切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 9075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, (H=7(  
    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, gN"Abc  
    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? ^uZ!e+   
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? &dA{<.  
    ?[Gj?D.Wc  
    8Ter]0M&  
    /eFudMl  
    <hG] f%  
    function z = zernfun(n,m,r,theta,nflag) ?r< F/$/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~Ey)9phZK  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gZ{q85C.>  
    %   and angular frequency M, evaluated at positions (R,THETA) on the a+wc"RQ |  
    %   unit circle.  N is a vector of positive integers (including 0), and fK-tvP0}*  
    %   M is a vector with the same number of elements as N.  Each element LojEJ  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) =lyP &u  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {~cG'S Y%  
    %   and THETA is a vector of angles.  R and THETA must have the same # MpW\yX  
    %   length.  The output Z is a matrix with one column for every (N,M) 'j6)5WL$  
    %   pair, and one row for every (R,THETA) pair. ">$.>sn{  
    % zpPzXQv]/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike I,rs&m?/m  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Glz yFj  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ^Ob#B!=  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, a04I.5!  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8Xo`S<8VS  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .)eJL  
    % 6x6xv:\  
    %   The Zernike functions are an orthogonal basis on the unit circle. ]m ED3#  
    %   They are used in disciplines such as astronomy, optics, and 52RFB!Z[  
    %   optometry to describe functions on a circular domain. =aL=SC+  
    % DM*GvBdR  
    %   The following table lists the first 15 Zernike functions. kTCWyc  
    % C3m](%?   
    %       n    m    Zernike function           Normalization kaKV{;UM  
    %       -------------------------------------------------- \W^+aNbv=8  
    %       0    0    1                                 1 d5b \kRr  
    %       1    1    r * cos(theta)                    2 (YOp  
    %       1   -1    r * sin(theta)                    2 jg,oGtRz  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ,7wxVR%Ys  
    %       2    0    (2*r^2 - 1)                    sqrt(3) $ U~3$*R  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) O(P ,!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ^N{Lau  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) gWqO5C~h  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) x+mf QcSD&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ZD)pdNX  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) oM')NIW@  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |G!PG6%1  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) {{3n">s}:  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rXortK#\%  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 83^|a5  
    %       -------------------------------------------------- muD7+rn?&  
    % K5oVB,z)  
    %   Example 1: dcK7Dd->  
    % GpW5)a  
    %       % Display the Zernike function Z(n=5,m=1) Obd};&6Q  
    %       x = -1:0.01:1; U}r^M( s!  
    %       [X,Y] = meshgrid(x,x); AX {~A:B  
    %       [theta,r] = cart2pol(X,Y); uTSTBI4t  
    %       idx = r<=1; C>1fL6ct  
    %       z = nan(size(X)); @A-*XJNS":  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); d;Uzl 1;  
    %       figure =Wb!j18]  
    %       pcolor(x,x,z), shading interp LTSoo.dE  
    %       axis square, colorbar ]+ \]2`?  
    %       title('Zernike function Z_5^1(r,\theta)') .:<-E%  
    % <Z8I#IPl  
    %   Example 2: ;k<n}shD  
    % 9`3%o9V9Y  
    %       % Display the first 10 Zernike functions Cfz020u`g  
    %       x = -1:0.01:1; 319 &:  
    %       [X,Y] = meshgrid(x,x); K1vm [Ne  
    %       [theta,r] = cart2pol(X,Y); d=q&UCC  
    %       idx = r<=1; <($'jlZ  
    %       z = nan(size(X)); P^1+;dL,D  
    %       n = [0  1  1  2  2  2  3  3  3  3]; evbqBb21b  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 6NvdFss'A{  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ("UzMr,  
    %       y = zernfun(n,m,r(idx),theta(idx)); c[/h7!/aH  
    %       figure('Units','normalized') x B%Felz  
    %       for k = 1:10 n+C,v.X  
    %           z(idx) = y(:,k); ~"oxytJ  
    %           subplot(4,7,Nplot(k)) eyx;8v cM  
    %           pcolor(x,x,z), shading interp U\_-GS;1  
    %           set(gca,'XTick',[],'YTick',[]) ]3+xJz~=  
    %           axis square q- U/JC  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) YW|KkHi*  
    %       end ql|ksios  
    % H*l2,0&W  
    %   See also ZERNPOL, ZERNFUN2. Z+mesj?.  
    yK1Z&7>J>  
    w(sD}YA)  
    %   Paul Fricker 11/13/2006 -I#]#i@gX  
    }'?N+MN  
    MZpG1  
    `%8byy@$  
    =Ws-s f]  
    % Check and prepare the inputs: HzW`j"\  
    % ----------------------------- 7 TTU&7l~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rA/jNX@S  
        error('zernfun:NMvectors','N and M must be vectors.') -SZW[T<N"  
    end \2F$FRWo  
    5`$.GV  
    rPK)=[MZ  
    if length(n)~=length(m) .?gpI Zv  
        error('zernfun:NMlength','N and M must be the same length.') a0vg%Z@!  
