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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 i{T mn  
    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! jd>ug=~x  
     
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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 D(X:dB50@  
    function z = zernfun(n,m,r,theta,nflag) V7S[rI<<r  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. f*%Y]XL;%  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &eA!h  
    %   and angular frequency M, evaluated at positions (R,THETA) on the )(/Bw&$  
    %   unit circle.  N is a vector of positive integers (including 0), and /s~(? =qYH  
    %   M is a vector with the same number of elements as N.  Each element 4{v?<x8  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) GEs5@EH  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, XI5TVxo(q  
    %   and THETA is a vector of angles.  R and THETA must have the same Jc=~BT_G  
    %   length.  The output Z is a matrix with one column for every (N,M) O)FkpZc@9c  
    %   pair, and one row for every (R,THETA) pair. >2^|r8l5  
    %  8MZ:=  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (ah^</  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &_1x-@oI2:  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral -J& b~t@  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7*MjQzg-P  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized eaWK2%v  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hy}n&h  
    % L> \/%x>Wx  
    %   The Zernike functions are an orthogonal basis on the unit circle. ^[=1J  
    %   They are used in disciplines such as astronomy, optics, and /EvnwYQy  
    %   optometry to describe functions on a circular domain. hpBn_  
    % $/)0iL{0  
    %   The following table lists the first 15 Zernike functions. XS_Ib\-50  
    % (>,}C/-UG  
    %       n    m    Zernike function           Normalization 4#Rq}/h  
    %       -------------------------------------------------- 8mn zxtk  
    %       0    0    1                                 1 zI& ).  
    %       1    1    r * cos(theta)                    2 X[E!q$ag  
    %       1   -1    r * sin(theta)                    2 ?y|8bw<  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3E$h W  
    %       2    0    (2*r^2 - 1)                    sqrt(3) FdE9k\E#/)  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) +\GuZ5`  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) gk^`-`P  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) @|;XDO`k;  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) EJm*L6>@R&  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ;kLp}CqV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 8eDKN9kq  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y{`hRz`  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) W*Gp0pX  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VD0U]~CWR  
    %       4    4    r^4 * sin(4*theta)             sqrt(10)  B@K =^77  
    %       -------------------------------------------------- JfVGs;_,  
    % _OY<Hb3%M  
    %   Example 1: Aw,#oG {N  
    % dMDSyd<(  
    %       % Display the Zernike function Z(n=5,m=1) FV>xAU$  
    %       x = -1:0.01:1;  $1.l|  
    %       [X,Y] = meshgrid(x,x); JrJTIUf_  
    %       [theta,r] = cart2pol(X,Y); @D2KDV3'  
    %       idx = r<=1; p}MH LM  
    %       z = nan(size(X)); #(dERET*  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); I`KBj6n  
    %       figure G&,2>qxK R  
    %       pcolor(x,x,z), shading interp `\Hs{t]  
    %       axis square, colorbar )A*Sl2ew  
    %       title('Zernike function Z_5^1(r,\theta)') jx-8%dxtZ  
    % K/D,sH!  
    %   Example 2: Y^ ti;:  
    % _/RP3"#  
    %       % Display the first 10 Zernike functions q,fk@GI'2  
    %       x = -1:0.01:1; :qxd s>Xm  
    %       [X,Y] = meshgrid(x,x); kOLS<>.  
    %       [theta,r] = cart2pol(X,Y); Yvxp(  
    %       idx = r<=1; 1+NmiGKg  
    %       z = nan(size(X)); fudLm  
    %       n = [0  1  1  2  2  2  3  3  3  3]; gt:Ot0\7  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Xb5 $ijH  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; S X6P>:`  
    %       y = zernfun(n,m,r(idx),theta(idx)); d A' h7D  
    %       figure('Units','normalized') OJ4-p&1  
    %       for k = 1:10 ~glFB`?[  
    %           z(idx) = y(:,k); BGZvgMxLJ  
    %           subplot(4,7,Nplot(k)) -"X} )N2  
    %           pcolor(x,x,z), shading interp n 7 m!   
