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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 W Q9Q:F2  
    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! eKo=g|D  
     
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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 7+IRI|d  
    function z = zernfun(n,m,r,theta,nflag) Plhakngj  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6V}xgfB  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N o^MoU2c  
    %   and angular frequency M, evaluated at positions (R,THETA) on the @8+v6z  
    %   unit circle.  N is a vector of positive integers (including 0), and [hk/Rp7{  
    %   M is a vector with the same number of elements as N.  Each element TJ_6:;4,|_  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) {`T^&b k  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [tElt4uG  
    %   and THETA is a vector of angles.  R and THETA must have the same LR\8M(rtvH  
    %   length.  The output Z is a matrix with one column for every (N,M) 5tzO=gO[  
    %   pair, and one row for every (R,THETA) pair. i[ws%GfEv  
    % 8OO[Le]1  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fO .=i1 E}  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m6]6 !_  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ll- KK`Ka  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7s!rer>  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ' I!/I  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. eT]*c?"  
    % 412E7   
    %   The Zernike functions are an orthogonal basis on the unit circle. zMBGpqdP  
    %   They are used in disciplines such as astronomy, optics, and :^xNHMp!  
    %   optometry to describe functions on a circular domain. M)AvcZNs  
    % &A`,hF8  
    %   The following table lists the first 15 Zernike functions. [9:";JSl"Y  
    % 3(vm'r&5n>  
    %       n    m    Zernike function           Normalization bd% M.,  
    %       -------------------------------------------------- +c, ^KHW  
    %       0    0    1                                 1 _-^mxC|M  
    %       1    1    r * cos(theta)                    2 |F<%gJ  
    %       1   -1    r * sin(theta)                    2 q^n LC6q  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) <*-8E(a  
    %       2    0    (2*r^2 - 1)                    sqrt(3) }gB^C3b6  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) %y*'bS  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) $b2~H+u(  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) V0&7MY*  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) kC6Y?g  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) yLK %lP  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ! hEZV&y  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "a33m:]J  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) [McqwU/Q  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5p5"3m;M7  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) W tHJG5  
    %       -------------------------------------------------- g)D@4RM  
    % *M0O&"~j  
    %   Example 1: 8bO+[" c  
    % bn5O2  
    %       % Display the Zernike function Z(n=5,m=1) pSIXv%1J  
    %       x = -1:0.01:1; Y9vVi]4  
    %       [X,Y] = meshgrid(x,x); 'zT7$ .L  
    %       [theta,r] = cart2pol(X,Y); ,:MUf]Ky  
    %       idx = r<=1; nn$^iw`  
    %       z = nan(size(X)); [KbLEMrPba  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); E}a.qM'  
    %       figure yf`_?gJ6d  
    %       pcolor(x,x,z), shading interp )  LTV+?  
    %       axis square, colorbar FeQo,a  
    %       title('Zernike function Z_5^1(r,\theta)') PYY<  
    % m qUDve(  
    %   Example 2: Fm6]mz%~u#  
    % 9F6dKPN:  
    %       % Display the first 10 Zernike functions -f1}N|hy  
    %       x = -1:0.01:1; ImH9 F\  
    %       [X,Y] = meshgrid(x,x); ]Y76~!N  
    %       [theta,r] = cart2pol(X,Y); _5O~ ]}  
    %       idx = r<=1; hN gT/y8  
    %       z = nan(size(X)); x_?K6[G&}  