    end $1Lm=2;U  
    OygR5s +  
    H.8f-c-4we  
    n = n(:); g3p*OYf  
    m = m(:); %fS__Tb#u  
    if any(mod(n-m,2)) f0 ;Fokt(  
        error('zernfun:NMmultiplesof2', ... [Rz9Di ;  
              'All N and M must differ by multiples of 2 (including 0).') 3Mvm'T:[  
    end -y8?"WB(b  
    =:T pH>f*  
    sqAZjfy@  
    if any(m>n) YTiXU Oj  
        error('zernfun:MlessthanN', ... P= e3f(M2  
              'Each M must be less than or equal to its corresponding N.') rKlu+/G  
    end Ms^U`P^V~P  
    {Z>OAR#   
    HG(J+ocn   
    if any( r>1 | r<0 ) +="?[:  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') &dqC =oK]  
    end S7tc  
    =WaZy>n}7  
    k<mfBNvuo  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I}5#!s< {&  
        error('zernfun:RTHvector','R and THETA must be vectors.') pi>,>-Z  
    end ogt<vng  
    8pc=Oor2Tv  
     !,rp|  
    r = r(:); xWY%-CWY.  
    theta = theta(:); ;\N{z6  
    length_r = length(r); \t LfB[S.5  
    if length_r~=length(theta) D49yV`  
        error('zernfun:RTHlength', ... Pt/dH+r`%  
              'The number of R- and THETA-values must be equal.') `QH-VR\_  
    end nf,R+oX  
    ar-N4+!@  
    ?Y:>Ouv*z'  
    % Check normalization: d ] J5c  
    % -------------------- [.M<h^xrB  
    if nargin==5 && ischar(nflag) >t-9yO1XQq  
        isnorm = strcmpi(nflag,'norm'); wS*An4%G  
        if ~isnorm |sf&t  
            error('zernfun:normalization','Unrecognized normalization flag.') -)biSU,  
        end -L;sv0  
    else 5PY,}1`  
        isnorm = false; _*d8:|qw  
    end f(Vr&X  
    uJQ#l\t  
    sW'SR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -O.q$D=as  
    % Compute the Zernike Polynomials 2!Bjs?K<bv  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .>4Zt'gCt  
    \'z&7;px  
    ('H[[YODh  
    % Determine the required powers of r: jV83%%e  
    % ----------------------------------- H Aq  
    m_abs = abs(m); 'CE3 |x\%K  
    rpowers = []; f+#^Lngo  
    for j = 1:length(n) `Sh#> Jp  
        rpowers = [rpowers m_abs(j):2:n(j)]; 1SddZ5  
    end $a'n{EP  
    rpowers = unique(rpowers); X,m6#vLK2  
    G}!dm0s$  
    _wMc7`6F  
    % Pre-compute the values of r raised to the required powers, n< npJ*  
    % and compile them in a matrix: W4 v/,g>  
    % ----------------------------- KI* erK [d  
    if rpowers(1)==0 jNKu5"HB  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ~s#vP<QHa  
        rpowern = cat(2,rpowern{:}); HYd&.*41rE  
        rpowern = [ones(length_r,1) rpowern]; ;?-A 4!V,  
    else ZCdlTdY   
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F:p'%#3rU/  
        rpowern = cat(2,rpowern{:}); 0L3v[%_j"  
    end 5](-(?k}~  
    a: C h"la  
    N3J T[7  
    % Compute the values of the polynomials: nnP] x [  
    % -------------------------------------- a?_!  
    y = zeros(length_r,length(n)); ]: VR3e"H  
    for j = 1:length(n) rCOH*m&  
        s = 0:(n(j)-m_abs(j))/2; O$<m(~[S  
        pows = n(j):-2:m_abs(j); 1y\ -Iz^  
        for k = length(s):-1:1 "pQFIV,  
            p = (1-2*mod(s(k),2))* ... ^T(v4'7  
                       prod(2:(n(j)-s(k)))/              ... xqP DL9\  
                       prod(2:s(k))/                     ... An cka  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ii< /!B(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); QqpXUyHp[  
            idx = (pows(k)==rpowers); 0l.\KF  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 3>Ne_kY  
        end dRl*rP/  
         |wef[|@%  
        if isnorm wrORyj  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); lCyBdY9n  
        end =f FTi1]/h  
    end XsOz {?G  
    % END: Compute the Zernike Polynomials &bh%>[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -SyQ`V)T7N  
    ,{tz%\, %  
    E5>y?N  
    % Compute the Zernike functions: bSK> p3  
    % ------------------------------ -w>2!@8  
    idx_pos = m>0; vvWje:H  
    idx_neg = m<0; 9E@}@ZV(  
    Z@Tb3N/[  
    \=3fO(  
    z = y; )GbVgYkk  
    if any(idx_pos) <i<[TPv";  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8`I/\8;H'p  
    end ;v}f7v '  
    if any(idx_neg) 0uw3[,I   
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); cJIA/HQe  
    end oRp;9   
    ;+86q"&n  
    #b^x!lR  
    % EOF zernfun rM|] }M=_V  
     
    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)  l<89[{9o  
    IwR/4LYI  
    DDE还是手动输入的呢? =Eh~ wm  
    3Dm`8Xt  
    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