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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, 5o5y3ibQ  
    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, F+_4Q  
    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? (KHTgZ6  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? h@T}WZv  
    A}sb 2P  
    iZQwo3"8r  
    Te~"\`omJ3  
    {hX. R  
    function z = zernfun(n,m,r,theta,nflag) 3C8'0DB  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5DfAL;o!  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X|H%jdta  
    %   and angular frequency M, evaluated at positions (R,THETA) on the gO?+:}!  
    %   unit circle.  N is a vector of positive integers (including 0), and pK#Ze/!  
    %   M is a vector with the same number of elements as N.  Each element oq=D9  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) O k_I}X  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [SgP1>M  
    %   and THETA is a vector of angles.  R and THETA must have the same 8f% @  
    %   length.  The output Z is a matrix with one column for every (N,M) SHPaSq'&N  
    %   pair, and one row for every (R,THETA) pair. 'z2}qJJ)  
    % _tL*sA>[~)  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )]!Ps` ,u  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PEoO s  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral =O w}MX  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, yE-&TW_q:>  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized J1Mm,LTO  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. YcGSZ0vQ  
    % pK4I?=A'  
    %   The Zernike functions are an orthogonal basis on the unit circle. 5B .+>u"e  
    %   They are used in disciplines such as astronomy, optics, and cn=~}T@~Z  
    %   optometry to describe functions on a circular domain. \w^iSK-  
    % Xd66"k\b+  
    %   The following table lists the first 15 Zernike functions. -[v:1\Vv  
    % y%=\E  
    %       n    m    Zernike function           Normalization ^v3ytS  
    %       -------------------------------------------------- 7(eWBJfTo  
    %       0    0    1                                 1 } O9q$-8!  
    %       1    1    r * cos(theta)                    2 1&Rz'JQ+  
    %       1   -1    r * sin(theta)                    2 M'W@K  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3`J?as@^8  
    %       2    0    (2*r^2 - 1)                    sqrt(3) U}6'_ PRQ  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) t qbS!r  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) FgNO#%  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) R* E/E  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 4>{q("r,  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;or(:Yoc-  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {LY$  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ? 8S0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) N6$pOQ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z}s0D]$+x  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8=T;R&U^M  
    %       -------------------------------------------------- vAq`*]W+  
    % V{$(#r  
    %   Example 1: 0X`Qt[  
    % Mvrc[s+o  
    %       % Display the Zernike function Z(n=5,m=1) S3:Pjz}t  
    %       x = -1:0.01:1; RqXcL,,9  
    %       [X,Y] = meshgrid(x,x); LCRreIIgZ  
    %       [theta,r] = cart2pol(X,Y); f$iv+7<B^  
    %       idx = r<=1; U{RW=sYB~9  
    %       z = nan(size(X)); ;) 5d wq  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Yp./3b VO  
    %       figure VCcLS3  
    %       pcolor(x,x,z), shading interp : +/V  
    %       axis square, colorbar NUEy0pLw  
    %       title('Zernike function Z_5^1(r,\theta)') 8Cs)_bj#!  
    % lOPCM1Se  
    %   Example 2: N/TU cG|m\  
    % $=4T# W=m  
    %       % Display the first 10 Zernike functions utQE$0F  
    %       x = -1:0.01:1; wZh&w<l'  
    %       [X,Y] = meshgrid(x,x); <O?iJ=$  
    %       [theta,r] = cart2pol(X,Y); bAeC=?U  
    %       idx = r<=1; Va\dMv-b  
    %       z = nan(size(X)); J8J~$DU\Gv  
    %       n = [0  1  1  2  2  2  3  3  3  3]; V? w;YTg  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; _,=A\C_b@  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; >,y291p2  
    %       y = zernfun(n,m,r(idx),theta(idx)); nyi}~sB  
    %       figure('Units','normalized') )(9>r /bq  
    %       for k = 1:10 4Ucg<Z&%  
    %           z(idx) = y(:,k); `ndesP  
    %           subplot(4,7,Nplot(k)) IwKhun  
    %           pcolor(x,x,z), shading interp PSI5$Vna4p  
    %           set(gca,'XTick',[],'YTick',[]) y!6B Gz  
    %           axis square H`njKKdR  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7!#x-KR~5  
    %       end {x W? v;  
    % 36*"oD=@  
    %   See also ZERNPOL, ZERNFUN2. @R_a'v-  
    Q'~kWmLf  
    &v Lz{  
    %   Paul Fricker 11/13/2006 (#BkL:dg  
    Y _m4:9p  
    _~&6Kb^*  
    A)kx,,[  
    8E&}+DR?  
