切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9506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, 9`vse>,-hg  
    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, r9u*c  
    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? 2f~s$I&l#  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 9Uk9TG5  
    12k)Ek9  
    g[Yok` e[  
    ,sJ{2,]~  
    '9RHwKu&s  
    function z = zernfun(n,m,r,theta,nflag) tU?lfU[7  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5a_K|(~3I  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N OO\UF6MCU  
    %   and angular frequency M, evaluated at positions (R,THETA) on the '3<YZWS  
    %   unit circle.  N is a vector of positive integers (including 0), and B|!YGf L  
    %   M is a vector with the same number of elements as N.  Each element [c3hwogf:  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) V:l; 2rW  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, }*+ca>K  
    %   and THETA is a vector of angles.  R and THETA must have the same UkeW2l`:  
    %   length.  The output Z is a matrix with one column for every (N,M) )DoY*'Cl  
    %   pair, and one row for every (R,THETA) pair. gE8>5_R|  
    % 242lR0#aY  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =P2T&Gb  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), v'Lckw@G4  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 6i&WF<%D  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, zzPgLE55  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized g:OVAA  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. BeplS  
    % `cVG_= 2  
    %   The Zernike functions are an orthogonal basis on the unit circle. B\N,%vsx#U  
    %   They are used in disciplines such as astronomy, optics, and ~omX(kPzK  
    %   optometry to describe functions on a circular domain. ;i,yT ?so  
    % Ba@UX(t  
    %   The following table lists the first 15 Zernike functions. Q@l3XNH|c  
    % ?2.< y_1  
    %       n    m    Zernike function           Normalization G =lC[i  
    %       --------------------------------------------------  BeP0lZ  
    %       0    0    1                                 1 sd#a_  
    %       1    1    r * cos(theta)                    2 -+c_TJ.dC  
    %       1   -1    r * sin(theta)                    2 rsiG]o=8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) YMm Fpy  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 9/Q5(P  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ];(w8l  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) /A{znE  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "9R3S[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Tw|=;m  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) PBkKn3P3  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) F#W'>WBU  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'fZHtnmc0  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 6B|IbQ^  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fq\E$'o$  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 9n44 *sZ  
    %       -------------------------------------------------- uv._N6mj  
    % B \[P/AC  
    %   Example 1: z^=9%tLJ  
    % 6kYn5:BhIi  
    %       % Display the Zernike function Z(n=5,m=1) 4. R >mN[  
    %       x = -1:0.01:1; ;Wb W\,P'  
    %       [X,Y] = meshgrid(x,x); )<jj O  
    %       [theta,r] = cart2pol(X,Y); ~hz]x^:  
    %       idx = r<=1; <BT}Tv9  
    %       z = nan(size(X)); Qv[@ioc  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); opdi5 e)jK  
    %       figure +ZXk0sP_<  
    %       pcolor(x,x,z), shading interp "EHwv2Hm>  
    %       axis square, colorbar Z\`uI+`  
    %       title('Zernike function Z_5^1(r,\theta)') 7pr@aA"vgj  
    % S,qsCnz  
    %   Example 2: yg/.=M  
    % 9<,\ +}^{  
    %       % Display the first 10 Zernike functions XCQ =`3f  
    %       x = -1:0.01:1; NcFHvK  
    %       [X,Y] = meshgrid(x,x); >CNH=  
    %       [theta,r] = cart2pol(X,Y); ~?S/0]?c  
    %       idx = r<=1; LXfDXXF  
    %       z = nan(size(X)); q=g;TAXZl  
    %       n = [0  1  1  2  2  2  3  3  3  3]; E}4R[6YD  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; lHr?sMt  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; c00a;=ji  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0FHN  
    %       figure('Units','normalized') >`\~=ivrD  
    %       for k = 1:10 YV 2T$#7u  
    %           z(idx) = y(:,k); qKZ~)B j  
    %           subplot(4,7,Nplot(k)) ZShRE"`  
    %           pcolor(x,x,z), shading interp ANi}q9SC  
    %           set(gca,'XTick',[],'YTick',[]) ,in`JM<o  
    %           axis square `3\5&Bf  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *|ubH?71%Y  
    %       end ~B|K]&/]  
    % ,Q2`N{f  
    %   See also ZERNPOL, ZERNFUN2. dk-Y!RfNx  
    D+#QQH  
    kf.w:X"i  
    %   Paul Fricker 11/13/2006 ]KLj Qpd  
    [y64%|m  
    7s1FJm=Y/  
    y kwS-e  
    Kcl>uAgU  
    % Check and prepare the inputs: l<! ?`V6}  
    % ----------------------------- *8t_$<'dQ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Gpo(Zf?  
        error('zernfun:NMvectors','N and M must be vectors.') p.gi8%f`  
    end ~Wf&$p<|  
    ixp(^>ZN  
    ##EMJi  
    if length(n)~=length(m) WEOW6UV(  
        error('zernfun:NMlength','N and M must be the same length.') DXsp 2  
    end j[ kg9z  
    9}:%CpD^~I  
    3g^_Fq'  
    n = n(:); M')f,5i&$  
    m = m(:); \Om.pOz  
    if any(mod(n-m,2)) i4JqU\((]  
        error('zernfun:NMmultiplesof2', ... I?EtU/AD  
              'All N and M must differ by multiples of 2 (including 0).') \l"1Io=  
    end O#sDZ.EL  
    edx-R-Dc-1  
    u @?n3l  
    if any(m>n) xOt%H\*k"  
        error('zernfun:MlessthanN', ... 71Q-_Hi  
              'Each M must be less than or equal to its corresponding N.') *[9FPya  
    end .|G([O^H  
    )C|[j@MD  
    PB(mUD2"r  
    if any( r>1 | r<0 ) !Z 3iu  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6f v{?0|  
    end ,Hlbl}.ls  
    x4r\cL1!  
