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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, %M&3VQ9w  
    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, \cQ .|S  
    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? 5)'P'kVi7.  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 5+P@s D  
    :EZQ'3X  
    \Hwg) Uc{  
    \iU]s\{).  
    hazq#J!  
    function z = zernfun(n,m,r,theta,nflag) Z0ReWrl;`  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ['[KR BJL  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8$vK5Dnn8  
    %   and angular frequency M, evaluated at positions (R,THETA) on the o>c ^aRZ{  
    %   unit circle.  N is a vector of positive integers (including 0), and d TGA5c  
    %   M is a vector with the same number of elements as N.  Each element ( 9$"#o  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ht[{ryTxu  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Dag`>|my  
    %   and THETA is a vector of angles.  R and THETA must have the same ;GsQR+en  
    %   length.  The output Z is a matrix with one column for every (N,M) {8@\Ij  
    %   pair, and one row for every (R,THETA) pair. }+R B=#~o  
    % # |^^K!%  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4 tXSYHd3  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lKKERO5+  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral |}[nH>  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EO)%UrWnC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized dDtFx2(R  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R1ktj  
    % (~s|=Hxq|-  
    %   The Zernike functions are an orthogonal basis on the unit circle. $h28(K%  
    %   They are used in disciplines such as astronomy, optics, and 5j^NV&/_  
    %   optometry to describe functions on a circular domain. 2~c~{ jl\  
    % S>Z|) I  
    %   The following table lists the first 15 Zernike functions.  k0H#:c}  
    % c ~F dx  
    %       n    m    Zernike function           Normalization -<N&0F4|*  
    %       -------------------------------------------------- I*\^,ow  
    %       0    0    1                                 1 M>l^%`  
    %       1    1    r * cos(theta)                    2 H?yE3 w  
    %       1   -1    r * sin(theta)                    2 2 x 4=  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `v nJ4*  
    %       2    0    (2*r^2 - 1)                    sqrt(3) S KXD^OH  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Vhg1/EgUr  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) oRq!=eUu_  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ohQAA h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) xxa} YIe8  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) qv+R:YYOq  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) .mxTfP=9  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) F#V q#|_)>  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) P7GRSjG  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) cd.brM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Qv#]81i(1  
    %       -------------------------------------------------- =_pwA:z"A  
    % 68t}w^=  
    %   Example 1: WZFH@I28  
    % 4]XI"-M^D  
    %       % Display the Zernike function Z(n=5,m=1) 6)*xU|fU  
    %       x = -1:0.01:1; >a>fb|r  
    %       [X,Y] = meshgrid(x,x); j"7 JLe*  
    %       [theta,r] = cart2pol(X,Y); 85]SC$  
    %       idx = r<=1; G'#41>q+  
    %       z = nan(size(X)); jO'|mGUM  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); K5"sj|d&  
    %       figure G<qIY&D'  
    %       pcolor(x,x,z), shading interp rPiNv 30L  
    %       axis square, colorbar q<{NO/Mm  
    %       title('Zernike function Z_5^1(r,\theta)') O\beKBT;  
    % z,f=}t[.Y  
    %   Example 2: cT'w=  
    % P-Su5F  
    %       % Display the first 10 Zernike functions E{Vo'!LY  
    %       x = -1:0.01:1; SUdm 0y  
    %       [X,Y] = meshgrid(x,x); RKkGITDk  
    %       [theta,r] = cart2pol(X,Y); K|^wc$  
    %       idx = r<=1; Ruaur]  
    %       z = nan(size(X)); sbsu(Sz+  
    %       n = [0  1  1  2  2  2  3  3  3  3]; f7<pEGb  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; "{BqtU*.  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ax<\jW<  
    %       y = zernfun(n,m,r(idx),theta(idx)); mLwY]2T"  
    %       figure('Units','normalized') sQ1jrkm  
    %       for k = 1:10 eaZQ2  
    %           z(idx) = y(:,k); Nhf~PO({&  
    %           subplot(4,7,Nplot(k)) l";'6;g  
    %           pcolor(x,x,z), shading interp +m$5a YX  
    %           set(gca,'XTick',[],'YTick',[]) 79 Bg]~}Z  
    %           axis square )pgrl  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) VcgBLkIF  
    %       end :@. ;  
    %  '3 ,\@4  
    %   See also ZERNPOL, ZERNFUN2. g`,AaWlF  
    oRY!\ADR  
    Q GPw2Q  
    %   Paul Fricker 11/13/2006 fEnQE EU~P  
    Tj`5L6N;8  
    Je7RrCz  
    vzR=>0#  
    Nw<P bklz  
    % Check and prepare the inputs: gA^q^>7  
    % ----------------------------- f} K`Jm_}?  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9'KonW  
        error('zernfun:NMvectors','N and M must be vectors.') I3y9:4  
    end tJD] (F  
    h'5Cp(G  
    XB\zkf_}Xc  
    if length(n)~=length(m) !-tz4vjw  
        error('zernfun:NMlength','N and M must be the same length.') yp]@^TN  
    end z@h~Vb&I  
    k*$3i  
    8<&EvOk  
    n = n(:); O6c\KFBSJ  
    m = m(:); ?b:_AO&  
    if any(mod(n-m,2)) PpOlt.yui  
        error('zernfun:NMmultiplesof2', ... @u9Mks|{  
              'All N and M must differ by multiples of 2 (including 0).') +"!aM?o  
    end CjZ2z%||=  
    l (kr'x  
    }C#3O{5  
    if any(m>n) H~fdbR  
        error('zernfun:MlessthanN', ... N}Vn;29  
              'Each M must be less than or equal to its corresponding N.') y\PxR708  
    end :L$4*8@`+  
    =!0I_L/  
    :`W|h E^  
    if any( r>1 | r<0 ) o$J6 ~dn  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') GESXc $E8  
    end f(Hu {c5yV  
    Fb_S&!  
