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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, -'j_JJ  
    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, ;b(*Bh<  
    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^OYgF  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 1}moT#  
    mVg$z  
    N3D{t\hg  
    Sn I-dXNF  
    }9&Z#1/  
    function z = zernfun(n,m,r,theta,nflag) tevB2'3^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. xz-z" 8d  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #1INOR9  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Ow0-}Im~  
    %   unit circle.  N is a vector of positive integers (including 0), and "f/Su(6{0  
    %   M is a vector with the same number of elements as N.  Each element O "jX|5  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Z /#&c  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Vv"JN?dHi  
    %   and THETA is a vector of angles.  R and THETA must have the same |i)7j G<  
    %   length.  The output Z is a matrix with one column for every (N,M) C #A sA  
    %   pair, and one row for every (R,THETA) pair. PSOW}Y|q  
    % ,_STt)  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 'W!N1W@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), T6gugDQ~.  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral pzX684  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1,  V Ae@P  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 1o&] =(  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RTPxAp+\5  
    % O~E6"v Q  
    %   The Zernike functions are an orthogonal basis on the unit circle. Q&zEa0^rG6  
    %   They are used in disciplines such as astronomy, optics, and DB1GW,  
    %   optometry to describe functions on a circular domain. D(EY"s37  
    % &d"c6il[  
    %   The following table lists the first 15 Zernike functions. AqPE.mf  
    % 5_bIc=L1  
    %       n    m    Zernike function           Normalization 'hTA O1n8  
    %       -------------------------------------------------- ,QDS_u$xi&  
    %       0    0    1                                 1 AOT +4*)%  
    %       1    1    r * cos(theta)                    2 4NY00d/R  
    %       1   -1    r * sin(theta)                    2 Y<~N x~w{  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) mN5`Fct*A>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) q|*}>=NX  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 8Iz-YG~%3  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) t<_Jx<{2  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) .~ )[>  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K"p$ga{  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) f.V1  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) >(v%"04|e  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >d.o1<  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~VNN  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4'&j<Ah[#  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) <ej Wl%4  
    %       -------------------------------------------------- S >E|A %  
    % JfJUOaL  
    %   Example 1: $U,`M"  
    % G8c 8`~t  
    %       % Display the Zernike function Z(n=5,m=1) (~YFm"S  
    %       x = -1:0.01:1; .rfufx9Sw  
    %       [X,Y] = meshgrid(x,x); KfC8~{O-  
    %       [theta,r] = cart2pol(X,Y); I\NiA>c  
    %       idx = r<=1; RR2Q  
    %       z = nan(size(X)); J. ]~J|K  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); +f{CfWIKs  
    %       figure Yzr RnVr  
    %       pcolor(x,x,z), shading interp :}r^sD  
    %       axis square, colorbar K<@gU\-!  
    %       title('Zernike function Z_5^1(r,\theta)') ;B%NFvG  
    % [ \I&/?On  
    %   Example 2: m$T?~o o  
    % h @{U>U7  
    %       % Display the first 10 Zernike functions P4"Pb\o*  
    %       x = -1:0.01:1; 'r KDw06/  
    %       [X,Y] = meshgrid(x,x); YkRv~bc1]  
    %       [theta,r] = cart2pol(X,Y); j@4 yRl ^  
    %       idx = r<=1; UQGOCP_  
    %       z = nan(size(X)); L nQm2uF  
    %       n = [0  1  1  2  2  2  3  3  3  3]; uZsm=('ww  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; :D-xa!7  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nC^|83  
    %       y = zernfun(n,m,r(idx),theta(idx)); C4Pi6.wf  
    %       figure('Units','normalized') F_8nxQ-  
    %       for k = 1:10 EJ$-  
    %           z(idx) = y(:,k); ;^5d^-T  
    %           subplot(4,7,Nplot(k)) l0c ws`V  
    %           pcolor(x,x,z), shading interp 4"$K66yk@  
    %           set(gca,'XTick',[],'YTick',[]) hFORs.L&G  
    %           axis square ahagt9[,:F  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) C -@  
    %       end %w <59d6  
    % al/3$0#U  
    %   See also ZERNPOL, ZERNFUN2. s1,kTde  
    *9"L?S(X#  
    7aH E:Dnwp  
    %   Paul Fricker 11/13/2006 I(6k.PQ  
    >QwZt  
    kyQUaFG  
    <VP@#  
    I!(.tu6u6c  
    % Check and prepare the inputs: ?b',kN,(  
    % ----------------------------- AXBv']Y  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2ql7*g?Uq@  
        error('zernfun:NMvectors','N and M must be vectors.') neQ2k=ao  
    end }D5*   
    ([loWr}QR  
    0- GA,I_  
    if length(n)~=length(m) ()=u#y  
        error('zernfun:NMlength','N and M must be the same length.') \>0F{-cR$  
    end ,BM6s,\  
    xk}(u`:.  
