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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, wpYk`L r  
    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, *l{epum;  
    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? 5v)bs\x6  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? m N}szW,  
    AK,'KO%{=  
    a{r"$>0  
    QK+,63@D\=  
    #f) TAA  
    function z = zernfun(n,m,r,theta,nflag) 7~QI4'e  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. s$>n U  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }}$@Tij19[  
    %   and angular frequency M, evaluated at positions (R,THETA) on the h#O9TB  
    %   unit circle.  N is a vector of positive integers (including 0), and jw=PeT|  
    %   M is a vector with the same number of elements as N.  Each element o zn&>k  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ceE]^X;p  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, eIalcBY  
    %   and THETA is a vector of angles.  R and THETA must have the same 5[SwF& zZ  
    %   length.  The output Z is a matrix with one column for every (N,M) y`buY+5l  
    %   pair, and one row for every (R,THETA) pair. O7VEyQqf5  
    % ').) 0;  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike CUI+@|]%  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .7^(~&5N  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral #O}}pF  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, $\h-F8|JMX  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized *PnO$q@`  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m].  &Q~W{.  
    % k*fU:q1  
    %   The Zernike functions are an orthogonal basis on the unit circle. WM ?a1j  
    %   They are used in disciplines such as astronomy, optics, and *"8Ls0!  
    %   optometry to describe functions on a circular domain. 4)8VmCW  
    % K-C,n~-  
    %   The following table lists the first 15 Zernike functions. (?\+  
    % 1Y'4 g3T  
    %       n    m    Zernike function           Normalization d6QrB"J`  
    %       -------------------------------------------------- }psRgF  
    %       0    0    1                                 1 v>} +->f  
    %       1    1    r * cos(theta)                    2 Blzvn19'h  
    %       1   -1    r * sin(theta)                    2 7:u+cv  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) /VT/KT{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ;z4F-SYQ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) h7"U1'b  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) !B%em%Tv  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Y\-xX:n.\  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ,sAAV%" >  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) bJ!\eI%ld  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 1}DA| !~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 11yXI[  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NAvR^"I~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \s5Uvws  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) V+ ("kz*  
    %       -------------------------------------------------- o2ggHZe/=@  
    % J/4T=:\  
    %   Example 1: XJ4f;U  
    % f*X CWr  
    %       % Display the Zernike function Z(n=5,m=1) 1z-.e$&z  
    %       x = -1:0.01:1; DQXUh#t\(]  
    %       [X,Y] = meshgrid(x,x); lWId 0eNS  
    %       [theta,r] = cart2pol(X,Y); ,D+ydr  
    %       idx = r<=1; [v"Z2F<.=  
    %       z = nan(size(X)); j1K3|E  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); SU~a()"  
    %       figure LBK{-(%  
    %       pcolor(x,x,z), shading interp 0 jth}\9  
    %       axis square, colorbar .r<a Py$  
    %       title('Zernike function Z_5^1(r,\theta)') ':wf%_Iw  
    % |qUGB.Q  
    %   Example 2: nTqU~'d'  
    % Pqomi!1  
    %       % Display the first 10 Zernike functions ^Qs}2%  
    %       x = -1:0.01:1; MuY:(zC%  
    %       [X,Y] = meshgrid(x,x); 'K,\  
    %       [theta,r] = cart2pol(X,Y); q`<:CfCt  
    %       idx = r<=1; yV{B,T`W  
    %       z = nan(size(X)); c1'@_Is  
    %       n = [0  1  1  2  2  2  3  3  3  3]; l'+3 6  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; wGArR7r  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; |RiJ>/ MK\  
    %       y = zernfun(n,m,r(idx),theta(idx)); QO'Hyf t  
    %       figure('Units','normalized') i?6&4  
    %       for k = 1:10 ,&t+D-s<f  
    %           z(idx) = y(:,k); i<Vc~ !pT  
    %           subplot(4,7,Nplot(k)) +FT c/r  
    %           pcolor(x,x,z), shading interp Y P2VSK2Q  
    %           set(gca,'XTick',[],'YTick',[]) lYx_8x2  
    %           axis square 03 @a G  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `>:5[Y  
    %       end A>@#eyB  
    % OM\J4"YV$  
    %   See also ZERNPOL, ZERNFUN2. t}q e_c  
    ;28d7e}  
    u;`]U$Qq9  
    %   Paul Fricker 11/13/2006 i1 E|lp)  
    (0$~T}lH  
    JmI%7bH@  
    B@,r8)D  
    o^"+X7)  
    % Check and prepare the inputs: Ma^jy.  
