切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9317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 正序阅读 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, /~*_x=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, Y!iZW  
    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? _@S`5;4x  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kmzH'wktt  
    W>-Et7&2  
    "&Po,AWa  
    ctE\ q  
    ^B8b%'\  
    function z = zernfun(n,m,r,theta,nflag) |5Xq0nvCe  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. =UyLk-P w  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lHgs;>U$  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ca+5=+X7  
    %   unit circle.  N is a vector of positive integers (including 0), and N F)~W#  
    %   M is a vector with the same number of elements as N.  Each element  : ]C~gc  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >EY3/Go>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !K|5bK  
    %   and THETA is a vector of angles.  R and THETA must have the same 6{ =\7AY  
    %   length.  The output Z is a matrix with one column for every (N,M) "DYJ21Ut4  
    %   pair, and one row for every (R,THETA) pair.  w@,zFV  
    % sQkhwMg  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5\z `-)  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]+X@ 7  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral T=ev[ mS  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 21"1NJzP  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Bve.C  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. gEjdN.  
    % F6z%VWU  
    %   The Zernike functions are an orthogonal basis on the unit circle. eio 4k-  
    %   They are used in disciplines such as astronomy, optics, and &Xf}8^T<V  
    %   optometry to describe functions on a circular domain. .SWlp2!M5  
    % A}l3cP; `#  
    %   The following table lists the first 15 Zernike functions. q@{Bt{$x  
    % v/_  
    %       n    m    Zernike function           Normalization uA< n  
    %       -------------------------------------------------- kDsFR#w&`  
    %       0    0    1                                 1 Z.Lc>7o  
    %       1    1    r * cos(theta)                    2 ^~etm  
    %       1   -1    r * sin(theta)                    2 ZP(f3X@  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) u ,KD4{!  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 5?x>9C a  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) iUN Ib  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) cz8T  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 'd9INz.  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ;>Ib^ov  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 1ukTA@Rj&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eceP0x  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JxM]9<a=4  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 7fZDs j:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nWw":K<@Q_  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) &OH={Au  
    %       -------------------------------------------------- K  &N  
    % 3EPv"f^V  
    %   Example 1: sYI-5D]  
    % X}Ai -D  
    %       % Display the Zernike function Z(n=5,m=1) QTk}h_<u  
    %       x = -1:0.01:1; wfLaRP  
    %       [X,Y] = meshgrid(x,x); 0AL=S$B)  
    %       [theta,r] = cart2pol(X,Y); *RJG!t*t  
    %       idx = r<=1; -&zZtDd F  
    %       z = nan(size(X)); s"r*YlSp"  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Ng2twfSl$  
    %       figure 52Z2]T c ,  
    %       pcolor(x,x,z), shading interp h-`?{k&e  
    %       axis square, colorbar *k.G5>@  
    %       title('Zernike function Z_5^1(r,\theta)') kTOzSiq  
    % KQ% GIz x  
    %   Example 2: Z>k#n'm^z  
    % ?]_$Dcmx  
    %       % Display the first 10 Zernike functions h+g_rvIG*  
    %       x = -1:0.01:1; l<58A7  
    %       [X,Y] = meshgrid(x,x); 6H.0vN&  
    %       [theta,r] = cart2pol(X,Y); Rq'S>#e  
    %       idx = r<=1; #wwH m3  
    %       z = nan(size(X)); XpB_N{v9w  
    %       n = [0  1  1  2  2  2  3  3  3  3]; a/4T> eC  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 'uS n}hm  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; +SR+gE\s0  
    %       y = zernfun(n,m,r(idx),theta(idx)); _^Ubs>d=*  
    %       figure('Units','normalized') ;#W2|'HD  
    %       for k = 1:10 24 'J  
    %           z(idx) = y(:,k); )4e.k$X^  
    %           subplot(4,7,Nplot(k)) |.: q  
    %           pcolor(x,x,z), shading interp cKca;SNql1  
    %           set(gca,'XTick',[],'YTick',[]) i &nSh ]KK  
    %           axis square i+ ?^8#  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) KxJ!,F{>H  
    %       end 3wF;GG  
    % )hsgC'H{~]  
    %   See also ZERNPOL, ZERNFUN2.  ,f%S'(>w  
    .5_2zat0H  
    <44G]eb  
    %   Paul Fricker 11/13/2006 N)X3XTY  
    Qz1E 2yJ  
    A:%`wX}  
    i>`%TW:g  
    q"lSZ; 'E  
    % Check and prepare the inputs: k(nW#*N_  
    % ----------------------------- /{ g>nzP  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z43M] P<  
        error('zernfun:NMvectors','N and M must be vectors.') [q[Y~1o/&H  
    end j3V -LnA  
    Ax7[;|2  
    %J?xRv!  
    if length(n)~=length(m) +~$ ]} %  
        error('zernfun:NMlength','N and M must be the same length.') YK'<NE3 4  
    end *i%.;Z"  
    T.BW H2gRP  
    f}P3O3Yv&  
    n = n(:); R 'zWYQ  
    m = m(:); ^\=`edN0  
    if any(mod(n-m,2)) "ze|W\Bv!  
