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    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

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    离线niuhelen
     
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    只看楼主 倒序阅读 楼主  发表于: 2011-03-12
    小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 nc.(bb),  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! CHojF+e  
     
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    离线phility
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    只看该作者 1楼 发表于: 2011-03-12
    可以用matlab编程,用zernike多项式进行波面拟合,求出zernike多项式的系数,拟合的算法有很多种,最简单的是最小二乘法,你可以查下相关资料,挺简单的
    离线phility
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    只看该作者 2楼 发表于: 2011-03-12
    泽尼克多项式的前9项对应象差的
    离线niuhelen
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    只看该作者 3楼 发表于: 2011-03-12
    回 2楼(phility) 的帖子
    非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 p#  4@  
    function z = zernfun(n,m,r,theta,nflag) 9wB}EDZ  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. @}r2xY1  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d${RZ}/  
    %   and angular frequency M, evaluated at positions (R,THETA) on the D rMG{Yiu  
    %   unit circle.  N is a vector of positive integers (including 0), and e]qbh_A  
    %   M is a vector with the same number of elements as N.  Each element KBO{ g:"  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ]-D&/88``  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, O*:8gu'Y2  
    %   and THETA is a vector of angles.  R and THETA must have the same OfAh? ^R  
    %   length.  The output Z is a matrix with one column for every (N,M) [Dv6z t>  
    %   pair, and one row for every (R,THETA) pair. VY#:IE:T  
    % |rhCQ"H  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike $zR[2{bg  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), p^(gXzW  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral bTrQ(qp  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,]\:]Y&?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized '(4#He?Gd  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. M.loG4r!  
    % V.f'Cw  
    %   The Zernike functions are an orthogonal basis on the unit circle. }p <p(  
    %   They are used in disciplines such as astronomy, optics, and -eA3o2'  
    %   optometry to describe functions on a circular domain. >.fN@8[  
    % ,O;+fhUJ(  
    %   The following table lists the first 15 Zernike functions. m K);NvJ!  
    % HfN:oww  
    %       n    m    Zernike function           Normalization +1] xmnts  
    %       -------------------------------------------------- 1,/L&_=_A  
    %       0    0    1                                 1 r8uc.z2%  
    %       1    1    r * cos(theta)                    2 , id`=L=  
    %       1   -1    r * sin(theta)                    2 bktw?{h  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) }$zJdf,\  
    %       2    0    (2*r^2 - 1)                    sqrt(3) vA(')"DDT  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) SjZ?keKZ  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) F9Bj$`#)  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) x9Qa.Jmj  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) GkutS.2G#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +TR#  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) R8ui LZd  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u\]aUP e  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YnCWmlC  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d:x=g i!  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =h"*1`  
    %       -------------------------------------------------- CL U[')H0  
    % ua'dm6",:  
    %   Example 1: gkN|3^  
    % dF- d  
    %       % Display the Zernike function Z(n=5,m=1) qZ:--,9+  
    %       x = -1:0.01:1; :<`hsKy&  
    %       [X,Y] = meshgrid(x,x); ke(LjRS  
    %       [theta,r] = cart2pol(X,Y); SLiQHWw*J  
    %       idx = r<=1; O 0lQ1<=  
    %       z = nan(size(X)); W9$mgs=S`E  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); |0wUOs*5  
    %       figure >H ,t^i}@  
    %       pcolor(x,x,z), shading interp 'yWv @)  
    %       axis square, colorbar bN#)F    
    %       title('Zernike function Z_5^1(r,\theta)') <AzM~]"3  
    % $jDp ^ -  
    %   Example 2: +bj[.  
