切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10958阅读
    • 9回复

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

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 z~#G6]M  
    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! 6XUcJ0  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 QE84l  
    function z = zernfun(n,m,r,theta,nflag) rmpJG |(  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. iN+Dmq5  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QKc3Q5)@j  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 6@g2v^ %  
    %   unit circle.  N is a vector of positive integers (including 0), and x68J [; jm  
    %   M is a vector with the same number of elements as N.  Each element 2,puu2F  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4Ub_;EI>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ESiNW&u2  
    %   and THETA is a vector of angles.  R and THETA must have the same l>h%J,W  
    %   length.  The output Z is a matrix with one column for every (N,M) n|lXBCY7K  
    %   pair, and one row for every (R,THETA) pair. ~!meO;|W  
    % qqT6C%Q`kG  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike K6~N{:.s  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), w_@N T}  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral (ZQ{%-i?qR  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]0by6hQ  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized iI+kZI-  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }cgEC-  
    % }|H]>U&  
    %   The Zernike functions are an orthogonal basis on the unit circle. n9gj{]%  
    %   They are used in disciplines such as astronomy, optics, and uljd)kLy4O  
    %   optometry to describe functions on a circular domain. M|?qSFv:  
    % g[*+R9'  
    %   The following table lists the first 15 Zernike functions. | ctGxS9  
    % RO'MFU<g  
    %       n    m    Zernike function           Normalization cZ \#074u/  
    %       -------------------------------------------------- l*HONl&j  
    %       0    0    1                                 1 c_}i(HQ  
    %       1    1    r * cos(theta)                    2 1""9+4  
    %       1   -1    r * sin(theta)                    2 |@]J*Kh  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) :N\*;>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Z}f$ KWj  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) i96Pel  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 9 H2^4D8  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Zw| IY9D  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) );}k@w fw)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) '?E^\\"*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) .oH0yNFX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Dk&cIZ43  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) G5ebb6[+  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E{^*^+c"h  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) =DvFY]9{  
    %       -------------------------------------------------- QOlm#S  
    % k^ J~l=?v  
    %   Example 1: 6/hY[a!  
    % $6XSW  
    %       % Display the Zernike function Z(n=5,m=1) &BqRyUM$F  
    %       x = -1:0.01:1; M A}=  
    %       [X,Y] = meshgrid(x,x); Z*.fSmT8)  
    %       [theta,r] = cart2pol(X,Y); qw&Wfk\}  
    %       idx = r<=1; ]7O)iq%  
    %       z = nan(size(X)); +Q If7=  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); Yb%H9A  
    %       figure w3PE.A"Q  
    %       pcolor(x,x,z), shading interp /S$p_7N  
    %       axis square, colorbar I.Co8is  
    %       title('Zernike function Z_5^1(r,\theta)') bRJYw6oA<  
    % _2q4Aaza  
    %   Example 2: t@ #sKdv  
    % dI5Z*"`R9  
    %       % Display the first 10 Zernike functions 2WIL0Siwl  
    %       x = -1:0.01:1; Um)0jT  
    %       [X,Y] = meshgrid(x,x); zKe&*tZ  
    %       [theta,r] = cart2pol(X,Y); dD1`[%  
    %       idx = r<=1; pM@|P,w {  
    %       z = nan(size(X)); XPd>DH(Yc  
    %       n = [0  1  1  2  2  2  3  3  3  3]; e-,U@_B  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !(*mcYA*W  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~7Kqc\/H&I  
    %       y = zernfun(n,m,r(idx),theta(idx)); m}T^rX%m_  
    %       figure('Units','normalized') %'kaNpBz  
    %       for k = 1:10 4 `Z@^W  
    %           z(idx) = y(:,k); ? 1?^>M  
    %           subplot(4,7,Nplot(k)) 3Ku!;uo!u  
    %           pcolor(x,x,z), shading interp '(5 &Sj/C  
    %           set(gca,'XTick',[],'YTick',[]) e7t).s)b{  
    %           axis square 8U/q3@EC  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) HD IB GG~  
    %       end _ YWw7q  
    % sT[)r]`T  
    %   See also ZERNPOL, ZERNFUN2. RU,f|hB 4  
    1Z'cL~9  
    %   Paul Fricker 11/13/2006 bESmKe(  
    a^ <  
    }IC$Du#  
    % Check and prepare the inputs: 4-eb&  
    % ----------------------------- ilw<Q-o4(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wp-5B= #:{  
        error('zernfun:NMvectors','N and M must be vectors.') v!E0/ gD  
    end $^.LZ1Jd  
    SDcxro|8i  
    if length(n)~=length(m) .6 !IO^`[  
        error('zernfun:NMlength','N and M must be the same length.') C?#if;c  
    end K7F uMB  
    F8;M++  
    n = n(:); Nv,[E+a2  
    m = m(:); O_ nk8  
    if any(mod(n-m,2)) b,Ed}Ir  
        error('zernfun:NMmultiplesof2', ... 3P6pQm'.f  
              'All N and M must differ by multiples of 2 (including 0).') P!,\V\TY]  
    end xrA(#\}f$  
    tE]g*]o  
    if any(m>n) x +! <_p  
        error('zernfun:MlessthanN', ... Dj;h!8t.  
