K'&(  
    %       -------------------------------------------------- hF3&i=;.  
    % Jdy <w&S  
    %   Example 1: 0)9"M.AIvo  
    % ;eigOU]  
    %       % Display the Zernike function Z(n=5,m=1) _ nP;Fx  
    %       x = -1:0.01:1; M+wt_ _vHf  
    %       [X,Y] = meshgrid(x,x); >QHo@Zqj(  
    %       [theta,r] = cart2pol(X,Y); m-T~fJ  
    %       idx = r<=1; Fg/dS6=n`?  
    %       z = nan(size(X)); DWt*jX*  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ED$DSz)x  
    %       figure 44\>gI<  
    %       pcolor(x,x,z), shading interp .[DthEF  
    %       axis square, colorbar i`)!X:j  
    %       title('Zernike function Z_5^1(r,\theta)') aFY_:.o2k`  
    % dSIH9D  
    %   Example 2: K?#]("De6  
    % X E}H3/2  
    %       % Display the first 10 Zernike functions b'ml=a#i 0  
    %       x = -1:0.01:1; 8*g ^o\M  
    %       [X,Y] = meshgrid(x,x); S bsouGD,{  
    %       [theta,r] = cart2pol(X,Y); ]%RNA:(F'  
    %       idx = r<=1; rZbEvS  
    %       z = nan(size(X)); Bn]K+h\E  
    %       n = [0  1  1  2  2  2  3  3  3  3]; %HtuR2#ca  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |m,VTViv;i  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ^TXfsQs  
    %       y = zernfun(n,m,r(idx),theta(idx)); R*1kR|*_)  
    %       figure('Units','normalized') j1Yq5`ia  
    %       for k = 1:10 ,]Zp+>{  
    %           z(idx) = y(:,k); Aox3s?  
    %           subplot(4,7,Nplot(k)) y?30_#[dN  
    %           pcolor(x,x,z), shading interp ,/&Zw01dGN  
    %           set(gca,'XTick',[],'YTick',[]) A1cb"N^  
    %           axis square ly4Qg\l  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *i$ePVU  
    %       end OySy6IN]q  
    % S"snB/  
    %   See also ZERNPOL, ZERNFUN2. cJn HW  
    ++[5q+b  
    xPmN},i'R$  
    %   Paul Fricker 11/13/2006 h3u1K>R)  
    eukA[nO7G  
    `GQ{*_-  
    OQlG+|  
    PfW|77  
    % Check and prepare the inputs: ]!YtH]}  
    % ----------------------------- 1M%S gV-#  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) KSs1CF'i  
        error('zernfun:NMvectors','N and M must be vectors.') 8{&["?  
    end H5wb_yBQ+  
    `?s.\Dh  
    CfT/R/L  
    if length(n)~=length(m) i 6no;}j  
        error('zernfun:NMlength','N and M must be the same length.') "c`xH@D  
    end +1{fzb>9_  
    9\O(n>  
    G`]w?Di4  
    n = n(:); _Sj}~ H  
    m = m(:); ~o15#Pfn/  
    if any(mod(n-m,2)) B0mLI%B  
        error('zernfun:NMmultiplesof2', ... OOy}]uYF`  
              'All N and M must differ by multiples of 2 (including 0).') =_=*OEgO]  
    end Ya4?{2h@+  
    eG] a zt  
    p6 xPheD  
    if any(m>n) 8<PKKDgbfd  
        error('zernfun:MlessthanN', ... Z>A{i?#m  
              'Each M must be less than or equal to its corresponding N.') setL dEi  
    end ~a+NJ6e1  
    y8s=\`~PR  
    LPE)  
    if any( r>1 | r<0 ) FRyPeZR  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') oNRG25  
    end a5wDm  
    -Wjh**  
    sk X]8  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )+~E8yK  
        error('zernfun:RTHvector','R and THETA must be vectors.') ,ECAan/@  
    end i2F(GH?p[  
    T)\NkM&  
    INNAYQ  
    r = r(:); &IQ%\W#aY  
    theta = theta(:); "n- pl  
    length_r = length(r); 8 $ ~3ra  
    if length_r~=length(theta) @FX{M..  
