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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 Q~Ay8L+  
    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! /@ y;iJk;  
     
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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 ( #* "c  
    function z = zernfun(n,m,r,theta,nflag) cFK @3a  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. GcT;e5D  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H$!+A  
    %   and angular frequency M, evaluated at positions (R,THETA) on the GF8 -_X  
    %   unit circle.  N is a vector of positive integers (including 0), and ;B~P>n}}_]  
    %   M is a vector with the same number of elements as N.  Each element (&jW}1D  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) zJ+3g!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s=D f `  
    %   and THETA is a vector of angles.  R and THETA must have the same u:@U $:sZ  
    %   length.  The output Z is a matrix with one column for every (N,M) i31<].|kA*  
    %   pair, and one row for every (R,THETA) pair. e+.\pe\  
    % DECB*9O ^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike q/NY72tj0  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), q*, Q5  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral |~WYEh  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, t6+YXjXK  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized !^e =P%S  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ytao"R/  
    % t V03+&jF  
    %   The Zernike functions are an orthogonal basis on the unit circle. SRZL\m}  
    %   They are used in disciplines such as astronomy, optics, and V|'1tB=;*1  
    %   optometry to describe functions on a circular domain. rAb&I"\ZY  
    % XM/vDdR  
    %   The following table lists the first 15 Zernike functions. "X04mQn15  
    % WNs}sNSf  
    %       n    m    Zernike function           Normalization i^)WPP>4Aw  
    %       -------------------------------------------------- KB!5u9  
    %       0    0    1                                 1 YuQ~AE'i  
    %       1    1    r * cos(theta)                    2 6.5wZN9<|  
    %       1   -1    r * sin(theta)                    2 Iy';x  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ?P/AC$:|I  
    %       2    0    (2*r^2 - 1)                    sqrt(3) +H_MV=A^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `S3>3  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) im]g(#GnKh  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) JN4fPGbV  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~=En +J}*  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 9M a0^_  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) O/Rhf[7v*  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ujr(K=E  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) tnz+bX26  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =WG=C1Z  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) c>HK9z{  
    %       -------------------------------------------------- fY,|o3#  
    % x[(?#  
    %   Example 1: geM6G$V&  
    %  fvEAIs  
    %       % Display the Zernike function Z(n=5,m=1) ;apzAF  
    %       x = -1:0.01:1; 8z2Rry w  
    %       [X,Y] = meshgrid(x,x); ?+0GfIV  
    %       [theta,r] = cart2pol(X,Y); e5?PkFV^a1  
    %       idx = r<=1; n6MM5h/#r  
    %       z = nan(size(X)); C [uOReo  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); g&Vcg`  
    %       figure uH@FU60  
    %       pcolor(x,x,z), shading interp y "w|g~x]c  
    %       axis square, colorbar +G F#?X0^  
    %       title('Zernike function Z_5^1(r,\theta)') Sv'y e  
    % I.'b'-^  
    %   Example 2: -%CoWcGP  
    % ]Mi.f3QlO6  
    %       % Display the first 10 Zernike functions yxt[= C  
    %       x = -1:0.01:1; @U{<a#  
    %       [X,Y] = meshgrid(x,x); EW<kI+0D  
    %       [theta,r] = cart2pol(X,Y); 2xwlKmI N  
    %       idx = r<=1; F {+`uG  
    %       z = nan(size(X)); FLZWZ;  
    %       n = [0  1  1  2  2  2  3  3  3  3]; $((6=39s  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Oakb'  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; O#a6+W"U  
    %       y = zernfun(n,m,r(idx),theta(idx)); D${={x  
    %       figure('Units','normalized') o|BP$P8V  
    %       for k = 1:10 3+Qxg+<  
    %           z(idx) = y(:,k); -r7]S  
    %           subplot(4,7,Nplot(k)) d XHB#  
    %           pcolor(x,x,z), shading interp 9BEFr/.  
    %           set(gca,'XTick',[],'YTick',[]) 5kypMHJm  
    %           axis square FQz?3w&ia  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +pm[f["C.  
