切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11136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 QcW8A ,\q  
    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! / De~K+w7o  
     
    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 .Q,"gsY  
    function z = zernfun(n,m,r,theta,nflag) !S':G  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. :rVR{,pL  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1+; bd'Ie  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Ak9{P`  
    %   unit circle.  N is a vector of positive integers (including 0), and p2Ep(0w,R5  
    %   M is a vector with the same number of elements as N.  Each element |l; Ot=C=  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Nh.+woFq4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9{jMO  
    %   and THETA is a vector of angles.  R and THETA must have the same Swhz\/u9  
    %   length.  The output Z is a matrix with one column for every (N,M) 9efDM  
    %   pair, and one row for every (R,THETA) pair. ]`&_!T  
    % 6(bN*.  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +?{"Q#.>;  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Cdz&'en^  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral JY#vq'dl|  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <eG|`  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized szsVk#p  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cmG27\cRO  
    % _YF>Y=D-  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?$f.[;mh  
    %   They are used in disciplines such as astronomy, optics, and Trv}YT.  
    %   optometry to describe functions on a circular domain. 5E'/8xpbB  
    % "/Qz?1>l+  
    %   The following table lists the first 15 Zernike functions. )}@D\(/@  
    % )j36Y =r3  
    %       n    m    Zernike function           Normalization ?Ij(B}D  
    %       -------------------------------------------------- f CU]  
    %       0    0    1                                 1 Zd[rn:9\  
    %       1    1    r * cos(theta)                    2 @sLN  
    %       1   -1    r * sin(theta)                    2 fs 'SCwx  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ; j!dbT~5  
    %       2    0    (2*r^2 - 1)                    sqrt(3) f8:nKb>nq$  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) e"S?qpJK  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) D;pI!S<#  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) odhS0+d^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) %;'~TtW5  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 6<];}M_{  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) v1OVrk>s>  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >3uNh:|>/  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Qo#]Lo> \g  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) BIWe Hx  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) yJ $6vmQ  
    %       -------------------------------------------------- |UXSUP @s  
    % [I *_0  
    %   Example 1: WywS1viD  
    % 9eMle?pF  
    %       % Display the Zernike function Z(n=5,m=1) DhyR  
    %       x = -1:0.01:1; n~I-mR)"  
    %       [X,Y] = meshgrid(x,x); Nm?^cR5r  
    %       [theta,r] = cart2pol(X,Y); qIi \[Ugh  
    %       idx = r<=1; :<J7g`f  
    %       z = nan(size(X)); -l= 4{^pK  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); %~!4DXrMk  
    %       figure Ek{QNlQ]4  
    %       pcolor(x,x,z), shading interp MGR:IOTa  
    %       axis square, colorbar >WSh)(Cg  
    %       title('Zernike function Z_5^1(r,\theta)') ;qWu8\T+  
    % ~[ufL25K  
    %   Example 2: |R}=HsYey  
    % cpdESc9W  
    %       % Display the first 10 Zernike functions S<0 &V  
    %       x = -1:0.01:1; <fUo@]Lv  
    %       [X,Y] = meshgrid(x,x);  q+L'h8  
    %       [theta,r] = cart2pol(X,Y); 8o~ NJ 6  
    %       idx = r<=1; [YOH'i&X  
    %       z = nan(size(X)); O4R\] B#Xu  
    %       n = [0  1  1  2  2  2  3  3  3  3]; lfgJQzi G  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; FzInIif  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; TjUwe@&Rw  
    %       y = zernfun(n,m,r(idx),theta(idx)); h&>3;Lj  
    %       figure('Units','normalized') ZNQ x;51  
    %       for k = 1:10 B>53+GyMV  
    %           z(idx) = y(:,k); LikcW#  