              'Each M must be less than or equal to its corresponding N.') D7X-|`kH  
    end U`,&Q ]  
    KunK.m  
    if any( r>1 | r<0 ) *;7&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') kWF4k  
    end hQ i[7r($8  
    ?Mp~^sgp'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) VBF3N5 ;W  
        error('zernfun:RTHvector','R and THETA must be vectors.') (s %T1 8  
    end V,<,;d fR  
    P|v;'9  
    r = r(:); iH9g5G`O  
    theta = theta(:); )?%FU?2jrn  
    length_r = length(r); "z69jxXo  
    if length_r~=length(theta) xp7,0'(;  
        error('zernfun:RTHlength', ... iVd*62$@$  
              'The number of R- and THETA-values must be equal.') f?dNTfQ3mi  
    end /R''R:j  
    @\i6m]\X  
    % Check normalization: rnIv|q6@  
    % -------------------- _0)#-L>xKF  
    if nargin==5 && ischar(nflag) yH|ucN~k5S  
        isnorm = strcmpi(nflag,'norm'); Mw?nIIu(@  
        if ~isnorm v>c[wg9P  
            error('zernfun:normalization','Unrecognized normalization flag.') ?#qA>:2,  
        end @ ~ N:F~  
    else 0Q;T <% U  
        isnorm = false; $ e+@9LNK  
    end 5w gtc~  
    ]"dZE2!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 022YuqL<v  
    % Compute the Zernike Polynomials +AZ=nMgW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gnl6>/L,  
    blid* @-  
    % Determine the required powers of r: DHbLS3-  
    % ----------------------------------- EQyRP. dq  
    m_abs = abs(m); x]euNa  
    rpowers = []; Ar'}#6  
    for j = 1:length(n) ,4NvD2Y  
        rpowers = [rpowers m_abs(j):2:n(j)]; HLN rI0  
    end "ltvD\  
    rpowers = unique(rpowers); enF.}fo]  
    RoxzCFsI\  
    % Pre-compute the values of r raised to the required powers, j5R= K*y  
    % and compile them in a matrix: p[0Ws460  
    % ----------------------------- Ufv{6"sH  
    if rpowers(1)==0 N Rcg~Nu  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !__f  
        rpowern = cat(2,rpowern{:}); !.+iA=K{  
        rpowern = [ones(length_r,1) rpowern]; `tVBV :4\  
    else K^J;iu4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N ]}Re$5  
        rpowern = cat(2,rpowern{:}); BNyDEFd  
    end (*/P~$xIj  
    K>RL  
    % Compute the values of the polynomials: yZb@  
    % -------------------------------------- u7^Z7; J  
    y = zeros(length_r,length(n)); cK(}B_D$  
    for j = 1:length(n) |O+R%'z'<  
        s = 0:(n(j)-m_abs(j))/2; r;y&Wa  
        pows = n(j):-2:m_abs(j); -gu)d5b  
        for k = length(s):-1:1 Fy`VQ\%7t  
            p = (1-2*mod(s(k),2))* ... c[sC 2  
                       prod(2:(n(j)-s(k)))/              ... Wfu%,=@,  
                       prod(2:s(k))/                     ... nkS6A}i3o  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... n j; KnZ  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ?b#/*T}ac  
            idx = (pows(k)==rpowers); A!Ng@r  
            y(:,j) = y(:,j) + p*rpowern(:,idx); xE9^4-Px*  
        end -3wg9uZ &  
         &VR<'^>  
        if isnorm 5irewh'R  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5;`([oX|_  
        end U(3LeS;mr  
    end ^P"t "  
    % END: Compute the Zernike Polynomials Lg9]kpOpa  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^[&*B#(  
    q1r\ 60M  
    % Compute the Zernike functions: `gfK#0x#  
    % ------------------------------ /J/r62  
    idx_pos = m>0; XwX1i!'54  
    idx_neg = m<0; ^nkwT~Bya  
    @F=ZGmq  
    z = y; 0 v/+%%4}  
    if any(idx_pos) O5_E"um  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m(B6FPjr  
    end J-Sf9^G  
    if any(idx_neg) m1\>v?=K  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -|J?-  
    end Qyt6+xL  
    RvDqo d  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) O~'1)k>  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. N#T MU  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated cTRCQ+W6:  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive H#w?$?nIWu  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Kz$Ijj  
