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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, I`H&b& .`  
    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, y<6c*e1  
    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? 2#sFY/@  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? $ P 5K   
    yF? O+9R A  
    !Q15qvRS  
    ]& ckq  
    T3@2e0u )  
    function z = zernfun(n,m,r,theta,nflag) HbI{Xf[6LP  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. wbrOL(q.m  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D 5n\h5  
    %   and angular frequency M, evaluated at positions (R,THETA) on the " nCK%w=  
    %   unit circle.  N is a vector of positive integers (including 0), and =eUKpYI  
    %   M is a vector with the same number of elements as N.  Each element ye9GBAj /  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) j?m(l,YD|*  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, L"vrX  
    %   and THETA is a vector of angles.  R and THETA must have the same uX3yq<lK"  
    %   length.  The output Z is a matrix with one column for every (N,M) O>qlWPht  
    %   pair, and one row for every (R,THETA) pair. ))f@9m  
    % I3Z?xsa@Z  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %W\NYSm  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :_6o|9J\t  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral kC%H E  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, * @]wT'  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Lw,}wM5X  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &uRT/+18W3  
    % ~6n|GxR.[  
    %   The Zernike functions are an orthogonal basis on the unit circle. i@nRZ$K  
    %   They are used in disciplines such as astronomy, optics, and zPp22  
    %   optometry to describe functions on a circular domain. #%k_V+o3  
    % iv_3R}IbX  
    %   The following table lists the first 15 Zernike functions. 9!n95  
    % s(3u\#P  
    %       n    m    Zernike function           Normalization _Sfu8k>):  
    %       -------------------------------------------------- -LRx}Mb9  
    %       0    0    1                                 1 F$tzsz,9n  
    %       1    1    r * cos(theta)                    2 ;&=CZ6vH  
    %       1   -1    r * sin(theta)                    2 >8I~i:hn  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ^6kl4:{idE  
    %       2    0    (2*r^2 - 1)                    sqrt(3) k1xx>=md|C  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) t;VMtIW+E  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) _]o7iqtv  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) WUie `p  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) KJoa^e;~  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 'uL$j=vB  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) W`9{RZ'  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g]L8Jli  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1q!k#Cliu  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) J' P:SC1  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) eK1l~W%  
    %       -------------------------------------------------- qzxWv5UH  
    % b}Gm{;s!  
    %   Example 1: rhPv{6Z|7  
    % D? %*L  
    %       % Display the Zernike function Z(n=5,m=1) >l)x~Bkf$j  
    %       x = -1:0.01:1; 2EpQ(G J  
    %       [X,Y] = meshgrid(x,x); (Ud"+a  
    %       [theta,r] = cart2pol(X,Y); ? 3fnt"  
    %       idx = r<=1; 5 ,MM`:{{  
    %       z = nan(size(X)); <Rw2F?S~)n  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Cj~e` VRhk  
    %       figure 81|[Y'f  
    %       pcolor(x,x,z), shading interp 3n7>qZ.d  
    %       axis square, colorbar I_8 n>\u  
    %       title('Zernike function Z_5^1(r,\theta)') RjUrpS[I  
    % ^^ix4[1$Z  
    %   Example 2: ,|$1(z*a{c  
    % SbUac<  
    %       % Display the first 10 Zernike functions Xmmj.ZUr  
    %       x = -1:0.01:1; KS5a8'U  
    %       [X,Y] = meshgrid(x,x); 8hww({S2  
    %       [theta,r] = cart2pol(X,Y); mm#UaEp  
    %       idx = r<=1; !sI^Lh,Y  
    %       z = nan(size(X)); VSUWX1k4%  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ImB5F'HI$  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; H'$H@Kn]-  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; S3HyB b  
    %       y = zernfun(n,m,r(idx),theta(idx)); e9'0CH<  
    %       figure('Units','normalized') X*7VDt=  
    %       for k = 1:10 3jfAv@I~  
    %           z(idx) = y(:,k); /: -&b#+  
    %           subplot(4,7,Nplot(k)) t; #@t/`  
    %           pcolor(x,x,z), shading interp be-HF;lZe'  
    %           set(gca,'XTick',[],'YTick',[]) >f&L7@  
    %           axis square \uo{I~Qd  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Zr6.Nw  
    %       end PL31(!`@d  
    % kene' aDm  
    %   See also ZERNPOL, ZERNFUN2. y~OP9Tg  
    Y>c+j  
    +:aNgO#e8  
    %   Paul Fricker 11/13/2006 mcQ A'  
    iSOyp\E|  
