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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 Xr@0RFdr[  
    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! mWZV O,t$  
     
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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 -|g~--@Q  
    function z = zernfun(n,m,r,theta,nflag) L\pe  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. XSXS;Fh)  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1I_(!F{Ho  
    %   and angular frequency M, evaluated at positions (R,THETA) on the EiSS_Lc  
    %   unit circle.  N is a vector of positive integers (including 0), and ~qs 97'  
    %   M is a vector with the same number of elements as N.  Each element p;g$D=2  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) sk 9*3d5I  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, WJ8i,7  
    %   and THETA is a vector of angles.  R and THETA must have the same < yBZsSj  
    %   length.  The output Z is a matrix with one column for every (N,M) JW (.,Ztm  
    %   pair, and one row for every (R,THETA) pair. Ao(Xz$cQfW  
    % K%O%#Kk  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^As^hY^p  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), qy"#XbBeV  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Bi9 S1 p  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ) m[0,  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized jUYb8:B  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "1t%J7c_  
    % V^[o{'+  
    %   The Zernike functions are an orthogonal basis on the unit circle. ng"R[/)In  
    %   They are used in disciplines such as astronomy, optics, and > T=($:n  
    %   optometry to describe functions on a circular domain. CtfI&rb[  
    % |#>\GU=!  
    %   The following table lists the first 15 Zernike functions. g?qm >X  
    % x[zt(kC0+  
    %       n    m    Zernike function           Normalization ?Mtd3F^o?  
    %       -------------------------------------------------- 'gI q_t|^  
    %       0    0    1                                 1 LY(YgqL  
    %       1    1    r * cos(theta)                    2 NJRk##Z  
    %       1   -1    r * sin(theta)                    2 *F[@lY\p  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) -A^18r  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Kf/1;:^  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) A!\ g!*  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Yj;KKgk  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) f%<kcM2  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) [@(M%  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ~bC{ R&p  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) h\k@7wgu  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O`<id+rx  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) `7[z%cuK  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tJ;<=.n  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %ukFn &-2@  
    %       -------------------------------------------------- >)\x\e  
    % CY"&@v1  
    %   Example 1: j51Wod<[  
    % 0]p! Bscaf  
    %       % Display the Zernike function Z(n=5,m=1) LQ(z~M0B  
    %       x = -1:0.01:1; Q8OA{EUtq  
    %       [X,Y] = meshgrid(x,x); kK\G+{z?  
    %       [theta,r] = cart2pol(X,Y); 6aRPm%  
    %       idx = r<=1; TrD2:N}dI  
    %       z = nan(size(X)); Myaj81  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); M$iDaEu-  
    %       figure CobMagPhr  
    %       pcolor(x,x,z), shading interp ++1<A& a  
    %       axis square, colorbar lV924mh  
    %       title('Zernike function Z_5^1(r,\theta)') O|wu;1pQ  
    % Ad$CHx-  
    %   Example 2: Va"H.]  
    % b|jdYJbol&  
    %       % Display the first 10 Zernike functions ,41Z_h  
    %       x = -1:0.01:1; ])Rs.Y{Q5  
    %       [X,Y] = meshgrid(x,x); PY.4J4nn|  
    %       [theta,r] = cart2pol(X,Y); ^K[WFiN}  
    %       idx = r<=1; Y%$@ZYW  
    %       z = nan(size(X)); I!LSD i3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ^jY/w>UdH  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; t 3LRmjL  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; p=13tQS<  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0 ]K\G55  
    %       figure('Units','normalized') o9GtS$ O\  
    %       for k = 1:10 EY+/ foP  
    %           z(idx) = y(:,k); Z/ w}so  
    %           subplot(4,7,Nplot(k)) 'DLgOUvh  
    %           pcolor(x,x,z), shading interp d}B_ wz'  
    %           set(gca,'XTick',[],'YTick',[]) "^gV.  