    %       n = [0  1  1  2  2  2  3  3  3  3]; A&%7Z^Pp  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; R~hIoaiN  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; _^zs(  
    %       y = zernfun(n,m,r(idx),theta(idx)); nA.U'=`  
    %       figure('Units','normalized') j \d)#+;  
    %       for k = 1:10 QR8]d1+GV  
    %           z(idx) = y(:,k); :eB+t`M  
    %           subplot(4,7,Nplot(k)) O&~ @ior  
    %           pcolor(x,x,z), shading interp nU\.`.39 +  
    %           set(gca,'XTick',[],'YTick',[]) B9cWxe4R#  
    %           axis square *ezft&{)`  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T?=]&9Y'  
    %       end -49I3&  
    % Z("N *`VP;  
    %   See also ZERNPOL, ZERNFUN2. GkU$Z @  
    ba ,n/yH  
    %   Paul Fricker 11/13/2006 ]W~M?1 }  
    H_Sv,lwz;c  
    e7&RZ+s#wZ  
    % Check and prepare the inputs: Sz')1<  
    % ----------------------------- )"M;7W?R0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w T_l>u  
        error('zernfun:NMvectors','N and M must be vectors.') l 6aD3?8LN  
    end BePb8 k<y  
    Dvl\o;  
    if length(n)~=length(m) RF4B ]Gqd  
        error('zernfun:NMlength','N and M must be the same length.') ;b=7m#5  
    end HJpx,NU'  
    w-v8 P`V  
    n = n(:); k*F9&-rtN  
    m = m(:); !,5qAGi0  
    if any(mod(n-m,2)) '}(Fj2P79  
        error('zernfun:NMmultiplesof2', ... ~ Hj c?*  
              'All N and M must differ by multiples of 2 (including 0).') JnnxXj30,  
    end l ^}5PHLd  
    r~fnK%|  
    if any(m>n) O~ x{p,s U  
        error('zernfun:MlessthanN', ... w Bm4~ ~_  
              'Each M must be less than or equal to its corresponding N.') Fy$ C._C$  
    end 7*Zm{r@u  
    kXUJlLod  
    if any( r>1 | r<0 ) wGIRRM !b  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ) R\";{`M  
    end Ep')@7^n  
    J\'f5)k  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h2:TbQ  
        error('zernfun:RTHvector','R and THETA must be vectors.') #,})N*7  
    end rfSEL 57'  
    %P#| }  
    r = r(:); vQ $"|8,  
    theta = theta(:); BZXee>3"  
    length_r = length(r); 9O^~l2`  
    if length_r~=length(theta) O]F(vHK\   
        error('zernfun:RTHlength', ... ATmyoN2@>  
              'The number of R- and THETA-values must be equal.') q%/.+g2-\  
    end AAB_Ytf  
    uOKdb6]r6  
    % Check normalization: 1UB.2}/:  
    % -------------------- Zx6h%l,%  
    if nargin==5 && ischar(nflag) "EWq{l_I5$  
        isnorm = strcmpi(nflag,'norm'); 9j5Z!Vsy  
        if ~isnorm jC?l :m?  
            error('zernfun:normalization','Unrecognized normalization flag.') BuC\Bd^0  
        end ]f wW dtz1  
    else ^d(gC%+!u  
        isnorm = false; Bw[IW[(~!  
    end Lc-Wf zT  
    S'@Ok=FSy  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4`RZ&w;1H2  
    % Compute the Zernike Polynomials X"HVK+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% { W5 _KX  
    |&bucG=  
    % Determine the required powers of r: 4)L};B=  
    % ----------------------------------- ;vpq0t`  
    m_abs = abs(m); "uyr@u0b  
    rpowers = []; V;~\+@  
    for j = 1:length(n) I;, n|o  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;MlPP)*k  
    end G2|G}#E  
    rpowers = unique(rpowers); #D >:'ezm  
    p2+K-/}ApP  
    % Pre-compute the values of r raised to the required powers, Ggv*EsN/cC  
    % and compile them in a matrix: #AO}JP  
    % ----------------------------- $"0`2C  
    if rpowers(1)==0 wg:\$_Og  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uOd1:\%*  
        rpowern = cat(2,rpowern{:}); Zl]@;*u  
        rpowern = [ones(length_r,1) rpowern]; x{rjngp2  
    else  8#1o  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); -|=)  
        rpowern = cat(2,rpowern{:}); ##1/{9ywy  
    end n+vv %  
    Mdu\ci)lr  
    % Compute the values of the polynomials: Sj8fo^K50  
    % -------------------------------------- C 8d9 (u  
    y = zeros(length_r,length(n)); jpMMnEVj6P  
    for j = 1:length(n) *Rc?rMF!  