        error('zernfun:NMmultiplesof2', ... ?+@?Up0wGO  
              'All N and M must differ by multiples of 2 (including 0).') .M%}X7  
    end 7R\<inCQ  
    ;?p>e'  
    sDlO#  
    if any(m>n) 3F2w-+L  
        error('zernfun:MlessthanN', ... VpDbHAg  
              'Each M must be less than or equal to its corresponding N.') 5U$0F$BBp  
    end m;QMQeGz  
    !Wnb|=j  
    8<Av@9 *}  
    if any( r>1 | r<0 ) 2FJ*f/  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '~=SzO  
    end b8 likP"T  
    vl:KF7:#m  
    %Q|Atgp  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?6WY:Zec@  
        error('zernfun:RTHvector','R and THETA must be vectors.') SO!8Di  
    end pW3^X=6  
    0 kW,I  
    X'iWJ8  
    r = r(:); vEJbA  
    theta = theta(:); 9\7en%(M  
    length_r = length(r); i9x+A/ o[  
    if length_r~=length(theta) Q^")jPd  
        error('zernfun:RTHlength', ... G4"F+%.  
              'The number of R- and THETA-values must be equal.') I; rGD^  
    end =dN@Sa/  
    utV_W&  
    EADqC>  
    % Check normalization: 0o&5 ]lEe  
    % -------------------- l*G[!u  
    if nargin==5 && ischar(nflag) 7@W>E;go  
        isnorm = strcmpi(nflag,'norm'); (#c:b  
        if ~isnorm vnuN6M{  
            error('zernfun:normalization','Unrecognized normalization flag.') JB<t6+"rD  
        end CU!Dhm/U  
    else El8,,E  
        isnorm = false; y?3; 06y|  
    end )vlhN2iv  
    wUJcmM;  
    k+*u/neh  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J#83 0r(-  
    % Compute the Zernike Polynomials [dz _R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2&cT~ZX&'  
    v`T c}c '  
    <1TAw.  
    % Determine the required powers of r: #KvlYZ+1  
    % ----------------------------------- 'V>-QD%1  
    m_abs = abs(m); uPvEwq* C  
    rpowers = []; CTmT@A{  
    for j = 1:length(n) Dw"\/p:-3  
        rpowers = [rpowers m_abs(j):2:n(j)]; Q/Rqa5LI:  
    end %BQ`MZ  
    rpowers = unique(rpowers); uXiN~j &Be  
    |I=T @1_D  
    gRzxLf`K  
    % Pre-compute the values of r raised to the required powers, !8 b ^,  
    % and compile them in a matrix: 3OB"#Ap8<  
    % ----------------------------- yf,z$CR  
    if rpowers(1)==0 ~}Pfu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Vjpy~iP4B  
        rpowern = cat(2,rpowern{:}); %z$#6?OK^  
        rpowern = [ones(length_r,1) rpowern]; !VzC&>'v^9  
    else "J1 4C9u   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1\.pMHv/  
        rpowern = cat(2,rpowern{:}); h2QmQ>y"  
    end CvdN"k  
    glw+l'@  
    /mZE/>&~ ,  
    % Compute the values of the polynomials: ),!qTjD  
    % -------------------------------------- =EsavN  
    y = zeros(length_r,length(n)); xyxy`qRA  
    for j = 1:length(n) _"{Xi2@H  
        s = 0:(n(j)-m_abs(j))/2; }-`4DHgq  
        pows = n(j):-2:m_abs(j); E" vS $  
        for k = length(s):-1:1 !n%j)`0M  
            p = (1-2*mod(s(k),2))* ... E*lxVua  
                       prod(2:(n(j)-s(k)))/              ... [N'h%1]\  
                       prod(2:s(k))/                     ... QsW/X0YBv  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jb)ZLA;L_c  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); y)<q /  
            idx = (pows(k)==rpowers); R|Q?KCI&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); phz&zl D  
        end `H+ lPM66  
         & nK<:^n  
        if isnorm P2nu;I_ &  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2Z%O7V~u  
        end J~- 4C)  
    end <oeIcN7d  
    % END: Compute the Zernike Polynomials *z2s$EZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LH6 vLuf  
    ~QVH<`sn  
    F:ELPs4"  
    % Compute the Zernike functions: wKHBAW[i]  
    % ------------------------------ Ir]\|t  
    idx_pos = m>0; :gC#hmm^  
    idx_neg = m<0; :v 4]D4\o  
    4GM6)"#d  
    V43H /hl  
    z = y; :Tq~8!s  
    if any(idx_pos) !!y a  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); =R\]=cRbg  
    end DTs;{c  
    if any(idx_neg) eDB;cN  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i6N',&jFU  
    end {>;R?TG]$  
    QSj]ZA  
    C7?/%7{  
    % EOF zernfun azU"G(6y?+  
     
    分享到
    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
    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)  igR";OQk  
    /RC7"QzL  
    DDE还是手动输入的呢? 46&/gehr  
    *ppffz  
    zygo和zemax的zernike系数,类型对应好就没问题了吧
    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)