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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 |qbCmsY5/  
    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! R_ J=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 >1:s.[&  
    function z = zernfun(n,m,r,theta,nflag) M xj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 'dM &~L SQ  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 6,M>'s,N  
    %   and angular frequency M, evaluated at positions (R,THETA) on the VpMpZ9oM<  
    %   unit circle.  N is a vector of positive integers (including 0), and * JGm  
    %   M is a vector with the same number of elements as N.  Each element b_ Sh#d&  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) >JS\H6  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, n"Ec%n  
    %   and THETA is a vector of angles.  R and THETA must have the same ba|x?kz  
    %   length.  The output Z is a matrix with one column for every (N,M) K,tmh1  
    %   pair, and one row for every (R,THETA) pair. %*OKhrM  
    % 4?M= ?K0  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 94I8~Jj4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &w:"e'FG`  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ^ef:cS$;  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mn\e(WoX  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized * b>W  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. z;1tJ  
    % {>OuxVl??k  
    %   The Zernike functions are an orthogonal basis on the unit circle. VY<v?Of i-  
    %   They are used in disciplines such as astronomy, optics, and liFNJd`|o+  
    %   optometry to describe functions on a circular domain. aW %ulZ  
    % ~$#DB@b  
    %   The following table lists the first 15 Zernike functions. hd9fD[5  
    % wM(!9Ws3  
    %       n    m    Zernike function           Normalization -Qo`UL.}  
    %       -------------------------------------------------- UY j  
    %       0    0    1                                 1 a}#[mw@m=  
    %       1    1    r * cos(theta)                    2  \A:m<::  
    %       1   -1    r * sin(theta)                    2 O<S*bN>BF  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 2tC ep  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 2f`u?T  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 4PTHUyX  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ,!kqEIp%  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ^C>i(j&  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) aMuc]Wy#  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 65N;PH59D  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) Rb<aCX  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =Xm [  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2uS&A \   
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;z#D%#Ztq  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) xBG&ZM4"^f  
    %       -------------------------------------------------- f'Wc_ L)  
    % 56u'XMB?  
    %   Example 1: =r+u!~%@''  
    % wED~^[]f  
    %       % Display the Zernike function Z(n=5,m=1) W>dS@;E  
    %       x = -1:0.01:1; Slq=;TDp  
    %       [X,Y] = meshgrid(x,x); Y {Klwn   
    %       [theta,r] = cart2pol(X,Y); a~OCo  
    %       idx = r<=1; ")ow,r^"  
    %       z = nan(size(X)); Sl^HMO  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); cG?RisSZ  
    %       figure s?=f,I  
    %       pcolor(x,x,z), shading interp KmZUDU%R  
    %       axis square, colorbar [[JwHM8H&  
    %       title('Zernike function Z_5^1(r,\theta)') 8_U*_I7(  
    % 9XF+? x  
    %   Example 2: !-x^b.${B  
    % eN>=x40  
    %       % Display the first 10 Zernike functions #1z}~1-  
    %       x = -1:0.01:1; {#=q[jVi%1  
    %       [X,Y] = meshgrid(x,x); -#3B>VY  
    %       [theta,r] = cart2pol(X,Y); Mz40([{  
    %       idx = r<=1; A[XEbfDO  
    %       z = nan(size(X));  tAP~  
    %       n = [0  1  1  2  2  2  3  3  3  3]; /,2Em>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; W3{k{~  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; !K'kkn,h  
    %       y = zernfun(n,m,r(idx),theta(idx)); &kXf)xc<~  
    %       figure('Units','normalized') !s\-i6S>  
    %       for k = 1:10 vwZ2kk!|i  
    %           z(idx) = y(:,k); ;. !AX|v  
    %           subplot(4,7,Nplot(k)) qQ/j+  
    %           pcolor(x,x,z), shading interp $4>K2  