    %           subplot(4,7,Nplot(k)) Scrj%h%[  
    %           pcolor(x,x,z), shading interp 6("_}9ZOc  
    %           set(gca,'XTick',[],'YTick',[]) xuioU  
    %           axis square P<PZ4hNx  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A=<7*E  
    %       end sINQ?4_8T  
    % xp^RAVXq`  
    %   See also ZERNPOL, ZERNFUN2. < z<>E1ZLI  
    h4;kjr}h}  
    %   Paul Fricker 11/13/2006 _**Nlp*%  
    6w^P{%ul  
    MAek856  
    % Check and prepare the inputs: FIq'W:q:  
    % ----------------------------- F&B\ X  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yK*vn]}  
        error('zernfun:NMvectors','N and M must be vectors.') %qc_kQ5%  
    end Kip&YB%rk  
    LF7- ?? '  
    if length(n)~=length(m) (]]hSkE  
        error('zernfun:NMlength','N and M must be the same length.') c*IrZm  
    end *[si!e%  
    Z [!"x&H]h  
    n = n(:); 0m_yW$w  
    m = m(:); J"r?F0  
    if any(mod(n-m,2)) BSm"]!D8*  
        error('zernfun:NMmultiplesof2', ... :33@y%>L  
              'All N and M must differ by multiples of 2 (including 0).') }N g P`m  
    end #mQ@4k9i  
    '_c/CNs  
    if any(m>n) ]\pi!oa  
        error('zernfun:MlessthanN', ... 6v)TCj/  
              'Each M must be less than or equal to its corresponding N.') bzi"7%c  
    end @v)Z>xv  
    Z[?n{vD7  
    if any( r>1 | r<0 ) yv,FzF}7  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') '09|Y#F  
    end Qq,2V  
    m{q'RAw  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ` Ig5*X4|  
        error('zernfun:RTHvector','R and THETA must be vectors.') h:4(Gm;  
    end \|HtE(uCM1  
    m+c-"arIpA  
    r = r(:); "^]gIQc  
    theta = theta(:); [q9B" @X  
    length_r = length(r); Hx.|5n,5  
    if length_r~=length(theta) !l[;,l   
        error('zernfun:RTHlength', ... 8C3k: D[  
              'The number of R- and THETA-values must be equal.') JxVGzb`8  
    end SzW;Yb"#^k  
    |*bUcS<S  
    % Check normalization: $TUYxf0q  
    % -------------------- x3O%W?5  
    if nargin==5 && ischar(nflag)  [Sm<X  
        isnorm = strcmpi(nflag,'norm'); R$&;  
        if ~isnorm NW\CEJV  
            error('zernfun:normalization','Unrecognized normalization flag.') VX)8 pV$  
        end Xh"9Bcjf  
    else 't<iB&wgF  
        isnorm = false; Sz0PZtJ  
    end qTuR[(  
    E+L7[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !JCs'?A  
    % Compute the Zernike Polynomials 5%,3)H{;t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u]*7",R uU  
    yT^2;/Z  
    % Determine the required powers of r: un "I  
    % ----------------------------------- KXt8IMP_"y  
    m_abs = abs(m); /M2in]oH  
    rpowers = []; iYXD }l;r  
    for j = 1:length(n) vXKL<  
        rpowers = [rpowers m_abs(j):2:n(j)]; 5:@bNNX'j  
    end C*Q7@+&  
    rpowers = unique(rpowers); 2!%)_<  
    O nXo0PV/(  
    % Pre-compute the values of r raised to the required powers, s$fM,l:!  
    % and compile them in a matrix: R_EU|a  
    % ----------------------------- H!;N0",]N  
    if rpowers(1)==0 8qe[x\,"8  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )\+Imn  
        rpowern = cat(2,rpowern{:}); <'\Nv._2a  
        rpowern = [ones(length_r,1) rpowern]; h"[B zX  
    else $0Y`> 3  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); X'OpR   
        rpowern = cat(2,rpowern{:}); 6))":<J  
    end kK5&?)3Y:  
    {K|?i9K  
    % Compute the values of the polynomials: @GQe-04W`  
    % -------------------------------------- hDXTC_^s  
    y = zeros(length_r,length(n)); t 24`*'  
    for j = 1:length(n) dS1HA>c)O  
        s = 0:(n(j)-m_abs(j))/2; 7C|AiSH  
        pows = n(j):-2:m_abs(j); P& 1$SWNyW  
        for k = length(s):-1:1 - (s0f  
            p = (1-2*mod(s(k),2))* ... ;@; a eu  
                       prod(2:(n(j)-s(k)))/              ... 2Bt/co-~4  
                       prod(2:s(k))/                     ... 1Ek3^TOv7  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ed'[_T}T3t  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); czRBuo+k+  
            idx = (pows(k)==rpowers); p[4 +`8  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ~(GvjB/C8  
        end :hICe+2ca  
         )"TVR{I%B  
        if isnorm =z}PR1X!  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); H&s`Xr  
        end YKe&Ph.  
