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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, %a%x`S3  
    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, c,a+u  
    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? t_HS0rxG  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛?  nN!/  
    <}S1ZEZcQ  
    LB}y,-vX>  
    s[h& Uv"G  
    9U1cH qV  
    function z = zernfun(n,m,r,theta,nflag) <Z%iP{  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ZS51QB  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N C2RR(n=N^  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 2_@vSwC  
    %   unit circle.  N is a vector of positive integers (including 0), and pp{Za@j  
    %   M is a vector with the same number of elements as N.  Each element ~e,k71  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) )SG+9!AbMZ  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 'V";"Ei  
    %   and THETA is a vector of angles.  R and THETA must have the same #~J)?JL  
    %   length.  The output Z is a matrix with one column for every (N,M) :A%|'HxH3  
    %   pair, and one row for every (R,THETA) pair. Vy-N3L  
    % /Po't(-x  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rblEyCR  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A<ca9g3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral f,GF3vu"  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _^cDB1I ?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8z&7wO  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~nk{\ rWO  
    % bQG2tDvu[  
    %   The Zernike functions are an orthogonal basis on the unit circle. t,#9i#q#  
    %   They are used in disciplines such as astronomy, optics, and ycAQHY~n  
    %   optometry to describe functions on a circular domain. wYnsd7@I  
    % 69{^Vfd;Y  
    %   The following table lists the first 15 Zernike functions. 3=w$1.B d  
    % [<m1xr4"k  
    %       n    m    Zernike function           Normalization .6Jo1$+  
    %       -------------------------------------------------- ,f0|eu>  
    %       0    0    1                                 1 g{K*EL <  
    %       1    1    r * cos(theta)                    2 (jYHaTL6Y'  
    %       1   -1    r * sin(theta)                    2 }C1&}hZ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 6XyhOs%/  
    %       2    0    (2*r^2 - 1)                    sqrt(3) II$B"-  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) _:oB#-0  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ara D_D  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 8>" vAEf  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 7UQFAt_r  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ~"eos~AuW  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 0M^7#),  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c@d[HstBJ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) TR:V7 d  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [@"~'fu0  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) UH=pQm ^W  
    %       -------------------------------------------------- u 0M[B7Q  
    % * SH5p  
    %   Example 1: ">='l9  
    % 5Vo8z8]t`  
    %       % Display the Zernike function Z(n=5,m=1) qN h:;`  
    %       x = -1:0.01:1; YTH3t] &  
    %       [X,Y] = meshgrid(x,x); :o$k(X7a  
    %       [theta,r] = cart2pol(X,Y); !{'C.sb?~  
    %       idx = r<=1; |F)BKo D  
    %       z = nan(size(X)); Rlc$2y@pU  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); $10"lM[  
    %       figure 5 [{l9  
    %       pcolor(x,x,z), shading interp LIfQh  
    %       axis square, colorbar jWHv9XtW  
    %       title('Zernike function Z_5^1(r,\theta)') Pf`HF|NI  
    % )C^ZzmB  
    %   Example 2: ]PWK^-4P  
    % F+yu[Dh:  
    %       % Display the first 10 Zernike functions bgD4;)?5b  
    %       x = -1:0.01:1; %j3XoRex><  
    %       [X,Y] = meshgrid(x,x); tkT:5O6  
    %       [theta,r] = cart2pol(X,Y); mS)|i+5  
    %       idx = r<=1; s~N WJ*i  
    %       z = nan(size(X)); +T]/4"^M  
    %       n = [0  1  1  2  2  2  3  3  3  3]; HCOv<k  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 1/b5i8I2 v  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; DIrQ5C  
    %       y = zernfun(n,m,r(idx),theta(idx)); IM-O<T6r[N  
    %       figure('Units','normalized') "+SnHpNx  
    %       for k = 1:10 $tKz|H)  
    %           z(idx) = y(:,k); (jj=CLe  
    %           subplot(4,7,Nplot(k)) "u#,#z_  
    %           pcolor(x,x,z), shading interp WdQR^'b$   
    %           set(gca,'XTick',[],'YTick',[]) n*twuB/P 1  
    %           axis square x-0O3IIE  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fpd4 v|(  
    %       end N]yh8"7X  
    % yU ?TdM\  
    %   See also ZERNPOL, ZERNFUN2. Er@'X0n  
     {yXpBS  
    b+b].,  
    %   Paul Fricker 11/13/2006 =i'APeNaQ  
    -^C^3pms  
    Cp!bsasj  
    V)|]w[(Y  
    "{TVd>9_  
    % Check and prepare the inputs: @\ udaZc  
    % ----------------------------- JDbRv'F:(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~w Ekbq=  
        error('zernfun:NMvectors','N and M must be vectors.') Epo/}y  
    end = Ob-'Syg>  
    pNt,RRoR  
    l~",<bTc  
    if length(n)~=length(m) MS7rD%(,'  
        error('zernfun:NMlength','N and M must be the same length.') a!?JVhD&  
    end 2~ [  
    VD.wO%9?)  
