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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, SJiQg-+<Uf  
    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, fu!T4{2  
    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? Jqxd92 bI  
    那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? dj0%?g>  
    Q:P)g#suc  
    `3\aX|4@  
    NJBSVC b  
    }d. X2?  
    function z = zernfun(n,m,r,theta,nflag) 2;Z 0pPR&  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. }d%CZnY&7  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m"!SyN}&9?  
    %   and angular frequency M, evaluated at positions (R,THETA) on the $@Vn+| Ix  
    %   unit circle.  N is a vector of positive integers (including 0), and V|YQhd0kv  
    %   M is a vector with the same number of elements as N.  Each element [5&k{*}}  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) nD5wN~[J  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ZUI6VM  
    %   and THETA is a vector of angles.  R and THETA must have the same eA& #33  
    %   length.  The output Z is a matrix with one column for every (N,M) ^Laqq%PI  
    %   pair, and one row for every (R,THETA) pair. #da{3>z:  
    % _$UJ'W})/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 7T/BzXr,B  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), BH'*I yv  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !.UE}^TV  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]Za[]E8MD  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized znrO~OK  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. WWp MuB_G  
    % xb\EJ1M>  
    %   The Zernike functions are an orthogonal basis on the unit circle. [63\2{_^v  
    %   They are used in disciplines such as astronomy, optics, and EV( F!&  
    %   optometry to describe functions on a circular domain. T`e`nQ0nn  
    % .2%t3ul[  
    %   The following table lists the first 15 Zernike functions. ]$2 yV&V&  
    % &5y  
    %       n    m    Zernike function           Normalization 1J[$f>%n]  
    %       -------------------------------------------------- gY7sf1\wX  
    %       0    0    1                                 1 LcGKYl(\K  
    %       1    1    r * cos(theta)                    2 cAN!5?D\  
    %       1   -1    r * sin(theta)                    2 4`8s]X  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) "mQp#d/'  
    %       2    0    (2*r^2 - 1)                    sqrt(3) eKOEOm+  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3VLwY!2:  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) T+L=GnYl  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) )h@PRDI_  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~HIj+kN  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) aV$kxzEc  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }.o.*N  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) t"B3?<?]  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) s~{rC{9X  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _. 9 5>`  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) +q pW"0[  
    %       -------------------------------------------------- '9!_:3[d\]  
    % =@d#@  
    %   Example 1: z I2DQ] 9  
    % E n7~wKF  
    %       % Display the Zernike function Z(n=5,m=1) lo!pslqsn  
    %       x = -1:0.01:1; 'd2 :a2C]  
    %       [X,Y] = meshgrid(x,x); deAV:c  
    %       [theta,r] = cart2pol(X,Y); MiZ<v/L2  
    %       idx = r<=1; ?1L<VL=b  
    %       z = nan(size(X)); ^mL X}E]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 7G+!9^  
    %       figure Gy \ ]j  
    %       pcolor(x,x,z), shading interp e.vt"eRB  
    %       axis square, colorbar poAJl;T  
    %       title('Zernike function Z_5^1(r,\theta)') l :{q I#Q  
    % Jk%5Fw0  
    %   Example 2: CzG[S\{+  
    % dm}1"BU<  
    %       % Display the first 10 Zernike functions a]*{!V{$i  
    %       x = -1:0.01:1; +d#8/S*  
    %       [X,Y] = meshgrid(x,x); _]@u)$  
    %       [theta,r] = cart2pol(X,Y); Lk|`\I T  
    %       idx = r<=1; oz=V|7,  
    %       z = nan(size(X)); }Hb0@ b_  
    %       n = [0  1  1  2  2  2  3  3  3  3]; HWV A5E[`Y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; O1~7#nJ*4[  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; EPJ>@A>;D  
    %       y = zernfun(n,m,r(idx),theta(idx)); Yeg<MrS4D  
    %       figure('Units','normalized') w<H2#d>5!@  
    %       for k = 1:10 sg=G<50i  
    %           z(idx) = y(:,k); "*HM8\  
    %           subplot(4,7,Nplot(k)) $e+4Kt ,  
    %           pcolor(x,x,z), shading interp Vz0(D  
    %           set(gca,'XTick',[],'YTick',[])  p0W<K  
    %           axis square ^.:&ZsqV  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) D SX%SE)  
    %       end 6:wk=#w  
    % |Ogh-<|<  
    %   See also ZERNPOL, ZERNFUN2. d;<'28A  
    Xps \+l%i  
    \ n 2MP  
    %   Paul Fricker 11/13/2006 gbYM1guiD  
    l@&-be  
    0 \ U*  
    6am6'_{  
    <pV8 +V)  
    % Check and prepare the inputs: )PvnB=wy  
    % ----------------------------- f/c&Ya(D~  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -ysNo4#e&  
        error('zernfun:NMvectors','N and M must be vectors.') Ej)7[  
    end 3\4e{3$  
    L+G0/G}O\  
    ^;ZpK@Luk  
    if length(n)~=length(m) uDND o  
        error('zernfun:NMlength','N and M must be the same length.') SW%}S*h  
    end kSiyMDY-  
    $1B?@~&  
    md<^x(h"<  
    n = n(:); 6O,k! y>  
    m = m(:); eH7x>[lH.  