    end ~<k>07  
    % END: Compute the Zernike Polynomials a8xvK;`  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x}2nn)fdZ  
    *(x.egORd  
    % Compute the Zernike functions: SGKAx<U  
    % ------------------------------ Oti;wf G7o  
    idx_pos = m>0; P#TPI*qw  
    idx_neg = m<0; ~ZafTCa;  
    4&E"{d >  
    z = y; EC,,l'%a|/  
    if any(idx_pos) :>!-[hfQ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 56C8)?  
    end *k(FbZ  
    if any(idx_neg) d- ZUuw  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]I*RuDv}  
    end jwwRejNV  
    mc]+j,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)  Ht.P670  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. A^}#  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated k*_Gg  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 9o,Eq x4J  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, o;c"-^>  
    %   and THETA is a vector of angles.  R and THETA must have the same <Ve0PhK  
    %   length.  The output Z is a matrix with one column for every P-value, DWtITO>  
    %   and one row for every (R,THETA) pair. 38sLyoG=i  
    % @Yt394gA%\  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike uWx<J3~q.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) qBF|' .$^  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 6!i`\>I]  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ((Av3{05H&  
    %   for all p. e oE)Mq  
    % ,~7~ S"  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 `Fcr`[  
    %   Zernike functions (order N<=7).  In some disciplines it is z1b@JCWE  
    %   traditional to label the first 36 functions using a single mode aMu6{u6  
    %   number P instead of separate numbers for the order N and azimuthal 2RZa}  
    %   frequency M. iUz?mt;k  
    % 7&,$  
    %   Example: b B#QIXY/L  
    % 0J?443A Y  
    %       % Display the first 16 Zernike functions io\t>_  
    %       x = -1:0.01:1; N?c~AEk9U  
    %       [X,Y] = meshgrid(x,x); U _pPI$ =  
    %       [theta,r] = cart2pol(X,Y); Lp%J:ogV`  
    %       idx = r<=1; p+Q9?9  
    %       p = 0:15; F u5zj\0J  
    %       z = nan(size(X)); ~NE`Ad.G  
    %       y = zernfun2(p,r(idx),theta(idx)); `i|!wD,=\  
    %       figure('Units','normalized') 0vEQgx>  
    %       for k = 1:length(p) K!AA4!eUzM  
    %           z(idx) = y(:,k); ~_0XG0oA  
    %           subplot(4,4,k) N5W!(h)  
    %           pcolor(x,x,z), shading interp u~,hT Y(%  
    %           set(gca,'XTick',[],'YTick',[])  !'!\>x$  
    %           axis square "KF]s.  
    %           title(['Z_{' num2str(p(k)) '}']) c)Ng9p  
    %       end a`:F07r  
    % !d 4DTo  
    %   See also ZERNPOL, ZERNFUN. >'#vC]@  
    .|CoueH  
    %   Paul Fricker 11/13/2006 'uzHI@i  
    HjzAFXRG  
    (mbm',%-(  
    % Check and prepare the inputs: .Erv\lv*  
    % ----------------------------- s/t,6-~EH  
    if min(size(p))~=1 `_.:O,^n^  
        error('zernfun2:Pvector','Input P must be vector.') z(,j)".  
    end -+i7T^@|  
    mS}.?[d"  
    if any(p)>35 "*HEXru#B  
        error('zernfun2:P36', ... $ r-rIW5\  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 6Ik v}q_j  
               '(P = 0 to 35).']) E3{kH 7_'\  
    end [T9]q8"  
    9s!R_R&W.  
    % Get the order and frequency corresonding to the function number: &hZ.K"@7{  
    % ---------------------------------------------------------------- >bI\pJ  
    p = p(:); ,Y| ;V  
    n = ceil((-3+sqrt(9+8*p))/2); OW6dK #CFt  
    m = 2*p - n.*(n+2); <}.!G>X  
    CXuMNa  
    % Pass the inputs to the function ZERNFUN: (I6Q"&h]  
    % ---------------------------------------- 9*~";{O.Oa  
    switch nargin jZ"j_ =o@  
        case 3 N2|NYDQs  
            z = zernfun(n,m,r,theta); )b%zYD9p  
        case 4 ,+Ocb-*  
            z = zernfun(n,m,r,theta,nflag); @K S.H  
        otherwise EqBTN07dZS  
            error('zernfun2:nargin','Incorrect number of inputs.') mm*nXJ  
    end g/FT6+&T.  