    % 4I[g{S nF  
    %       % Display the first 10 Zernike functions !u} }V  
    %       x = -1:0.01:1; ^ H,oI*  
    %       [X,Y] = meshgrid(x,x); `GG PkTN  
    %       [theta,r] = cart2pol(X,Y); U73{Uv  
    %       idx = r<=1; #hBDOXHPf  
    %       z = nan(size(X)); ={a8=E!;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; \\qw"w9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; y3 {om^ f  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; hE-u9i  
    %       y = zernfun(n,m,r(idx),theta(idx)); }tIIA"dZ  
    %       figure('Units','normalized') d45JT?qg&  
    %       for k = 1:10 <3!jra,h  
    %           z(idx) = y(:,k); ^[d|^fRH Q  
    %           subplot(4,7,Nplot(k)) C?FUc cI  
    %           pcolor(x,x,z), shading interp Ef;OrE""  
    %           set(gca,'XTick',[],'YTick',[]) |7jUf$Q\p  
    %           axis square !2('Cq_^  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +^c;4-X 0  
    %       end Y dgaZJs  
    % t._W643~  
    %   See also ZERNPOL, ZERNFUN2. mn=G6h T}W  
    /CtR|~wL  
    %   Paul Fricker 11/13/2006 ACg5"  
    r+BPz%wM=O  
    OG_2k3v  
    % Check and prepare the inputs: @x>J-Owd]J  
    % ----------------------------- 'w+T vOB  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Q<y&*o3YF|  
        error('zernfun:NMvectors','N and M must be vectors.') =$B:i>z<  
    end %2<G3]6^U  
    +3 2"vq)_  
    if length(n)~=length(m) su}> >07  
        error('zernfun:NMlength','N and M must be the same length.') t Ztyx;EP  
    end Z[baQO  
    +_8*;k@F'  
    n = n(:); 4Lx#5}P  
    m = m(:); *8zn\No<,  
    if any(mod(n-m,2)) VP$`.y  
        error('zernfun:NMmultiplesof2', ... f$x\~y<[  
              'All N and M must differ by multiples of 2 (including 0).') 1{oq8LB  
    end Y5~_y?BX  
    i#t)tM"  
    if any(m>n) Qa nE]  
        error('zernfun:MlessthanN', ... @<ba+z>"~4  
              'Each M must be less than or equal to its corresponding N.') ZGHkW9b&  
    end 2 $^n@<uZ@  
    A0SEzX({[  
    if any( r>1 | r<0 ) M@rknq@  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') \N\Jny  
    end nf5Ld"|%9  
    n>tYeN)F<  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :7t~p&J  
        error('zernfun:RTHvector','R and THETA must be vectors.') R 2uo ZA,  
    end 'aQ"&GX@  
    Si#b"ls'  
    r = r(:); 1&~u:RUXe  
    theta = theta(:); :,$:@  
    length_r = length(r); 9-Bp=M  
    if length_r~=length(theta) i0 ax`37  
        error('zernfun:RTHlength', ... @, D 3$P8}  
              'The number of R- and THETA-values must be equal.') 33lD`4i+  
    end >A#wvQl7   
    9 ve q  
    % Check normalization: gaaW:**y  
    % -------------------- Kc+;"4/#q  
    if nargin==5 && ischar(nflag) < @9p|[!  
        isnorm = strcmpi(nflag,'norm'); 'dYjbQ}~;  
        if ~isnorm s+>VqyHgf  
            error('zernfun:normalization','Unrecognized normalization flag.') iN0gvjZ  
        end q;a`*gX^  
    else j?ihUNY!+  
        isnorm = false; C2;qSKG3{m  
    end "q(#,,_  
    JPQ[JD^]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <o^_il$W  
    % Compute the Zernike Polynomials 7a Fvj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6D,xs}j1  
    $\l7aA5~  
    % Determine the required powers of r: =e/{fUg8f  
    % ----------------------------------- nS0K&MH6B  
    m_abs = abs(m); a;J{'PHu  
    rpowers = []; i $H aE)qZ  
    for j = 1:length(n) je1f\N45  
        rpowers = [rpowers m_abs(j):2:n(j)]; wkK61a h6  
    end [H5TtsQ[  
    rpowers = unique(rpowers); sw{,l"]<  
    PDaHY  
    % Pre-compute the values of r raised to the required powers, f?T6Ne'  
    % and compile them in a matrix: LC/9)Sh_n  
    % ----------------------------- N!>Gg|@~  
    if rpowers(1)==0 |e@9YDZ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pqO}=*v@  
        rpowern = cat(2,rpowern{:}); !uLW-[F,  
        rpowern = [ones(length_r,1) rpowern]; 8Czy<}S<G  
    else A-e#&pJ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ru,]!YPJE2  
        rpowern = cat(2,rpowern{:}); "h'0&ZP~_  
    end Hzs]\%"  
    O;c;>x_dA  
    % Compute the values of the polynomials: 0UeDM*  
    % -------------------------------------- @EH:4~  
    y = zeros(length_r,length(n)); n<6p0w  
    for j = 1:length(n) s0"S;{_#  
        s = 0:(n(j)-m_abs(j))/2; u1a5Vtel  
        pows = n(j):-2:m_abs(j); m`!C|?hu  
        for k = length(s):-1:1 }R:eKj  
            p = (1-2*mod(s(k),2))* ... 57e'a&}e  
                       prod(2:(n(j)-s(k)))/              ... =s`\W7/;{-  
                       prod(2:s(k))/                     ... } 5i0R  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .a\b_[+W  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); w(pLU$6X  
            idx = (pows(k)==rpowers); $lmbeW[0  
            y(:,j) = y(:,j) + p*rpowern(:,idx); S0nBX"$u  
        end [8AGW7_  
         >=-w2&  
        if isnorm MVU5+wX  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); [=079UN-X  
        end l-4T Tg  
    end I`kaAOe  
    % END: Compute the Zernike Polynomials I=X-e#HM?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /gh=+;{  
    Qi`Lj5;\F  
    % Compute the Zernike functions: yS0YWqv]6@  
    % ------------------------------ (yWU9q)5  
    idx_pos = m>0; w!o[pvyR$  
    idx_neg = m<0; {LfVV5?  