    \TzBu?,v8  
    NuF?:L[  
    ^u90N>Dvq  
    % Check and prepare the inputs: yfqe6-8U  
    % ----------------------------- \AI-x$5R*  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) c*<BU6y  
        error('zernfun:NMvectors','N and M must be vectors.') M;AvOk|&  
    end ?%ltoezf  
    -~J5aG[@~>  
    rR{KnM  
    if length(n)~=length(m) "85)2*+  
        error('zernfun:NMlength','N and M must be the same length.') 6e.l# c!1}  
    end 'O2/PU2_  
    %)d7iT~M  
    =?]S8cth  
    n = n(:); ZhRdml4U2  
    m = m(:); Hd-g|'^K  
    if any(mod(n-m,2)) m#_M"B.cm  
        error('zernfun:NMmultiplesof2', ... OM7AK B=S  
              'All N and M must differ by multiples of 2 (including 0).')  N?,  
    end lc [)Ev  
    H<nA*Zf2@R  
    yP` K [/  
    if any(m>n) C(>g4.-p8  
        error('zernfun:MlessthanN', ... T~ XKV`LQ  
              'Each M must be less than or equal to its corresponding N.') `|92!Ej  
    end ZcHIk{|  
    (6 }7z+  
    9G7Brs:  
    if any( r>1 | r<0 ) @x[A ^  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \j.l1O  
    end Y>8JHoV  
    ]70ZerQ~L  
    oxnI/Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |,H 2ge  
        error('zernfun:RTHvector','R and THETA must be vectors.') F]<2nb7  
    end dxS5-aWy9w  
    ,E%O_:}R  
    b8%TwYp  
    r = r(:); at\u7>;.^k  
    theta = theta(:); >-3>Rjo>  
    length_r = length(r); Ll#W:~  
    if length_r~=length(theta) 4}*.0'Hz  
        error('zernfun:RTHlength', ... +.rOqkxJ  
              'The number of R- and THETA-values must be equal.') L0{ [L  
    end &?xtmg<d  
    0#m=76[b  
    !`W0;0'Zg  
    % Check normalization: Gv#bd05X  
    % -------------------- nC?Lz1re  
    if nargin==5 && ischar(nflag) 7G Erh,  
        isnorm = strcmpi(nflag,'norm'); x",ktE>9  
        if ~isnorm +`$$^x  
            error('zernfun:normalization','Unrecognized normalization flag.') w7q6v>  
        end zyP/'X_~:  
    else ,S`F xJcE  
        isnorm = false; ~oK0k_{~  
    end +nB0O/m'U  
    23'{{@30  
    $Tt.r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {(t R<z)  
    % Compute the Zernike Polynomials /+ais 3  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QOPh3+.5  
    \;Q!}_ K  
    5'`DrTOA  
    % Determine the required powers of r: *.D{d0A  
    % ----------------------------------- -Oz! GX  
    m_abs = abs(m); !\Cu J5U  
    rpowers = []; ,R7j9#D  
    for j = 1:length(n) ZnuRy:  
        rpowers = [rpowers m_abs(j):2:n(j)]; MJH>rsTQ  
    end @`^Z5n.4  
    rpowers = unique(rpowers); \F+".X#jh  
    Yn?Xo_Y  
    <*/Z>Z_c2  
    % Pre-compute the values of r raised to the required powers, 2FO<Z %Y  
    % and compile them in a matrix: pu ?CO A  
    % ----------------------------- XgeUS;qtta  
    if rpowers(1)==0 hKnV=Ha(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 7*WO9R/  
        rpowern = cat(2,rpowern{:}); x'6i9]+r  
        rpowern = [ones(length_r,1) rpowern]; bwszfPM  
    else W?ghG  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W(-son~I  
        rpowern = cat(2,rpowern{:}); z 9WeOs  
    end Y9st3  
    +;oR_]l  
    &wu1Zz[qcz  
    % Compute the values of the polynomials: 3.ShAL  
    % -------------------------------------- Xw|-v$'y  
    y = zeros(length_r,length(n)); #i.BOQxS  
    for j = 1:length(n) uI9+@oV  
        s = 0:(n(j)-m_abs(j))/2; _oefp*iWS  
        pows = n(j):-2:m_abs(j); WZTv  
        for k = length(s):-1:1 ,u- 9e4  
            p = (1-2*mod(s(k),2))* ... NH=@[t) P,  
                       prod(2:(n(j)-s(k)))/              ... MFWkJbZV  
                       prod(2:s(k))/                     ... 2$o\`^dy  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... f3.oc9G  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 2<'ol65/c  
            idx = (pows(k)==rpowers); K05T`+N,  
            y(:,j) = y(:,j) + p*rpowern(:,idx); L i 9$N"2  
        end  >Af0S;S  
         ol {N^fi K  
        if isnorm ?UeV5<TewS  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qn}VW0!  
        end h^14/L=|  
    end ;.R) uCd{=  
    % END: Compute the Zernike Polynomials mW,b#'hy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +-5YmN'  
    + kF%>F]  
    y T&#k1  
    % Compute the Zernike functions: rx<P#y]3)  
    % ------------------------------ :n%&  
    idx_pos = m>0; Vk[M .=J  
    idx_neg = m<0; g$/7km{TP  
    Bc b '4*:  
    N6%L4v8-}X  
    z = y; ^L.'At  
    if any(idx_pos) A2P.5EN  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0]nveC$  
    end ZcTjOy?  
    if any(idx_neg) .O&YdUo  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |S:erYE,G  
    end iYlkc  
    t/3qD7L  
    G)o:R iq  
    % EOF zernfun W!+=`[Ff  
     
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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)  =-qYp0sVP  
    rX)_!mR  
    DDE还是手动输入的呢? v] m`rV8S[  
    0n={Mb  
    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