        s = 0:(n(j)-m_abs(j))/2; E?Qg'|+_  
        pows = n(j):-2:m_abs(j); Uqly|FS &n  
        for k = length(s):-1:1 !y2yS/  
            p = (1-2*mod(s(k),2))* ... V*@&<x"E  
                       prod(2:(n(j)-s(k)))/              ... : ' pK  
                       prod(2:s(k))/                     ... Ngm/5Lc  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]2[\E~^KU  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); XuU>.T$]c  
            idx = (pows(k)==rpowers); Z 2$S'}F  
            y(:,j) = y(:,j) + p*rpowern(:,idx); IiX2O(*ZE  
        end ~BnmAv$m[  
         m/,8\+  
        if isnorm OE}c$!@  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Kc>Rd  
        end rDc$#  
    end lg^Lk\Y+re  
    % END: Compute the Zernike Polynomials cf%2A1I2W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `bd9N !K  
    %PK(Z*>  
    % Compute the Zernike functions: (^<skx>  
    % ------------------------------ _m%Ab3iT~  
    idx_pos = m>0;  y'^b{q@  
    idx_neg = m<0; Qv8 =CnuOT  
    "|PX5  
    z = y; +NOq>kH@  
    if any(idx_pos) yv$hIU2X  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 91k-os(4]  
    end JbXi|OS/  
    if any(idx_neg) K>-01AGHL  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); /d$kz&aIV  
    end A[:(#iR5-E  
     ]l  
    % 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) =n8M'  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. VpED9l]y  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ,lb}&uZo  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive L#!m|_Mz  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, WkPT6d  
    %   and THETA is a vector of angles.  R and THETA must have the same )X8N|W>vh  
    %   length.  The output Z is a matrix with one column for every P-value, t&_X{!1X"w  
    %   and one row for every (R,THETA) pair. x l=i_  
    % 0XA0 b1VX  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike `9|Uu#x  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ]?Q<lMG  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) K4OiKYq  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 j%8 1q  
    %   for all p. Qzv&  
    % nrbP3sf*  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ( F4c0  
    %   Zernike functions (order N<=7).  In some disciplines it is $JiypX^DOP  
    %   traditional to label the first 36 functions using a single mode [|(=15;  
    %   number P instead of separate numbers for the order N and azimuthal #E_<}o  
    %   frequency M. bb-u'"5^]  
    % s$^2Qp  
    %   Example: Lw<.QMN%f  
    % CKC5S^Mx  
    %       % Display the first 16 Zernike functions OLqynY  
    %       x = -1:0.01:1; yI%q3lB}^  
    %       [X,Y] = meshgrid(x,x); XS.*CB_m_  
    %       [theta,r] = cart2pol(X,Y); KD- -w(4  
    %       idx = r<=1; zqp>Xw  
    %       p = 0:15; y-"QY[  
    %       z = nan(size(X)); ,MG`} *N}  
    %       y = zernfun2(p,r(idx),theta(idx)); r5N H*\Q  
    %       figure('Units','normalized') t8*NldC  
    %       for k = 1:length(p) x1}Ono3"T  
    %           z(idx) = y(:,k); v'r)d-T   
    %           subplot(4,4,k) 6wZ)GLW[  
    %           pcolor(x,x,z), shading interp D?4bp'0 3  
    %           set(gca,'XTick',[],'YTick',[]) `^h:} V  
    %           axis square Hk=HO|&<XB  
    %           title(['Z_{' num2str(p(k)) '}']) Jw{ duM;]  
    %       end wGx H  
    % j@{dsS: 6  
    %   See also ZERNPOL, ZERNFUN. Wmx3@]<  
    [c v!YE  
    %   Paul Fricker 11/13/2006 NnaO!QW%  
    wNmC1HOh  
    d;{k,rP6  
    % Check and prepare the inputs: Bi>]s%zp  
    % ----------------------------- amWKykVS5  
    if min(size(p))~=1 FwD q@Oj  
        error('zernfun2:Pvector','Input P must be vector.') uJ0Wb$%  
    end g2A#BMe'.$  
    Rgl cd  
    if any(p)>35 1X9J[5|ll  
        error('zernfun2:P36', ... UKPr[  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... t=fP^bJ  
               '(P = 0 to 35).']) @|e we. r  
    end 3jHg9M23[^  
    '~1Zr uO  
    % Get the order and frequency corresonding to the function number: *eI{g  
    % ---------------------------------------------------------------- M4% 3a j  
    p = p(:); lr@w1*  
    n = ceil((-3+sqrt(9+8*p))/2); `g0^ W/ j  
    m = 2*p - n.*(n+2); "F4 3q8P  
    A8Km8"  
    % Pass the inputs to the function ZERNFUN: g1(5QWb  
    % ---------------------------------------- Hx!eCTO:*  
    switch nargin 5hTScnL%  
        case 3 kfZ(:3W$  
            z = zernfun(n,m,r,theta); <2~DI0pp(  
        case 4 *vq75k$7  
            z = zernfun(n,m,r,theta,nflag); ;<"V}, C  
        otherwise 1qBE|PwBp  
            error('zernfun2:nargin','Incorrect number of inputs.') q+cD  
    end G\^<MR|  
    Mc$rsqDz  
