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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  4V 5  
    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! Y7@$#/1  
     
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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 UN Kr FYl  
    function z = zernfun(n,m,r,theta,nflag) H6%%n X  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. V.kRV{43  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LHgEb9\Q  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ~"#[<d  
    %   unit circle.  N is a vector of positive integers (including 0), and ^Y+P(o$HM  
    %   M is a vector with the same number of elements as N.  Each element Kv>P+I'|r  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) C.q4rr  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, H D{2nZT  
    %   and THETA is a vector of angles.  R and THETA must have the same Ld:U~M-  
    %   length.  The output Z is a matrix with one column for every (N,M) H.]rH,8  
    %   pair, and one row for every (R,THETA) pair. ~jn~M_}K  
    % ).9m6.%Uk  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gKWsmx!["  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2<8JY4]!]  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral u40<>A  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B@v"giJgr  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized A&/ YnJ"  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UU" '  
    % `oGL==  
    %   The Zernike functions are an orthogonal basis on the unit circle. c*HWH$kB  
    %   They are used in disciplines such as astronomy, optics, and 1|/]bffg!c  
    %   optometry to describe functions on a circular domain. KO5! (vi@  
    % ;ax%H @o  
    %   The following table lists the first 15 Zernike functions. S{F'k;x/5  
    % [BzwQ 4  
    %       n    m    Zernike function           Normalization byetbt(IF  
    %       -------------------------------------------------- )r.4`5Rc  
    %       0    0    1                                 1 Ht=h9}x"g  
    %       1    1    r * cos(theta)                    2 E\dJb}"x %  
    %       1   -1    r * sin(theta)                    2 A/w7 (  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ,"EgYd8-'  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 1/% g VB8  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) lzup! `g  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) =E10j.r  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9L9+zs3 k  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) T+U,?2nF:  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) @fO[{V  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) EQ> ]~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U>=& 2Z2?  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) F>/"If#  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) lD\vq2  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ud,=O X q  
    %       -------------------------------------------------- , UiA?7k  
    % 3}9c0%}F  
    %   Example 1: [/IN820t  
    % ?A`8c R=)I  
    %       % Display the Zernike function Z(n=5,m=1) l0-zu6i w  
    %       x = -1:0.01:1; 5svM3  #  
    %       [X,Y] = meshgrid(x,x); `37$YdX  
    %       [theta,r] = cart2pol(X,Y); iX\]-_D  
    %       idx = r<=1; :#&Y  
    %       z = nan(size(X)); 0$A7"^]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); A4`3yy{0-  
    %       figure .1#G*A|  
    %       pcolor(x,x,z), shading interp .*W_;Fo  
    %       axis square, colorbar *N!>c&8  
    %       title('Zernike function Z_5^1(r,\theta)') 7r,h[9~e  
    % Qq*Ks 5   
    %   Example 2: )CM3v L {  
    % (Ceq@eAlT  
    %       % Display the first 10 Zernike functions $:D-dUr1  
    %       x = -1:0.01:1; (Y>|P  
    %       [X,Y] = meshgrid(x,x); $>=?'wr  
    %       [theta,r] = cart2pol(X,Y); BA(PWX`H  
    %       idx = r<=1; O{w'i|  
    %       z = nan(size(X)); "Q <  
    %       n = [0  1  1  2  2  2  3  3  3  3]; k2Y *  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; w:+wx/\  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; # )]L3H<  
    %       y = zernfun(n,m,r(idx),theta(idx)); 7;x}W-`iF  
    %       figure('Units','normalized') M:QM*?+)  
    %       for k = 1:10 8^>qzaf 8  
    %           z(idx) = y(:,k);  mX&!/U  
    %           subplot(4,7,Nplot(k)) NUp,In_  
    %           pcolor(x,x,z), shading interp oW\kJ>!  