    %           axis square {9mXJu$cc  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) o}^/K m+t  
    %       end pX 4:WV  
    % s0D,n1x  
    %   See also ZERNPOL, ZERNFUN2. ppYIVI  
    |aWeo.;c  
    %   Paul Fricker 11/13/2006 AlPk o($E*  
    ]\TYVv)  
    /i !3Fr"  
    % Check and prepare the inputs: *;N6S~_'Y  
    % ----------------------------- dio<?6ZD9P  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cpY'::5.%  
        error('zernfun:NMvectors','N and M must be vectors.')  <xn96|$  
    end ;pH&YBY  
    O8\>?4)  
    if length(n)~=length(m) 3P}^Wu  
        error('zernfun:NMlength','N and M must be the same length.') 2D'b7zPJ3  
    end HLL:nczj  
    }^b7x;O|  
    n = n(:); `qXCY^BH2  
    m = m(:); 7A,QA5G ]C  
    if any(mod(n-m,2)) dx.,  
        error('zernfun:NMmultiplesof2', ... 6_rgj{L  
              'All N and M must differ by multiples of 2 (including 0).') *- S/{ .&  
    end Gl!fT1zh0  
    ,V`zW<8  
    if any(m>n) QXaE2}}P  
        error('zernfun:MlessthanN', ... r}>q*yx:  
              'Each M must be less than or equal to its corresponding N.') '!V5 #J  
    end nuvRjd^N  
    t%k1=Ow5i  
    if any( r>1 | r<0 ) $T#yxx  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') - WEEnwZ  
    end #&$a7L}  
    7RpAsLH=  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \c1NIuJR  
        error('zernfun:RTHvector','R and THETA must be vectors.') 2SABu796j  
    end =cP7"\  
     0m&  
    r = r(:); f+c<|"we  
    theta = theta(:); !dcG Bj  
    length_r = length(r); X:/Y^Xu  
    if length_r~=length(theta) dv7IHUFf  
        error('zernfun:RTHlength', ... QIb4ghm,  
              'The number of R- and THETA-values must be equal.') .dE2,9{Z  
    end s/+k[9l2  
    Fv!KLw@  
    % Check normalization: <+r<3ZBA  
    % -------------------- cUDo}Yu  
    if nargin==5 && ischar(nflag) o$XJSz|6  
        isnorm = strcmpi(nflag,'norm'); Cg]Iz< <bE  
        if ~isnorm #"PRsMUw  
            error('zernfun:normalization','Unrecognized normalization flag.') {>]7xTpwZ  
        end x$gVEh*k  
    else wOg?.6<Kxa  
        isnorm = false; )=9EShz!  
    end %~{G*%:  
    OS{j5o  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $LXz Q>w9  
    % Compute the Zernike Polynomials l,w$!FnmR  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]+)cXJ}6#  
    %uUQBZ4  
    % Determine the required powers of r: OZCbMeB{+J  
    % ----------------------------------- ]A.tauSW  
    m_abs = abs(m); p]^?4  
    rpowers = []; 3[T<pAZ  
    for j = 1:length(n) [@4.<4Y  
        rpowers = [rpowers m_abs(j):2:n(j)]; %J b/HWC[  
    end wMx# dP4W8  
    rpowers = unique(rpowers); dU<qFxW  
    2<X.kM?N{B  
    % Pre-compute the values of r raised to the required powers, Jmcf9g  
    % and compile them in a matrix: 2$@N4  
    % ----------------------------- 24; BY'   
    if rpowers(1)==0 QVq+';cG  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uB*Y}"Fn  
        rpowern = cat(2,rpowern{:}); {wS)M  
        rpowern = [ones(length_r,1) rpowern]; $ KAOJc4<  
    else 5;4bZ3e,0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6 h%%?  
