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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, {r].SrW9s9  
    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, O{B e )E~  
    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? :Hf0Qx6  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? <h@z=ijN  
    +i`Q 7+d  
    SA(UD   
    >Z2,^5P{  
    yK&* ,J |  
    function z = zernfun(n,m,r,theta,nflag) o1#:j?sN  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. E&];>3C  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /J[H5uA  
    %   and angular frequency M, evaluated at positions (R,THETA) on the E/dO7I`B   
    %   unit circle.  N is a vector of positive integers (including 0), and gP %|:"  
    %   M is a vector with the same number of elements as N.  Each element L*UV  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) U7]<U-.&  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, S[L#M;n  
    %   and THETA is a vector of angles.  R and THETA must have the same I NPYJ#%  
    %   length.  The output Z is a matrix with one column for every (N,M) 2GiUPtO&Gj  
    %   pair, and one row for every (R,THETA) pair. &'huS?g A9  
    % 9b"9m*gC  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike S7UZGGjTk  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 62MRI    
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral YH'$_,8peM  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mZbWRqP[|_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `\/toddUh[  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P>{US1t  
    % J+}+ "h~.  
    %   The Zernike functions are an orthogonal basis on the unit circle. Z@uTkqG)  
    %   They are used in disciplines such as astronomy, optics, and >k&8el6h  
    %   optometry to describe functions on a circular domain. UK"}}nO@e  
    % Z p7yaz3y  
    %   The following table lists the first 15 Zernike functions. a@fE46o6<  
    % *?^Z)C>  
    %       n    m    Zernike function           Normalization 3C rQBIj1  
    %       -------------------------------------------------- Wa[x`:cT?u  
    %       0    0    1                                 1 S]e j=6SP  
    %       1    1    r * cos(theta)                    2 +9CEC1-l  
    %       1   -1    r * sin(theta)                    2 B]^>GH  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 4?>18%7&  
    %       2    0    (2*r^2 - 1)                    sqrt(3) XOysgX0g  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) * MSBjH|  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9^ >M>f"  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ]g;^w?9h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Sc1+(z  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) :W.jNV{e\F  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) {J,6iP{>ZN  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -,~;qSs  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f {y]  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <`R|a *  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2PVx++*]C  
    %       -------------------------------------------------- |'V DI]p&  
    %  SwdC,  
    %   Example 1: E /fw?7eQ  
    % ]ZzoJ7lr  
    %       % Display the Zernike function Z(n=5,m=1) ^Yj"RM$;N  
    %       x = -1:0.01:1; K-J|/eB  
    %       [X,Y] = meshgrid(x,x); ="uKWt6n'  
    %       [theta,r] = cart2pol(X,Y); _\ .  
    %       idx = r<=1; cS<TmS!  
    %       z = nan(size(X)); V#ndyUM;  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); PUbaS{J7  
    %       figure X}oj_zsy;^  
    %       pcolor(x,x,z), shading interp 7"ylN"syZ  
    %       axis square, colorbar iD>G!\&  
    %       title('Zernike function Z_5^1(r,\theta)') )Vwj9WD  
    % "| K f'/r  
    %   Example 2: `9.dgV  
    % 6m4Te|  
    %       % Display the first 10 Zernike functions F,*2#:Ki  
    %       x = -1:0.01:1; ]>tq|R78  
    %       [X,Y] = meshgrid(x,x); %mY|  
    %       [theta,r] = cart2pol(X,Y); z^4KU\/JK  
    %       idx = r<=1; 9<xTu>7J  
    %       z = nan(size(X)); M[ x_#m|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; F\>oxttS1  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; `kv1@aQPL  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Q`H# fS~  
    %       y = zernfun(n,m,r(idx),theta(idx)); blJIto '  
    %       figure('Units','normalized') )-=2w-ZX  
    %       for k = 1:10  X ?tj$  
    %           z(idx) = y(:,k); ]EB6+x!G  
    %           subplot(4,7,Nplot(k)) {IJ-4>  
    %           pcolor(x,x,z), shading interp tf4*R_6;1$  
    %           set(gca,'XTick',[],'YTick',[]) g[/^cJHQ  
    %           axis square >@^<S_KVh  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >&bv\R/  
    %       end l`SK*Bm~<  
    % 9160L qY  
    %   See also ZERNPOL, ZERNFUN2. <5dH *K  
    _1sP.0 t  
    |5W8Q|>%  
    %   Paul Fricker 11/13/2006 i-`,/e~XT  
    nz^nptw  
    h ~ $&  
    }04Dg '  
    "X`RQ6~]>  