    %   and THETA is a vector of angles.  R and THETA must have the same [jAhw>  
    %   length.  The output Z is a matrix with one column for every P-value, Q=uwmg86  
    %   and one row for every (R,THETA) pair. ;pAkdX&b  
    % v^,A~oe`t  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike `e`4[I  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) pKr3(5~  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) HK2`.'D  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 $?*+P``  
    %   for all p. Ls|;gewp  
    % ;GH(A=}/Y  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 zoUW}O  
    %   Zernike functions (order N<=7).  In some disciplines it is t{?_]2vl  
    %   traditional to label the first 36 functions using a single mode R L)'m  
    %   number P instead of separate numbers for the order N and azimuthal K''b)v X4  
    %   frequency M. !Y\hF|[z  
    % r>z8DX@  
    %   Example: Ty@&s 58a  
    % ef!I |.FW  
    %       % Display the first 16 Zernike functions 0n(Q@O  
    %       x = -1:0.01:1; T}u'  
    %       [X,Y] = meshgrid(x,x); >"`:w  
    %       [theta,r] = cart2pol(X,Y); {/n$Y|TIQt  
    %       idx = r<=1; AiO,zjM=  
    %       p = 0:15; N>]u;HjH  
    %       z = nan(size(X)); RKj A`cJ  
    %       y = zernfun2(p,r(idx),theta(idx)); YI\^hP#  
    %       figure('Units','normalized') Y~Jq!  
    %       for k = 1:length(p) YVs{\1|'  
    %           z(idx) = y(:,k); 4pc=MR  
    %           subplot(4,4,k) >fb*X'Zi%  
    %           pcolor(x,x,z), shading interp L G=Q  
    %           set(gca,'XTick',[],'YTick',[]) vW]BOzK  
    %           axis square F6q}(+9i  
    %           title(['Z_{' num2str(p(k)) '}']) @|d|orMC  
    %       end .(`u'G=  
    % !r[uwJ=  
    %   See also ZERNPOL, ZERNFUN. 9cj=CuE  
    kkd<CEz2IM  
    %   Paul Fricker 11/13/2006 j?.VJ^Ff/u  
    34P? nW(  
    lJU[9)Q_  
    % Check and prepare the inputs: '4O1Y0K  
    % ----------------------------- V}aXS;(r%  
    if min(size(p))~=1 i<@|+*>M  
        error('zernfun2:Pvector','Input P must be vector.') ^8fO3<Jg  
    end >+$1 p_  
    C w$y  
    if any(p)>35 ~L}0) FZ\9  
        error('zernfun2:P36', ... e@I?ESZ5  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... fY-{,+ `'  
               '(P = 0 to 35).']) I[F.M}5:z  
    end RS$:]hxd>_  
    ,:;_j<g`e  
    % Get the order and frequency corresonding to the function number: gbSZ- ej  
    % ---------------------------------------------------------------- x$A5Ved  
    p = p(:); HPt"  
    n = ceil((-3+sqrt(9+8*p))/2); 7v=Nh  
    m = 2*p - n.*(n+2); nQ/El&{  
    T=>&`aZH  
    % Pass the inputs to the function ZERNFUN: zIm$S/Qe*  
    % ---------------------------------------- 5HJ6[.HO  
    switch nargin {@C+Js5  
        case 3 yfaXScbE  
            z = zernfun(n,m,r,theta); %q:V  
        case 4 8-+IcyUza  
            z = zernfun(n,m,r,theta,nflag); vU~#6sl  
        otherwise ;=_KLG <  
            error('zernfun2:nargin','Incorrect number of inputs.') 3RGVH,  
    end G>H&M#7K  
    KDCq::P<  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) p+yU!Qj  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. A$%%;O   
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of V! ~uGf  
    %   order N and frequency M, evaluated at R.  N is a vector of J *^|ojX  
    %   positive integers (including 0), and M is a vector with the {{giSW'  
    %   same number of elements as N.  Each element k of M must be a s8 3_Bd  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) r@iGM Jx$  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is l-M~e]  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix .F> c Z,  
    %   with one column for every (N,M) pair, and one row for every DJ=miJI'  
    %   element in R. pn'*w 1i  