    %           set(gca,'XTick',[],'YTick',[]) SPY4l*kX  
    %           axis square d){Al(/  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }RY&f4&GV,  
    %       end x|IG'R1:Y  
    % CJ 9tO#R  
    %   See also ZERNPOL, ZERNFUN2. Bl8&g]dk  
    wA>bLPTw  
    %   Paul Fricker 11/13/2006 bcy( ?(  
    "K$ y(}C  
    o]@g%_3X  
    % Check and prepare the inputs: :fE*fU@  
    % ----------------------------- h | +(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O-K!Bv^ Q  
        error('zernfun:NMvectors','N and M must be vectors.') +gsk}>"  
    end 8L}N,6gC4_  
    s7#|'jhZt  
    if length(n)~=length(m) rXR}]|;>  
        error('zernfun:NMlength','N and M must be the same length.') R@H}n3,  
    end )gq(  
    Y2Y!^A89  
    n = n(:); )B' U_*  
    m = m(:); ;o0o6pF  
    if any(mod(n-m,2)) *tZ#^YG{(  
        error('zernfun:NMmultiplesof2', ... -?AaRwZ,  
              'All N and M must differ by multiples of 2 (including 0).') m%?b"kxL[  
    end tXIre-. 2}  
    C JNz J(  
    if any(m>n) 4D\+_Ic3  
        error('zernfun:MlessthanN', ... P!)k4n  
              'Each M must be less than or equal to its corresponding N.') %C8fv|@:f  
    end D3emO'`gQ  
    XT5Vo  
    if any( r>1 | r<0 ) {\HE'C/?  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') FE/2.!]&o  
    end ^D0BGC&&  
    NR)[,b\v  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :4D#hOI  
        error('zernfun:RTHvector','R and THETA must be vectors.') !jDqRXi(  
    end ?ixzlDto\  
    UVD::  
    r = r(:); 9/k?Lv  
    theta = theta(:); !u#o"e<qh  
    length_r = length(r); IBzHXa>75  
    if length_r~=length(theta) kty,hAXe  
        error('zernfun:RTHlength', ... }PY? ZG  
              'The number of R- and THETA-values must be equal.') K,IPVjS  
    end ]41G!'E=  
    V8xv@G{;  
    % Check normalization: ka&-tGg  
    % -------------------- \g}FoN&  
    if nargin==5 && ischar(nflag) Hvq< _&2  
        isnorm = strcmpi(nflag,'norm'); */L;6_  
        if ~isnorm u0J+Nj9  
            error('zernfun:normalization','Unrecognized normalization flag.') yf=ek= =  
        end A{E0 a:v  
    else `Vwj|[0k  
        isnorm = false; "A:wWb<m  
    end [VPqI~u5)  
    7,e=|%7.  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vAJfMUlP  
    % Compute the Zernike Polynomials V_(?mC  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ':!;6v|L  
    J 6S  
    % Determine the required powers of r: ,9#G/nF  
    % ----------------------------------- cQv*lvG9>  
    m_abs = abs(m); =fHt|}.K  
    rpowers = []; M{7EFTy!y  
    for j = 1:length(n) -c=IO(B/  
        rpowers = [rpowers m_abs(j):2:n(j)]; qgca4VV|z  
    end Y#6@0Nn[G  
    rpowers = unique(rpowers); I01On>"@7  
    N_VAdNJ^:  
    % Pre-compute the values of r raised to the required powers, .@APxeU  
    % and compile them in a matrix: %p2C5z?  
    % ----------------------------- 3a{QkVeV7  
    if rpowers(1)==0 ~pj9_I  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &/\0_CoTR\  
        rpowern = cat(2,rpowern{:}); "eQ96^'J  
        rpowern = [ones(length_r,1) rpowern]; zPV/{)S  
    else <UQ:1W8>B  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $vy.BY Fm  
        rpowern = cat(2,rpowern{:}); W{;!JI7;z  
    end mc(&'U8R0I  
    oT|E\wj  
    % Compute the values of the polynomials: VUF7-C*  
    % -------------------------------------- -"a+<(Y  
    y = zeros(length_r,length(n)); i}<R >]S  
    for j = 1:length(n) e`$v\7K  
        s = 0:(n(j)-m_abs(j))/2; {=g-zsc]K  
        pows = n(j):-2:m_abs(j); #K*d:W3C  
        for k = length(s):-1:1 K)Db3JIIk  
            p = (1-2*mod(s(k),2))* ... 5Cy)#Z{  
                       prod(2:(n(j)-s(k)))/              ... x\Sp~]o3C  
                       prod(2:s(k))/                     ... 2z[Pw0#V  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... wOi>i`D&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); %k$C   
            idx = (pows(k)==rpowers); Ya9uu@F  
            y(:,j) = y(:,j) + p*rpowern(:,idx); xJ&StN/'  
        end u khI#:[  
         "W+4`A(/l  
        if isnorm WejY b;KS  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,XA;S5FE  
        end 0KDDAkR5R  
    end II<<-Y6  
    % END: Compute the Zernike Polynomials N{ 9<Tf*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OC>" +  
    ]owH [wvX  
    % Compute the Zernike functions: K5.C*|w  
    % ------------------------------ Dea;9O  
    idx_pos = m>0; . t3@86xTJ  
    idx_neg = m<0; nlY ^  
    ev?>Nq+Z  
    z = y; q$t& *O_  
