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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 T\6Qr$t  
    就是我用ansys计算出了镜面的面型的数据,怎样可以得到zernike多项式系数,然后用zemax各阶得到像差!谢谢啦! uyj!$}4  
     
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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 ~<[;6&{  
    function z = zernfun(n,m,r,theta,nflag) 2a@X-Di  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. r/h\>s+N  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >MYxj}I4{z  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 7w73,r/D8A  
    %   unit circle.  N is a vector of positive integers (including 0), and $1=7^v[U  
    %   M is a vector with the same number of elements as N.  Each element 9/"&6,  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) dv. 77q  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, =eA|gt  
    %   and THETA is a vector of angles.  R and THETA must have the same `0upm%A  
    %   length.  The output Z is a matrix with one column for every (N,M) fw%p_Cm  
    %   pair, and one row for every (R,THETA) pair. Q<>u) %92@  
    % 'D W|a  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ivo3 pibk%  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L|[i<s;  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 3Ei^WDJ  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, EWJB /iED  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized DN^+"_:TB  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &>-'|(m+2  
    % GW {tZaB  
    %   The Zernike functions are an orthogonal basis on the unit circle. cc${[yj)  
    %   They are used in disciplines such as astronomy, optics, and #X] *kxQ<  
    %   optometry to describe functions on a circular domain. ]Zb9F[  
    % u?>},M/  
    %   The following table lists the first 15 Zernike functions. } W]A`-Jv  
    % w&@tP^`  
    %       n    m    Zernike function           Normalization ,DEq"VW_  
    %       -------------------------------------------------- <.`i,|?MHS  
    %       0    0    1                                 1 2IJniS=[>  
    %       1    1    r * cos(theta)                    2 /CALX wL  
    %       1   -1    r * sin(theta)                    2 #>yOp *  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) A^lm0[3q  
    %       2    0    (2*r^2 - 1)                    sqrt(3) x|Uwk=;X|s  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Kmx4bp4  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ;)ay uS sQ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) %lbvK^  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) H@- GYX"4  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) '&Ur(axs  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) uP4yJ/]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) l_k:OZ  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 9ad`q+kY  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Vu_oxL}  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) W. d',4)  
    %       -------------------------------------------------- B\D)21Ik}%  
    % Z7wl~Hk  
    %   Example 1: )4fQ~)  
    % ](I||JJa9f  
    %       % Display the Zernike function Z(n=5,m=1) ?u CL[  
    %       x = -1:0.01:1; y ;mk]  
    %       [X,Y] = meshgrid(x,x); t=lDN'\P  
    %       [theta,r] = cart2pol(X,Y); <uU<qO;6  
    %       idx = r<=1; h[! @8  
    %       z = nan(size(X)); ( x% 4*  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); bD)"Jy  
    %       figure m p_7$#{l  
    %       pcolor(x,x,z), shading interp lDBAei3iB  
    %       axis square, colorbar 'Rnzu0<lF  
    %       title('Zernike function Z_5^1(r,\theta)') = 1veO0  
    % Ot.v%D`e 5  
    %   Example 2: xd `MEOY  
    % UNSXr`9  
    %       % Display the first 10 Zernike functions L5UZ@R,  
    %       x = -1:0.01:1; RKrNmD*rk*  
    %       [X,Y] = meshgrid(x,x); #6~KO7}  
    %       [theta,r] = cart2pol(X,Y); ai d1eF  