    %           set(gca,'XTick',[],'YTick',[]) Ia!B8$$'RP  
    %           axis square ^DH*\ee  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?&rt)/DV,  
    %       end ;2%8tV$V  
    % 9w:9XziT  
    %   See also ZERNPOL, ZERNFUN2. ;r.0=Uo9]  
    NGq@x%T  
    %   Paul Fricker 11/13/2006 (HD8Mm  
    Tw+V$:$$  
    $$f89, h  
    % Check and prepare the inputs: 2SV}mK U  
    % ----------------------------- b^q8s4(   
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %Z]c[V.  
        error('zernfun:NMvectors','N and M must be vectors.') |O4LR,{G.w  
    end 3]cW08"c  
    P'Diie  
    if length(n)~=length(m) ILyI%DA&  
        error('zernfun:NMlength','N and M must be the same length.') {Ne5*HFV  
    end i4s_:%+  
    Gw1Rp  
    n = n(:); $3c9iVK~_  
    m = m(:); q\]"}M 8  
    if any(mod(n-m,2)) S<nf"oy_K  
        error('zernfun:NMmultiplesof2', ... xN CU5  
              'All N and M must differ by multiples of 2 (including 0).') f<;w1sM\  
    end Y6w7sr_R  
    = iXHu *g  
    if any(m>n) xixdv{M<FF  
        error('zernfun:MlessthanN', ... 'Tbdo >y  
              'Each M must be less than or equal to its corresponding N.') XSoHh-  
    end N3$%!\~O  
    V N<omi+4  
    if any( r>1 | r<0 ) ^<OcbOn;O  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') %`)lCK)2  
    end `%ulorS  
    U6x$R O!  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) KbTd`AIL  
        error('zernfun:RTHvector','R and THETA must be vectors.') ,:=g}i  
    end 7GG:1:2+>  
    Q@0Zh, l  
    r = r(:); PL|zm5923  
    theta = theta(:); Sk7sxy<F'  
    length_r = length(r); gUWW}*\ U  
    if length_r~=length(theta) "OWW -m  
        error('zernfun:RTHlength', ... %yPjPUHy  
              'The number of R- and THETA-values must be equal.') G5,g$yNs  
    end qac8zt#2 C  
    M7gb3gw6  
    % Check normalization: K'\Jnn  
    % -------------------- 'dvi@Jx  
    if nargin==5 && ischar(nflag) Kv ~'*A)d  
        isnorm = strcmpi(nflag,'norm'); Z66h  
        if ~isnorm 1G<S'd+N  
            error('zernfun:normalization','Unrecognized normalization flag.') U~I y),5  
        end .NSV%I  
    else FaQz03N\  
        isnorm = false; aE 2=  
    end M.3ULt8  
    {!>'# F^e  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^@HWw@GA  
    % Compute the Zernike Polynomials 51gSbkVX  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UZRN4tru6  
    Hj97&C{Q^  
    % Determine the required powers of r: {M ^5w  
    % ----------------------------------- '_B;e=v`  
    m_abs = abs(m); >qS2ha  
    rpowers = []; >UnLq:G  
    for j = 1:length(n) :j&-Lc  
        rpowers = [rpowers m_abs(j):2:n(j)]; nq qqP  
    end !uW;Ea?  
    rpowers = unique(rpowers); 8DkZ @}  
    p\22_m_wd  
    % Pre-compute the values of r raised to the required powers, *?rO@sQy]  
    % and compile them in a matrix: "h7Np/ m3  
    % -----------------------------  {HbSty  
    if rpowers(1)==0 IC:>60A,]  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ok1-`c P  
        rpowern = cat(2,rpowern{:}); K1CgM1v  
        rpowern = [ones(length_r,1) rpowern]; 45Lzq6  
    else BG_6$9y  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4w#:?Y _\[  
        rpowern = cat(2,rpowern{:}); )(+q~KA}  
    end U8OVn(qV  
    95mwDHbA  
    % Compute the values of the polynomials: {[~dI ~  
    % -------------------------------------- 6 \}.l  
    y = zeros(length_r,length(n)); $6]1T>  
    for j = 1:length(n) :u`gjj$:s  
        s = 0:(n(j)-m_abs(j))/2; dlH&8  
        pows = n(j):-2:m_abs(j); :%<'('S |  
        for k = length(s):-1:1 "#P#;]\`  
            p = (1-2*mod(s(k),2))* ... 0-:dzf  
                       prod(2:(n(j)-s(k)))/              ... ?tkl cYB  
                       prod(2:s(k))/                     ... [&sabM`Ul  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... H"c2kno9  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); L KLLBrm:  
            idx = (pows(k)==rpowers); 49=L9:  
            y(:,j) = y(:,j) + p*rpowern(:,idx); rN'8,CV  
        end C9 j{:&  
         g>QN9v})  
        if isnorm tuJ{IF  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); L),r\#Y(v  
        end D< 0))r  
    end =klfCFwP  
    % END: Compute the Zernike Polynomials xoQ(GrBY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LKgo(&mY  
    {7Hc00FM  
    % Compute the Zernike functions: nd"$gi  
    % ------------------------------ "~q~)T1Z  
    idx_pos = m>0; @<koL  
    idx_neg = m<0; |3BxNFe`%  
     0:$pJtx"  
    z = y; e4FR)d0x  
    if any(idx_pos) <B!DwMk;.  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X/h|;C* 9  
    end ;Irn{O  
    if any(idx_neg) U+[h^M$U  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <vt}+uMzXv  
    end Ro=dgQ0:t  
    '4}8WYKQ  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) rDGrq9  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. lUdk^7:M  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated n. vrq-  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive u/V&1In  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, q2/kegAT  
    %   and THETA is a vector of angles.  R and THETA must have the same qMw_`dC  
    %   length.  The output Z is a matrix with one column for every P-value, _na/&J 6  
    %   and one row for every (R,THETA) pair. (gIFuOGi>  
    % sQ+s3x1y  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Tj}%G  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 4'td6F  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 53>(2 _/[r  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 YF>1 5{H  
    %   for all p. p0PK-e`@:  
    % bXA%|7*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 RK p9[^/?  