    if any(idx_pos) -%N (X8  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); cn\_;TYiJ  
    end 1OGlD+f  
    if any(idx_neg) Z9sg6M@s  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); {[9^@k  
    end gvU6p[D  
    V+Tj[:ok  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) g/@CESfm'  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. )b7mzDp(  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated i Bi/9  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive UB1/FM4~  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, z]WT>4  
    %   and THETA is a vector of angles.  R and THETA must have the same ec0vg.>p  
    %   length.  The output Z is a matrix with one column for every P-value, TB[vpTC9)  
    %   and one row for every (R,THETA) pair. OiOL 4}5(  
    % Ya. $x~  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Lf-8G5G  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 'IP!)DS  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) aj,ZM,Ad  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 b?^n'0  
    %   for all p. +?<jSmGW  
    % QCo^#-   
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 @$*c0 . |z  
    %   Zernike functions (order N<=7).  In some disciplines it is m\>x_:sE  
    %   traditional to label the first 36 functions using a single mode Z~|J"2.  
    %   number P instead of separate numbers for the order N and azimuthal '!I?C/49k  
    %   frequency M. oN$ZZk R  
    % }cT}G;L'-  
    %   Example: FX^E |  
    % <(MFEIt  
    %       % Display the first 16 Zernike functions Q.\>+4]1&&  
    %       x = -1:0.01:1; %FU[ j^  
    %       [X,Y] = meshgrid(x,x); qcot T\rq  
    %       [theta,r] = cart2pol(X,Y); 1fy{@j(W  
    %       idx = r<=1; Esjv^* v9-  
    %       p = 0:15; F"=MU8  
    %       z = nan(size(X)); LZVO9e]  
    %       y = zernfun2(p,r(idx),theta(idx)); [jw o D  
    %       figure('Units','normalized') 8z CAy@u  
    %       for k = 1:length(p) >+#[O"  
    %           z(idx) = y(:,k); JK(&E{80  
    %           subplot(4,4,k) $ZU(bEUOG  
    %           pcolor(x,x,z), shading interp W24bO|>D  
    %           set(gca,'XTick',[],'YTick',[]) rYJ ))@  
    %           axis square ,7(/Il9  
    %           title(['Z_{' num2str(p(k)) '}']) b {5|2&=  
    %       end W!k6qTz)  
    % 3$8}%?i  
    %   See also ZERNPOL, ZERNFUN. 'dzp@-\  
    6`C27  
    %   Paul Fricker 11/13/2006 7N"Bbl  
    WFd2_oAT  
    K*9b `%  
    % Check and prepare the inputs: m}9V@@  
    % ----------------------------- ?N ga  
    if min(size(p))~=1 4L e5Ms/  
        error('zernfun2:Pvector','Input P must be vector.') "{_"Nj H  
    end 1Tq$E[  
    c)8wO=!  
    if any(p)>35 1,T9HpM  
        error('zernfun2:P36', ... mz*z1`\7v\  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 7gwZ9Fob  
               '(P = 0 to 35).']) |^Es6 .~  
    end F47n_JV!d  
    Z?.*.<"Sj  
    % Get the order and frequency corresonding to the function number: 6iG<"{/U5  
    % ---------------------------------------------------------------- )^N8L<   
    p = p(:); |S{P`)z%f  
    n = ceil((-3+sqrt(9+8*p))/2); bJm0  
    m = 2*p - n.*(n+2); L{f>;[FR  
    Wts{tb  
    % Pass the inputs to the function ZERNFUN: Wu( 8 G  
    % ---------------------------------------- $O'2oeM  
    switch nargin _,FoXf7  
        case 3 yk<jlVF$j  
            z = zernfun(n,m,r,theta); )6&\WNL-x  
        case 4 +{%(_ <  
            z = zernfun(n,m,r,theta,nflag); _oTT3[7P  
        otherwise nZCpT |M5  
            error('zernfun2:nargin','Incorrect number of inputs.') 0 '7s  
    end ^P/D8cXa4  
    hPz df*(8  
    % 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) 9c{%m4  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. n>:c}QAJH  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of kg I=0W>  
    %   order N and frequency M, evaluated at R.  N is a vector of mK40 f  
    %   positive integers (including 0), and M is a vector with the f=_?<I{  
    %   same number of elements as N.  Each element k of M must be a d]ZC8<`w  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) l~4e2xoT  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is e4q k>Cw  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix x_y>j)  
    %   with one column for every (N,M) pair, and one row for every S~rVRC"<xo  
    %   element in R. ( ssH=a  
    %  K-5"#  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 7lDaok  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is {<}I9D5  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to O\?5#.   