    )O~LXK=b  
    z = y; (y%}].[bB  
    if any(idx_pos) <wUDcF  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');  b=v  
    end z/u;afB9q  
    if any(idx_neg) cmF&1o3_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $A\fm`  
    end 1P(rgn:8e  
    ;Ut0tm  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) (@*[^@ipV  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. eAN]*: ]g  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated yi*)g0M  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive )*@n G$i99  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, !Kr|04Qp#x  
    %   and THETA is a vector of angles.  R and THETA must have the same asqbLtQ  
    %   length.  The output Z is a matrix with one column for every P-value, p\OUxAm  
    %   and one row for every (R,THETA) pair. @qk$ 6X  
    % jY'svD~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike D@ut -J(.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) \U $'3M  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;Z|X` <6g  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 I$!rNfrs  
    %   for all p. `s93P^%  
    % 0zfh:O  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 -Yx'qz@  
    %   Zernike functions (order N<=7).  In some disciplines it is 8&?Kg>M  
    %   traditional to label the first 36 functions using a single mode N>##} i  
    %   number P instead of separate numbers for the order N and azimuthal ZGgKCCt  
    %   frequency M. 9x@( K|  
    % 0nUcUdIf+  
    %   Example: l&l&e OE  
    % rOd<nP^`\  
    %       % Display the first 16 Zernike functions ?145^ w  
    %       x = -1:0.01:1; 5v6 x  
    %       [X,Y] = meshgrid(x,x); 94 58.!3  
    %       [theta,r] = cart2pol(X,Y); BM5+;h !  
    %       idx = r<=1; _/Ky;p.  
    %       p = 0:15; `|?K4<5|  
    %       z = nan(size(X)); ax$ashFO/!  
    %       y = zernfun2(p,r(idx),theta(idx)); 4FURm@C6  
    %       figure('Units','normalized') 9IJBK  
    %       for k = 1:length(p) o1C1F}gxU  
    %           z(idx) = y(:,k); ZXV_Dc   
    %           subplot(4,4,k) J%ng8v5ex  
    %           pcolor(x,x,z), shading interp -xs @rV`  
    %           set(gca,'XTick',[],'YTick',[]) {I'8+~|pZL  
    %           axis square #M kXio; h  
    %           title(['Z_{' num2str(p(k)) '}']) MH@=Qqx#=t  
    %       end 8,iBG! RF  
    % 278:5yC  
    %   See also ZERNPOL, ZERNFUN. iAD'MB  
    D2N| A  
    %   Paul Fricker 11/13/2006 u^.7zL+  
    Qj;wk lq  
    Fy:CG6@X  
    % Check and prepare the inputs: ROcI.tL  
    % ----------------------------- IoO tn  
    if min(size(p))~=1 n N.6?a  
        error('zernfun2:Pvector','Input P must be vector.') x(oL\I_Z  
    end ,z<J`n  
    LsaE-l  
    if any(p)>35 |f5WN&c  
        error('zernfun2:P36', ... UrtA]pc3L  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... zq]I"0Bi.  
               '(P = 0 to 35).']) :/YHU3~Y  
    end .. jc^'L  
    qw^kA?  