    % y 37n~~%  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- HnU}Lhjzj  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is jcevpKkRG  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Ro? 4tGn  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 kOJs;k  
    %   for all [n,m]. g$$uf[A-SL  
    % V,&s$eQC  
    %   The radial Zernike polynomials are the radial portion of the `MEH/  
    %   Zernike functions, which are an orthogonal basis on the unit hPhN7E03  
    %   circle.  The series representation of the radial Zernike du`],/ 6  
    %   polynomials is Xgop1  
    % "r.pU(uxt  
    %          (n-m)/2 uKhfZSx0 w  
    %            __ t0Ec` +)  
    %    m      \       s                                          n-2s 5C2 *f 4|  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r  27w]Q_C  
    %    n      s=0 gdHPi;  
    % ?hsOhUs(5  
    %   The following table shows the first 12 polynomials. Z]"ktb;+[  
    % |67<h5Q1  
    %       n    m    Zernike polynomial    Normalization !.x(lOqf  
    %       --------------------------------------------- :DQHb"(  
    %       0    0    1                        sqrt(2) -1Tws|4gc  
    %       1    1    r                           2 (hdP(U77  
    %       2    0    2*r^2 - 1                sqrt(6) O"_FfwO a  
    %       2    2    r^2                      sqrt(6) +9,"ne1'e  
    %       3    1    3*r^3 - 2*r              sqrt(8) 3'*SSZmnOB  
    %       3    3    r^3                      sqrt(8) IjJ3./L!5  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10)  |o=eS&)  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) HWD  
    %       4    4    r^4                      sqrt(10) #OH# &{H  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) QvjsI;CQ-  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) '$4o,GA8  
    %       5    5    r^5                      sqrt(12) [C/h{WPC-  
    %       --------------------------------------------- uppA`>  
    % VA.:'yQtJ  
    %   Example: ~Ui<y=d  
    % wD}[XE?S  
    %       % Display three example Zernike radial polynomials VO[s:e9L  
    %       r = 0:0.01:1; uu]<R@!J  
    %       n = [3 2 5]; !<@k\~9^D  
    %       m = [1 2 1]; C6:<.`iD87  
    %       z = zernpol(n,m,r); SJj0*ry:  
    %       figure ImyB4welo  
    %       plot(r,z) OB l-6W  
    %       grid on >*{\N^:z  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') $w#C;2k]N  
    % h#'(UZ  
    %   See also ZERNFUN, ZERNFUN2. q_ ']i6  
    5;C+K~Y  
    % A note on the algorithm. %/-Z1Nv*#  
    % ------------------------ r9z/hm}E  
    % The radial Zernike polynomials are computed using the series IHMZE42  
    % representation shown in the Help section above. For many special doVBVTk^  
    % functions, direct evaluation using the series representation can FC/m,D50oI  
    % produce poor numerical results (floating point errors), because * t{A=Wk  
    % the summation often involves computing small differences between `A$yF38!  
    % large successive terms in the series. (In such cases, the functions xG\&QE  
    % are often evaluated using alternative methods such as recurrence Bp>%'L  
    % relations: see the Legendre functions, for example). For the Zernike "JKrbgN@;L  
    % polynomials, however, this problem does not arise, because the ld$LG6[PA  
    % polynomials are evaluated over the finite domain r = (0,1), and OGrp {s  
    % because the coefficients for a given polynomial are generally all ={YW*1Xw  
    % of similar magnitude. 0;} 9XZ  
    % b.Z K1  
    % ZERNPOL has been written using a vectorized implementation: multiple F+v?2|03  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] lm o>z'<  
    % values can be passed as inputs) for a vector of points R.  To achieve >}43MxU?  
    % this vectorization most efficiently, the algorithm in ZERNPOL K{t7_i#tv  
    % involves pre-determining all the powers p of R that are required to qun#z$  
    % compute the outputs, and then compiling the {R^p} into a single l7 j3;Ly  
    % matrix.  This avoids any redundant computation of the R^p, and w*uHB;?  
