切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9099阅读
    • 5回复

    [讨论]如何从zernike矩中提取出zernike系数啊 [复制链接]

    上一主题 下一主题
    离线jssylttc
     
    发帖
    25
    光币
    13
    光券
    0
    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, 4H'9y3dk  
    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, /2c?+04+  
    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? JSM{|HJxh  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? _+GCd8d  
    qVf~\H@  
    #8'%CUF*<8  
    D,2,4h!ka  
    {YkW5zC(L  
    function z = zernfun(n,m,r,theta,nflag) tw;`H( UZ^  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. qYE-z( i  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N (t <Um Vd  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 1:-$mt_*  
    %   unit circle.  N is a vector of positive integers (including 0), and f@ySTz;u  
    %   M is a vector with the same number of elements as N.  Each element "*UHit;"+{  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) :U~[%]  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, VB Ce=<  
    %   and THETA is a vector of angles.  R and THETA must have the same J &c}z4  
    %   length.  The output Z is a matrix with one column for every (N,M) r8mE   
    %   pair, and one row for every (R,THETA) pair. Es?~Dd  
    % PS>k67sI  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !=ZbBUJF  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), aFLm,  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~q<U E\H  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, IE3GM^7\  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized il*bsnwpZv  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. c1c0b|B!U  
    % `jP6;i  
    %   The Zernike functions are an orthogonal basis on the unit circle. P",53R+"  
    %   They are used in disciplines such as astronomy, optics, and rXA7<_Vg  
    %   optometry to describe functions on a circular domain. ]R0^ }sI  
    % R!:1{1  
    %   The following table lists the first 15 Zernike functions. gbF.Q7?$u  
    % )=~1m85+5B  
    %       n    m    Zernike function           Normalization 8G9V8hS1#B  
    %       -------------------------------------------------- =_,w<  
    %       0    0    1                                 1 $"sf%{~  
    %       1    1    r * cos(theta)                    2 <#:"vnm$j  
    %       1   -1    r * sin(theta)                    2 /QTGZ b  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) qUCiB}  
    %       2    0    (2*r^2 - 1)                    sqrt(3) <MY_{o8d  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4rv3D@E  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) .a$][Jny  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) t0/fF'GZD  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .x}ImI  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ^}9Aq $R  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !IP[C?(nB  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9v^MZ ^Y{  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NX$$4<A1  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;gf^;%FK  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) #qHo+M$"  
    %       -------------------------------------------------- UAa2oY&  
    % i4AmNRs  
    %   Example 1: lv,<[Hw1  
    % &AC-?R|Dp  
    %       % Display the Zernike function Z(n=5,m=1) an.)2*u  
    %       x = -1:0.01:1; "#(]{MY  
    %       [X,Y] = meshgrid(x,x); U1dz:OG>  
    %       [theta,r] = cart2pol(X,Y); Z|E( !"zE9  
    %       idx = r<=1; )'92{-A0  
    %       z = nan(size(X)); j&ddpS(s  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); haS`V  
    %       figure /8lGP! z  
    %       pcolor(x,x,z), shading interp \#  
    %       axis square, colorbar r'-)@|  
    %       title('Zernike function Z_5^1(r,\theta)') t[%9z6t  
    % ^BW V6  
    %   Example 2: ]e 81O#t3  
    % Bx2E9/S3  
    %       % Display the first 10 Zernike functions }wz )"  
    %       x = -1:0.01:1; u.R:/H<>~  
    %       [X,Y] = meshgrid(x,x); J=5G<  
    %       [theta,r] = cart2pol(X,Y); J %URg=r  
    %       idx = r<=1; $}N'm  
    %       z = nan(size(X)); %=%jy  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ` Y"Rh[C  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; p<tj6O  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; '3aDvV0  
    %       y = zernfun(n,m,r(idx),theta(idx)); uG~%/7Qt{  
    %       figure('Units','normalized') IYb@@Jzo  
    %       for k = 1:10 XV]`?  
    %           z(idx) = y(:,k); i e%ZX  
    %           subplot(4,7,Nplot(k)) d2Bn`VI  
    %           pcolor(x,x,z), shading interp 0~Z2$`(  
    %           set(gca,'XTick',[],'YTick',[]) 5,k&^CK}  
    %           axis square b2 duC  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9-I;'  
    %       end 3@_je)s  
    % "EDn;l-Q  
    %   See also ZERNPOL, ZERNFUN2. {C[<7r uF  
    JmtU>2z\  
    }r9f}yX9Q  
    %   Paul Fricker 11/13/2006 JEGcZeq)  
    %BC*h}KGH  
    ySL 31%  
    RV.*_FG  
    -%NT)o  
    % Check and prepare the inputs: xLPyV&j-  
    % ----------------------------- ;q59Cr75  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ay22-/C|@  