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ONN{4&7@<  
    %   for all [n,m]. 3TiXYH  
    % &lh_-@Xz  
    %   The radial Zernike polynomials are the radial portion of the _c9 WWp?  
    %   Zernike functions, which are an orthogonal basis on the unit J4Z<Yt/  
    %   circle.  The series representation of the radial Zernike pm W6~%}*  
    %   polynomials is udEb/7ZL  
    % }8V;s-1  
    %          (n-m)/2 I "Q9W|J_&  
    %            __ 1s} ``1>  
    %    m      \       s                                          n-2s p=-B~:  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r h`EH~W0:z  
    %    n      s=0 !v5sWVVR  
    % eW"x%|/Q7  
    %   The following table shows the first 12 polynomials. R!/,E  
    % ~e`;"n@4  
    %       n    m    Zernike polynomial    Normalization oX 2DFgz  
    %       --------------------------------------------- Z;Ir>^<  
    %       0    0    1                        sqrt(2) c ;21i;&,9  
    %       1    1    r                           2 [2QY  
    %       2    0    2*r^2 - 1                sqrt(6) D2TXOPH  
    %       2    2    r^2                      sqrt(6) tLXn?aNY  
    %       3    1    3*r^3 - 2*r              sqrt(8) t|#NMRz  
    %       3    3    r^3                      sqrt(8) ;ad9{":J#B  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) /md Q(Dm  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) yo?g"vbE  
    %       4    4    r^4                      sqrt(10) ^C:{z)"h  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Pzk[^z$C  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 2yJ7]+Jd7Y  
    %       5    5    r^5                      sqrt(12) A| gs Uh  
    %       --------------------------------------------- E2qB:  
    % xyS2_Q  
    %   Example: '#McY'.D T  
    % %|`:5s-T%  
    %       % Display three example Zernike radial polynomials 2w x[D  
    %       r = 0:0.01:1; cy&  
    %       n = [3 2 5]; <nOuyGIZ  
    %       m = [1 2 1]; zfP[1  
    %       z = zernpol(n,m,r); *C3uMiz  
    %       figure n~"g'Y  
    %       plot(r,z) |~/3u/  
    %       grid on U9@t?j_#X{  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') pu2wEQ  
    % f*|8n$%   
    %   See also ZERNFUN, ZERNFUN2. ]} 5I>l  
    8g# c%eZ  
    % A note on the algorithm. )I Y 5Y  
    % ------------------------ +Dq|l}  
    % The radial Zernike polynomials are computed using the series HC9vc,Fp  
    % representation shown in the Help section above. For many special F,L82N6\U  
    % functions, direct evaluation using the series representation can 1*!c X  
    % produce poor numerical results (floating point errors), because 45r]wT(C   
    % the summation often involves computing small differences between @H3s2|  
    % large successive terms in the series. (In such cases, the functions E !Oz|q  
    % are often evaluated using alternative methods such as recurrence nxn[ ~~  
    % relations: see the Legendre functions, for example). For the Zernike 7GBZA=J  
    % polynomials, however, this problem does not arise, because the nJldz;  
    % polynomials are evaluated over the finite domain r = (0,1), and H7z>S G0  
    % because the coefficients for a given polynomial are generally all YZ"+c&V"  
    % of similar magnitude. Pg*?[^*  
    % a2c x  
    % ZERNPOL has been written using a vectorized implementation: multiple =RW* %8C  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] wc6 E- rB  
    % values can be passed as inputs) for a vector of points R.  To achieve 7~!I2DV_  
    % this vectorization most efficiently, the algorithm in ZERNPOL @MN}^umx`  
    % involves pre-determining all the powers p of R that are required to 6o[0sM_];  
    % compute the outputs, and then compiling the {R^p} into a single o ;[C(OS  
    % matrix.  This avoids any redundant computation of the R^p, and vCy.CN$  
    % minimizes the sizes of certain intermediate variables. \yJZvhUk  
    % u|_LR5S!j  
    %   Paul Fricker 11/13/2006 "fX_gN?  