    %       end 8.J( r(;>  
    % cO2& VC  
    %   See also ZERNPOL, ZERNFUN2. AK!hK>u`  
    oR1^/e  
    %   Paul Fricker 11/13/2006 Y-mK+1 2  
    &MZ$j46  
    lv&mp0V+  
    % Check and prepare the inputs: O,2~"~kF  
    % ----------------------------- G!N{NCq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) B/JO~;{  
        error('zernfun:NMvectors','N and M must be vectors.') {66sB{P  
    end tR0pH8?e"  
    H5CR'Rp  
    if length(n)~=length(m) mW2,1}Jv  
        error('zernfun:NMlength','N and M must be the same length.') '_\;jFAM  
    end "\W-f  
    Uxfl_@lJ  
    n = n(:); 7uorQfR?  
    m = m(:); kd`0E-QU  
    if any(mod(n-m,2)) :<Y}l-x  
        error('zernfun:NMmultiplesof2', ... j$UV/tp5T  
              'All N and M must differ by multiples of 2 (including 0).') Q5{Pv}Jx  
    end aI(7nJ=R  
    %3q0(Xl  
    if any(m>n) X,aYK;q%z  
        error('zernfun:MlessthanN', ... 4/kv3rv  
              'Each M must be less than or equal to its corresponding N.') ?bZovRx  
    end p(;U@3G  
    {rfF'@[  
    if any( r>1 | r<0 ) 2kAx>R  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') YJg,B\z}  
    end GZS1zTwBL  
    H1GRMDNXOA  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Pg9hW  
        error('zernfun:RTHvector','R and THETA must be vectors.') Oa;X +  
    end NjPDX>R\K  
    a,F&`Wg  
    r = r(:); ;*ix~taL%  
    theta = theta(:); ^, l_{  
    length_r = length(r); |Fm6#1A@  
    if length_r~=length(theta) \^(0B8|w  
        error('zernfun:RTHlength', ... NNhL*C[_7  
              'The number of R- and THETA-values must be equal.') >3 yk#U|7}  
    end 09A X-JP  
    ETp%s{8  
    % Check normalization: iwz  
    % -------------------- /525w^'pd  
    if nargin==5 && ischar(nflag) gBT2)2]  
        isnorm = strcmpi(nflag,'norm'); !USd9  
        if ~isnorm du$|lxC  
            error('zernfun:normalization','Unrecognized normalization flag.') g  %K>  
        end Om{l>24i.\  
    else {3})=>u:S  
        isnorm = false; L9pvG(R%  
    end l4n)#?Q?  
    qq)0yyL r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m)V/L]4  
    % Compute the Zernike Polynomials AL$&|=C-$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N#lDW~e'  
    XwV'Ha  
    % Determine the required powers of r: `V)Z)uN{0  
    % ----------------------------------- 0 a]/%y3V  
    m_abs = abs(m); z <mK>$  
    rpowers = []; yc|VJ2R*  
    for j = 1:length(n) %WqNiF0-  
        rpowers = [rpowers m_abs(j):2:n(j)]; vR0 ];{  
    end 8Ll[ fJZA  
    rpowers = unique(rpowers); pg]BsJN  
    ^pM+A6 XY  
    % Pre-compute the values of r raised to the required powers, 98 8]}{w  
    % and compile them in a matrix: Oj<S.fi  
    % ----------------------------- 2[0JO.K 4  
    if rpowers(1)==0 P oEqurH0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); I`z@2Z+pJ  
        rpowern = cat(2,rpowern{:}); u77E! z4Uz  
        rpowern = [ones(length_r,1) rpowern]; 7~#:>OjW  
    else ?"?6,;F(4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s@MYc@k  
        rpowern = cat(2,rpowern{:}); VqL.iZ-  
    end {3N'D2N  
    %OgS^_tu  
    % Compute the values of the polynomials: @ HZKc\1  
    % -------------------------------------- E}%hz*Q)(  
    y = zeros(length_r,length(n)); uEc<}pV  
    for j = 1:length(n) $gBd <N9|c  
        s = 0:(n(j)-m_abs(j))/2; Y(.OF Q  
        pows = n(j):-2:m_abs(j); .z13 =yv  
        for k = length(s):-1:1 :eo  
            p = (1-2*mod(s(k),2))* ... ~=R SKyzt  
                       prod(2:(n(j)-s(k)))/              ... eNiaM6(J  
                       prod(2:s(k))/                     ... &rkEK4  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... (C]o,7cYS  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); i#%aTRKHd6  
            idx = (pows(k)==rpowers); A(]H{>PMy  
            y(:,j) = y(:,j) + p*rpowern(:,idx); r\nx=  
        end mS k5u7  
         5k|9gICyd*  
        if isnorm /b|0PMX  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <0S=,!  
        end iAa;6mH  
    end eAPXWWAZJ1  
    % END: Compute the Zernike Polynomials )Ud-}* g  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W2uOR{ '?  