    %       idx = r<=1; U=%(kOx  
    %       z = nan(size(X)); @?s>oSyV  
    %       n = [0  1  1  2  2  2  3  3  3  3]; x+~!M:fAc9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; h [Sd3Z*  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; X<_HQ  
    %       y = zernfun(n,m,r(idx),theta(idx)); XR VZU~ZV  
    %       figure('Units','normalized') `]]5!U2  
    %       for k = 1:10 6yYjZ<  
    %           z(idx) = y(:,k); v? 8i;[  
    %           subplot(4,7,Nplot(k)) ?@in($67  
    %           pcolor(x,x,z), shading interp IjOBY  
    %           set(gca,'XTick',[],'YTick',[]) 6 o   
    %           axis square f5M;q;  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *]/iL#  
    %       end Yt=)=n  
    % }>y !I5O  
    %   See also ZERNPOL, ZERNFUN2. 3ouy-SQ  
    x?A<X2  
    %   Paul Fricker 11/13/2006 qh W]Wd" g  
    ?=)lbSu K  
    dHAT($QG  
    % Check and prepare the inputs: H9'psv  
    % ----------------------------- LV1drc  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?zP 2   
        error('zernfun:NMvectors','N and M must be vectors.') s[eSPSFZ  
    end vC1fKo\p  
    yX*$PNL5w  
    if length(n)~=length(m) 3st?6?7|  
        error('zernfun:NMlength','N and M must be the same length.') GwXhn2  
    end '+l"zK ]L-  
    2Y9u9;ah  
    n = n(:); {d#sZT  
    m = m(:); Yx,E5}-  
    if any(mod(n-m,2)) #mJRL[V5^  
        error('zernfun:NMmultiplesof2', ...  96;5  
              'All N and M must differ by multiples of 2 (including 0).') %R?WkG  
    end ]jI<Js* F  
    2::YR?  
    if any(m>n) I*N v|HST  
        error('zernfun:MlessthanN', ... /? d)01  
              'Each M must be less than or equal to its corresponding N.') I.Catm2  
    end D=%1?8K  
    ?Hdu=+ZV  
    if any( r>1 | r<0 ) MBjAe!,-  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') (#6Fg|f4Y  
    end 6DU(KYN  
    yMyvX_UNI  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o,?G(  
        error('zernfun:RTHvector','R and THETA must be vectors.') <L*`WO]\l  
    end wjH1Ombt  
    yK&  
    r = r(:); 7*M-?  
    theta = theta(:); *pD|N  
    length_r = length(r); %A3m%&(m&%  
    if length_r~=length(theta) ,) dlL tUm  
        error('zernfun:RTHlength', ... #Vmf 6  
              'The number of R- and THETA-values must be equal.') IS!OO<  
    end iO Z#}"  
    QL7.QG  
    % Check normalization: `YwJ.E  
    % -------------------- 5|r*,! CF  
    if nargin==5 && ischar(nflag) .9Cy<z  
        isnorm = strcmpi(nflag,'norm'); >r\GB#\5  
        if ~isnorm 1MO-60  
            error('zernfun:normalization','Unrecognized normalization flag.') j `!Ge  
        end 7BINqVS&  
    else co\Il]`R/  
        isnorm = false; N.q*jY= X|  
    end sm Ql^ 6a  
    .vy@uT,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `9^+KK"  
    % Compute the Zernike Polynomials \1<|X].jNY  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M?My+ oT  
    s I\-0og  
    % Determine the required powers of r: DrioBb@  
    % ----------------------------------- opm_|0  
    m_abs = abs(m); &b^~0Z  
    rpowers = []; (K8Ob3zN_  
    for j = 1:length(n) )=iv3nF?6N  
        rpowers = [rpowers m_abs(j):2:n(j)]; ?ZGsh7<k  
    end {PxFG<^U  
    rpowers = unique(rpowers); k]$oir  
    z7sDaZL?_  
    % Pre-compute the values of r raised to the required powers, VJTO:}Q  
    % and compile them in a matrix: 7$g$p&,VX  
    % -----------------------------  yZ[g2*1L  
    if rpowers(1)==0 ^dk$6%0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J]Z~.f="  
        rpowern = cat(2,rpowern{:}); U+>M@!=  
        rpowern = [ones(length_r,1) rpowern]; O<V 4j,  
    else #| ,cy,v4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); flC%<V%'-  
        rpowern = cat(2,rpowern{:}); li\=mH,Wr  
    end #O;JV}y  
    \5!7zPc  
    % Compute the values of the polynomials: o<3$|`S&  
    % -------------------------------------- 6YNL4HE?  