    %           set(gca,'XTick',[],'YTick',[]) + ?*,J=/  
    %           axis square zjM+F{P8  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5Tb93Q@c  
    %       end `P)atQ  
    % 8NPt[*  
    %   See also ZERNPOL, ZERNFUN2. #`); UAf  
    cQu1WgQ G  
    %   Paul Fricker 11/13/2006 Th`IpxV  
    P et0yH  
    /0!6;PC<  
    % Check and prepare the inputs: _tb)F"4V  
    % ----------------------------- fph*|T&R  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d;:+Xd`  
        error('zernfun:NMvectors','N and M must be vectors.') vxZvK0b620  
    end 7>wSbAR<  
    d#vq+wR  
    if length(n)~=length(m) _&.CI6  
        error('zernfun:NMlength','N and M must be the same length.') tE9%;8;H  
    end _yJd@  
    Q6RBZucv  
    n = n(:); j*q]-$2E  
    m = m(:); #";(&|7  
    if any(mod(n-m,2)) K S,X$)9  
        error('zernfun:NMmultiplesof2', ... 2y,NT|jp  
              'All N and M must differ by multiples of 2 (including 0).') 7zgU>$i  
    end '?v.O}  
    hR[Qdu6r  
    if any(m>n) 9-Qu b+0o  
        error('zernfun:MlessthanN', ... W _yVVr  
              'Each M must be less than or equal to its corresponding N.') ]EE}ax%#aq  
    end Av _1cvR:  
    "DjD"?/b  
    if any( r>1 | r<0 ) Tr(w~et  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') * "~^k^_b}  
    end %=]~5a9  
    1$q SbQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ds4ERe /  
        error('zernfun:RTHvector','R and THETA must be vectors.') 71@V|$Dy  
    end Hp8)-eT  
    x!tCK47Yq  
    r = r(:); <lB^>Hfu  
    theta = theta(:); T,!?+#  
    length_r = length(r); {&4+W=0 n  
    if length_r~=length(theta) hJkIFyQ{j  
        error('zernfun:RTHlength', ... P,j)m\|  
              'The number of R- and THETA-values must be equal.') A>bo Xcr  
    end :jT1=PfL  
    Hb#8?{  
    % Check normalization: wg<DV!GZ  
    % -------------------- ]Yp;8#:1  
    if nargin==5 && ischar(nflag) V'mQ {[{R  
        isnorm = strcmpi(nflag,'norm'); t1 OnA#]/_  
        if ~isnorm #:v|/2   
            error('zernfun:normalization','Unrecognized normalization flag.') E-MEMran4  
        end =BMON{K  
    else ss-{l+Z5  
        isnorm = false; qYl%v  
    end 2x"&8Bg3  
    ido'<;4>  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W+cmn)8  
    % Compute the Zernike Polynomials }~:`9PV)Z%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MIsjTKE  
    ^}a..@|%W  
    % Determine the required powers of r: ^$FHI_  
    % ----------------------------------- =d!3_IZ  
    m_abs = abs(m); !.?2zp~  
    rpowers = []; w +fsw@dK&  
    for j = 1:length(n) VWj]X7v  
        rpowers = [rpowers m_abs(j):2:n(j)]; XPBKQm_}  
    end Z_zN:BJ8L  
    rpowers = unique(rpowers); 0/6f9A  
    }:])1!a  
    % Pre-compute the values of r raised to the required powers, MD1n+FgTu  
    % and compile them in a matrix: }G]6Rip 3  
    % ----------------------------- U6t>UE6k  
    if rpowers(1)==0 Ovxs+mQ  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4[44Eku\  
        rpowern = cat(2,rpowern{:}); Kyq/'9`  
        rpowern = [ones(length_r,1) rpowern]; [6`8^-}?  
    else @!=q.4b  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jL8.*pfv  
        rpowern = cat(2,rpowern{:}); XT9]+b8(M  
    end % r`hW \4{  
    A_tdtN<  
    % Compute the values of the polynomials: \uQ yp*P1s  
    % -------------------------------------- p9 <XaJ}   
    y = zeros(length_r,length(n)); 8d?r )/~  
    for j = 1:length(n) 6ey{+8  
        s = 0:(n(j)-m_abs(j))/2; --6C>iY[&u  
        pows = n(j):-2:m_abs(j); !i,Eo-[Z  
        for k = length(s):-1:1 z\Hg@J&#  
            p = (1-2*mod(s(k),2))* ... }^ +E S^~  
                       prod(2:(n(j)-s(k)))/              ... 7hQXGY,q  
                       prod(2:s(k))/                     ... 2Nrb}LH  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... P(a!I{A(  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); h6Ovl  
            idx = (pows(k)==rpowers); 0/5 a3-3{  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2w_[c.  
        end R.@I}>  
         Hb55RilC  
        if isnorm hfE5[  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); " R!,5HQF;  
        end uH="l.u  
    end ^SM>bJ1Z_  
    % END: Compute the Zernike Polynomials NX%"_W/W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $ @g\wz  
    1Bp?HyCR  
    % Compute the Zernike functions: fUx;_GX?  