    % 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) emB<{kOkw  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Ge7B%p8  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of tmoaa!yRnT  
    %   order N and frequency M, evaluated at R.  N is a vector of 8=zREt<Se  
    %   positive integers (including 0), and M is a vector with the G;EJ\J6@Yw  
    %   same number of elements as N.  Each element k of M must be a uX]]wj-R3  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ]'w5s dP  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is %b2Hm9r+  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix zQ<;3+*  
    %   with one column for every (N,M) pair, and one row for every k 8%@PC$  
    %   element in R. Sw5:T  
    % c27(en(  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .rnT'""i<5  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is gsl_aW!  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to .w'b%M  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 t1yOAbI  
    %   for all [n,m]. .GL@`7"  
    % &\b(  
    %   The radial Zernike polynomials are the radial portion of the O'{kNr{u  
    %   Zernike functions, which are an orthogonal basis on the unit `AvK=]  
    %   circle.  The series representation of the radial Zernike A|YgA66M  
    %   polynomials is 'cQ,;y  
    % $)BPtGMGo  
    %          (n-m)/2 NJVkn~<  
    %            __ {9.UeVz  
    %    m      \       s                                          n-2s FK94CI  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r u3E =r  
    %    n      s=0 `%"x'B`mM  
    % ,v#n\LD`  
    %   The following table shows the first 12 polynomials. Ei\>gXTH1-  
    % g j]8/~lr  
    %       n    m    Zernike polynomial    Normalization AO|1m$xf  
    %       --------------------------------------------- -KH"2q  
    %       0    0    1                        sqrt(2) jZ:/d!$S  
    %       1    1    r                           2 ! Vlx  
    %       2    0    2*r^2 - 1                sqrt(6) N:'!0|6?x-  
    %       2    2    r^2                      sqrt(6) 5 6.JB BZZ  
    %       3    1    3*r^3 - 2*r              sqrt(8) B3u/ y  
    %       3    3    r^3                      sqrt(8) dNF_ T?E\  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X(rXRP#  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 9=}[~V n  
    %       4    4    r^4                      sqrt(10) z8]@Gh+ (  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,S(s  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) gA}<Y  
    %       5    5    r^5                      sqrt(12) -.ZP<,?@F  
    %       --------------------------------------------- s S#/JLDx]  
    % ZkQ6~cM  
    %   Example: MI^$df  
    % b-#lKW so  
    %       % Display three example Zernike radial polynomials 4cM0f,nc+  
    %       r = 0:0.01:1; HW,v"  
    %       n = [3 2 5]; BHYguS^qz  
    %       m = [1 2 1]; -!O8V  
    %       z = zernpol(n,m,r); +zq"dj_  
    %       figure 0Q?%B6g$m[  
    %       plot(r,z) aR('u:@jHi  
    %       grid on (_CvN=A  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 3 H5  
    % &=$f\O1Ty  
    %   See also ZERNFUN, ZERNFUN2. b6sf1E  
    e84%Y8,0  
    % A note on the algorithm. dv3u<XM~  
    % ------------------------ 6 w{_+=T  
    % The radial Zernike polynomials are computed using the series jw {B8<@s  
    % representation shown in the Help section above. For many special Az8ZA~Op=  
    % functions, direct evaluation using the series representation can DI2e%`$  
    % produce poor numerical results (floating point errors), because I"x|U[*B  
    % the summation often involves computing small differences between &GJVFr~z  
    % large successive terms in the series. (In such cases, the functions JMo r[*  
    % are often evaluated using alternative methods such as recurrence c$L1aZo  
    % relations: see the Legendre functions, for example). For the Zernike GEh(pJ  
    % polynomials, however, this problem does not arise, because the Z f<T`'_d  
    % polynomials are evaluated over the finite domain r = (0,1), and $x]/|u/9  
    % because the coefficients for a given polynomial are generally all "J2q|@.  
    % of similar magnitude. ]?wz.  
    % CI$z+ zN  
    % ZERNPOL has been written using a vectorized implementation: multiple yt="kZ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] knph549  
    % values can be passed as inputs) for a vector of points R.  To achieve ~u2f`67{  
    % this vectorization most efficiently, the algorithm in ZERNPOL alHA&YC{K  
    % involves pre-determining all the powers p of R that are required to -T{2R:\{  
    % compute the outputs, and then compiling the {R^p} into a single j>:N0:  
    % matrix.  This avoids any redundant computation of the R^p, and 5;p|iT  
    % minimizes the sizes of certain intermediate variables. |3!)  