    +MG(YP/ l  
    n = n(:); ;IhkGPpWP  
    m = m(:); bP;cDQ(g  
    if any(mod(n-m,2)) zx7*Bnu0  
        error('zernfun:NMmultiplesof2', ... {7^7)^@  
              'All N and M must differ by multiples of 2 (including 0).') . e2qa  
    end ?#@JH  
    H-%)r&"vn  
    *&X.  
    if any(m>n) {p -b,J9~a  
        error('zernfun:MlessthanN', ... Y21,!$4gb  
              'Each M must be less than or equal to its corresponding N.') vt`hY4  
    end .Z=D|&!  
    cnAwoTt4  
    {r2-^Q HF  
    if any( r>1 | r<0 ) Zmf\A  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') O@U[S.IK  
    end lhm=(7Y  
    {Y{*(5YV  
    HjTK/x'_'L  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y$3H$F.+  
        error('zernfun:RTHvector','R and THETA must be vectors.') <wwcPe}  
    end Q2;zve&Dl  
    x;NCW  
    ^ri?eKy.-g  
    r = r(:); qX{m7  
    theta = theta(:); M70Xdn  
    length_r = length(r); $rf4h]&<  
    if length_r~=length(theta) jRXpEiM  
        error('zernfun:RTHlength', ... fRo_rj _  
              'The number of R- and THETA-values must be equal.') X&._<2  
    end [T', ZLR|  
    ;czMsHu0X  
    C)> ])'S  
    % Check normalization: <Vp7G%"'W  
    % -------------------- 3=xb%Upw  
    if nargin==5 && ischar(nflag) T*>n a8W  
        isnorm = strcmpi(nflag,'norm'); zECdj'/  
        if ~isnorm gkUG*Zw  
            error('zernfun:normalization','Unrecognized normalization flag.')  $3](6  
        end ShanwaCDqv  
    else G.K3'^_  
        isnorm = false; \ief [  
    end ^=Rqa \;  
    ,)A^3Q*  
    $NP5Z0v7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b7>^w<ki  
    % Compute the Zernike Polynomials R}4o{l6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ug!DL=ZW  
    .E|Hk,c9  
    "|pNS)  
    % Determine the required powers of r: -}k'a{sj=  
    % ----------------------------------- a#W:SgE?Y  
    m_abs = abs(m); DsY-JBDvoz  
    rpowers = []; tYyva  
    for j = 1:length(n) }NPF]P;  
        rpowers = [rpowers m_abs(j):2:n(j)]; ((rk)Q+;v  
    end vTYI ez`g  
    rpowers = unique(rpowers); 8Dpf{9Y-E  
    #V[ ?puE@  
    -CW&!oW  
    % Pre-compute the values of r raised to the required powers, Lys4l$J]  
    % and compile them in a matrix: }gL9G  
    % ----------------------------- xd8UdQ, lt  
    if rpowers(1)==0 s)<#a(!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $DW3H1iW  
        rpowern = cat(2,rpowern{:}); &NV[)6!  
        rpowern = [ones(length_r,1) rpowern]; sChMIbq!Av  
    else /h%<e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L1*P<Cb  
        rpowern = cat(2,rpowern{:}); 9BB<. p  
    end xbrxh-gV  
    3ydOBeY  
    Fa^5.p  
    % Compute the values of the polynomials: (+>~6SE  
    % -------------------------------------- hb9X<N+p  
    y = zeros(length_r,length(n)); i7 `dY {p7  
    for j = 1:length(n) V ?3>hQtB  
        s = 0:(n(j)-m_abs(j))/2; }[Uh4k8P  
        pows = n(j):-2:m_abs(j); e;pVoRI  
        for k = length(s):-1:1 ]9)pFL  
            p = (1-2*mod(s(k),2))* ... TCp!4-~,  
                       prod(2:(n(j)-s(k)))/              ... m}0US;c#f  
                       prod(2:s(k))/                     ... ayyn6a8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BQ[1,\>  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 'nI2RX  
            idx = (pows(k)==rpowers); 2;%DE<Z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); >]Hz-2b  
        end z h0m3|9O  
         I#@iA!  
        if isnorm "^gZh3  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); FCQoz"M  
        end 3tI=? E#  
    end r9@O`i  
    % END: Compute the Zernike Polynomials .%`|vGF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Uq9C-Iu  
    wNDLN`,^H  
    `|wH=  
    % Compute the Zernike functions: mp0p#8txi  
    % ------------------------------ JU:!lyd  
    idx_pos = m>0; zB\g'F/  
    idx_neg = m<0; KgVit+4u/  
    ]>/YU*\  
    [y}/QPR  
    z = y; ]R}#3(]1  
    if any(idx_pos) y#HD1SZ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');  O/gok+K  
    end R B.j@*  
    if any(idx_neg) rMSB|*_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c a_N76o!  
    end 4 C[,S|J  
    gB&]kHLO  
    z=!$3E ecr  
    % EOF zernfun x@2rfs  
     
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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)  :+R5"my  
    Y\op9 Fw  
    DDE还是手动输入的呢? |HG%o 3E]  
    2X*epU_1h  
    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