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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, ?#]K54?  
    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, AfvTStwr  
    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? ;aYPv8s~,:  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? sQW$P9s c  
    6suB!XF;  
     N3^pFy`  
    'qLk"   
    z79L2lJn  
    function z = zernfun(n,m,r,theta,nflag) Avw"[~Xd  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. uE:#m.Q  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +T[3wL~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ><iEVrpN  
    %   unit circle.  N is a vector of positive integers (including 0), and G 8|[.n  
    %   M is a vector with the same number of elements as N.  Each element 8 g'9( )&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) bj` cYL%  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, l/OG 79qq  
    %   and THETA is a vector of angles.  R and THETA must have the same v}dt**l  
    %   length.  The output Z is a matrix with one column for every (N,M) L]0+ u\(  
    %   pair, and one row for every (R,THETA) pair. RLY Ae  
    % "d'xT/l "  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (omdmT%D  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9\TvX!)h  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral en7i})v\".  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, xt +fu L  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized "Ks%!  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (]j*)~=V  
    % S6}_Z  
    %   The Zernike functions are an orthogonal basis on the unit circle. 3hR7 . /  
    %   They are used in disciplines such as astronomy, optics, and G/(oQA  
    %   optometry to describe functions on a circular domain. )?'sw5C  
    % O60jC;{F  
    %   The following table lists the first 15 Zernike functions. `ZN@L<I6  
    % u]E%R&  
    %       n    m    Zernike function           Normalization $Z 10Zf=  
    %       -------------------------------------------------- FVG|5'V^  
    %       0    0    1                                 1 a[s%2>e  
    %       1    1    r * cos(theta)                    2 Cd#*Wp)s  
    %       1   -1    r * sin(theta)                    2 |NtT-T)7  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #Vn=(U4}!_  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 23+6u{   
    %       2    2    r^2 * sin(2*theta)             sqrt(6) : ` F>B  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) L3q)j\ ls  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e^~t52]  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 9 )B>|#\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) BO[Q"g$Kon  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 2EE/xnwX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ] >ipC,v  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) &:]_a?|*S  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oZ6xHdPc4  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =i%2/kdi0b  
    %       -------------------------------------------------- Fh v)  
    % qCgP8U/jv  
    %   Example 1: NL&g/4A[a  
    % R$,`}@VqZ3  
    %       % Display the Zernike function Z(n=5,m=1) 2!68W X  
    %       x = -1:0.01:1; C==tJog[  
    %       [X,Y] = meshgrid(x,x); 9[T#uh!DC  
    %       [theta,r] = cart2pol(X,Y); 1b3Lan_2  
    %       idx = r<=1; | nry^zb  
    %       z = nan(size(X)); q*{"6"4(  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Cy6[p  
    %       figure 3{MIBMA  
    %       pcolor(x,x,z), shading interp @T/C<-/:  
    %       axis square, colorbar n^&QOII@>  
    %       title('Zernike function Z_5^1(r,\theta)') -<z'f){gb  
    % gK)B3dH*&  
    %   Example 2: qwFn(pK[  
    % NBMY1Xgj  
    %       % Display the first 10 Zernike functions $<s@S;Ri  
    %       x = -1:0.01:1; <S$y=>.9  
    %       [X,Y] = meshgrid(x,x); aE{b65'Dt  
    %       [theta,r] = cart2pol(X,Y); =j;o, J:(  
    %       idx = r<=1; y_$^Po  
    %       z = nan(size(X)); *y(2BrL>  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8-?n<h%8E  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; n+uq|sYVa  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; )0}obPp  
    %       y = zernfun(n,m,r(idx),theta(idx)); H8\{ GGg  
    %       figure('Units','normalized') mz\ m^g3  
    %       for k = 1:10 y Fp1@*ef  
    %           z(idx) = y(:,k); bjT0Fi0-  
    %           subplot(4,7,Nplot(k)) 8#Z$}?W  
    %           pcolor(x,x,z), shading interp +'#d*r91@  
    %           set(gca,'XTick',[],'YTick',[]) ZN4&:9M  
    %           axis square cQ+, F2  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Be]o2N;J  
    %       end r1?LKoJOn  
    % K4R jGSaF  
    %   See also ZERNPOL, ZERNFUN2. HYg _{  
    HKxrBQr78  
    :sA-$*&x  
    %   Paul Fricker 11/13/2006 uwQ4RYz  
    fZ %ZV  
    IB;y8e,  
    \pPq ]k  
    O0$ijJa|  
    % Check and prepare the inputs: wy -!1wd  
    % ----------------------------- IS=)J( 0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?5GjH~  
        error('zernfun:NMvectors','N and M must be vectors.') 3K0J6/mc  
    end iTK1I0  
    qob!!A14p  
    Ahwu'mgnC  
    if length(n)~=length(m) jgMWjM6.  