    % Get the order and frequency corresonding to the function number: >YG1sMV-J  
    % ---------------------------------------------------------------- KnL-qc  
    p = p(:); #r5IwyL  
    n = ceil((-3+sqrt(9+8*p))/2);  NGQBOV  
    m = 2*p - n.*(n+2); 7{jB!Xj  
    -u)f@e  
    % Pass the inputs to the function ZERNFUN: )~W 35  
    % ---------------------------------------- sBUK v(U)  
    switch nargin \dvzL(,  
        case 3 dH]0 (aJ  
            z = zernfun(n,m,r,theta); U\OfB'Dn  
        case 4 z+3G zDLy  
            z = zernfun(n,m,r,theta,nflag); r~f*aD  
        otherwise 'T=$Q%Qv  
            error('zernfun2:nargin','Incorrect number of inputs.') k;EG28   
    end ,Dh+-}  
    0,a/t jSr  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) {T;A50  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 3K)12x$.K  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of j%w}hGW%,  
    %   order N and frequency M, evaluated at R.  N is a vector of =yT3#A~<G  
    %   positive integers (including 0), and M is a vector with the ^wnlZ09J  
    %   same number of elements as N.  Each element k of M must be a AQ}(v,DOb  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) gc5u@(P"  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is p~v rr 5  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix :eHD{=  
    %   with one column for every (N,M) pair, and one row for every n,,hE_  
    %   element in R. 4k$i:st;  
    % [u?*' c{  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- D./!/>@f  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is mBAI";L3  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to v w.rkAGY  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Kp]\r-5UD>  
    %   for all [n,m]. 2Ra}&ie  
    % _1>Xk_  
    %   The radial Zernike polynomials are the radial portion of the v8{ jEAK  
    %   Zernike functions, which are an orthogonal basis on the unit #\P\(+0K  
    %   circle.  The series representation of the radial Zernike  _%r+?I  
    %   polynomials is ?$chO|QY  
    % ! .q,m>?+  
    %          (n-m)/2 \r.{Ru  
    %            __ NE~R&ym9  
    %    m      \       s                                          n-2s tVUC@M>'  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Y5dt/8Jo  
    %    n      s=0 $Gy&  
    % ?'xwr )v  
    %   The following table shows the first 12 polynomials. yuef84~  
    % 6np  
    %       n    m    Zernike polynomial    Normalization ^X?D4a|;#g  
    %       --------------------------------------------- cr wui8  
    %       0    0    1                        sqrt(2) pq T+lai)#  
    %       1    1    r                           2 yG v7^d  
    %       2    0    2*r^2 - 1                sqrt(6) fen~k#|l  
    %       2    2    r^2                      sqrt(6) 6@rebe!&=  
    %       3    1    3*r^3 - 2*r              sqrt(8) DqH?:`G  
    %       3    3    r^3                      sqrt(8) },]G +L;R  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) qj.>4d  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) L2 ybL#dz  
    %       4    4    r^4                      sqrt(10) /W|=Or2oR  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) %&Fsk]T%:  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) hx.ln6=4  
    %       5    5    r^5                      sqrt(12) ^ ,cwm:B@  
    %       --------------------------------------------- xs}3=&c(  
    % |_h$}~ ;  
    %   Example: @$*LU:[  
    % VG=mA4Dd  
    %       % Display three example Zernike radial polynomials u4KP;_,m  
    %       r = 0:0.01:1; &^7^7:Y=?  
    %       n = [3 2 5]; 4Ou5Vp&y  
    %       m = [1 2 1]; >N bb0T  
    %       z = zernpol(n,m,r); sw\O\%^  
    %       figure cWU9mzsE  
    %       plot(r,z) h>a/3a$g  
    %       grid on W2`/z)[*>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') G u4mP  
    % pYBY"r  
    %   See also ZERNFUN, ZERNFUN2. Cu $mb}@  
    =i1+t"=  
    % A note on the algorithm. 'JpCS  
    % ------------------------ F,.dC&B  
    % The radial Zernike polynomials are computed using the series O{{\jn|lR  
    % representation shown in the Help section above. For many special uE=pq<  
    % functions, direct evaluation using the series representation can _!ITCkBj  
    % produce poor numerical results (floating point errors), because lP;X=X>  
    % the summation often involves computing small differences between n5U-D0/Q  
    % large successive terms in the series. (In such cases, the functions -Pt']07E  
    % are often evaluated using alternative methods such as recurrence {/2 _"H3:  
    % relations: see the Legendre functions, for example). For the Zernike EpCT !e  
    % polynomials, however, this problem does not arise, because the DkA@KS1Dq  
    % polynomials are evaluated over the finite domain r = (0,1), and 1w$X;q"  
    % because the coefficients for a given polynomial are generally all GF/!@N  
    % of similar magnitude. - %'ys  
    % 1 k\~%  
    % ZERNPOL has been written using a vectorized implementation: multiple /l b"g_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] R^f-j-$o]  
    % values can be passed as inputs) for a vector of points R.  To achieve [dIlt"2fV  
    % this vectorization most efficiently, the algorithm in ZERNPOL 0_f6Qrcj  
    % involves pre-determining all the powers p of R that are required to T*"*##c  
    % compute the outputs, and then compiling the {R^p} into a single AM:lU  
    % matrix.  This avoids any redundant computation of the R^p, and Tya[6b!8  
    % minimizes the sizes of certain intermediate variables. (Wu J9  
    % ^fqco9^;  
    %   Paul Fricker 11/13/2006 #\T5r*W  
    zf.&E3Sn  
    YcdT/  
    % Check and prepare the inputs: gNaB^IY  
    % ----------------------------- ~a06x^=j  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) n:P++^ j  
        error('zernpol:NMvectors','N and M must be vectors.') 9k*1_  
    end qZB}}pM#  
    ><DXT nt'x  
    if length(n)~=length(m) 1=Y pNXX  
        error('zernpol:NMlength','N and M must be the same length.') TD^w|U.  