    %   Zernike functions (order N<=7).  In some disciplines it is 5n1`$T.WG  
    %   traditional to label the first 36 functions using a single mode n2|@Hz_  
    %   number P instead of separate numbers for the order N and azimuthal >">Xd@Wk  
    %   frequency M. r;p@T8k  
    % pu_?) U  
    %   Example: @$nh6l>i  
    % ^^< C9  
    %       % Display the first 16 Zernike functions MI-S}Qoe  
    %       x = -1:0.01:1; +S'm<}"1  
    %       [X,Y] = meshgrid(x,x); y}?PyPz  
    %       [theta,r] = cart2pol(X,Y); 4*inN~cU  
    %       idx = r<=1; C-g,uARX(r  
    %       p = 0:15; Yj'"Wg  
    %       z = nan(size(X)); 0O>M/ *W  
    %       y = zernfun2(p,r(idx),theta(idx)); pE%*r@p4&4  
    %       figure('Units','normalized') ^XG$?2<U  
    %       for k = 1:length(p) q>wO=qWx  
    %           z(idx) = y(:,k); |oSqy  
    %           subplot(4,4,k) 3i\Np =  
    %           pcolor(x,x,z), shading interp F'CJN$6Mw/  
    %           set(gca,'XTick',[],'YTick',[]) hj{)6dBX%  
    %           axis square Wf-XH|j[  
    %           title(['Z_{' num2str(p(k)) '}']) JSID@ n<b?  
    %       end gk;hpO  
    % CugZ!>;^  
    %   See also ZERNPOL, ZERNFUN. YT,yRV9#  
    /qMiv7m~Q  
    %   Paul Fricker 11/13/2006 ].2t7{64  
    'z{|#zd9  
    eI-SWwmv/u  
    % Check and prepare the inputs: My Ky*wD  
    % ----------------------------- ,*YmXR-"  
    if min(size(p))~=1 vif)g6,  
        error('zernfun2:Pvector','Input P must be vector.') u~>G8y)k9O  
    end L=fy!R  
    s0:M'wA  
    if any(p)>35 1hSV/%v_  
        error('zernfun2:P36', ... mU"Am0Bdjq  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ,In%r`{i  
               '(P = 0 to 35).']) FBvh7D.hV  
    end f~/hsp~Hp  
    ijvDFyN>  
    % Get the order and frequency corresonding to the function number: 8 nL9#b  
    % ---------------------------------------------------------------- D u T6Od/f  
    p = p(:); f=VlO d  
    n = ceil((-3+sqrt(9+8*p))/2); g-B{K "z  
    m = 2*p - n.*(n+2); .lM]>y)  
    Ul6|LTY  
    % Pass the inputs to the function ZERNFUN: ef_H*e  
    % ---------------------------------------- 7U?#Xi5  
    switch nargin o|q5eUh=EY  
        case 3 (:O6sTx-hE  
            z = zernfun(n,m,r,theta); )WW*X6[k  
        case 4 ZI1*Cb  
            z = zernfun(n,m,r,theta,nflag); <QC7HR  
        otherwise lpS v  
            error('zernfun2:nargin','Incorrect number of inputs.') QgQclML1|  
    end M d8(P23hS  
    +u*Pi  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) _+nk3-yQw  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. NZ&ZK@h}.  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 6R^32VeK($  
    %   order N and frequency M, evaluated at R.  N is a vector of <|.]$QSi  
    %   positive integers (including 0), and M is a vector with the <66%(J>  
    %   same number of elements as N.  Each element k of M must be a 54JZOtC3~  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 'q_Z dw%  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &-p~UZy  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix /; /:>c  
    %   with one column for every (N,M) pair, and one row for every .-![ ra  
    %   element in R. db8vm4  
    % ujkWVE'  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- U`8Er48X  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is q_`j-!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to S[yrGX8lu  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i+in?!@G:  
    %   for all [n,m]. T3<1{"&  
    % [! wJIy?,  
    %   The radial Zernike polynomials are the radial portion of the K]Vp! G  
