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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, Q8;x9o@p  
    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, DGa#d_I  
    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? DU/9/ I?~  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? IL+#ynC  
    r%g <h T 8  
    (!3Yc:~RE  
    |MOn0 *  
    vF.?] u  
    function z = zernfun(n,m,r,theta,nflag) .KT 7le<Zm  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?VMi!-POE  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;-3h~k  
    %   and angular frequency M, evaluated at positions (R,THETA) on the rt5oRf:wY  
    %   unit circle.  N is a vector of positive integers (including 0), and W\I$`gyC/  
    %   M is a vector with the same number of elements as N.  Each element jsk:fh0~M  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) qO:U]\P  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, a^RZsR  
    %   and THETA is a vector of angles.  R and THETA must have the same fap|SMGt  
    %   length.  The output Z is a matrix with one column for every (N,M) 07$/]eO%C  
    %   pair, and one row for every (R,THETA) pair. D 7Gd%  
    % hr J$%U  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike SjZd0H0  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), w;N{>)hv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [=XZza.z  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Nf=C?`L  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ]h #WkcXQ  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. sl~b\j  
    % 20 jrv'f  
    %   The Zernike functions are an orthogonal basis on the unit circle. '4af ],  
    %   They are used in disciplines such as astronomy, optics, and '4J&Gpx  
    %   optometry to describe functions on a circular domain. aj&\CJ  
    % +V2C}NQ5R  
    %   The following table lists the first 15 Zernike functions. UqD5 A~w  
    % $@_YdZ!  
    %       n    m    Zernike function           Normalization =L:[cIRrT;  
    %       -------------------------------------------------- l)}<#Ri  
    %       0    0    1                                 1 8&?^XcJ*x  
    %       1    1    r * cos(theta)                    2 5?^]1P_  
    %       1   -1    r * sin(theta)                    2 t@X M /=d  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) "EJ\]S]$X  
    %       2    0    (2*r^2 - 1)                    sqrt(3) V78Mq:7d  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) j1'\R+4U  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) -s{R/6 :  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) jeY4yM  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) g* %bzfk=|  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) H!p!sn  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) $B<~0'6}  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q?W r7  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) TLy ;4R2Nn  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4\v~HFsv  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) /#29Y^Z)=  
    %       -------------------------------------------------- ',DeP>'%>  
    % c qv .dC  
    %   Example 1: 6 tX.(/+L  
    % tAaYL \~  
    %       % Display the Zernike function Z(n=5,m=1) 6{L F-`S%  
    %       x = -1:0.01:1; ma3Qi/  
    %       [X,Y] = meshgrid(x,x); T)`gm{T  
    %       [theta,r] = cart2pol(X,Y); R;%^j=Q  
    %       idx = r<=1; bH_I7G&m  
    %       z = nan(size(X)); v Xc!Zg~  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); U8E0~[y'  
    %       figure CK=ARh#|  
    %       pcolor(x,x,z), shading interp [ ynuj3G V  
    %       axis square, colorbar )4PB<[u  
    %       title('Zernike function Z_5^1(r,\theta)') V w7WK  
    % #T[%6(QW  
    %   Example 2: 3=IG#6)~C  
    % 6I"C~&dt  
    %       % Display the first 10 Zernike functions Bf/ |{@  
    %       x = -1:0.01:1; >n(F4C-pl  
    %       [X,Y] = meshgrid(x,x); SGQD ro=l  
    %       [theta,r] = cart2pol(X,Y); RTZ:U@  
    %       idx = r<=1; TKd6MZhT  
    %       z = nan(size(X)); \PzN XQ$  
    %       n = [0  1  1  2  2  2  3  3  3  3];  C[R`Ml  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; \,hrk~4U;(  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; \d:h$  
    %       y = zernfun(n,m,r(idx),theta(idx)); / xs9.w8-  
    %       figure('Units','normalized') Ep<YCSQy$i  
    %       for k = 1:10 J,9%%S8/C  
    %           z(idx) = y(:,k); {-J:4*`  
    %           subplot(4,7,Nplot(k)) 1c / X  
    %           pcolor(x,x,z), shading interp qZ&a76t  
    %           set(gca,'XTick',[],'YTick',[]) -nOq\RYV  
    %           axis square /e .D /;]  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .8:+MW/  
    %       end svqvG7  
    % "U*5Z:8?9  
    %   See also ZERNPOL, ZERNFUN2. i0iez9B  
    I.-v?1>,  
    v[smQO  
    %   Paul Fricker 11/13/2006 HZ{n&iJ  
    Hk~k@Wft  
    yZ5 x8 8>  
    4QO/ff[ o  
    F(;jM(  
    % Check and prepare the inputs: X5 j=C]  
    % ----------------------------- .> wFztK  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4T%cTH:.9N  
        error('zernfun:NMvectors','N and M must be vectors.') dHq#  
    end Li]k7w?H  
    Gnk|^i;t  
    G0pBR]_5z$  
    if length(n)~=length(m) UUH;L  
        error('zernfun:NMlength','N and M must be the same length.') $ o " L;j  
    end MUB37  
    wA631kr  
    <ZVZ$ZW~D  
    n = n(:); |by@ :@*y  
    m = m(:); P.h.M A]  
    if any(mod(n-m,2)) rd" &QB{  
        error('zernfun:NMmultiplesof2', ... 9ad6uTc  
              'All N and M must differ by multiples of 2 (including 0).') 6rT4iC3Q{  
    end ~z`/9 ;  
    RG&6FRoq  
    B4^`Sw  
    if any(m>n) _ eiF@G  
        error('zernfun:MlessthanN', ... ;_N"Fdl  
              'Each M must be less than or equal to its corresponding N.') g|4w8ry  
    end a,cC!   