    H}&JrT95  
    % 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) Z;tWV%F5  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #}Xsi&:XU  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of *2.h*y'u  
    %   order N and frequency M, evaluated at R.  N is a vector of YUTh*`1k<  
    %   positive integers (including 0), and M is a vector with the WAtv4  
    %   same number of elements as N.  Each element k of M must be a vxi_Y\r=T  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) '~7zeZ'  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is lqO>Q1_{K  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix * RX^ z6  
    %   with one column for every (N,M) pair, and one row for every $Fi1Bv)  
    %   element in R. (7&b)"y  
    %  > T:0  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- &Cm]*$?  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is oLq N  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ~e)"!r  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 GjN6Af~}  
    %   for all [n,m]. I#]pk!  
    % De{ZQg)  
    %   The radial Zernike polynomials are the radial portion of the X f;R'a,$  
    %   Zernike functions, which are an orthogonal basis on the unit 0DnOO0Nc  
    %   circle.  The series representation of the radial Zernike ~>_UTI  
    %   polynomials is zK_P3r LsS  
    % py%~Qz%  
    %          (n-m)/2 C1l'<  
    %            __ "j_cI-@6  
    %    m      \       s                                          n-2s 1D!MXYgm1b  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r WWOt>C~zV  
    %    n      s=0 i6P$>8jBQ-  
    % Wl+spWqW  
    %   The following table shows the first 12 polynomials. )%kiM<})  
    % \hEIQjfi  
    %       n    m    Zernike polynomial    Normalization #_K<-m%9  
    %       --------------------------------------------- eJ ^I+?h  
    %       0    0    1                        sqrt(2) Akws I@@  
    %       1    1    r                           2 YdIZikF#  
    %       2    0    2*r^2 - 1                sqrt(6) z;/8R7L&  
    %       2    2    r^2                      sqrt(6) 1_;{1O+B  
    %       3    1    3*r^3 - 2*r              sqrt(8) mH\2XG8nV  
    %       3    3    r^3                      sqrt(8) x&+&)d  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) G;[O~N3n.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 4b, +;  
    %       4    4    r^4                      sqrt(10) Hr7pcz/#l  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) r1}1lJ>7H  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 9HPwl  
    %       5    5    r^5                      sqrt(12) MR5[|kHJT  
    %       --------------------------------------------- .RAyi>\e  
    % xsy45az<ip  
    %   Example: Bc-/s(/Eq  
    % =1VZcLNt  
    %       % Display three example Zernike radial polynomials M)Z!W3  
    %       r = 0:0.01:1; S,avvY.U\  
    %       n = [3 2 5]; \!w |  
    %       m = [1 2 1]; P*U^,Jh<  
    %       z = zernpol(n,m,r); >M##q?.  
    %       figure KDV.ZSF7  
    %       plot(r,z) V,8Z!.MG  
    %       grid on cW"DDm g  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') !"-.D4*r  
    % _2}~Vqb+  
    %   See also ZERNFUN, ZERNFUN2. |;d#k+/;  
    |YV> #l  
    % A note on the algorithm. h^1 !8oOYD  
    % ------------------------ G+k wG)K  
    % The radial Zernike polynomials are computed using the series ;KEie@Ry  
    % representation shown in the Help section above. For many special =w"Kkj>%oh  
    % functions, direct evaluation using the series representation can OA} r*Wz  
    % produce poor numerical results (floating point errors), because SXvflr] =m  
    % the summation often involves computing small differences between s aHY9{)  
    % large successive terms in the series. (In such cases, the functions 8K8jz9.s  
    % are often evaluated using alternative methods such as recurrence WB<MU:.Vc  
    % relations: see the Legendre functions, for example). For the Zernike 7fSNF7/+  
    % polynomials, however, this problem does not arise, because the m I:^lp  
    % polynomials are evaluated over the finite domain r = (0,1), and BpX`49  
    % because the coefficients for a given polynomial are generally all 0@y`iZ] 1S  
    % of similar magnitude. d+ZXi'  
    % cD)9EFo  
    % ZERNPOL has been written using a vectorized implementation: multiple bu $u@:q 6  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] R?2HnJh  
    % values can be passed as inputs) for a vector of points R.  To achieve TXf60{:f  
    % this vectorization most efficiently, the algorithm in ZERNPOL x'OP0],#  
    % involves pre-determining all the powers p of R that are required to .c@Y ?..+  
    % compute the outputs, and then compiling the {R^p} into a single {{>,c}O /  
    % matrix.  This avoids any redundant computation of the R^p, and }QQ 7jE  
    % minimizes the sizes of certain intermediate variables. x(4"!#  
    % 3c(mZ   
    %   Paul Fricker 11/13/2006 VZ">vIRyi|  
    utl-#Wwt/  
    0S'@(p[A  
    % Check and prepare the inputs: =VT\$ 5A  
    % ----------------------------- ![fNlG!r  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o`7 Z<HF  
        error('zernpol:NMvectors','N and M must be vectors.') 7sWe32  
    end qdmAkYUC  
    ""|;5kJS4  
    if length(n)~=length(m) :=5X)10  
        error('zernpol:NMlength','N and M must be the same length.') 1w7XM0SHcn  
    end ~Lg ;7i1L  
    B*Om\I  
    n = n(:); ".N{v1  
    m = m(:); YK$[)x\S  
    length_n = length(n); q bCU&G|)  
    w:iMrQeJg  
    if any(mod(n-m,2)) >}2 ,2  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') mO(Y>|mm  
    end 1;i|GXY:h  
    B4*y-Q.*  
    if any(m<0) j{2 0  
        error('zernpol:Mpositive','All M must be positive.') mW+5I-~  
    end k'PvQl"I  
    >H5t,FfQL  
    if any(m>n) C]l)Pz$  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ;T8(byH ?  