    if any(mod(n-m,2)) bD=H$)  
        error('zernfun:NMmultiplesof2', ... sN8pwRjb  
              'All N and M must differ by multiples of 2 (including 0).') \]4EAKJE  
    end . q -: 3b  
    HI{q#  
    Uhr2"Nuuy  
    if any(m>n) 0?oL zw&  
        error('zernfun:MlessthanN', ... y;CX )!8  
              'Each M must be less than or equal to its corresponding N.') ;o'r@4^&$R  
    end !&Q?ASJH  
    I5E =Ujc_  
    59{X;  
    if any( r>1 | r<0 ) ECi;o1hda  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3#$X  
    end ECA<%'$?E  
    oh:9v+  
    [ KgO:},c  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) n%29WF6Zf  
        error('zernfun:RTHvector','R and THETA must be vectors.') a9;KS>~bq  
    end 5- GS@fY  
    @Ol(:{<  
    ,vmn{gz  
    r = r(:); WPsfl8@D  
    theta = theta(:); ~5N oR  
    length_r = length(r); p~3 x=X4  
    if length_r~=length(theta) o}G`t Bz  
        error('zernfun:RTHlength', ... bwhH2^ !  
              'The number of R- and THETA-values must be equal.') nK03xYA  
    end q/zU'7%@  
    >U`G3(#7S  
    L!*+: L DL  
    % Check normalization: w!H(zjv&(  
    % -------------------- B(1-u!pz  
    if nargin==5 && ischar(nflag) [m{sl(Q  
        isnorm = strcmpi(nflag,'norm'); VO eVS&}  
        if ~isnorm !@ ]IJ"\  
            error('zernfun:normalization','Unrecognized normalization flag.') "G%</G8M  
        end izcaWt3 a  
    else XxMZU(5  
        isnorm = false; <-?C\c~G@  
    end B VeMV4  
    UA*VqK)Y  
    ws9IO ?|&G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o/7u7BQl2  
    % Compute the Zernike Polynomials 3%5YUG@  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FfM^2`xP  
    6Jq[]l"v  
    bKmwXDv'  
    % Determine the required powers of r: N8pL2y:R[P  
    % ----------------------------------- 5z 0VMt  
    m_abs = abs(m); +={K -g7U  
    rpowers = []; YhV<.2^k  
    for j = 1:length(n) qJ`:$U  
        rpowers = [rpowers m_abs(j):2:n(j)]; 131(0nl)=I  
    end C^L xuUW  
    rpowers = unique(rpowers); Y&yfm/Ru  
    K [DpH&  
    }r@dZ Bp:  
    % Pre-compute the values of r raised to the required powers, & V>rq'~;  
    % and compile them in a matrix: WqF,\y%W*  
    % ----------------------------- zsJ# CDm  
    if rpowers(1)==0 *'{-!Y  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G*+^b'7  
        rpowern = cat(2,rpowern{:}); T%)E!:}v  
        rpowern = [ones(length_r,1) rpowern]; lvWwr!w  
    else exhU!p8  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !\4B.  
        rpowern = cat(2,rpowern{:}); 1X5g(B  
    end VSY  p  
    h97#(_wV>  
    SdYf^@%}F  
    % Compute the values of the polynomials: IyHbl_ P ^  
    % -------------------------------------- V_gKl;Kfe8  
    y = zeros(length_r,length(n)); x ']'ODs  
    for j = 1:length(n) `5@F'tKQ  
        s = 0:(n(j)-m_abs(j))/2; 5_'lu  
        pows = n(j):-2:m_abs(j); J;obh.}u"{  
        for k = length(s):-1:1 Z,#H\1v3lB  
            p = (1-2*mod(s(k),2))* ... ;9k>; g3m  
                       prod(2:(n(j)-s(k)))/              ... [o#% Eg;  
                       prod(2:s(k))/                     ... 2.z-&lFBZ  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...  eo9/  
                       prod(2:((n(j)+m_abs(j))/2-s(k)));  %nY\"  
            idx = (pows(k)==rpowers); L_!ShE  
            y(:,j) = y(:,j) + p*rpowern(:,idx); CfU|]<  
        end =lJ ?yuc  
         ;Z{D@g+  
        if isnorm p5#x7*xR6  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); p@G7}'|eyA  
        end x>[]Qk^?q  
    end Y/.C+wW2  
    % END: Compute the Zernike Polynomials y,nmPX?]n  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4uIYX  
    2; ^ME\  
    h) Wp  
    % Compute the Zernike functions: 2DCQ5XewYe  
    % ------------------------------ .}!.4J%q2  
    idx_pos = m>0; Gc|)4c  
    idx_neg = m<0; *z0d~j*W;  
    gY~r{  
    uMg\s\Z  
    z = y; GkJcd;  
    if any(idx_pos) [Iks8ZWr_  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !e|\1v'0  
    end Tsg9,/vXM  
    if any(idx_neg) a<G&}|6  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Q|AZv>'!  
    end cFL~< [>_  
    uatUo  
    SL4?E<Jb  
    % EOF zernfun Q6Gw!!Z5EA  
     
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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)  .Mt3e c<  
    R@vcS=m7  
    DDE还是手动输入的呢? fr@F7s5}  
    BrV{X&>[i  
    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