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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 Hs}3c R}  
    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! =UGyZV:z5  
     
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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 CXI%8eFXe$  
    function z = zernfun(n,m,r,theta,nflag) |\# ~  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. \LN!k-c  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _l{`lQ}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the &U.U<  
    %   unit circle.  N is a vector of positive integers (including 0), and ?RP&XrD  
    %   M is a vector with the same number of elements as N.  Each element -Lo3@:2i  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) !_yWe  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, |~+i=y  
    %   and THETA is a vector of angles.  R and THETA must have the same R [qfG! "  
    %   length.  The output Z is a matrix with one column for every (N,M) uK6'TJ  
    %   pair, and one row for every (R,THETA) pair. 43'!<[?x  
    % 3Fu5,H EJ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fTq/9=Rq4  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )z" .lw  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral V_x8 Q+~?  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SsY :gp_  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized h/i L/Q=  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. <n;9IU  
    % pO_$8=G+  
    %   The Zernike functions are an orthogonal basis on the unit circle. J,W<vrKOcN  
    %   They are used in disciplines such as astronomy, optics, and z^FJ  
    %   optometry to describe functions on a circular domain. )/p=ZH0[  
    % iaV%*  
    %   The following table lists the first 15 Zernike functions. Os rHA  
    % ^b;3Jj  
    %       n    m    Zernike function           Normalization u3G.xlHH[  
    %       -------------------------------------------------- +jPJv[W  
    %       0    0    1                                 1 X-_ $jKfM  
    %       1    1    r * cos(theta)                    2 _+aMP=H  
    %       1   -1    r * sin(theta)                    2 -$A >b8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) +I <^w)  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ]4X08Cm^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) @'> Ul!.]  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) u]766<Z  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Hz >_tA"^T  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) !q8"Q t  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <K:L.c!  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) !>8/Xz~-  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gj@>9  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1_B;r9x  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f;`7}7C  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) Fy#7 <Hp  
    %       -------------------------------------------------- k^{}p8;3  
    % uBUT84i  
    %   Example 1: @UK%l :L  
    % W[G5+*i  
    %       % Display the Zernike function Z(n=5,m=1) n w  
    %       x = -1:0.01:1; ]}Jb'(gMO4  
    %       [X,Y] = meshgrid(x,x); \gW6E^  
    %       [theta,r] = cart2pol(X,Y); O4g2s8k  
    %       idx = r<=1; :5#iVa#<  
    %       z = nan(size(X)); BGrV,h^  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); p6&6^v\  
    %       figure CxV$_J  
    %       pcolor(x,x,z), shading interp Maw$^Tz,  
    %       axis square, colorbar +UX~TT:  
    %       title('Zernike function Z_5^1(r,\theta)') +=Y$v2BZA3  
    % -%_vb6u  
    %   Example 2: 3n)\D<f]#  
    % #PGpB5vnaA  
    %       % Display the first 10 Zernike functions ?~9o2[  
    %       x = -1:0.01:1; AT -  
    %       [X,Y] = meshgrid(x,x); Fp(-&,L0fc  
    %       [theta,r] = cart2pol(X,Y); l|S_10x5  
    %       idx = r<=1; Ru~;awV?  
    %       z = nan(size(X)); vWZ?*0^  
    %       n = [0  1  1  2  2  2  3  3  3  3]; nhLw&V3y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; @M)"  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; QR\2 %}9b  
    %       y = zernfun(n,m,r(idx),theta(idx)); 8=,?B h".  
    %       figure('Units','normalized') ~(-df>  
    %       for k = 1:10 +ZJ1> n  
    %           z(idx) = y(:,k); [l*;+N+  
    %           subplot(4,7,Nplot(k)) iTVepYv4m  
    %           pcolor(x,x,z), shading interp y(yBRR  
    %           set(gca,'XTick',[],'YTick',[]) _X~xfmU  
    %           axis square c{{RP6o/j=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y?4N%c_;  
    %       end fU>4Ip1?y/  
    % -1%AM40j  
    %   See also ZERNPOL, ZERNFUN2. wqF_hs(O  
    P0l.sVqL  
    %   Paul Fricker 11/13/2006 G DwijZw  
    CP LsSv5  
    3Lm7{s?=Z-  
    % Check and prepare the inputs: |o#pd\  
    % ----------------------------- @0D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) a\xf\$Ym  
        error('zernfun:NMvectors','N and M must be vectors.') yaK4% k  
    end {S"!c.  