    PZ OKrW  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) v 81rfB5  
        error('zernfun:RTHvector','R and THETA must be vectors.') F[E? A95W  
    end > <Z'D  
    J=}F2C   
    ?0vNEz[  
    r = r(:); Ijo(^v@  
    theta = theta(:); bLS&H[f K  
    length_r = length(r); v'9m7$  
    if length_r~=length(theta) 2{o10 eL  
        error('zernfun:RTHlength', ... RU_L<Lpi  
              'The number of R- and THETA-values must be equal.') Mq\~`8V  
    end %a 8&W  
    r6Nm!Bq7  
     s>[{}7ca  
    % Check normalization: C{m&}g`  
    % -------------------- la, h  
    if nargin==5 && ischar(nflag) fI:H8  
        isnorm = strcmpi(nflag,'norm'); vr IV%l=  
        if ~isnorm %e=!nRc  
            error('zernfun:normalization','Unrecognized normalization flag.') |*\C{b  
        end ElR)Gd_8  
    else =ApY9`  
        isnorm = false; `,#!C`E 9  
    end +{-]P\oc  
    8wFn}lw&  
    XB/'u39  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pj|X]4?wdI  
    % Compute the Zernike Polynomials ;z>p8N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8pe0$r`b  
    nQLs<]h1  
    p\D >z("  
    % Determine the required powers of r: 9k>uRV6  
    % ----------------------------------- -Ktwo_ V*  
    m_abs = abs(m); =AkX4k  
    rpowers = []; u0]q`u/ T  
    for j = 1:length(n) qgexb\x\4  
        rpowers = [rpowers m_abs(j):2:n(j)]; Eo=HNe  
    end 0XIxwc0Iw  
    rpowers = unique(rpowers); z8iENECwj  
    T$c+m\j6  
    pxplWP,  
    % Pre-compute the values of r raised to the required powers, -!R l(if  
    % and compile them in a matrix: r8v:|Q1"  
    % ----------------------------- <\D Uo0]J  
    if rpowers(1)==0 JDhwN<0R  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Xb<)LHA~3  
        rpowern = cat(2,rpowern{:}); ,nYZxYLf+  
        rpowern = [ones(length_r,1) rpowern]; T`Hw49  
    else (02g#A`  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); PqfVX8/q0  
        rpowern = cat(2,rpowern{:}); <}2A=~ _  
    end 9dYOH)f  
    \=g!$  
    }td6fj_{  
    % Compute the values of the polynomials: X_?%A54z?  
    % -------------------------------------- ?>?ZAr  
    y = zeros(length_r,length(n)); D_ ug-<QT  
    for j = 1:length(n) UK:M:9  
        s = 0:(n(j)-m_abs(j))/2; RKk"  
        pows = n(j):-2:m_abs(j); i'HST|!j  
        for k = length(s):-1:1 vnZ/tF  
            p = (1-2*mod(s(k),2))* ... "m>};.lj  
                       prod(2:(n(j)-s(k)))/              ... Xz* tbW#  
                       prod(2:s(k))/                     ... |"\lL9CT  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8b~7~VCk  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Y3M','H([  
            idx = (pows(k)==rpowers); 2'dG7lLu4  
            y(:,j) = y(:,j) + p*rpowern(:,idx); mxhW|}_-j  
        end AeQC:  
         rgn|24x  
        if isnorm Te}IMi:  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i*A$SJ:}  
        end f#c BQ~  
    end Cha?7F[xL  
    % END: Compute the Zernike Polynomials -faw:  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [Ekgft&  
    LOt#1Qv  
    6\mC$:F  
    % Compute the Zernike functions: f>4+,@G   
    % ------------------------------ %Fm`Y .l  
    idx_pos = m>0; hhj ,rcsi  
    idx_neg = m<0; )SD_}BY%k  
    8fEAYRGd  
    W7]mfy^  
    z = y; dcR6KG8  
    if any(idx_pos) 3]7ipwF2q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6(sfpK'  
    end (ai72#nFtb  
    if any(idx_neg) cnYYs d{  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E =  ^-Z  
    end "mG!L$  
    8ZzU^x  
    -KA4Inn]5  
    % EOF zernfun `F@f?*s:  
     
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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)  : .UX[!^  
    U)T/.L{0i  
    DDE还是手动输入的呢? X(0:zb,#G*  
    PLY-,Q&'  
    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