    % ----------------------------- vhrf89-q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DB'0  
        error('zernfun:NMvectors','N and M must be vectors.') 8MJJ w;  
    end Q]k< Y  
    t%=7v)IOE  
    yE$PLM  
    if length(n)~=length(m) j_8 YFz5  
        error('zernfun:NMlength','N and M must be the same length.') 5PeS/%uT@  
    end }%< ?]  
    boo361L  
    hg)Xr5>  
    n = n(:); VdHT3r  
    m = m(:); NdXHpq;  
    if any(mod(n-m,2)) DSrU7#  
        error('zernfun:NMmultiplesof2', ...  U4!bW  
              'All N and M must differ by multiples of 2 (including 0).') RM2Ik_IH[l  
    end \((iR>^|  
    clE9I<1v  
    Ni_H1G  
    if any(m>n) Xoe|]@U`  
        error('zernfun:MlessthanN', ... ]*2),H1 c  
              'Each M must be less than or equal to its corresponding N.') ~MG6evm &  
    end 1W USp;JMl  
    h3MdQlJ&  
    TDh)}Ms  
    if any( r>1 | r<0 ) "Lp.*o  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'n &p5%  
    end t>bzo6cj  
    iQG!-.aX  
    x93@[B*%  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .n 9.y8C  
        error('zernfun:RTHvector','R and THETA must be vectors.') P3oYk_oW  
    end PQHztS"  
    GkAd"<B  
    c1H.v^Y5  
    r = r(:); *lfjsrPu  
    theta = theta(:);  {53FR  
    length_r = length(r); CmU@8-1  
    if length_r~=length(theta) K9<8FSn  
        error('zernfun:RTHlength', ... 9jal D X  
              'The number of R- and THETA-values must be equal.') JYdb^j2c  
    end _J,**AZ~z  
    49qa  
    l)u%`Hcn  
    % Check normalization: dwA"QVp{  
    % -------------------- }z]d]  
    if nargin==5 && ischar(nflag) mF6-f#t>H+  
        isnorm = strcmpi(nflag,'norm'); /X}1%p  
        if ~isnorm HhbBt'fH  
            error('zernfun:normalization','Unrecognized normalization flag.') RoqkT|#$  
        end bmT%?it  
    else !?,, ZD  
        isnorm = false; N_%@_$3G]  
    end 4H8r[  
    }&v}S6T  
    Qf:e;1F!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t>[QW`EeP  
    % Compute the Zernike Polynomials (kL"*y/"p  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P]+B}))  
    ~kc#"^s J  
    !'$*Z(  
    % Determine the required powers of r: =ejcP&-V/  
    % ----------------------------------- uP9b^LEoN  
    m_abs = abs(m); Bc=(1ty)  
    rpowers = []; O"\4[HE^  
    for j = 1:length(n) |!oC7!+0^  
        rpowers = [rpowers m_abs(j):2:n(j)]; l$u52e!7  
    end $QiMA,  
    rpowers = unique(rpowers); -jjB2xP  
    %|jS`kj  
    3W'fEh5  
    % Pre-compute the values of r raised to the required powers, r/h\>s+N  
    % and compile them in a matrix: >MYxj}I4{z  
    % ----------------------------- 7w73,r/D8A  
    if rpowers(1)==0 $1=7^v[U  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); FBE|pG7  
        rpowern = cat(2,rpowern{:}); MR "f)  
        rpowern = [ones(length_r,1) rpowern]; =eA|gt  
    else EW$drY@  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); A!Tl  
        rpowern = cat(2,rpowern{:}); ^!tX+`,6^  
    end aZf/WiR2  
    a"s2N%{  
    bU gg2iFS  
    % Compute the values of the polynomials: :$I "n\  
    % -------------------------------------- *twGIX  
    y = zeros(length_r,length(n)); =p|IWn{P  
    for j = 1:length(n) @<K<"`~H  
        s = 0:(n(j)-m_abs(j))/2; CC^D4]ug  
        pows = n(j):-2:m_abs(j); \d:Q%S  
        for k = length(s):-1:1 T4x%3-4 ;  
            p = (1-2*mod(s(k),2))* ... O+!4KNN.-  
                       prod(2:(n(j)-s(k)))/              ... 05F/&+V  
                       prod(2:s(k))/                     ... !>(uhuTBF  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >V.?XZ nt  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); %)i&|AV"  
            idx = (pows(k)==rpowers); LR&MhG7  
            y(:,j) = y(:,j) + p*rpowern(:,idx); :r{-:   
        end Ry[7PLn]  
         Q`i@['?p  
        if isnorm vU *: M8k  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6$#,$aO  
        end .i\ FK@2  
    end c Lyf[z)W  
    % END: Compute the Zernike Polynomials $.C\H,H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FqyxvL.  
    {]Mwuqn  
    n\9IRuYO  
    % Compute the Zernike functions: Pjq'c+4.yL  
    % ------------------------------ T6y~iNd<  
    idx_pos = m>0; gZHgL7@  
    idx_neg = m<0; p#c41_?'e  
    4UbqYl3 |a  
    P^o@x,V!&  
    z = y; jR\pYRK  
    if any(idx_pos) 5[2kk5,  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;(mNjxA  
    end p` ~=v4;b  
    if any(idx_neg) }#g]qK  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <Hv/1:k}  
    end 5_A*I C]  
    O<fy^[r:`  
    CeU=A9  
    % EOF zernfun b~ )@e9  
     
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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)  N)D+FV29y  
    ?zC{T*a  
    DDE还是手动输入的呢? u^MRKLn  
    qe(gKKA%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