    % ------------------------------ .;}vp*  
    idx_pos = m>0; NXo$rf:  
    idx_neg = m<0; 0`UI^Y~Q  
    QiC}hj$  
    z = y; ##!idcC  
    if any(idx_pos) o5LyBUJ  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;}1O\nngR  
    end uE] HU  
    if any(idx_neg) xl2;DFiYt  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Oxsx\f_  
    end |`eHUtjH  
    1i3;P/  
    % 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) fVf @Ngvu  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. O]_a$U*6  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Br4[hUV/  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {,aX|*1Ku~  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, HOt,G _{  
    %   and THETA is a vector of angles.  R and THETA must have the same 4j|IG/m  
    %   length.  The output Z is a matrix with one column for every P-value, ?}g^/g !  
    %   and one row for every (R,THETA) pair. QNbV=*F?  
    % ,="hI:*<  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Th_PmkvC  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) BSH2Kq  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 2ieyU5q7#  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 |P0!dt7sQ  
    %   for all p. ZSWZz8  
    % L:j3  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 a\-AGG{2/X  
    %   Zernike functions (order N<=7).  In some disciplines it is =E.!Ff4~(  
    %   traditional to label the first 36 functions using a single mode =xw+cs1,x  
    %   number P instead of separate numbers for the order N and azimuthal JAx0(MZO  
    %   frequency M. 9Js+*,t  
    % HK NT. a  
    %   Example: rMWJ  
    % 3_bqDhVI5  
    %       % Display the first 16 Zernike functions "UX/yLc3(  
    %       x = -1:0.01:1; ]A%]W^G  
    %       [X,Y] = meshgrid(x,x); mUj_V#v  
    %       [theta,r] = cart2pol(X,Y); -*A1[Z ?  
    %       idx = r<=1; E$.fAIt  
    %       p = 0:15; .8wf {y  
    %       z = nan(size(X)); sZx`u+  
    %       y = zernfun2(p,r(idx),theta(idx)); ZyM7)!+kPa  
    %       figure('Units','normalized') 9;7Gzr6A"  
    %       for k = 1:length(p) brCXimG&jo  
    %           z(idx) = y(:,k); :6MV@{;PJ  
    %           subplot(4,4,k) v-Tkp Yn  
    %           pcolor(x,x,z), shading interp nuH=pIq6x  
    %           set(gca,'XTick',[],'YTick',[]) =(+]ee!Ti  
    %           axis square Al1_\vx7  
    %           title(['Z_{' num2str(p(k)) '}']) f$76p!pDa  
    %       end C(8VXtx_  
    % E+ctiVL  
    %   See also ZERNPOL, ZERNFUN. !>\&*h-Cm#  
    3xk_ZK82  
    %   Paul Fricker 11/13/2006 dGglt Y  
    GKc?  
    D V\7KKJE  
    % Check and prepare the inputs: QJ&]4*>a  
    % ----------------------------- vHZq z<  
    if min(size(p))~=1 U&i#cF   
        error('zernfun2:Pvector','Input P must be vector.') >AFQm  
    end wBDHhXi0  
     !2kM  
    if any(p)>35 5tyA{&Ao  
        error('zernfun2:P36', ... Xdi<V_!BC-  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... + -uQ] ^n  
               '(P = 0 to 35).']) -T}r$A  
    end /qKA1-R}4  
    Wv|CJN;4  
    % Get the order and frequency corresonding to the function number: mqHcD8X  
    % ---------------------------------------------------------------- uI$n7\G!  
    p = p(:); Atb`Q'Yrw  
    n = ceil((-3+sqrt(9+8*p))/2); xax[# Vl4  
    m = 2*p - n.*(n+2); SwsJ<Dq^z  
    _aYhW{wW  
    % Pass the inputs to the function ZERNFUN: L3w.<h  
    % ---------------------------------------- wz1nV}  
    switch nargin No"i6R+  
        case 3 p5jR;nOZ%l  
            z = zernfun(n,m,r,theta); X::@2{-@y  
        case 4  )ut$644R  
            z = zernfun(n,m,r,theta,nflag); XHxJzYMc  
        otherwise vh.-9eD  
            error('zernfun2:nargin','Incorrect number of inputs.') BTD_j&+(  
    end ;vneeW4|  
    >fMzUTJ4  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Fm=jgt3wv8  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. %X's/;(Lx`  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of r&Nh>6<&/  
    %   order N and frequency M, evaluated at R.  N is a vector of Ux1j+}y  
    %   positive integers (including 0), and M is a vector with the w>8HS+  
    %   same number of elements as N.  Each element k of M must be a sZ~03QvkT  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) +_ /ys!  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is w,X)g{^T  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix )Nqx=ms[(!  