    %   Zernike functions, which are an orthogonal basis on the unit wB' !@>db  
    %   circle.  The series representation of the radial Zernike %4F\#" A  
    %   polynomials is [tJn! cMs  
    % OFmHj]I7=  
    %          (n-m)/2 xV`l6QS  
    %            __ On~KTt3Mp  
    %    m      \       s                                          n-2s [7~AWZU3  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r +9|0\Q  
    %    n      s=0 ZzTkEz >  
    % V*fv>f:Yv  
    %   The following table shows the first 12 polynomials. Vy[xu$y  
    % \P9ms?((A  
    %       n    m    Zernike polynomial    Normalization |<,0*2  
    %       --------------------------------------------- ~_"V7  
    %       0    0    1                        sqrt(2) ^BRqsVw9  
    %       1    1    r                           2 "*j8G8  
    %       2    0    2*r^2 - 1                sqrt(6) @Lf&[_  
    %       2    2    r^2                      sqrt(6) @x}^2FE  
    %       3    1    3*r^3 - 2*r              sqrt(8) :[(%4se  
    %       3    3    r^3                      sqrt(8) @RszPH1B  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 0A~UuH0.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) cN?/YkW?]  
    %       4    4    r^4                      sqrt(10) j<~T:Tk  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) D}X6I#U'/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) $ (xdF  
    %       5    5    r^5                      sqrt(12) yEbo`/ ]b  
    %       --------------------------------------------- 4%8den,|  
    % iymN|KdpaZ  
    %   Example: %C qp88]  
    % <]KQ$8dtD  
    %       % Display three example Zernike radial polynomials 1L'Q;?&2H,  
    %       r = 0:0.01:1; yp^k;G?_d  
    %       n = [3 2 5]; na8`V`77  
    %       m = [1 2 1]; tJ6Q7 J;n  
    %       z = zernpol(n,m,r); LL (TD&  
    %       figure 8q{|nH  
    %       plot(r,z) %`T}%B  
    %       grid on IvkYM`%  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') GiM-8y~  
    % M&29J  
    %   See also ZERNFUN, ZERNFUN2. ];6955I!  
    czu9a"M>X  
    % A note on the algorithm. SJh~4R\  
    % ------------------------ k[ D,du')  
    % The radial Zernike polynomials are computed using the series ( >ze{T|  
    % representation shown in the Help section above. For many special ?<W|Ya  
    % functions, direct evaluation using the series representation can I;E?;i  
    % produce poor numerical results (floating point errors), because ^M q@} 0  
    % the summation often involves computing small differences between /ILd|j(e  
    % large successive terms in the series. (In such cases, the functions {5X,xdzR  
    % are often evaluated using alternative methods such as recurrence $H)!h^7^9  
    % relations: see the Legendre functions, for example). For the Zernike )fIG4#%\  
    % polynomials, however, this problem does not arise, because the ^t7u4w!  
    % polynomials are evaluated over the finite domain r = (0,1), and .uS`RS8JM  
    % because the coefficients for a given polynomial are generally all ^< cJ;u*0  
    % of similar magnitude. n-jPb064  
    % *w _o8!3-  
    % ZERNPOL has been written using a vectorized implementation: multiple =M*pym]QSY  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] wL, -"  
    % values can be passed as inputs) for a vector of points R.  To achieve et)n`NlcK  
    % this vectorization most efficiently, the algorithm in ZERNPOL GwP!:p|  
    % involves pre-determining all the powers p of R that are required to : Bo  
    % compute the outputs, and then compiling the {R^p} into a single 1 /{~t[*.  