    % Check and prepare the inputs: $/Gvz)M  
    % ----------------------------- @ JZ I  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cNtGjLpx;  
        error('zernfun:NMvectors','N and M must be vectors.') zu5'Ex`gQa  
    end A`T VV  
    UZi^ &  
    C~ t?<  
    if length(n)~=length(m) L)a8W   
        error('zernfun:NMlength','N and M must be the same length.') bTHKMaGWC  
    end {^i73}@O  
    yMq&9R9F  
    gD3s,<>o  
    n = n(:); =MEv{9_  
    m = m(:); dFS>uIT7X  
    if any(mod(n-m,2)) 5B#q/d1/a  
        error('zernfun:NMmultiplesof2', ... i6?,2\K  
              'All N and M must differ by multiples of 2 (including 0).') l)[\TD  
    end <{bQl L  
    swYlp  
    ;n%SjQ'%  
    if any(m>n) ];Z)=y,vM  
        error('zernfun:MlessthanN', ... :'91qA%Wr  
              'Each M must be less than or equal to its corresponding N.') :6S!1roi  
    end !Y>lAxd  
    <k<K"{  
    .+MJ' bW  
    if any( r>1 | r<0 ) |!E>I  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') CL.JalR`b  
    end &PaqqU.  
    ns[v.YDL  
    eqU2>bI f  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SeN4gr*  
        error('zernfun:RTHvector','R and THETA must be vectors.') (9% ki$=}+  
    end M$~3`n*^  
    f uQbDb&  
    3('=+d[}Vw  
    r = r(:); Ni#!C:q  
    theta = theta(:); Aayh'xQ  
    length_r = length(r); <nlZ?~%}  
    if length_r~=length(theta) 11[[Hk X@  
        error('zernfun:RTHlength', ... ZQXv-"  
              'The number of R- and THETA-values must be equal.') oW(lQ'"  
    end {STOWuY  
    0]4kR8R3[  
    ?%% 'GX  
    % Check normalization: d9>*a$x;/  
    % -------------------- o(w!x!["  
    if nargin==5 && ischar(nflag) 5LdVcXf  
        isnorm = strcmpi(nflag,'norm'); (|)`~z  
        if ~isnorm |z\5Ik!fF]  
            error('zernfun:normalization','Unrecognized normalization flag.') w F6ywr  
        end ;*1bTdB5a  
    else G6(k wv4  
        isnorm = false; [ -"o5!0<  
    end d0Xb?- }3M  
    =F'p#N0_2  
    yI/2 e[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /_<`#?5T(  
    % Compute the Zernike Polynomials fZ1v|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oNQ;9&Z,^2  
    W&CQ87b  
    ,Tc3koi  
    % Determine the required powers of r: oJa6)+b(3  
    % ----------------------------------- bwo-9B  
    m_abs = abs(m); x2x) y08  
    rpowers = []; w}No ^.I*4  
    for j = 1:length(n) cpvN }G  
        rpowers = [rpowers m_abs(j):2:n(j)]; J@D5C4>i  
    end mkgGX|k;  
    rpowers = unique(rpowers); Mx<z34(T  
    pYZ6-s  
    y_EkW f  
    % Pre-compute the values of r raised to the required powers, rE0?R( _  
    % and compile them in a matrix: aEU[k>&  
    % ----------------------------- BCsz8U!  
    if rpowers(1)==0 ,<?iL~> %  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .{sKEVK  
        rpowern = cat(2,rpowern{:});  R}Pw#*B  
        rpowern = [ones(length_r,1) rpowern]; w}+#w8hu  
    else S^q)DuF5!  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >dKK [E/[d  
        rpowern = cat(2,rpowern{:}); j1 _ E^  
    end 7pMl:\  
    r@N 0%JZZ  
    n][/c_]q  
    % Compute the values of the polynomials: !Ic;;<  
    % -------------------------------------- x g=}MoX  
    y = zeros(length_r,length(n)); ].F7. zi  
    for j = 1:length(n) J-*&&  
        s = 0:(n(j)-m_abs(j))/2; vSty.:bY\p  
        pows = n(j):-2:m_abs(j); }s)MDq9  
        for k = length(s):-1:1 b`"E(S/  
            p = (1-2*mod(s(k),2))* ... Q#C;4)e  
                       prod(2:(n(j)-s(k)))/              ... 272j$T  
                       prod(2:s(k))/                     ... L9tjH C]  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,M2u (9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); XMhDx  
            idx = (pows(k)==rpowers); @X`~r8&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); K&FGTS,  
        end GMmz`O XN  
         VBc[(8o  
        if isnorm LhM{LUi  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v|5:;,I  
        end D|-^}I4  
    end f[,9WkC  
    % END: Compute the Zernike Polynomials ?^Sk17G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !iKR~&UpAL  
    y,qP$ 5xiq  
    5dffF e  
    % Compute the Zernike functions: rM<lPMr1*  
    % ------------------------------ 1I({2@C  
    idx_pos = m>0; }e3M5LI1L  
    idx_neg = m<0; ~wnTl[:  
    W{E2 2J}  
    Pn@k)g  
    z = y; p7(Pymkd  
    if any(idx_pos) /dTy%hZC}  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^NJ]~h{n$  
    end Xx{ho 4qq  
    if any(idx_neg) ""Ul6hRgv  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); dz/' m7  
    end vW4~\]  
    `@GqD  
    !_GY\@}  
    % EOF zernfun )6|7L)Dk  
     
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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)  qq-&z6;$  
    =khjD[muC  
    DDE还是手动输入的呢? 6uDA{[OH  
    ]wne2WXE  
    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