    ,Tvfn`;(  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j(AN] g:  
        error('zernfun:RTHvector','R and THETA must be vectors.') Cd?a C  
    end -iLp3m<ai  
    F\^9=}b_i  
    XJ^dX]4  
    r = r(:); /5b,&  
    theta = theta(:); fzT|{vG8  
    length_r = length(r); wrSw>sE"  
    if length_r~=length(theta) ,qz$6oxh\  
        error('zernfun:RTHlength', ... 3WHj|ENW  
              'The number of R- and THETA-values must be equal.') |_x U{Pu  
    end nYhI0q  
    7k.=_Tl  
    k)U9 %Pr  
    % Check normalization: outAZy=R;  
    % -------------------- b= amd*  
    if nargin==5 && ischar(nflag) "j#;MOK  
        isnorm = strcmpi(nflag,'norm'); {ss^L  
        if ~isnorm (S3\O `5  
            error('zernfun:normalization','Unrecognized normalization flag.') FZf{kWH  
        end ;~CAHn|Fe  
    else :08b&myx  
        isnorm = false; U$-Gc[=|  
    end j?<>y/IR  
    l.[S.@\=.  
    U.g7'`Z<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }#]2u| G  
    % Compute the Zernike Polynomials <]1Z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BC.~wNz6  
    }TfZ7~o[  
    9f1,E98w_  
    % Determine the required powers of r: uJhB>/Og  
    % ----------------------------------- Y_'3pX,  
    m_abs = abs(m); %P@V7n  
    rpowers = []; )nE=H,U?y  
    for j = 1:length(n) HG kL6o=  
        rpowers = [rpowers m_abs(j):2:n(j)]; U?]}K S;6  
    end wyWe2d  
    rpowers = unique(rpowers); jNV)=s^ed[  
    Z'=:Bo{  
    c"F3[mrff  
    % Pre-compute the values of r raised to the required powers, t*J *?Ma  
    % and compile them in a matrix: "Bn8WT2?  
    % ----------------------------- m ioNMDG  
    if rpowers(1)==0 2aj9:S  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w1>uD]  
        rpowern = cat(2,rpowern{:}); &gGh%:`B  
        rpowern = [ones(length_r,1) rpowern]; 9vX~gh{]~  
    else A><w1-X&=o  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @s7ZfV??  
        rpowern = cat(2,rpowern{:}); P?WS=w*O0  
    end f0!i<9<  
    'Z ;8-1M?O  
    )w/ #T  
    % Compute the values of the polynomials: B L^?1x  
    % -------------------------------------- 1V/?p<A  
    y = zeros(length_r,length(n)); ': fq/k3;&  
    for j = 1:length(n) u_31Db<  
        s = 0:(n(j)-m_abs(j))/2; K3g<NC  
        pows = n(j):-2:m_abs(j); naOCa  
        for k = length(s):-1:1 MuI>ZoNF  
            p = (1-2*mod(s(k),2))* ... ZhvZe/  
                       prod(2:(n(j)-s(k)))/              ... nLvF^%P8  
                       prod(2:s(k))/                     ... 4zo^ b0v  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Pk{eGG<F$  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ECW=865jL  
            idx = (pows(k)==rpowers); P6G&3yPt  
            y(:,j) = y(:,j) + p*rpowern(:,idx); L'A9TW2  
        end WlJ=X$  
         `|rF^~6(dR  
        if isnorm (Mzv"FN]  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); r3OR7f[  
        end )/87<Y;o  
    end ~9ZW~z'  
    % END: Compute the Zernike Polynomials rm}%C(C{J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IJ[r!&PY  
    =(aA`:Nl  
    qnc?&f  
    % Compute the Zernike functions: a%g|E'\Jw  
    % ------------------------------ \5 S^~(iL  
    idx_pos = m>0; 7oWT6Qa5  
    idx_neg = m<0; >(.GIR  
    bfV&z+Rv-5  
    IoAG!cS  
    z = y; NpPuh9e{  
    if any(idx_pos) S&JsDPzSd  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6< hE]B)  
    end Ga$J7 R  
    if any(idx_neg) dilom#2l  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); VY1&YR}Y  
    end yw@kh^L  
    *r90IS}A$2  
    V9:Jz Q=?`  
    % EOF zernfun '73g~T%$^*  
     
    分享到
    离线phoenixzqy
    发帖
    4352
    光币
    5479
    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
    960
    光币
    1088
    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)  #h}a   
    yu}yON  
    DDE还是手动输入的呢? e<$s~ UXv  
    *cP(3n3]R  
    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