    "xe7Dl  
    dJdD"xj  
    % Check and prepare the inputs: ]97Xu_  
    % ----------------------------- 26\HV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wo7N7R5  
        error('zernpol:NMvectors','N and M must be vectors.') %gV)arwK  
    end =R8f)UQYx  
    4)z3X\u|Z2  
    if length(n)~=length(m) jsk:fh0~M  
        error('zernpol:NMlength','N and M must be the same length.') 4 o3)*  
    end {&\J)oZ  
     uD.  
    n = n(:); BpQ;w,sefq  
    m = m(:); =,&u_>Dp  
    length_n = length(n); $\0cJCQ3  
    o :.~X  
    if any(mod(n-m,2)) "?oo\op  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ppwd-^f3j  
    end |QnUK5D$  
    17V\2=Io  
    if any(m<0) t Y:G54d=_  
        error('zernpol:Mpositive','All M must be positive.') lEO?kn.:z  
    end ;el]LnV!O  
    iyA*J CD  
    if any(m>n) ~hS .\h  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') w"fCI 13  
    end [=XZza.z  
    u~#%P&3 _W  
    if any( r>1 | r<0 ) pj!k|F9  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') + 6x"trC  
    end IT8B~I\OY  
    [TRGIGtq  
    if ~any(size(r)==1) WafdE  
        error('zernpol:Rvector','R must be a vector.') ?~F]@2)5w  
    end b*lKT]D,  
    R19'| TJ  
    r = r(:); ::p-9F  
    length_r = length(r); !d:tIu{)  
    Sp<hai  
    if nargin==4 er(8}]X8Q  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); pWMiCXnW  
        if ~isnorm GYFgEg}  
            error('zernpol:normalization','Unrecognized normalization flag.') o8w-$ Qb  
        end 1t0b Uf;(M  
    else re7!p(W?,  
        isnorm = false; Ro$j1Aw(  
    end y.jS{r".  
    5\uNEs$T  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9U!JK3d  
    % Compute the Zernike Polynomials 2;&mkc K'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \U==f &G?J  
    TAXd,z N  
    % Determine the required powers of r: OZ eiH X!  
    % ----------------------------------- V78Mq:7d  
    rpowers = []; 2}D,df'W4  
    for j = 1:length(n) w+Ad$4Pf"  
        rpowers = [rpowers m(j):2:n(j)]; gs$3)t  
    end cL6 6gOEL  
    rpowers = unique(rpowers); (&q@~ dJ  
    ?(]a*~rx  
    % Pre-compute the values of r raised to the required powers, 4; ?1Kb#  
    % and compile them in a matrix: |__d 8a  
    % ----------------------------- >r~0SMQr  
    if rpowers(1)==0 fwRGT|":B  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); %wOOzp`  
        rpowern = cat(2,rpowern{:}); ,xYg  
        rpowern = [ones(length_r,1) rpowern]; #g,H("Qy({  
    else x<8\-  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &q.)2o#Q.  
        rpowern = cat(2,rpowern{:}); "K 8nxnq  
    end C=8H)Ef,l  
    O2C6V>Q;  
    % Compute the values of the polynomials: MY&<)|v\  
    % -------------------------------------- ^n t~-%  
    z = zeros(length_r,length_n); RvWFF^,.  
    for j = 1:length_n L%f-L.9`u  
        s = 0:(n(j)-m(j))/2; 2%bhW,?I  
        pows = n(j):-2:m(j); ,Ak ^nX  
        for k = length(s):-1:1 =1O?jrl~q  
            p = (1-2*mod(s(k),2))* ... ~Hs=z$  
                       prod(2:(n(j)-s(k)))/          ... &.hoC Po$  
                       prod(2:s(k))/                 ... 6{L F-`S%  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... E`~i-kf  
                       prod(2:((n(j)+m(j))/2-s(k))); *`%4loW  
            idx = (pows(k)==rpowers); <Uf|PFVj$  
            z(:,j) = z(:,j) + p*rpowern(:,idx); dZF8 R  
        end Yi[4DfA  
         NOV.Bs{ yL  
        if isnorm "=FIFf  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); +5#x6[  
        end }&mj.hGv  
    end wI*Y{J  
    t`uc3ta"9  
    % 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)  ~$Y|ca  
    +aj^Cs1$  
    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。 `.[ 8$  
    ;:T9IL  
    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)