        rpowern = cat(2,rpowern{:}); !o{>[  
    end AzlZe\V?)~  
    qTV;L-  
    % Compute the values of the polynomials: ] l@Mo7|w  
    % -------------------------------------- gOSFvH8FU  
    y = zeros(length_r,length(n)); D>>?8a  
    for j = 1:length(n) GyP.;$NHa[  
        s = 0:(n(j)-m_abs(j))/2; R4 x!b`:i  
        pows = n(j):-2:m_abs(j); XqxmvN  
        for k = length(s):-1:1 -&UP[Mq  
            p = (1-2*mod(s(k),2))* ... !$1'q~sO  
                       prod(2:(n(j)-s(k)))/              ... W_\~CntyZ  
                       prod(2:s(k))/                     ... +/|;<K5_LI  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )B1gX>J\8  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); NAnccB D!{  
            idx = (pows(k)==rpowers); #@^mA{Dt5  
            y(:,j) = y(:,j) + p*rpowern(:,idx); s/cclFji]  
        end BJ,D1E  
         Z H1UAf  
        if isnorm f?P>P23  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); O3];1ud  
        end M0$wTmXM  
    end .9'bi#:Cw  
    % END: Compute the Zernike Polynomials {!]7=K)W9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K>_~zWnc  
    G-#]|)  
    % Compute the Zernike functions: !YZ$WiPl  
    % ------------------------------ 552U~t  
    idx_pos = m>0; ~REP@!\r^  
    idx_neg = m<0; .r4M]1Of  
     gK Uci  
    z = y; >x0)  
    if any(idx_pos) >v9@p7Dn  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6%Ws>H4@|  
    end CG397Y^  
    if any(idx_neg) M>m+VsJV  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Ku3/xcu:My  
    end V#-\ 4`c  
    X`' @ G  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) %!HBPLk  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. WN`|5"?$  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated @9ndr$t  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive PD$@.pib  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, V 'e _gH  
    %   and THETA is a vector of angles.  R and THETA must have the same g*a|QBj%  
    %   length.  The output Z is a matrix with one column for every P-value, KsR^:_e  
    %   and one row for every (R,THETA) pair. xTV3U9 v  
    % [:xpz,  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike b$O1I[o  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) \Ng|bWR>LQ  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) `j1(GQt  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 %*Aq%,.={  
    %   for all p. tLc 9-  
    % x}(p\Efx  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ~P5;k_&  
    %   Zernike functions (order N<=7).  In some disciplines it is tJgo% P1  
    %   traditional to label the first 36 functions using a single mode Z<M?_<3  
    %   number P instead of separate numbers for the order N and azimuthal T 7EkRcb  
    %   frequency M. pjaDtNb  
    % ) HPe}(ypt  
    %   Example: !W?6,i-]  
    % !hS~\+E  
    %       % Display the first 16 Zernike functions R3\oLT4  
    %       x = -1:0.01:1; 6Q}WX[| tQ  
    %       [X,Y] = meshgrid(x,x); /QT"5fxKJ  
    %       [theta,r] = cart2pol(X,Y); S{.G=O  
    %       idx = r<=1; ^9wQl!e ob  
    %       p = 0:15; z:a%kZQ!0  
    %       z = nan(size(X)); ;E0aTV)Zp  
    %       y = zernfun2(p,r(idx),theta(idx)); [fl x/E  
    %       figure('Units','normalized') C_q@ixF{  
    %       for k = 1:length(p) <~ 9a3c?  
    %           z(idx) = y(:,k); 8e{S(FZ7Ed  
    %           subplot(4,4,k) AQR/nWwx  
    %           pcolor(x,x,z), shading interp s4uYp  
    %           set(gca,'XTick',[],'YTick',[]) Wd78 bu|  
    %           axis square [l%6wIP&{  
    %           title(['Z_{' num2str(p(k)) '}']) Ed$;#4  
    %       end $CgR~D2G  
    % Bk)*Z/1<x  
    %   See also ZERNPOL, ZERNFUN. Tu*"+*r>s  
    hNWZ1r~_  
    %   Paul Fricker 11/13/2006 AFN"#M  
    vD@ =V#T  
    [n!5!/g>j  
    % Check and prepare the inputs: ^_C]?D?  
    % ----------------------------- z&x3":@u<  
    if min(size(p))~=1 U`w `Cr  
        error('zernfun2:Pvector','Input P must be vector.') <{xU.zp'  
    end kl/eJN'S  
     WPnw  
    if any(p)>35 Fu;\t 0  
        error('zernfun2:P36', ... e <]^7pz  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... THcK,`lX@  
               '(P = 0 to 35).']) We51s^(  
    end 52H'aHO1  
    /yhGc}h  
    % Get the order and frequency corresonding to the function number: g(`m#&P>G  
    % ---------------------------------------------------------------- $22_>OsA  
    p = p(:); A.@/~\  
    n = ceil((-3+sqrt(9+8*p))/2); a"6AZT"8  
    m = 2*p - n.*(n+2); aUVJ\ ;V  
    zUNWcv!& "  
    % Pass the inputs to the function ZERNFUN: \4qw LM?E^  
    % ---------------------------------------- 5&QDZnsl  
    switch nargin oMNgyAp^  
        case 3 dd{pF\a  
            z = zernfun(n,m,r,theta); Hvj1R.I/  
        case 4 _${//`ia=  
            z = zernfun(n,m,r,theta,nflag); |y T-N3H@  
        otherwise zVL"$ )  
            error('zernfun2:nargin','Incorrect number of inputs.') ) }.<lSw  
    end H,I k&{@j  
    g82_KUkB  
    % 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) LEJ8 .z6$  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. h_{//W[  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 2wKW17wj,  
    %   order N and frequency M, evaluated at R.  N is a vector of .&R j2d  
    %   positive integers (including 0), and M is a vector with the &N^^[ uG  
    %   same number of elements as N.  Each element k of M must be a iR`c/  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) %?`TyVt&0  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is qDzd_E@aR  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 3$E\B=7/U  
    %   with one column for every (N,M) pair, and one row for every XX@@tzN  
    %   element in R. p~h)@  
    % afJ`1l  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- X?Yp=%%  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Ry+Ax4#+(y  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to NE'4atQ|  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 }RT#V8oc  
    %   for all [n,m]. P{{pp<tX*&  
    % Z\@m_ /g  
    %   The radial Zernike polynomials are the radial portion of the :^Ouv1!e1  
    %   Zernike functions, which are an orthogonal basis on the unit 8&0+Az"{O  
    %   circle.  The series representation of the radial Zernike '&<T;V%  
    %   polynomials is  b}eBy  
    % HU='Hk!  