        error('zernfun:RTHlength', ... |>utWT]S  
              'The number of R- and THETA-values must be equal.') J|j;g!fK  
    end .hz2&9Ow  
    /7p>7q 9g  
    O~'FR[J  
    % Check normalization: %Y',|+Arx  
    % -------------------- z\Ui8jo:;  
    if nargin==5 && ischar(nflag) cf*zejbw  
        isnorm = strcmpi(nflag,'norm'); dB)9K)  
        if ~isnorm sc xLB;  
            error('zernfun:normalization','Unrecognized normalization flag.') ^5)_wUf  
        end x;U|3{I o  
    else jH0Bo;  
        isnorm = false; 1X:&* a"5  
    end ?`. XK}  
    fOBN=y6x  
    BED@?:U#h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VK4/82@5  
    % Compute the Zernike Polynomials pG28M]\  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "?H+ u/8$  
    (Jpm KO  
    ~07RFR  
    % Determine the required powers of r: 8A/>JD3^  
    % ----------------------------------- oFyeH )!  
    m_abs = abs(m); qy9i9$8  
    rpowers = []; -A;w$j6*  
    for j = 1:length(n) gb_X?j%p7  
        rpowers = [rpowers m_abs(j):2:n(j)]; JN^bo(kb  
    end cHEz{'1m  
    rpowers = unique(rpowers); Z3`2-r_=  
    \3j)>u,r  
    #~e9h9  
    % Pre-compute the values of r raised to the required powers, {G.jB/  
    % and compile them in a matrix: |Mlh;  
    % ----------------------------- \\s?B K  
    if rpowers(1)==0 {rfte'4;=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); A;t zRe  
        rpowern = cat(2,rpowern{:}); ,RN|d0dE  
        rpowern = [ones(length_r,1) rpowern]; T/Q==Q{W:  
    else L]>4Nd  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3{q[q#"  
        rpowern = cat(2,rpowern{:}); <?4cWp|i  
    end AA.Ys89V  
    L5C2ng>  
    MLeX;He  
    % Compute the values of the polynomials: g-eq&#  
    % -------------------------------------- WVkG 2  
    y = zeros(length_r,length(n)); &%:*\_2s  
    for j = 1:length(n) -fQX4'3R  
        s = 0:(n(j)-m_abs(j))/2; 3.~h6r5-  
        pows = n(j):-2:m_abs(j); x Ty7lfSe  
        for k = length(s):-1:1 N1s.3`  
            p = (1-2*mod(s(k),2))* ... #'iPDRYy  
                       prod(2:(n(j)-s(k)))/              ... c.-cpFk^L&  
                       prod(2:s(k))/                     ... oB}K[3uB:t  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... '2xcce#  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 4JSZ0:O  
            idx = (pows(k)==rpowers); &/DOO ^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); o oDdV >  
        end 8.-S$^hj~6  
         &58 {  
        if isnorm rFO_fIJno  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;x16shH  
        end K+-zY[3  
    end {70 Ou}*  
    % END: Compute the Zernike Polynomials 9PCa*,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uYabJqV  
    *{Yi}d@h(  
    +/(|?7i@  
    % Compute the Zernike functions: i.F8  
    % ------------------------------ i<Q& D\Pv  
    idx_pos = m>0; iA&oLu[y3  
    idx_neg = m<0; !^]q0x  
    /t$*W\PL@  
    q$|0)}  
    z = y; >^ ;(c4C  
    if any(idx_pos) (< :mM  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "B~WcC  
    end yW{mK  
    if any(idx_neg) NQg'|Pt(%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &b!vWX1N  
    end U-1VnX9m  
    f@h2;An$w  
    QYH."7X >  
    % EOF zernfun u*U_7Uw$  
     
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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)  TMnT#ypf<5  
    B`'}&6jr.  
    DDE还是手动输入的呢? ;XD>$t@  
    AxG?zBTFx  
    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