    rVcBl4&1*g  
    fhr-Y'  
    if any( r>1 | r<0 ) o:9$UV[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') iadkH]w  
    end (u9Zk~)F  
    a:b^!H>#  
    Pr/]0<s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Yf= FeH7"  
        error('zernfun:RTHvector','R and THETA must be vectors.') 4TVwa(cB  
    end @[v8}D  
    KuXkI;63J>  
    )*D'csGc  
    r = r(:); 2yxi= XWZ  
    theta = theta(:); ,ux+Qz5(  
    length_r = length(r); y<*-tZV[  
    if length_r~=length(theta) wDw<KU1UK  
        error('zernfun:RTHlength', ... @c]Xh:I  
              'The number of R- and THETA-values must be equal.') 6pm~sD  
    end |[LE9Lq/  
    8[R1A  
    Q.ukY@L.'  
    % Check normalization: C{&)(#*L  
    % -------------------- g`3H(PVg  
    if nargin==5 && ischar(nflag) ._,trb>o  
        isnorm = strcmpi(nflag,'norm'); "i%jQL'.  
        if ~isnorm e8q4O|I_  
            error('zernfun:normalization','Unrecognized normalization flag.') 'hIU_  
        end <+q$XL0  
    else @n@g)`  
        isnorm = false; s5A gsMq  
    end |X3">U +-  
     5~s{N  
    s0lYj@E'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |_nC6 ;  
    % Compute the Zernike Polynomials wv^b_DR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @|=UrKAN  
    ! Rvn'|!  
    1R^4C8*B  
    % Determine the required powers of r: c/'M#h)"  
    % ----------------------------------- 5Eal1Qu  
    m_abs = abs(m); r0Z+ RB^I  
    rpowers = []; aTClw<6}  
    for j = 1:length(n) GX5W^//}  
        rpowers = [rpowers m_abs(j):2:n(j)]; #_fY4vEO  
    end EneAX&SG  
    rpowers = unique(rpowers); S&01SX6  
    KZ  )Ys  
    \ 3G*j`  
    % Pre-compute the values of r raised to the required powers, MS{{R +&  
    % and compile them in a matrix: :o$@F-$k  
    % ----------------------------- "kr,x3 =  
    if rpowers(1)==0 L#ZLawG  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "mt p0  
        rpowern = cat(2,rpowern{:}); 7E\gxQ(vU  
        rpowern = [ones(length_r,1) rpowern]; )S Q('vwg  
    else pYh!]0n  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m}pL`:e!  
        rpowern = cat(2,rpowern{:}); Mj'lASI  
    end Q c3?}os2  
    X HQh4W3  
    {MxnIg7'  
    % Compute the values of the polynomials: Bk@WW#b  
    % -------------------------------------- 9A+M|;O  
    y = zeros(length_r,length(n)); =qX*]  
    for j = 1:length(n) p%8 v`  
        s = 0:(n(j)-m_abs(j))/2; !qaDn.9  
        pows = n(j):-2:m_abs(j); _.=`>%,  
        for k = length(s):-1:1 b^Z$hnh]S  
            p = (1-2*mod(s(k),2))* ... LU( %K{9  
                       prod(2:(n(j)-s(k)))/              ... ~d>uXrb  
                       prod(2:s(k))/                     ... ;dOs0/UM&  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <soj&f+  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 6l[G1KkV  
            idx = (pows(k)==rpowers); r{Z[xWIX  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [Auc*@  
        end c _mq  
         I+~bCcgPi  
        if isnorm 9gR.RwR X  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /x/4NeD  
        end B@-"1m~la?  
    end ob]dZ  
    % END: Compute the Zernike Polynomials IXJ6PpQLv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZOn_dYjC  
    avBua6i'  
    M5 `m.n<  
    % Compute the Zernike functions: LfllO  
    % ------------------------------ gLx/w\l6  
    idx_pos = m>0; 4oN${7k0  
    idx_neg = m<0; `oVB!eapl  
    [?I/Uo8  
    (Com,  
    z = y; f8#*mQ  
    if any(idx_pos) 7t3X`db  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); z^3Q.4Qc6^  
    end o$\tHzB9!A  
    if any(idx_neg) UM`nq;>  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]hKgA~;  
    end >[8#hSk  
    Gql`>~  
    6y9C@5p}B  
    % EOF zernfun e2bLkb3c  
     
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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)  ipfm'aQ  
    k-io$  
    DDE还是手动输入的呢? 1 iquHn  
    9*f2b.Aj  
    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