    % matrix.  This avoids any redundant computation of the R^p, and zz(|V  
    % minimizes the sizes of certain intermediate variables. )~R[aXkvY  
    % |,oLZC Na  
    %   Paul Fricker 11/13/2006 T"za|Fo  
    fi*b]a\'  
    ,C><n kx  
    % Check and prepare the inputs: |X A0F\  
    % ----------------------------- e CN:  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $<2d|;7r  
        error('zernpol:NMvectors','N and M must be vectors.') g&F$hm  
    end a$Ud"  
     I2i'  
    if length(n)~=length(m) .|go$}Fk  
        error('zernpol:NMlength','N and M must be the same length.') YQHpW>z  
    end ^L d5<  
    _^(1Qb[  
    n = n(:); 3ddw'b'aQ  
    m = m(:); \ZV>5N3hS  
    length_n = length(n); ZpOME@9,  
    &a=rJvnIO&  
    if any(mod(n-m,2)) I)` +:+P  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') nyR<pnuC'  
    end `3VI9GmQ  
    o jxK8_kl  
    if any(m<0) =Jw*T[E  
        error('zernpol:Mpositive','All M must be positive.') 71AYDO  
    end @Z'i7Z  
    }`QZV_  
    if any(m>n) l]wLQqoO  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') p\;8?x  
    end 3]JJCaf  
    FqbGT(QB0  
    if any( r>1 | r<0 ) }MMKOr(  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') S{&%tj~U  
    end {vW0O&[  
    Pi?G:IF  
    if ~any(size(r)==1) GU/P%c/V  
        error('zernpol:Rvector','R must be a vector.') -A<@Pg  
    end Myg;2.  
    Q) aZ0 Pt  
    r = r(:); Ieq_XF]U  
    length_r = length(r); c\rP -"C  
    ?K2EK'-q  
    if nargin==4 ,ps?@lD  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); lv!j  
        if ~isnorm 9RJ#zUK  
            error('zernpol:normalization','Unrecognized normalization flag.') C)yw b6  
        end j96}E/gF  
    else m5N,[^-  
        isnorm = false; r7_%t_O|IL  
    end mUP!jTF  
    RiR],Sj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fyw X  
    % Compute the Zernike Polynomials "y7\F9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %2I>-0]B  
    %Ul,9qG+  
    % Determine the required powers of r: ydOG8EI  
    % ----------------------------------- |+HJ>xA4I  
    rpowers = []; 9fbbJ"I+  
    for j = 1:length(n) ,Il) tH  
        rpowers = [rpowers m(j):2:n(j)]; `0U\|I#  
    end S`  U,  
    rpowers = unique(rpowers); AHuIA{AdUR  
    2Uf/'  
    % Pre-compute the values of r raised to the required powers, Y{dX[^[  
    % and compile them in a matrix: ;J+iwS*Z  
    % ----------------------------- T&6>Eb0{  
    if rpowers(1)==0 1CZO+MB&"$  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); MYVVI1A  
        rpowern = cat(2,rpowern{:}); fNx!'{o"  
        rpowern = [ones(length_r,1) rpowern]; Ue;Z)}  
    else a;; Es  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @?]>4+Oa0  
        rpowern = cat(2,rpowern{:}); .6rbn8h  
    end *mj=kJ7(  
    D[]0/+,  
    % Compute the values of the polynomials: j`@`M*)GB  
    % -------------------------------------- G^h:#T  
    z = zeros(length_r,length_n); Tzj v-9^V  
    for j = 1:length_n v]Pyz<+  
        s = 0:(n(j)-m(j))/2; g.62XZF@  
        pows = n(j):-2:m(j); 7;TMxO=bra  
        for k = length(s):-1:1 NA0Z~Ug>  
            p = (1-2*mod(s(k),2))* ... b5%<},ySq  
                       prod(2:(n(j)-s(k)))/          ... u Uq= L  
                       prod(2:s(k))/                 ... SnUR?k1  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... abS~'r14  
                       prod(2:((n(j)+m(j))/2-s(k))); & uwOyb  
            idx = (pows(k)==rpowers); 7>r[.g  
            z(:,j) = z(:,j) + p*rpowern(:,idx); cQd?,B3#F  
        end #M!{D  
         jbZTlG  
        if isnorm {*8G<&  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); C`qV+pV  
        end 6Ktq7'Z@  
    end j$6Q]5KdoS  
    :F[s  
    % 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)  cQBc6eAi  
    Sx0{]1J  
    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。 <1* \ ~CX  
    T VSCjI  
    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)