    % Check and prepare the inputs: aiYo8+{!#  
    % ----------------------------- _*Pfp+if  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) l1&5uwuF  
        error('zernfun:NMvectors','N and M must be vectors.') ~%`EeJwT  
    end cn$5:%IK  
    Zb]/nP1P  
    bZiyapM  
    if length(n)~=length(m) !~WZ_z  
        error('zernfun:NMlength','N and M must be the same length.') ugno]5Ni  
    end pjACFVMFX  
    sH%&+4!3  
    s3seK6x'  
    n = n(:); d>&\V)E  
    m = m(:); V{!lk]p}a  
    if any(mod(n-m,2)) W+8^P( K  
        error('zernfun:NMmultiplesof2', ... %*6RzJO6  
              'All N and M must differ by multiples of 2 (including 0).') ' PELf P8  
    end *|oPxQCtK  
    ~x'zX-@rC  
    EJ G2^DSS  
    if any(m>n) D ZVXz|g  
        error('zernfun:MlessthanN', ... l8^y]M  
              'Each M must be less than or equal to its corresponding N.') CJp-Y}fGEA  
    end :<|Z.4}kJb  
    |~eY%LB  
    @l{I[pp  
    if any( r>1 | r<0 ) }wfI4?}j}  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') WHP;Neb6  
    end AuAT]`  
    ABcBEv3  
    L?HF'5o  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 0(8gQ 2n  
        error('zernfun:RTHvector','R and THETA must be vectors.') Ah (iE  
    end vO]J]][  
     //<:k8  
    ;}D-:J-z_  
    r = r(:); JA<~xo[Q9  
    theta = theta(:); Pg Syt  
    length_r = length(r); ugI#ZFjJWE  
    if length_r~=length(theta) KSc~GP _  
        error('zernfun:RTHlength', ... ^sV|ck  
              'The number of R- and THETA-values must be equal.') zks#EzQ  
    end )!eEO [\d  
    ENq"mwV|  
    ds]?;l"  
    % Check normalization: ^>^ \CP]  
    % -------------------- Jn*Nao_)  
    if nargin==5 && ischar(nflag) g5}lLKT  
        isnorm = strcmpi(nflag,'norm'); VHW`NP 5Jl  
        if ~isnorm ,Aj }]h\L  
            error('zernfun:normalization','Unrecognized normalization flag.') xQo~%wW,?  
        end <(YF5Xm6$h  
    else WNa3^K/W{  
        isnorm = false; IcFK,y%1  
    end K6hfauWd[  
    :CTL)ad2  
    p![&8i@ym  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~ M*gsW$  
    % Compute the Zernike Polynomials j&CZ=?K^c  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C`0%C7  
    O: JPJ"!  
    .E$q&7@/j  
    % Determine the required powers of r:  2:'lZQ  
    % ----------------------------------- C2G  |?=  
    m_abs = abs(m); 4%7s259%  
    rpowers = []; +9zA^0   
    for j = 1:length(n) TV=c,*TV  
        rpowers = [rpowers m_abs(j):2:n(j)]; rz.IoQo  
    end u s`}  
    rpowers = unique(rpowers); N@()F&e  
    -NzTqLBn  
    1Nj=B_T  
    % Pre-compute the values of r raised to the required powers, fa{@$ppx  
    % and compile them in a matrix: uN bIX:L,  
    % ----------------------------- &SmXI5>Bo0  
    if rpowers(1)==0 EwQae(PpA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .&iN(Bd  
        rpowern = cat(2,rpowern{:}); ltSh'w0  
        rpowern = [ones(length_r,1) rpowern]; h<% U["   
    else .S_QQM}Q  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); L/x(RCD  
        rpowern = cat(2,rpowern{:}); Dtt-|_EMS  
    end yW("G-Nm  
    <d"Gg/@a  
    XWtiwf'K  
    % Compute the values of the polynomials: 7Z0/(V.-  
    % -------------------------------------- SF< [FM%1  
    y = zeros(length_r,length(n)); \Y e%o}.{  
    for j = 1:length(n) 4SR(->@  
        s = 0:(n(j)-m_abs(j))/2; J3=BE2L  
        pows = n(j):-2:m_abs(j); $<OhGk-  
        for k = length(s):-1:1 e$|VG* d  
            p = (1-2*mod(s(k),2))* ... Wc|z7P~',%  
                       prod(2:(n(j)-s(k)))/              ... .zS D`v@[  
                       prod(2:s(k))/                     ... |I^y0Q:K  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... G),db%,X2  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); B 8{ uR  
            idx = (pows(k)==rpowers); dy:d=Z  
            y(:,j) = y(:,j) + p*rpowern(:,idx); /{X_ .fv<v  
        end w$>3pQ8d  
         H$tb;:  
        if isnorm KlU qoJ;"  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Rla4L`X;  
        end O]qPmEj  
    end bulboyA&#  
    % END: Compute the Zernike Polynomials  $Nu)E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u D(t`W"  
    L~eAQR  
    |zpx)8Q  
    % Compute the Zernike functions: S$O,] @)  
    % ------------------------------ <xlm K(  
    idx_pos = m>0; :woa&(wN;1  
    idx_neg = m<0; @~o`#$*|  
    U3F3((EYJ  
     %+wF"  
    z = y; cy1jZ1)  
    if any(idx_pos) z*LiweR-  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $Ha%Gr  
    end 9=$ !gC)  
    if any(idx_neg) ;+`uER  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); gX,9Gh  
    end U9#WN.noG  
    2OalAY6RS  
    :3? |VE F  
    % EOF zernfun p4wr`" Zz  
     
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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)
    在线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)  PqEAqP  
    mJMq{6;  
    DDE还是手动输入的呢? E`)Qs[?Gk  
    dAxp ,):&J  
    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