    t $u.  
    if length(n)~=length(m) `##^@N<P  
        error('zernfun:NMlength','N and M must be the same length.') I^?hVH  
    end )E}eK-Yu  
    ,h },jkY4  
    n = n(:); roNs~]6  
    m = m(:); rds0EZ4W  
    if any(mod(n-m,2)) 4Ep6vm X  
        error('zernfun:NMmultiplesof2', ... dG%{&W9  
              'All N and M must differ by multiples of 2 (including 0).') ?Vc/mO2X  
    end t|v_[Za}Z  
    {KqERS& g  
    if any(m>n) Jzj>=jWX@  
        error('zernfun:MlessthanN', ... +|.6xC7U  
              'Each M must be less than or equal to its corresponding N.') g]PC6xr38  
    end 6G;t:[H G  
    :Y[?@/m4  
    if any( r>1 | r<0 ) <*+Y]=  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') VcORRUp  
    end %!V=noo  
    ?dQ#%06mn  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) PHg(O:3WG  
        error('zernfun:RTHvector','R and THETA must be vectors.') g acE?bW'  
    end 7DB!s@"  
    X~rHNRIU  
    r = r(:); PaBqv]  
    theta = theta(:); F=V_ACU  
    length_r = length(r);  m8z414o  
    if length_r~=length(theta) FfibR\dhY  
        error('zernfun:RTHlength', ... T#=&oy7  
              'The number of R- and THETA-values must be equal.') `YK%I8  
    end V w5@)l*f  
    .!Q?TSQ+{!  
    % Check normalization: `E5vO1Pl  
    % -------------------- FSyeDC^@  
    if nargin==5 && ischar(nflag) e%v0EJ},  
        isnorm = strcmpi(nflag,'norm'); lKLb\F%  
        if ~isnorm V6tUijz  
            error('zernfun:normalization','Unrecognized normalization flag.') #yR@.&P  
        end )Zit6I  
    else 8@BN6  
        isnorm = false; S3Sn_zqG  
    end  F!&_  
    9 p`|~^X  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d<>jhp5el  
    % Compute the Zernike Polynomials $6yr:2Xvt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hG>3y\!#  
    RiCzH  
    % Determine the required powers of r: XFcIBWS  
    % ----------------------------------- E@S5|CM  
    m_abs = abs(m); :~B'6b  
    rpowers = []; b`X"yg+  
    for j = 1:length(n) \I~9%QJ>  
        rpowers = [rpowers m_abs(j):2:n(j)]; mx")cGGQ  
    end KI8Q =*  
    rpowers = unique(rpowers); m|cT)-  
    .="[In '  
    % Pre-compute the values of r raised to the required powers, D3kx&AR  
    % and compile them in a matrix: 6)Dp2  
    % ----------------------------- q)KLf\  
    if rpowers(1)==0 I DtGtkF  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); x\!Uk!fM  
        rpowern = cat(2,rpowern{:}); .5YIf~!59  
        rpowern = [ones(length_r,1) rpowern]; t 4tXLI;'  
    else G ~|Z (}H  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #e(P~'A0  
        rpowern = cat(2,rpowern{:}); X~5kgq0"  
    end h?2:'Vu]  
    T0Zv.  
    % Compute the values of the polynomials: A]CO Ysc  
    % -------------------------------------- ]Qb85;0)  
    y = zeros(length_r,length(n)); -~ 5|_G2Y"  
    for j = 1:length(n) F!qt#Sw!\  
        s = 0:(n(j)-m_abs(j))/2; aBx8wl*Vm  
        pows = n(j):-2:m_abs(j); YF(TG]?6  
        for k = length(s):-1:1 ]aVFWzey  
            p = (1-2*mod(s(k),2))* ... )a'c_ 2[  
                       prod(2:(n(j)-s(k)))/              ... UkV{4*E  
                       prod(2:s(k))/                     ... D_4UM#Tw  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~LuR)T=%es  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); pCm|t!,  
            idx = (pows(k)==rpowers); =lqBRut  
            y(:,j) = y(:,j) + p*rpowern(:,idx); =/]d\JSp  
        end 3~Vo]wv  
         SUQk0 (M  
        if isnorm *1fZcw'C.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qX?k]m   
        end v3{[rK}  
    end Z )f\^  
    % END: Compute the Zernike Polynomials fb||q-E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !O~5<tA[#1  
    V=|X=:fuih  
    % Compute the Zernike functions: @(_M\>!%M  
    % ------------------------------ :6Oh?y@  
    idx_pos = m>0; =2yg:D  
    idx_neg = m<0; drZ1D s  
    ".R5K ?  