    TR7j`?  
    n = n(:); 0j\} @  
    m = m(:); W}6OMAbsE;  
    if any(mod(n-m,2)) qDlh6W?}k  
        error('zernfun:NMmultiplesof2', ... t%S2D  
              'All N and M must differ by multiples of 2 (including 0).') G;jX@XqZ  
    end 7+'&(^c  
    $kAal26z  
    SN#Cnu}  
    if any(m>n) !xD$U/%c  
        error('zernfun:MlessthanN', ... }0okyGg>q  
              'Each M must be less than or equal to its corresponding N.') rt8"U <~  
    end zWO!z =  
    ~DJILc  
    _P}wO8  
    if any( r>1 | r<0 ) {JGXdp:SB  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') _&SST)Y|  
    end ^55q~DP}>  
    '&LH9r  
    rbw~Ml0  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .ri?p:a}w  
        error('zernfun:RTHvector','R and THETA must be vectors.') tO}Y=kZa{  
    end ']C" 'b  
    P*!~Z *"  
    ^ }kqAmr  
    r = r(:); VX6M4<8  
    theta = theta(:); *L{^em#b  
    length_r = length(r); j=kz^o~mH  
    if length_r~=length(theta) !Bu=?gf  
        error('zernfun:RTHlength', ... k*u4N  
              'The number of R- and THETA-values must be equal.') f^]^IXzXw.  
    end w+][L||4c  
    werTwe2Q  
    WF_24Mw  
    % Check normalization: wl N l|+ K  
    % -------------------- INNTp[  
    if nargin==5 && ischar(nflag) {>h,@  
        isnorm = strcmpi(nflag,'norm'); ]|8*l]oc  
        if ~isnorm FT;I|+H*P  
            error('zernfun:normalization','Unrecognized normalization flag.') !*!i&0QC~R  
        end *|B5,Ey  
    else j V'~>  
        isnorm = false; 2{A/Fbk  
    end dF+R q|n{  
    GLiD,QX<  
    u`gY/]y!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M?v`C>j  
    % Compute the Zernike Polynomials zbZN-j#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j&l2n2z  
    }>yQ!3/i  
    ;mauA#vd  
    % Determine the required powers of r: 7Um3m yXU  
    % ----------------------------------- ;\54(x}|K  
    m_abs = abs(m); S{S.H?{F  
    rpowers = []; k/m-jm_h  
    for j = 1:length(n) S]<%^W'  
        rpowers = [rpowers m_abs(j):2:n(j)]; rPx:o}&<  
    end |bX{MF  
    rpowers = unique(rpowers); eMOnzW|h  
    K!O7q~s[D  
    C<E;f]d  
    % Pre-compute the values of r raised to the required powers, ^$;5ZkQy  
    % and compile them in a matrix: {SwvUWOf"  
    % ----------------------------- JL=s=9N;3  
    if rpowers(1)==0 +GlG.6  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J%1 2Ey@6  
        rpowern = cat(2,rpowern{:}); iu+rg(*%  
        rpowern = [ones(length_r,1) rpowern]; _xdFQ  
    else W~?mr! `  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m%.7l8vT  
        rpowern = cat(2,rpowern{:}); 9;L50q>s  
    end osPrr QoH  
    %&&;06GU}  
    ~+anI  
    % Compute the values of the polynomials: MB"<^ZX  
    % -------------------------------------- te b/  
    y = zeros(length_r,length(n)); F2C v,&'  
    for j = 1:length(n) KF f6um  
        s = 0:(n(j)-m_abs(j))/2; &-(p~[|  
        pows = n(j):-2:m_abs(j); e) kVS}e?  
        for k = length(s):-1:1 q:3HU<  
            p = (1-2*mod(s(k),2))* ... o0FVVSl  
                       prod(2:(n(j)-s(k)))/              ... (E<QA  
                       prod(2:s(k))/                     ... 89l{h8R  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .WpvDDUK3  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); r=:o$e  
            idx = (pows(k)==rpowers); }Oe9Zq  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 5 u^;71  
        end 1'YksuYx6f  
         $LJCup,1"  
        if isnorm KO&oT#S  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w<G'gi]  
        end A9C  
    end ">'`{mXew  
    % END: Compute the Zernike Polynomials H<C+ rAIb  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PP!} w  
    PXDwTuyc  
    lPOcX'3\  
    % Compute the Zernike functions: @ >Ul0&Mf?  
    % ------------------------------ p WLFJH}N  
    idx_pos = m>0; I;3Uzv  
    idx_neg = m<0; D",~?  
    +EP=uV9t  
    Cl'3I%$8K  
    z = y; sI#r3:?i  
    if any(idx_pos) ;&U! g&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 93fClF|@  
    end $S{]` +  
    if any(idx_neg) V0a)9\x(\  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $ZfoJR]%  
    end '(&,i/O  
    XdGA8%^cY  
    F<|x_6a\  
    % EOF zernfun =d`/BDD  
     
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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)  dY'mY~Tv  
    ub* j&L=  
    DDE还是手动输入的呢? s/"?P/R  
    UFLN/  
    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