    % lqJ92vi6Q  
    %   Paul Fricker 11/13/2006 Fb8d= Zc  
    ~n%Lo3RiP  
    X#JUorGp  
    % Check and prepare the inputs: 4 l-Urn Z  
    % ----------------------------- j3/6hE>  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Og1vD5a  
        error('zernpol:NMvectors','N and M must be vectors.') NFx%e  
    end hCr,6ncC  
    \gPMYMd  
    if length(n)~=length(m) U.P1KRY|=  
        error('zernpol:NMlength','N and M must be the same length.') 0:u:#))1  
    end V,d\Wkk/  
    {j]cL !Od  
    n = n(:); JW^ ${4  
    m = m(:); JJ_ Z{  
    length_n = length(n); w?|qKO  
    6Z J-oT!.  
    if any(mod(n-m,2)) M."/"hV`-  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ,@?9H ~\  
    end un-%p#  
    )lS04|s  
    if any(m<0) e"eIQI|N  
        error('zernpol:Mpositive','All M must be positive.') 2z;3NUL$n  
    end 7]T(=gg /  
    ux(~+<k  
    if any(m>n) M kJBKS  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') GF% /q:9  
    end ~//E'V-  
    4}/gV)  
    if any( r>1 | r<0 ) ppvlU H5;  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ly[d V.<P  
    end Aixe?A_x  
    NFEr ,n  
    if ~any(size(r)==1) jmaw-Rx  
        error('zernpol:Rvector','R must be a vector.') vCJa%}  
    end @!! u>1  
    b5^>QzgD  
    r = r(:); Er~KX3vF  
    length_r = length(r); H8 ? Y{H  
    uZrp ^  
    if nargin==4 } f&=}  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); kG!hqj  
        if ~isnorm d!R+-Fp  
            error('zernpol:normalization','Unrecognized normalization flag.') sV{\IgH/x  
        end +<F3}]]  
    else i^.eX VV/  
        isnorm = false; a4~B  
    end y _"V=:  
    M NwY   
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (E00T`@t0i  
    % Compute the Zernike Polynomials t7x<=rW7u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W5`pQdk  
    k@|px#kq  
    % Determine the required powers of r: $RYGAh  
    % ----------------------------------- b:Zh|-  
    rpowers = []; ]3I a>i  
    for j = 1:length(n) qQ3Q4R\  
        rpowers = [rpowers m(j):2:n(j)]; \l /}` w  
    end FauASu,A  
    rpowers = unique(rpowers); Fd<Ouyxqe  
    8o%Vn'^t  
    % Pre-compute the values of r raised to the required powers, rY^uOrR>j*  
    % and compile them in a matrix: Z@Q*An  
    % ----------------------------- g&2g>]  
    if rpowers(1)==0 T&pCLvkz  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }O@>:?U  
        rpowern = cat(2,rpowern{:}); ANw1P{9*  
        rpowern = [ones(length_r,1) rpowern]; ^"?a)KC  
    else ~s HdOMw  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0:[A4S`X  
        rpowern = cat(2,rpowern{:}); _^GBfM.  
    end /Ls|'2J<$  
    }'x)e  
    % Compute the values of the polynomials: $aJay]F  
    % -------------------------------------- %+j/nA1%S  
    z = zeros(length_r,length_n); Fh)xm* u(  
    for j = 1:length_n wQy~5+LE  
        s = 0:(n(j)-m(j))/2; `Ze$Bd\  
        pows = n(j):-2:m(j); ig.Z,R3@r  
        for k = length(s):-1:1 :3Q:pKg  
            p = (1-2*mod(s(k),2))* ... vkGF_aenk  
                       prod(2:(n(j)-s(k)))/          ... Ep./->fOA  
                       prod(2:s(k))/                 ... \os"w "  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... nN<,rN{ :  
                       prod(2:((n(j)+m(j))/2-s(k))); b; C}=gg  
            idx = (pows(k)==rpowers); ?B ,<gen  
            z(:,j) = z(:,j) + p*rpowern(:,idx); %4!^AA%  
        end :~8@fEKb{  
         us|Hb  
        if isnorm sd%)g<t  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); d9TTAaf  
        end (jU_lsG  
    end Ss 5@n  
    '1b8>L  
    % 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)  UGM:'xa<T  
    d!<>Fh^6,  
    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。 & ;5f/  
    KQ9w>!N[  
    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)