    % minimizes the sizes of certain intermediate variables. [ % KBc}  
    % b~y1'|}g  
    %   Paul Fricker 11/13/2006 sq*d?<:3  
    "Z#97Jc+J  
    Fpzps!(;=  
    % Check and prepare the inputs: _t 'Kj \  
    % ----------------------------- n!~{4 uUW  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) e(O"V3wq*6  
        error('zernpol:NMvectors','N and M must be vectors.') '9H7I! L@  
    end m .le' &  
    ;vc$;54K  
    if length(n)~=length(m) ,3!l'|0jJ  
        error('zernpol:NMlength','N and M must be the same length.') 3)Wi? -  
    end )PVX)2P_C  
    0<Vw0%!  
    n = n(:); My&h{Qk  
    m = m(:); r8pTtf#Q  
    length_n = length(n); *ukE"Aj  
    kQRNVdiz  
    if any(mod(n-m,2)) m%X~EwFc.  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') F'|D  
    end /Uz2.Ua=  
    O|O#T.Tg  
    if any(m<0) 8Tyf#`'I  
        error('zernpol:Mpositive','All M must be positive.') @=E@ *@g  
    end 9e@Sx{?r  
    dUk^DI,:l  
    if any(m>n) aqK<}jy  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') l[fU0;A  
    end .-k\Q} D  
    6-{QU] #  
    if any( r>1 | r<0 ) L!|c: 8  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 2v0lWO~c7z  
    end KcF2}+iM   
    MW &iNioX  
    if ~any(size(r)==1) _Y40a+hk]  
        error('zernpol:Rvector','R must be a vector.') _8)9I?jH  
    end ^PMP2\JQA  
    N5Eb.a9S  
    r = r(:); WFocA:  
    length_r = length(r); }b ~;x6  
    l}&egq DC  
    if nargin==4 \h~;n)FI  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); N1jj\.nB  
        if ~isnorm -5l74f!i  
            error('zernpol:normalization','Unrecognized normalization flag.') ?_3K]i1IS  
        end w&8gA[y*u  
    else cfyN)#9  
        isnorm = false; DB-4S-2  
    end $cHA_$ `  
    d}GO(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@9#od O  
    % Compute the Zernike Polynomials +O`0Mc$%'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \::<]  
    mWZoo/xtT  
    % Determine the required powers of r: $ &M"Ji  
    % ----------------------------------- }Y~o =3-  
    rpowers = []; D:sQHJ. y  
    for j = 1:length(n) q %i2' yE  
        rpowers = [rpowers m(j):2:n(j)]; &EYO[~D06  
    end 7Q7z6p/\v  
    rpowers = unique(rpowers); #>g]CRN  
    ev7Y^   
    % Pre-compute the values of r raised to the required powers, ,~`R{,N`  
    % and compile them in a matrix: ,kfUlv=  
    % ----------------------------- ZC'(^liAp  
    if rpowers(1)==0 a$.(Zl  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }@_F( B  
        rpowern = cat(2,rpowern{:}); 6H\3  
        rpowern = [ones(length_r,1) rpowern]; J~9l+?  
    else ABvB1[s#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I:Wrwd  
        rpowern = cat(2,rpowern{:}); J'`,];su  
    end R R<92R  
    Zmw'.hL  
    % Compute the values of the polynomials: ";AM3  
    % -------------------------------------- -&oJ@Aa  
    z = zeros(length_r,length_n); :jKD M  
    for j = 1:length_n Z.Z+cFi  
        s = 0:(n(j)-m(j))/2; h1} x2  
        pows = n(j):-2:m(j); hVo]fD|W  
        for k = length(s):-1:1  T},Nqt<  
            p = (1-2*mod(s(k),2))* ... {.v-  
                       prod(2:(n(j)-s(k)))/          ... 73OFFKbsk  
                       prod(2:s(k))/                 ... f mILkXKz  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ^5x\cR  
                       prod(2:((n(j)+m(j))/2-s(k))); HWG5Ghu8,)  
            idx = (pows(k)==rpowers); $q);xs  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 4zX@TI>j  
        end _@wXh-nc  
         UmZ#Cm  
        if isnorm gF+Uj( d  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); */fmy|#   
        end &$ZJfHD@  
    end 9ar+Ph@*  
    gf7%vyMo$  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  W%&gvZre.  
    5I`_S Oa!  
    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。 }F B]LLi  
    ]?un'$%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)