    HHqwq.zIy  
    % Compute the Zernike functions: !|c|o*t{  
    % ------------------------------ Ts~L:3oaQ  
    idx_pos = m>0; l }XU 59  
    idx_neg = m<0; ja=F7Usb  
    xq"Jy=4Q*  
    z = y; xC C:BO`pw  
    if any(idx_pos) |d6T/Uxo  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |p$spQ  
    end 43V}# DA@  
    if any(idx_neg) mDZ*E!B  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,^icPQSwc  
    end DNP13wp@  
    ? `J[[",  
    % 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) P'Q+GRpSw  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /rSH"$  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated SM@QUAXO  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive tnLAJ+ -M  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, z5TuGY b<  
    %   and THETA is a vector of angles.  R and THETA must have the same {/pm<k=  
    %   length.  The output Z is a matrix with one column for every P-value, =N 5z@;!  
    %   and one row for every (R,THETA) pair. yv)ux:P&+  
    % 4V~?.  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike YtO|D  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) aN(|'uO@  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) /a6Xa&(B  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ES40?o*]x  
    %   for all p.  rb{P :MX  
    % +>4;Zd!@d  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 O`vTnrY  
    %   Zernike functions (order N<=7).  In some disciplines it is *YlV-C<}W"  
    %   traditional to label the first 36 functions using a single mode 6S~sVUL9`  
    %   number P instead of separate numbers for the order N and azimuthal Uo2GK3nT  
    %   frequency M. ^i:B+ rl  
    % h>Hb `G<  
    %   Example: ~RWktv  
    % 'MY/*k7:  
    %       % Display the first 16 Zernike functions xp Og8u5  
    %       x = -1:0.01:1; i E CrI3s  
    %       [X,Y] = meshgrid(x,x); R"K#7{p9  
    %       [theta,r] = cart2pol(X,Y); +o9":dl  
    %       idx = r<=1; ]Zmj4vK J  
    %       p = 0:15; MQ"xOcD*F  
    %       z = nan(size(X)); NB<A>baL*  
    %       y = zernfun2(p,r(idx),theta(idx)); B,{K*-7)MX  
    %       figure('Units','normalized') -I=l8m6L  
    %       for k = 1:length(p) JY6 Q p  
    %           z(idx) = y(:,k); #UbF9})q  
    %           subplot(4,4,k) {P*m;a`}  
    %           pcolor(x,x,z), shading interp L5,NP5RC  
    %           set(gca,'XTick',[],'YTick',[]) nR`ov1RH  
    %           axis square QcpXn4/*  
    %           title(['Z_{' num2str(p(k)) '}']) QV\eMuNy  
    %       end aE2.L;Tk?  
    % NQ6sGL  
    %   See also ZERNPOL, ZERNFUN. NC38fiH_N  
    >;[*!<pfK5  
    %   Paul Fricker 11/13/2006 ,TFIG^Dvq  
    y:6; LZ9[  
    KGg3 !jY  
    % Check and prepare the inputs: Z4\=*ic@  
    % ----------------------------- QqU!Najf  
    if min(size(p))~=1 r-<F5<H+K@  
        error('zernfun2:Pvector','Input P must be vector.') LGtIm7  
    end h2D>;k  
    Ng_!zrx04  
    if any(p)>35 ye MB0Z*r  
        error('zernfun2:P36', ... 6H7],aMg$A  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 5;HH4?]p  
               '(P = 0 to 35).']) mWvl 38  
    end ynrT a..  