        error('zernfun:NMvectors','N and M must be vectors.') W1iKn  
    end $*{PUj  
    /)e&4.6  
    ~W_m<#K(  
    if length(n)~=length(m) Q9]7.^l  
        error('zernfun:NMlength','N and M must be the same length.') 2(Vm0E  
    end ; P&K a  
    y/'2WO[  
    0,{Dw9W:  
    n = n(:); HFB2ep7N  
    m = m(:); Zm4IN3FGLv  
    if any(mod(n-m,2)) ?S36)oZzg  
        error('zernfun:NMmultiplesof2', ... gQCkoQi:j  
              'All N and M must differ by multiples of 2 (including 0).') i\ X Ok!  
    end uL1e?  
    3W5|Y@0  
    pdngM 8n  
    if any(m>n) b(&2/|hd  
        error('zernfun:MlessthanN', ... j_H{_Ug  
              'Each M must be less than or equal to its corresponding N.') k^:$ETW2 D  
    end *}$T:kTH  
    <}L`d(E@f  
    s0~a5Ti3  
    if any( r>1 | r<0 ) kVCS FF*  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') u I}S9  
    end Eg FV  
     I QS|  
    f nX!wN  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) DvKM[z3j  
        error('zernfun:RTHvector','R and THETA must be vectors.') ;oH17  
    end HpC|dtro  
    U"v(9m@  
    T2AyQ~5~  
    r = r(:); }5z6b>EI9a  
    theta = theta(:); FVPhk2  
    length_r = length(r); C7dy{:y`  
    if length_r~=length(theta) $6L gaz  
        error('zernfun:RTHlength', ... h \hQ  
              'The number of R- and THETA-values must be equal.') >JkQ U e  
    end rUvqAfE&+  
    1Thr74M  
    (wdE@/V  
    % Check normalization: nU-.a5  
    % -------------------- Qx1ZxJz #  
    if nargin==5 && ischar(nflag) W/<]mm~95  
        isnorm = strcmpi(nflag,'norm');  Jx9S@L`  
        if ~isnorm Og4 X3QG  
            error('zernfun:normalization','Unrecognized normalization flag.') KdHR.;*  
        end "WdGY*r  
    else R] tHd=kf  
        isnorm = false; _ r0oOpE  
    end 4_Tx FulX.  
    E3<jH  
    22"M#:r$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,A[40SZA  
    % Compute the Zernike Polynomials 1mm/Ssw:C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0(VH8@h`O  
    `C%,Nj  
    %<6oKE  
    % Determine the required powers of r: 8xJdK'  
    % ----------------------------------- ^3B{|cqf  
    m_abs = abs(m); FbO-K-  
    rpowers = []; d8`^;T ;}d  
    for j = 1:length(n) BG_m}3j  
        rpowers = [rpowers m_abs(j):2:n(j)]; z6#N f,  
    end uc<XdFcu  
    rpowers = unique(rpowers); 6Xb\a^ q  
    ]:(>r&'  
    FY)vrM*yh  
    % Pre-compute the values of r raised to the required powers, Q:& ,8h[  
    % and compile them in a matrix: D|/Azy.[  
    % ----------------------------- <mjH#aSy  
    if rpowers(1)==0 \:mx Ri  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); VI,z7 \  
        rpowern = cat(2,rpowern{:}); yw^t6E  
        rpowern = [ones(length_r,1) rpowern]; %Qgo0  
    else 4- ^|e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kbJ/7  
        rpowern = cat(2,rpowern{:}); C(Ujx=G+3  
    end @ +h2R  
    QDYS}{A:V  
    QMea2q|3$  
    % Compute the values of the polynomials: g6o-/A!Q3  
    % -------------------------------------- O6LZ<}oUR  
    y = zeros(length_r,length(n)); [X0Wfb}{  
    for j = 1:length(n) ]`0(^)U &  
        s = 0:(n(j)-m_abs(j))/2; rVowHP  
        pows = n(j):-2:m_abs(j); (C*G)Aj7  
        for k = length(s):-1:1 BoYWx^VHx^  
            p = (1-2*mod(s(k),2))* ... V|zzj[c  
                       prod(2:(n(j)-s(k)))/              ... +Gqh  
                       prod(2:s(k))/                     ... H$au02dpU  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... rQg7r>%Q  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); O9wZx%<  
            idx = (pows(k)==rpowers); ?6+GE_VZ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); Rcs7 'q5  
        end +6@".<  
         8fFURk  
        if isnorm Ay;=1g)8+f  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u6IEBYG ((  
        end 9-[g/qrF  
    end ]^$&Ejpe#  
    % END: Compute the Zernike Polynomials A1e|Y  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H>AQlO+J  
    >e :&kp  
    c) Zid1  
    % Compute the Zernike functions: oNY;z-QK  
    % ------------------------------ }C!N$8d,  
    idx_pos = m>0; | VPs5  
    idx_neg = m<0; g#ubxC7t<  
    z #c)Q  
    9:"%j  
    z = y; Zm& X $U  
    if any(idx_pos) ^]o]'  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); b<};"H0a  
    end (.4mX t  
    if any(idx_neg) +Rn]6}5m\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ; S7 %  
    end fQRGz\r*k  
    ^d Fdw\  
    !:t}8  
    % EOF zernfun fC+<n{"C  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    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
    发帖
    959
    光币
    1087
    光券
    1
    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
    发帖
    25
    光币
    13
    光券
    0
    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)  ^f][;>c  
    'u"r^o?  
    DDE还是手动输入的呢? es]S]}JV  
    ErZYPl  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    离线jssylttc
    发帖
    25
    光币
    13
    光券
    0
    只看该作者 4楼 发表于: 2012-05-14
    顶顶·········
    离线18257342135
    发帖
    51
    光币
    1518
    光券
    0
    只看该作者 5楼 发表于: 2016-12-13
    支持一下,慢慢研究