    %          (n-m)/2 Ba]J3Yp,z  
    %            __ mV58&SZT  
    %    m      \       s                                          n-2s Oy,`tG0  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @&7|Laa  
    %    n      s=0 [kjmEMF9i  
    % /1Q i9uit  
    %   The following table shows the first 12 polynomials. p? q~.YY  
    % :mpR}.^hv  
    %       n    m    Zernike polynomial    Normalization ND3(oes+;K  
    %       --------------------------------------------- S<+/Ep 2  
    %       0    0    1                        sqrt(2) 6N^sUc0s  
    %       1    1    r                           2 $Bd13%>)  
    %       2    0    2*r^2 - 1                sqrt(6) < mb.F-8  
    %       2    2    r^2                      sqrt(6) $xWebz0  
    %       3    1    3*r^3 - 2*r              sqrt(8) 6tKm'`^z4  
    %       3    3    r^3                      sqrt(8) x-Cy,d:YX  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) /61P`1y(J  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ~MOab e  
    %       4    4    r^4                      sqrt(10) |6mDooTy  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) w,'"2^Cwy  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) 3O W) %  
    %       5    5    r^5                      sqrt(12) m ^O9G?  
    %       --------------------------------------------- f8m%T%]f  
    % r-ldqj  
    %   Example: kCq]#e~wq  
    % \8I>^4t'/  
    %       % Display three example Zernike radial polynomials _B5t)7I  
    %       r = 0:0.01:1; e>:bV7h j~  
    %       n = [3 2 5]; -}h+hS50F  
    %       m = [1 2 1]; d_0r  
    %       z = zernpol(n,m,r); j\ dY  
    %       figure 7:Jyu/*]  
    %       plot(r,z) h7EKb-@  
    %       grid on c]y"5;V8  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <N Lor55.]  
    % #\Q{?F!4  
    %   See also ZERNFUN, ZERNFUN2. d]v4`nc  
    S;582H9D  
    % A note on the algorithm. UP@a ?w  
    % ------------------------ q66+x)  
    % The radial Zernike polynomials are computed using the series 1>doa1  
    % representation shown in the Help section above. For many special f-V8/  
    % functions, direct evaluation using the series representation can ?Q~6\xA  
    % produce poor numerical results (floating point errors), because l>Nz]Ul%{  
    % the summation often involves computing small differences between :oH~{EQ  
    % large successive terms in the series. (In such cases, the functions A1zqm_X5)P  
    % are often evaluated using alternative methods such as recurrence 1-Fz#v7p  
    % relations: see the Legendre functions, for example). For the Zernike 31w9$H N  
    % polynomials, however, this problem does not arise, because the 0]F'k8yLN  
    % polynomials are evaluated over the finite domain r = (0,1), and q;))3aQe  
    % because the coefficients for a given polynomial are generally all V7zF5=w  
    % of similar magnitude. x<0-'EF/S  
    % )-_NtMr~`!  