    z = y; d 9n{jv|  
    if any(idx_pos) EO[UezuU  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p|b&hgA  
    end M&5;Qeoiv  
    if any(idx_neg) ZT;:Hxv0N  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7iJ=~po:o  
    end NFQR  
    \x_fP;ma=_  
    % 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) 68^5X"OGF  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. J0hY~B~X  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated m8|&z{  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive #iot.alNA  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, *&vySyt  
    %   and THETA is a vector of angles.  R and THETA must have the same +'NiuN  
    %   length.  The output Z is a matrix with one column for every P-value, G'>z~I]6S  
    %   and one row for every (R,THETA) pair. %7$oig\wE  
    % 'e(`2  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Tmo+I4qoL  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) mne=9/sE"  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 2_4m}T3   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 cQaEh1n  
    %   for all p. DfCo=  
    % sH>Z{xjr  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 H...!c1M@  
    %   Zernike functions (order N<=7).  In some disciplines it is wb>>bV+U  
    %   traditional to label the first 36 functions using a single mode 2_x~y|<9  
    %   number P instead of separate numbers for the order N and azimuthal hkO)q|1  
    %   frequency M. U-$ B"w&  
    % % DQ.f*%  
    %   Example: cXO_g!&2A  
    % Qhd~4  
    %       % Display the first 16 Zernike functions Z81{v<c;  
    %       x = -1:0.01:1; EuAJ.n  
    %       [X,Y] = meshgrid(x,x); C<.t'|  
    %       [theta,r] = cart2pol(X,Y); .kT5 4U;{  
    %       idx = r<=1; 3f{%IU(z  
    %       p = 0:15;  4^L+LY  
    %       z = nan(size(X)); \@kY2,I V  
    %       y = zernfun2(p,r(idx),theta(idx)); su`] l"[,]  
    %       figure('Units','normalized') $?OuY*ZeY9  
    %       for k = 1:length(p) HHbkR2H1  
    %           z(idx) = y(:,k); R,hX *yVq  
    %           subplot(4,4,k) ?D#]g[6  
    %           pcolor(x,x,z), shading interp 9's/~T  
    %           set(gca,'XTick',[],'YTick',[]) Sdc;jK 9d!  
    %           axis square 4=H/-v'&  
    %           title(['Z_{' num2str(p(k)) '}']) W`C&$v#  
    %       end >mT< AQ  
    % VY'#>k} }  
    %   See also ZERNPOL, ZERNFUN. EiY i<Z_S  
    ]FLuiC  
    %   Paul Fricker 11/13/2006 2'"$Y'  
    t/,k{5lX  
    )T6+}   
    % Check and prepare the inputs: JF>mybB  
    % ----------------------------- =;{vfjj  
    if min(size(p))~=1 ,=tVa])  
        error('zernfun2:Pvector','Input P must be vector.') '|e5cW6z  
    end &Jj^)GBU  
    x C'>W"pY  
    if any(p)>35 Ed|7E_v  
        error('zernfun2:P36', ... /q %TjQ}F  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... " [=Ee[/  
               '(P = 0 to 35).']) cI3uH1;#  
    end yZaQ{]"  
    GYiUne $  
    % Get the order and frequency corresonding to the function number: ,\5]n&T;r  
    % ---------------------------------------------------------------- l_LfVON  
    p = p(:); J=/|iW  
    n = ceil((-3+sqrt(9+8*p))/2); RIUJ20PfYQ  
    m = 2*p - n.*(n+2); (jkjj7a  
    P%smX`v  
    % Pass the inputs to the function ZERNFUN: wX7B&w8wV  
    % ---------------------------------------- Q]7Q4U  
    switch nargin n<7#?X7  
        case 3 =o=)EU{~  
            z = zernfun(n,m,r,theta); o([+Pp  
        case 4 8oY0?|_Bx  
            z = zernfun(n,m,r,theta,nflag); Gq;0j:?CC  
        otherwise <;x+ ?j  
            error('zernfun2:nargin','Incorrect number of inputs.') hNFMuv  
    end x>5#@SX J  
    MQ"<r,o?:  
    % 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) &cDnZ3Q;  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. *8"5mC ;"  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of <H)h+?&~d  
    %   order N and frequency M, evaluated at R.  N is a vector of $K\\ 8$Z  
    %   positive integers (including 0), and M is a vector with the Qd]-i3^0  
    %   same number of elements as N.  Each element k of M must be a ` M"Zq  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) =g/K>B  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is !e.@Xk.P6  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix W3 2mAz;  