    %   with one column for every (N,M) pair, and one row for every iZ>P>x\  
    %   element in R. n-2!<`UFX  
    % DLP@?]BBOA  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 1X2|jj  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Vpp$yM&?  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to W4$aX5ow$  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ZV:df 6S  
    %   for all [n,m]. hxj\  
    % x&^Xgi?  
    %   The radial Zernike polynomials are the radial portion of the ]]_5_)"4  
    %   Zernike functions, which are an orthogonal basis on the unit }cI-]|)|2  
    %   circle.  The series representation of the radial Zernike 2+I5VPf  
    %   polynomials is L-)ZjXzk  
    % sxA]o|  
    %          (n-m)/2 ;~DrsQb  
    %            __ eI:x4K,#  
    %    m      \       s                                          n-2s %TRJ  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r [T4{K &  
    %    n      s=0 WMnSkO  
    % mi$C%~]5m  
    %   The following table shows the first 12 polynomials. kssRwe%>;  
    % BJ]L@L%  
    %       n    m    Zernike polynomial    Normalization Y'jgp Vt  
    %       --------------------------------------------- zRmVV}b  
    %       0    0    1                        sqrt(2) x0>N{ADXQ  
    %       1    1    r                           2 /s%-c!o^  
    %       2    0    2*r^2 - 1                sqrt(6) G /$+e  
    %       2    2    r^2                      sqrt(6) :R=7dH~r  
    %       3    1    3*r^3 - 2*r              sqrt(8) ern\QAhXX  
    %       3    3    r^3                      sqrt(8) f+ZOE?"  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) fd #QCs  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 9{U@s  
    %       4    4    r^4                      sqrt(10) -(e=S^36  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) GOGS"q  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) wLiPkW  
    %       5    5    r^5                      sqrt(12) ~8 UMwpl-  
    %       --------------------------------------------- Nt_sV7zzb  
    % KPDJ$,:  
    %   Example: @aN~97 H\  
    % cAGM|%  
    %       % Display three example Zernike radial polynomials S&-F(#CF^  
    %       r = 0:0.01:1; #g@4c3um|  
    %       n = [3 2 5]; L4T\mP7D7*  
    %       m = [1 2 1]; = 03G~7B>  
    %       z = zernpol(n,m,r); +w(6#R8u5  
    %       figure N-b'O`C  
    %       plot(r,z) Mv/ SU">F  
    %       grid on o<p4r}*AVJ  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 1c @S[y  
    % RTvOaZ  
    %   See also ZERNFUN, ZERNFUN2. bC"h7$3  
    pg!oi?Jn  
    % A note on the algorithm. k<j]b^jbz  
    % ------------------------ x5xMr.vm  
    % The radial Zernike polynomials are computed using the series Y@q9   
    % representation shown in the Help section above. For many special /=l!F'  
    % functions, direct evaluation using the series representation can "[k>pzl6  
    % produce poor numerical results (floating point errors), because op2Zf?Bx{+  
    % the summation often involves computing small differences between jj;TS%  
    % large successive terms in the series. (In such cases, the functions <KtL,a=2+  
    % are often evaluated using alternative methods such as recurrence Het>G{  
    % relations: see the Legendre functions, for example). For the Zernike 6Y6t.j0vN.  
    % polynomials, however, this problem does not arise, because the gBWr)R  
    % polynomials are evaluated over the finite domain r = (0,1), and _*g.U=u  
    % because the coefficients for a given polynomial are generally all 3TeRZ=2:*x  
    % of similar magnitude. ZybfqBTD&c  
    % :6%ivS  
    % ZERNPOL has been written using a vectorized implementation: multiple <h+@;/v:  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] {/N8[?zML  
    % values can be passed as inputs) for a vector of points R.  To achieve LkK&<z  
    % this vectorization most efficiently, the algorithm in ZERNPOL DzA'MX  
    % involves pre-determining all the powers p of R that are required to 8 l= EL7  
    % compute the outputs, and then compiling the {R^p} into a single T*Ge67  
    % matrix.  This avoids any redundant computation of the R^p, and A.7lo  
    % minimizes the sizes of certain intermediate variables. P0_Ymn=&  
    % 3LJ\y  
    %   Paul Fricker 11/13/2006 xT* 3QwK  
    SYQP7oG9oQ  
    lb*;Z7fx<'  
    % Check and prepare the inputs: ^jb;4nf  
    % ----------------------------- xzfugW  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) bQ 0Ab"+D  
        error('zernpol:NMvectors','N and M must be vectors.') Uc ,..  