    K1T4cUo  
    % Get the order and frequency corresonding to the function number: 6AhM=C  
    % ---------------------------------------------------------------- <%" b9T`'  
    p = p(:); d m`E!R_  
    n = ceil((-3+sqrt(9+8*p))/2); lg&t8FHa;  
    m = 2*p - n.*(n+2); qo|WXwP2  
    ~Rr~1I&mR,  
    % Pass the inputs to the function ZERNFUN: 4H/fP]u  
    % ---------------------------------------- ,l)^Ft`5  
    switch nargin zOiu5  
        case 3 ?[ lV-  
            z = zernfun(n,m,r,theta); g?ULWeZg5  
        case 4 >m$ 1+30X  
            z = zernfun(n,m,r,theta,nflag); =z /dcC$r  
        otherwise &mx)~J^m  
            error('zernfun2:nargin','Incorrect number of inputs.') .*)2SNH  
    end 9_5ow  
    ;-qO'V:;  
    % 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) +O?KNZ  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. jIyB  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of OL^l 3F  
    %   order N and frequency M, evaluated at R.  N is a vector of $[a8$VY^Cm  
    %   positive integers (including 0), and M is a vector with the }|8_9Rx0*  
    %   same number of elements as N.  Each element k of M must be a ybKWOp:O  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k)  UWo]s.  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ][p>Y>:b-  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix :BV6y|J9O^  
    %   with one column for every (N,M) pair, and one row for every yvO{:B8%  
    %   element in R.   t!_<~  
    % 8p:e##%  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ) u`[6,d  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is @X;!92i  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to E;R n`oxk  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 7\s"o&G  
    %   for all [n,m].  KJaXg;,H  
    % 4p,EBn9(  
    %   The radial Zernike polynomials are the radial portion of the =E#%'/ A;c  
    %   Zernike functions, which are an orthogonal basis on the unit J`].:IOh  
    %   circle.  The series representation of the radial Zernike }%{LJ}\Px  
    %   polynomials is DrY:9[LP  
    % 2Tp1n8FV  
    %          (n-m)/2 ?Yth0O6?sb  
    %            __ Ay0U=#XP  
    %    m      \       s                                          n-2s r w2arx  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Ssou  
    %    n      s=0 '9 [vDG~  
    % jk[1{I/  
    %   The following table shows the first 12 polynomials. &&8IU;J  
    % zGkS^Z=(  
    %       n    m    Zernike polynomial    Normalization QLiu2U o  
    %       --------------------------------------------- 'R'*kxf  
    %       0    0    1                        sqrt(2) nz=G lO'[  
    %       1    1    r                           2 b)qoh^  
    %       2    0    2*r^2 - 1                sqrt(6) `-J%pEIza  
    %       2    2    r^2                      sqrt(6) i/`m`qdg  
    %       3    1    3*r^3 - 2*r              sqrt(8) qGB{7-ru  
    %       3    3    r^3                      sqrt(8) lJ}_G>GJ  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ?IqQ-C)6D  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) R\G0'?h >  
    %       4    4    r^4                      sqrt(10) sHt].gZ  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 2q=AEv/  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) zck#tht4 n  
    %       5    5    r^5                      sqrt(12) g4=pnK8  
    %       --------------------------------------------- aJbO((%$|u  
    % :*Z4yx  
    %   Example: J\:R|KaP<p  
    % kwo3`b  
    %       % Display three example Zernike radial polynomials - -HZX  
    %       r = 0:0.01:1; c4^ks&)'  
    %       n = [3 2 5]; F@'Jbd`   
    %       m = [1 2 1]; T?tgd J  
    %       z = zernpol(n,m,r); p'*>vk  
    %       figure >,$_| C  
    %       plot(r,z) NV72  
    %       grid on "$+Jnc!!  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') /v1Q4mq  
    % ff,pvk8N5  
    %   See also ZERNFUN, ZERNFUN2. ;o2$ Q  
    1{ ~#H<K  
    % A note on the algorithm. H8Bs<2  
    % ------------------------ +./H6!  
    % The radial Zernike polynomials are computed using the series )NXmn95  
    % representation shown in the Help section above. For many special %et } A93  
    % functions, direct evaluation using the series representation can a!7A_q8M  
    % produce poor numerical results (floating point errors), because ;g5m0l5  
    % the summation often involves computing small differences between c[wla<dO*  
    % large successive terms in the series. (In such cases, the functions (2J: #  
    % are often evaluated using alternative methods such as recurrence Xqg@ e:g  
    % relations: see the Legendre functions, for example). For the Zernike )wam8k5  
    % polynomials, however, this problem does not arise, because the PV'x+bN5  
    % polynomials are evaluated over the finite domain r = (0,1), and B}Z63|/N  
    % because the coefficients for a given polynomial are generally all q<[P6}.  