    end p->b Vt  
    ;hZ^zL  
    n = n(:); ;NPbEPL[5  
    m = m(:); (&@,ZI;  
    length_n = length(n); ~&%&Z  
    Z~Vups#+f  
    if any(mod(n-m,2)) m[$pj~<\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') %^LwLyoVM  
    end l}& &f8n  
    *Hed^[sO  
    if any(m<0) E,gpi  
        error('zernpol:Mpositive','All M must be positive.') @q++eGm\Q  
    end Uy;e5<<  
    .?dYY;P  
    if any(m>n) Kq&JvY^  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') }WM!e"  
    end K0-AP $  
    Xd@_:ds  
    if any( r>1 | r<0 ) 9^2l<4^Z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') `CqF&b  
    end )D8V;g(7F  
    $(N+E,XB  
    if ~any(size(r)==1) `S:LuU8e  
        error('zernpol:Rvector','R must be a vector.')  .H7xG'$  
    end %O02xr=  
    o`%;*tx  
    r = r(:); B nu5\P  
    length_r = length(r); 6$ x9@x8  
    5 K[MKfT  
    if nargin==4 9 =zZ,dg  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); v7#`b}'W  
        if ~isnorm 'E_~>  
            error('zernpol:normalization','Unrecognized normalization flag.') Tp&7CNl|  
        end $Y=xu2u)  
    else Ek,$XH  
        isnorm = false; } U_z XuUz  
    end ?a{es!  
    |L%d^m  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .EdQ]c-E=  
    % Compute the Zernike Polynomials g>;u} +lO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]v|n'D-?  
    z z2'h>  
    % Determine the required powers of r: f;cY&GC  
    % -----------------------------------  zPW_  
    rpowers = []; ~F^7L5d}C  
    for j = 1:length(n) "S^ ""5  
        rpowers = [rpowers m(j):2:n(j)]; 6sz:rv}  
    end OTV$8{  
    rpowers = unique(rpowers); bO6LBSZx]  
     mm9xO%  
    % Pre-compute the values of r raised to the required powers,  ;)ji3M  
    % and compile them in a matrix: {qbx iL-  
    % ----------------------------- (^9dp[2  
    if rpowers(1)==0 1b`WzoJgH  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,); -v4$  
        rpowern = cat(2,rpowern{:}); ^rssZQKY[  
        rpowern = [ones(length_r,1) rpowern]; =, G^GMi'  
    else -YJ4-]Z  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )L/o|%r!  
        rpowern = cat(2,rpowern{:}); ;wKsi_``@  
    end #"KaRh  
    ,;k`N`#'  
    % Compute the values of the polynomials: >A ?{cbJ  
    % -------------------------------------- 1`v$R0 `!  
    z = zeros(length_r,length_n); vD3j(d  
    for j = 1:length_n ,l7',@6Y  
        s = 0:(n(j)-m(j))/2; B0dv_'L}L  
        pows = n(j):-2:m(j); Pt[ b;}  
        for k = length(s):-1:1 ~%d*#Yxq  
            p = (1-2*mod(s(k),2))* ... mz?1J4rt  
                       prod(2:(n(j)-s(k)))/          ... @8"cT-  
                       prod(2:s(k))/                 ... X3l? YA  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Z[G:  
                       prod(2:((n(j)+m(j))/2-s(k))); wLOS , =  
            idx = (pows(k)==rpowers); IZ 3e:  
            z(:,j) = z(:,j) + p*rpowern(:,idx); M TOZ:b  
        end ~?<VT k  
         =5u;\b>*  
        if isnorm S?e*<s9k  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); >@uFye$  
        end %IH ra6  
    end SJ).L.Cm6  
    ,;~@t:!c  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  j6#Vwcr  
    _ pO1XM  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 r Fhi:uRV  
    ]:svR@E  
    07年就写过这方面的计算程序了。
    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)