        error('zernfun:NMlength','N and M must be the same length.') S7SPc   
    end x)Th2es\  
    U)l>#gf8  
    rU~"A  
    n = n(:); CNN?8/u!@  
    m = m(:); ?PQiVL  
    if any(mod(n-m,2)) EwOTG Y{0p  
        error('zernfun:NMmultiplesof2', ... ;;`KkNys m  
              'All N and M must differ by multiples of 2 (including 0).') g,W#3b6>j  
    end d z\b]H]  
    &a(w0<  
    ~,guw7F  
    if any(m>n) 02+^rqIx5  
        error('zernfun:MlessthanN', ... mcR!P~"i  
              'Each M must be less than or equal to its corresponding N.') @v'<~9vG  
    end ]E3g8?L  
    ?nn,RBS-  
    "l@~WE  
    if any( r>1 | r<0 ) (J;?eeP  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') =1JRu[&]8  
    end 6x7=0}'  
    h7w<.zwu t  
    TDseWdA  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .5z|g@ 6  
        error('zernfun:RTHvector','R and THETA must be vectors.') tsa6: D  
    end u,]yd*  
    oy'+n-  
    x SUR<  
    r = r(:); ~iSW^mi  
    theta = theta(:); Af%?WZlOq  
    length_r = length(r); eyG.XAP  
    if length_r~=length(theta) [/kO >  
        error('zernfun:RTHlength', ... V:+bq`  
              'The number of R- and THETA-values must be equal.') S`^W#,rj  
    end iUKj:q:  
     (M=Br  
    2u:j6ic  
    % Check normalization: !(~>-;A8  
    % -------------------- h ^c'L=dR  
    if nargin==5 && ischar(nflag) `sXx,sV?B  
        isnorm = strcmpi(nflag,'norm'); C G7 LF  
        if ~isnorm f:SF&t*  
            error('zernfun:normalization','Unrecognized normalization flag.') u rOGOa$  
        end @W,Y_8:  
    else r/v&tU  
        isnorm = false; ^/uGcz|.  
    end Y^G3<.B  
    g=(+oK?  
    oGqv,[$qN  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #+G2ZJxL|  
    % Compute the Zernike Polynomials n\YxRs7 hF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vB{b/xmah  
    aFym&n\  
    {Vm36/a  
    % Determine the required powers of r: MD)"r>k  
    % ----------------------------------- X3nhqQTZ  
    m_abs = abs(m); LA+MX 0*  
    rpowers = []; 1`t?5|s>  
    for j = 1:length(n) Uu+C<j&-  
        rpowers = [rpowers m_abs(j):2:n(j)]; a3 x~B=E  
    end <7^~r(DP  
    rpowers = unique(rpowers); bij?q\  
    &^H "T6  
    ;cr6Xop#?  
    % Pre-compute the values of r raised to the required powers,  R'/wOE2  
    % and compile them in a matrix: fz3*oJ'  
    % ----------------------------- >C[1@-]G%7  
    if rpowers(1)==0 :ct+.#  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DE ws+y-*  
        rpowern = cat(2,rpowern{:}); VZoOdR:d  
        rpowern = [ones(length_r,1) rpowern]; A& F4;>dms  
    else G#: !wI  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Oy&'zigJ  
        rpowern = cat(2,rpowern{:}); <^Tj}5 )n  
    end ^Q>*f/.KN  
    F21[r!3  
    t] wM_]+  
    % Compute the values of the polynomials: 6hK"k  
    % -------------------------------------- gpWS_Dw9  
    y = zeros(length_r,length(n)); @E2nF|N  
    for j = 1:length(n) %b;+/s2W  
        s = 0:(n(j)-m_abs(j))/2; =fG8YZ(  
        pows = n(j):-2:m_abs(j); LDeVNVM  
        for k = length(s):-1:1 E+zn\v  
            p = (1-2*mod(s(k),2))* ... .M2&ad :  
                       prod(2:(n(j)-s(k)))/              ... SZ{cno1`  
                       prod(2:s(k))/                     ... GuWBl$|+b  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... XB-|gPk  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); E{ s|#  
            idx = (pows(k)==rpowers); QtQ^"d65  
            y(:,j) = y(:,j) + p*rpowern(:,idx); =bWq 3aP)P  
        end QJWES%m`  
         |:+pPh!-  
        if isnorm o$VH,2 QF  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3gy;$}Lq T  
        end *^6xt7  
    end +c`C9RXk  
    % END: Compute the Zernike Polynomials "NH+qQhs  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~q(C j"7  
    R"gm]SQ/  
    tQ JH'YV  
    % Compute the Zernike functions: ~#_$?_/(  
    % ------------------------------ HF+fk*_Q  
    idx_pos = m>0; gsWlTI  
    idx_neg = m<0; 3b@1Zahz  
    IQ=|Kj9h  
    ' ,`4 U F  
    z = y; [KI`e  
    if any(idx_pos) y~c[sW   
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8;\tP29  
    end ;n{j,HB  
    if any(idx_neg) ysJhP .  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); X]MM7hMuR  
    end }|"*"kxi!  
    rqe_zyc&  
    5z w23!  
    % EOF zernfun Qfu*F}  
     
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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)  XC/]u%n8](  
    |!xfIR>=F  
    DDE还是手动输入的呢? H6PXx  
    TH(Lzrbg  
    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