    % of similar magnitude. LrM=*R h,O  
    % ]@j*/IP  
    % ZERNPOL has been written using a vectorized implementation: multiple 4B =7:r  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ~:kZgUP_f  
    % values can be passed as inputs) for a vector of points R.  To achieve rb5~XnJk  
    % this vectorization most efficiently, the algorithm in ZERNPOL QdH\LL^8R4  
    % involves pre-determining all the powers p of R that are required to -3t7*  
    % compute the outputs, and then compiling the {R^p} into a single Xx."$l  
    % matrix.  This avoids any redundant computation of the R^p, and 0%&1\rm+j  
    % minimizes the sizes of certain intermediate variables. R]c+?4J  
    % 591>rh)  
    %   Paul Fricker 11/13/2006 DBW[{D E  
    :mh_G  
    C%$edEi  
    % Check and prepare the inputs: Q('r<v96  
    % ----------------------------- TyD4|| %  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) QUq_:t+Dv  
        error('zernpol:NMvectors','N and M must be vectors.') l2zFKCGF(  
    end q>_/u"  
    5{|7$VqPF  
    if length(n)~=length(m) >Ea8G,  
        error('zernpol:NMlength','N and M must be the same length.') j"ThEx0  
    end @| M|+k3  
    A-H&  
    n = n(:); mXRB7k  
    m = m(:); [-65PC4aN  
    length_n = length(n); W98i[Q9A7  
    <r .)hT"0  
    if any(mod(n-m,2)) tX9{hC^  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') {@H6HqD  
    end ,X[kt z  
    wa<MRt W=  
    if any(m<0) BWeA@v  
        error('zernpol:Mpositive','All M must be positive.') Tzt8h\Q^z  
    end fM]+SMZy  
    `YFtL  
    if any(m>n) 3EV;LH L  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') zvYq@Mhr  
    end 0LPig[  
    w j*,U~syB  
    if any( r>1 | r<0 ) )IP,;<  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ciFmaM.  
    end V/%>4GYnC  
    ^ZvWR%  
    if ~any(size(r)==1) ;kFDMuuO  
        error('zernpol:Rvector','R must be a vector.') :#LLo}LKp  
    end ' KWyx  
    *?5*m+  
    r = r(:); qW$<U3u}  
    length_r = length(r); }6p@lla,%]  
    F|d\k Q  
    if nargin==4 i2@VB6]?  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); #+:9T /*>0  
        if ~isnorm h%o%fH&F!  
            error('zernpol:normalization','Unrecognized normalization flag.') RHaI~jb  
        end .GsV>H  
    else <Y*+|T+&d  
        isnorm = false; (_niMQtF}  
    end 8|):`u  
    k52/w)Ro,$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qi ua  
    % Compute the Zernike Polynomials Y'c>:;JEe  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KK1 gNC4R  
    q 0$,*[PH  
    % Determine the required powers of r: G<At_YS  
    % ----------------------------------- Uddr~2%(  
    rpowers = []; 4iqoR$3Fc  
    for j = 1:length(n) j5K]CTz#  
        rpowers = [rpowers m(j):2:n(j)]; I!^;8Pg  
    end gwOa$f%O  
    rpowers = unique(rpowers); dU6ou'p f  
    ta35 K"  
    % Pre-compute the values of r raised to the required powers, H2&@shOOQJ  
    % and compile them in a matrix: OP~HdocB  
    % ----------------------------- I3=%h  
    if rpowers(1)==0 Ov};e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); SFB~ ->db  
        rpowern = cat(2,rpowern{:}); I~q#eO)  
        rpowern = [ones(length_r,1) rpowern]; aDq5C-MzG  
    else 1%EBd%`#  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w:%o?pKet1  
        rpowern = cat(2,rpowern{:}); A'j;\ `1  
    end $LKIT0  
    ~?D4[D|sB  
    % Compute the values of the polynomials: Te.Y#lCT$  
    % -------------------------------------- m`v2: S}  
    z = zeros(length_r,length_n); PpGL/,]X  
    for j = 1:length_n EqyeJq .  
        s = 0:(n(j)-m(j))/2; V `b2TS  
        pows = n(j):-2:m(j); Qt iDTr  
        for k = length(s):-1:1 {!.(7wV\  
            p = (1-2*mod(s(k),2))* ... 2>|dF~"  
                       prod(2:(n(j)-s(k)))/          ... 0@ yXi  
                       prod(2:s(k))/                 ... ?i)f^O  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... $56Z#'(D  
                       prod(2:((n(j)+m(j))/2-s(k))); Fgkajig  
            idx = (pows(k)==rpowers); vqnw#U4`  
            z(:,j) = z(:,j) + p*rpowern(:,idx); Ao&\EcIOT  
        end -u&6X,Oq\u  
         n1qQ+(xC  
        if isnorm D;oe2E{I  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); x4g3 rmp  
        end O?NeSx 1  
    end 3!3xCO  
    3 j!3E  
    % 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)  M7+nW ; e%  
    ._8KsuJG  
    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。 Vnx,5E&  
    (WK&^,zQn  
    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)