    %   with one column for every (N,M) pair, and one row for every ^`=Z=C$fj  
    %   element in R. o?^j1\^  
    % )y.J2_lI8  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- rtf>\j+  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is @<},-u  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 1NN99^ q  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ^5u}   
    %   for all [n,m]. _V6;`{$WK  
    % GC$Hp!H  
    %   The radial Zernike polynomials are the radial portion of the @?*26}qp  
    %   Zernike functions, which are an orthogonal basis on the unit (sO;etW  
    %   circle.  The series representation of the radial Zernike Y.F:1<FAtf  
    %   polynomials is :(A]Bm3  
    % lGjmw"/C  
    %          (n-m)/2 athU  
    %            __ bbiDY  
    %    m      \       s                                          n-2s GIo&zPx  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r vYmRW-1Zxq  
    %    n      s=0 "z< =S  
    % Gt\F),@  
    %   The following table shows the first 12 polynomials. 04:^<n+{  
    % .0.Ha}{6b  
    %       n    m    Zernike polynomial    Normalization z9&$Xao  
    %       --------------------------------------------- \|DcWH1  
    %       0    0    1                        sqrt(2) gjL>FOe8u  
    %       1    1    r                           2 N$>g)Ml?  
    %       2    0    2*r^2 - 1                sqrt(6) o*I=6`j  
    %       2    2    r^2                      sqrt(6) ./[%%"  
    %       3    1    3*r^3 - 2*r              sqrt(8) #<4h Y7/  
    %       3    3    r^3                      sqrt(8) h3>/..l  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) l5D8DvJCj  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) t%q@W,2J  
    %       4    4    r^4                      sqrt(10)  U`IDZ{g  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) *8,]fBUq  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 8WZM}3x$f{  
    %       5    5    r^5                      sqrt(12) ,V.X-`Y  
    %       --------------------------------------------- !4]w b!F  
    % /V2 ^/`&;a  
    %   Example: *hAq]VC})  
    % #r#UO  
    %       % Display three example Zernike radial polynomials a0CmCv2#  
    %       r = 0:0.01:1; 5Ee%!Pk  
    %       n = [3 2 5]; vM1f-I-  
    %       m = [1 2 1]; g15e|y)th  
    %       z = zernpol(n,m,r); P8).Qn  
    %       figure Ngi$y>{Sq  
    %       plot(r,z) DE^{8YX,  
    %       grid on 3iR;(l}  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 6i(nyA 2!  
    % ,g 6w2y7 ]  
    %   See also ZERNFUN, ZERNFUN2. 4}!riWR   
    AnP7KSN[\  
    % A note on the algorithm. qOV#$dkY  
    % ------------------------ x =5k74  
    % The radial Zernike polynomials are computed using the series k)1K6ug  
    % representation shown in the Help section above. For many special F%+/j5~^  
    % functions, direct evaluation using the series representation can $cSrT)u :  
    % produce poor numerical results (floating point errors), because \=XAl >}\  
    % the summation often involves computing small differences between }tua0{N:z  
    % large successive terms in the series. (In such cases, the functions 1ePZs$  
    % are often evaluated using alternative methods such as recurrence ]xCJ3.9  
    % relations: see the Legendre functions, for example). For the Zernike M`9qo8zCi  
    % polynomials, however, this problem does not arise, because the k/u6Cw0/  
    % polynomials are evaluated over the finite domain r = (0,1), and KArR.o }  
    % because the coefficients for a given polynomial are generally all 4T{+R{_Y1  
    % of similar magnitude. tUDOL-Tv  
    % |K,9EM3  
    % ZERNPOL has been written using a vectorized implementation: multiple ^j0Mu.+_  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] B<I%:SkF@  
    % values can be passed as inputs) for a vector of points R.  To achieve u-qg9qXJb  
    % this vectorization most efficiently, the algorithm in ZERNPOL *rXESw]BR  
    % involves pre-determining all the powers p of R that are required to ZXY5Xvt:v  
    % compute the outputs, and then compiling the {R^p} into a single C;1A$]bk  
    % matrix.  This avoids any redundant computation of the R^p, and z+@aQ@75  
    % minimizes the sizes of certain intermediate variables. | Z0?  