    end FqGMHM\J  
    ~#VDJ[Z  
    if length(n)~=length(m) 7@e}rh?N-|  
        error('zernpol:NMlength','N and M must be the same length.') kef% 5B  
    end 7I]?:%8 h  
    g2^{+,/^K  
    n = n(:); 2h]CZD4  
    m = m(:); (M u;U!M"P  
    length_n = length(n); VK,{Mu=.9  
    r~7}w4U  
    if any(mod(n-m,2)) `HYj:4v'  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') .q:6F*,1M  
    end  @e\ @EW  
    lfd-!(tXD  
    if any(m<0) T%Cj#J&L  
        error('zernpol:Mpositive','All M must be positive.') ?UIW&*h}  
    end 8'qlg|{!~  
    3fX _XH1Q  
    if any(m>n) .V}bfd[k$  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') rK}sQ4z=  
    end aR@+Qf  
    \Nf#{  
    if any( r>1 | r<0 ) VG$;ri>  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') kz("LI]  
    end *wd=&Z^19  
    e0ni  
    if ~any(size(r)==1) *:un+k  
        error('zernpol:Rvector','R must be a vector.') v_v>gPl,  
    end { ] 0T  
    FjiIB1 T  
    r = r(:); 7i02M~*uS  
    length_r = length(r); Qgf|obrEi6  
    }vgM$o  
    if nargin==4 :M`~9MCRf  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); l g ,%  
        if ~isnorm >dw 0@T&p  
            error('zernpol:normalization','Unrecognized normalization flag.') e}7!A  
        end ;Oq>c=9%  
    else 3A~<|<}t  
        isnorm = false; ]-a/)8  
    end []yIz1P=j  
    ux6)K= ]  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +<V$G/"  
    % Compute the Zernike Polynomials )#hR}|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <m{#u4FC'  
    DR]oK_  
    % Determine the required powers of r: $ rbr&TJ  
    % ----------------------------------- R*k;4*1u  
    rpowers = []; $/(``8li_  
    for j = 1:length(n) Hv:~)h$  
        rpowers = [rpowers m(j):2:n(j)]; )Wt&*WMFXl  
    end  Yy`A0v  
    rpowers = unique(rpowers); CQ Ei(ty  
    u$ o 19n  
    % Pre-compute the values of r raised to the required powers, --c)!Vxzx  
    % and compile them in a matrix: Z?9G2<i  
    % ----------------------------- Hl{ul'o  
    if rpowers(1)==0 *J': U>p  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); /S^>06{-+  
        rpowern = cat(2,rpowern{:}); ,Tx38  
        rpowern = [ones(length_r,1) rpowern]; i\.(6hf+  
    else G@T_o4t  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hM="9] i.  
        rpowern = cat(2,rpowern{:}); If'N0^'W  
    end :iQJ9Hdz  
    TC=>De2;  
    % Compute the values of the polynomials: #KHj.Vg  
    % -------------------------------------- E0!0 uSg&  
    z = zeros(length_r,length_n); _o+OkvhU  
    for j = 1:length_n P-yVc2YH  
        s = 0:(n(j)-m(j))/2; !Zc#E,  
        pows = n(j):-2:m(j); -sDl[  
        for k = length(s):-1:1 GH3RRzp r  
            p = (1-2*mod(s(k),2))* ... ka(3ONbG  
                       prod(2:(n(j)-s(k)))/          ... Y(T$k9%}+  
                       prod(2:s(k))/                 ... ,LLx&jS  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... #BH]`A J  
                       prod(2:((n(j)+m(j))/2-s(k))); I?\P^f  
            idx = (pows(k)==rpowers); AxO.adQE%  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ]S@DVXH  
        end wsAb8U C_  
         BPOT!-  
        if isnorm Y$|KY/)H)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1));  3(*vZ  
        end m|]"e@SF2  
    end dV*9bDkM/  
    h*Mi/\  
    % 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)  1'1>B  
    AN)r(86L  
    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 @Vd'k<  
    NIp]n[ =.q  
    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)