    % ZERNPOL has been written using a vectorized implementation: multiple  M>mk=-l  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 1I?D$I>CV  
    % values can be passed as inputs) for a vector of points R.  To achieve q/o|uAq  
    % this vectorization most efficiently, the algorithm in ZERNPOL @$5GxIw<l  
    % involves pre-determining all the powers p of R that are required to MI|51&m  
    % compute the outputs, and then compiling the {R^p} into a single k~(j   
    % matrix.  This avoids any redundant computation of the R^p, and =sqh PS<>  
    % minimizes the sizes of certain intermediate variables. AnbY<&OC1  
    % B%v2)+?@  
    %   Paul Fricker 11/13/2006 3~e"CKD>  
    <p48?+K9  
    :qV}v2  
    % Check and prepare the inputs: N0@&eX|$i4  
    % ----------------------------- sckyG  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -fl?G%:(!0  
        error('zernpol:NMvectors','N and M must be vectors.') @#T*OH  
    end $B6"fYiDk  
    rf+'U9  
    if length(n)~=length(m) wW3fsXu  
        error('zernpol:NMlength','N and M must be the same length.') c2}?[\U]  
    end C043h?x  
    z$3 3NM  
    n = n(:); iOB]72dh  
    m = m(:); 73$^y)AvY  
    length_n = length(n); H61 ,pr>  
    m6a q_u{W  
    if any(mod(n-m,2)) _P!J0  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') yZ[H&>  
    end Y[AL!h  
    360V  
    if any(m<0) h[D"O6 y  
        error('zernpol:Mpositive','All M must be positive.') |Ire#0Nwx  
    end &qki NS  
    &zsaVm8  
    if any(m>n) %nJ^0X_]  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') `}1IQ.3  
    end #zC_;u$  
    $_-f}E  
    if any( r>1 | r<0 ) #>-_z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') QE&rpF7l{  
    end r;`6ML[5Vx  
    AZ)H/#be  
    if ~any(size(r)==1) mie<jha  
        error('zernpol:Rvector','R must be a vector.') !>B|z=  
    end ;z;O}<8s  
    LL5n{#)N  
    r = r(:); 8(UUc>g  
    length_r = length(r); 7:U^Ki  
    JcEPwF.  
    if nargin==4 |3f?1:"Z  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); :E/]Bjq$;  
        if ~isnorm  ?[Od.  
            error('zernpol:normalization','Unrecognized normalization flag.') <d,Qi.G4  
        end 6[kp#  
    else sQrP,:=r#  
        isnorm = false; fYF\5/_  
    end `;-K/)/x  
    * B!uYP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'qS&7 W(  
    % Compute the Zernike Polynomials !Q!= =*1H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &g R+D  
    YaJ[39V  
    % Determine the required powers of r: <rc?EV  
    % ----------------------------------- <Q'J=;vV  
    rpowers = []; 4y&%YLMpl  
    for j = 1:length(n) T]\_[e:'  
        rpowers = [rpowers m(j):2:n(j)]; Zr,:i MPZ  
    end 12;"=9e!  
    rpowers = unique(rpowers); Tj}H3/2  
    =$6z1] ;3  
    % Pre-compute the values of r raised to the required powers, RiC1lCE  
    % and compile them in a matrix: won;tO]\;@  
    % ----------------------------- tgy*!B6a~  
    if rpowers(1)==0 b f.__3{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X TpYf  
        rpowern = cat(2,rpowern{:}); Q?KWiFA}'  
        rpowern = [ones(length_r,1) rpowern]; /k3v\Jq{  
    else 7Z0fMk  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); H(U`S  
        rpowern = cat(2,rpowern{:}); m.ev~Vv~  
    end !E<y:$eH:  
    4$ LVl  
    % Compute the values of the polynomials: 9 |v3lGK(  
    % -------------------------------------- B2Qt tcJ  
    z = zeros(length_r,length_n); SW|{)L,  
    for j = 1:length_n F)'kN2  
        s = 0:(n(j)-m(j))/2; fB+4mEG@  
        pows = n(j):-2:m(j); .$5QM&  
        for k = length(s):-1:1 4{JoeIRyz  
            p = (1-2*mod(s(k),2))* ... _]btsv\)f  
                       prod(2:(n(j)-s(k)))/          ... 7M*&^P\}es  
                       prod(2:s(k))/                 ... XlP q>@4p  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... o.IJ4'}aN  
                       prod(2:((n(j)+m(j))/2-s(k)));  }t}y  
            idx = (pows(k)==rpowers); (27bNKr  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ygoA/*s  
        end KECo7i=e  
         K TJm[44  
        if isnorm 9?$Qk0jc  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7 lc -  
        end 8."B  
    end "NR`{1f:O  
    ~R-P%l P  
    % 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)  bdYx81  
    |!6<L_31%  
    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。 D/oO@;`'c  
    aGs\zCAP  
    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)