    end R#8cOmZ  
    ) j&khHD  
    if any( r>1 | r<0 ) *QI Yq  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') v6[VdWOx5  
    end 8O60pB;4  
    i_*.  
    if ~any(size(r)==1) @p}_"BHYWt  
        error('zernpol:Rvector','R must be a vector.') B!8X?8D  
    end 1^V.L+0s]  
    >&R@L KP  
    r = r(:); |%fNLUJ)  
    length_r = length(r); S'w}Ir  
    05\0g9  
    if nargin==4 C!9mygI  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); b`j9}t Z  
        if ~isnorm f\Hw Y)^>  
            error('zernpol:normalization','Unrecognized normalization flag.') Nh/i'q/  
        end Kng=v~)N'  
    else 8;c\} D  
        isnorm = false; O@W/s!&lFa  
    end 6#K.n&=*  
    P>)J:.tr0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VAUd^6Xdwx  
    % Compute the Zernike Polynomials &2[Xu4*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #R31V QwK5  
    T /IX(b'<  
    % Determine the required powers of r: 2EN}"Du]mj  
    % ----------------------------------- {hN<Ot  
    rpowers = []; &y|PseH"  
    for j = 1:length(n) ycki0&n3  
        rpowers = [rpowers m(j):2:n(j)]; P$N5j~*  
    end Mqk|H~l5c  
    rpowers = unique(rpowers); * a1q M?  
    "lC>_A  
    % Pre-compute the values of r raised to the required powers, F2_'U' a  
    % and compile them in a matrix: ;;XY&J  
    % ----------------------------- 9=/4}!.  
    if rpowers(1)==0 ?p 4iXHE  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); s'l|Ii  
        rpowern = cat(2,rpowern{:}); llh +r?  
        rpowern = [ones(length_r,1) rpowern]; kTT%< e  
    else u*uHdV5  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); nnE'zk<"  
        rpowern = cat(2,rpowern{:}); )+8r$ i  
    end V EsM  
    b\o>4T  
    % Compute the values of the polynomials: c9Cc%EK  
    % -------------------------------------- *)I^+zN  
    z = zeros(length_r,length_n); ].aFdy  
    for j = 1:length_n ht>/7.p]  
        s = 0:(n(j)-m(j))/2;  iycceZ  
        pows = n(j):-2:m(j); yD.(j*bMK;  
        for k = length(s):-1:1 Jg{K!P|i  
            p = (1-2*mod(s(k),2))* ... E]g6|,4~-  
                       prod(2:(n(j)-s(k)))/          ... @p^EXc*|  
                       prod(2:s(k))/                 ... _5(p=Zc  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... h"Wpb}FT  
                       prod(2:((n(j)+m(j))/2-s(k))); `'3 De(  
            idx = (pows(k)==rpowers); 5WxNH}{  
            z(:,j) = z(:,j) + p*rpowern(:,idx); w2/3[VZ}l  
        end fO^s4gWTg  
         /38I (0  
        if isnorm YPq:z"`-y4  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $3&XM  
        end 'NfsAE  
    end tSoF!@6  
    @"/H er  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  XK@&$~iA3  
    r@vt.t0#  
    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。 j&Xx{ 4v  
    UpE +WzY  
    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)