    % SWN i@  
    %   Paul Fricker 11/13/2006 F@& R"-  
    \|F4@  
    E*:!G  
    % Check and prepare the inputs: 26G2. /**<  
    % ----------------------------- %FDi7Rx  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C5CUMYU  
        error('zernpol:NMvectors','N and M must be vectors.') E5~HH($b  
    end JN .\{ Y  
    'nz;|6uC  
    if length(n)~=length(m) 0~iC#lHO  
        error('zernpol:NMlength','N and M must be the same length.') }/nbv;)  
    end &na#ES $X,  
    %g5TU 6WP  
    n = n(:); j&6,%s-M`a  
    m = m(:); D^baXp8  
    length_n = length(n); Kyt.[" p  
    5bYU(]  
    if any(mod(n-m,2)) $3[IlQ?  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') y< W?hE[  
    end CC0@RU  
    `MA ee8u'  
    if any(m<0) w},' 1  
        error('zernpol:Mpositive','All M must be positive.') g{.>nE^Sc5  
    end ! @{rk p  
    6}='/d-[  
    if any(m>n) [^EU'lewnW  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') )@09Y_9r  
    end -wH#B<'  
    iS&~oj_-%  
    if any( r>1 | r<0 ) ,24NMv7  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') kTb.I;S  
    end #s$b\"4  
    bZHuEh2w  
    if ~any(size(r)==1) }__g\?Yf  
        error('zernpol:Rvector','R must be a vector.') 7!+kyA\}r^  
    end M$DJ$G|Z  
    VUC  
    r = r(:); h!5^d!2,  
    length_r = length(r); :y==O4  
    Z4A a  
    if nargin==4 #4%4iR5%  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ]W7(}~m  
        if ~isnorm 3UU]w`At  
            error('zernpol:normalization','Unrecognized normalization flag.') I+Qv$#S/  
        end IMIZ#/  
    else t j&+HC  
        isnorm = false; !sQ$a#Ea  
    end ^h{A AS>  
    7F;"=DarOE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r=Z#"68$  
    % Compute the Zernike Polynomials gP"p7\ (  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /%EKq+ZP  
    7Fw`s@/%  
    % Determine the required powers of r: %y)5:]  
    % ----------------------------------- AFcsbw  
    rpowers = []; iDt^4=`  
    for j = 1:length(n) @qmONQ eb  
        rpowers = [rpowers m(j):2:n(j)]; P*oKcq1R  
    end s `HSTq2  
    rpowers = unique(rpowers); o90SXa&l/  
    Zx,R6@l  
    % Pre-compute the values of r raised to the required powers, G;_QE<V~_  
    % and compile them in a matrix: [SgWUP*  
    % ----------------------------- m(>_C~rGN  
    if rpowers(1)==0 gvvl3`S{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |};]^5s9  
        rpowern = cat(2,rpowern{:}); Xoyk 'T] -  
        rpowern = [ones(length_r,1) rpowern]; oJe9H<  
    else i#$9>X  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); LI_>fuv"8  
        rpowern = cat(2,rpowern{:}); MzT#1~  
    end ,cPNZ-%  
    "'h?O*V]u{  
    % Compute the values of the polynomials: 4J0{$Xuu 0  
    % -------------------------------------- J;h4)w~9H3  
    z = zeros(length_r,length_n); z"*X/T  
    for j = 1:length_n XIh2Y\33ys  
        s = 0:(n(j)-m(j))/2; LWHP31{R  
        pows = n(j):-2:m(j); })!d4EcZf  
        for k = length(s):-1:1 +]uW|owxo  
            p = (1-2*mod(s(k),2))* ... 1RM;"b/  
                       prod(2:(n(j)-s(k)))/          ... Mnyg:y*=  
                       prod(2:s(k))/                 ... H1_XEcaM+*  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... TWYz\Hmw  
                       prod(2:((n(j)+m(j))/2-s(k))); DrLNY"Zq  
            idx = (pows(k)==rpowers); bhe~ekb  
            z(:,j) = z(:,j) + p*rpowern(:,idx); t5mI)u  
        end 3#huC=zbf  
         Q W#]i  
        if isnorm Cbm  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); U^vQr%ha  
        end VvN52 qeL  
    end V#[I/D  
    J01w\#62pQ  
    % 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)  S;NChu?8  
    OzBo *X/p  
    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。 RL~|Kr<7J  
    %8`zaa  
    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)