    y = zeros(length_r,length(n)); a,S;JF)v  
    for j = 1:length(n) M.s'~S7y  
        s = 0:(n(j)-m_abs(j))/2; q!'p   
        pows = n(j):-2:m_abs(j); ihwJBN>(  
        for k = length(s):-1:1 9`N5$;NzY  
            p = (1-2*mod(s(k),2))* ... $F# 5/gDVQ  
                       prod(2:(n(j)-s(k)))/              ... g;p} -=  
                       prod(2:s(k))/                     ... [yk-<}#B  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... I_z(ft.  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7XyCl&Dc:  
            idx = (pows(k)==rpowers); 4LB8p7$|a3  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 7pY :.iVO  
        end wxc#)W  
         h ':ZF  
        if isnorm Mhti  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3Y2~HuM  
        end }kr?+)wB  
    end 0stc$~~v  
    % END: Compute the Zernike Polynomials HLwMo&*rA  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zM=MFKhi ~  
    b\`S[  
    % Compute the Zernike functions: Pb8@owG8  
    % ------------------------------ ^c.D&y%5  
    idx_pos = m>0; [F-GaaM  
    idx_neg = m<0; JJtx `@Bc  
    n8F5z|/  
    z = y; =gQ9>An  
    if any(idx_pos) -GCo`PR?b  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Su2{nNC>  
    end ;mk[!  
    if any(idx_neg) wTa u.Bo  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Yjjh}R#  
    end ]1<GZ`  
    [DM0'4  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) Y"MHs0O5>  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. MY@&^71i4  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated zd=O;T;.  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive Xv-p7$?f  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Y/FPkH4  
    %   and THETA is a vector of angles.  R and THETA must have the same  L\PmT  
    %   length.  The output Z is a matrix with one column for every P-value, c[,h|~K/_?  
    %   and one row for every (R,THETA) pair. 2aM7zP[Z  
    % u##th8h4U  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike m|aK_  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) D=#RQ-  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) cQK-Euum  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 :D)(3U5  
    %   for all p. A#=TR_@:  
    % 3x0t[{l  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 sF{aG6u   
    %   Zernike functions (order N<=7).  In some disciplines it is EsMX #1>/m  
    %   traditional to label the first 36 functions using a single mode wfmM`4Y   
    %   number P instead of separate numbers for the order N and azimuthal =n5'~1?X?  
    %   frequency M. kUf i  
    % \#_ymM0  
    %   Example: ?Afe }  
    % gA!@oiq@  
    %       % Display the first 16 Zernike functions "Wwu Ty|  
    %       x = -1:0.01:1; 4#B'pJMw9  
    %       [X,Y] = meshgrid(x,x); S<"Fp1#"l  
    %       [theta,r] = cart2pol(X,Y); $8=|<vt  
    %       idx = r<=1; R c+olJ^5  
    %       p = 0:15; SMX]JZmH  
    %       z = nan(size(X)); Y_JQPup  
    %       y = zernfun2(p,r(idx),theta(idx)); e7RgA1  
    %       figure('Units','normalized') c1yRy|  
    %       for k = 1:length(p) J-v1"7[2GC  
    %           z(idx) = y(:,k); )XN_|zCk  
    %           subplot(4,4,k) !ZYPz}&N_  
    %           pcolor(x,x,z), shading interp =&bI-  
    %           set(gca,'XTick',[],'YTick',[]) S(zp_  
    %           axis square }5;4'l8  
    %           title(['Z_{' num2str(p(k)) '}']) 6:ettdj  
    %       end /4&gA5BS]  
    % -]Z7^  
    %   See also ZERNPOL, ZERNFUN. R~\R>\  
    [7Lr"  
    %   Paul Fricker 11/13/2006 QqA=QTZ}  
    (~GQncqa  
    uuC ["Z  
    % Check and prepare the inputs: tVAi0`DV  
    % -----------------------------  w4U,7%V  
    if min(size(p))~=1 AW]\n;f  
        error('zernfun2:Pvector','Input P must be vector.') v675C#l(  
    end .kT]^rv ;  
    6 c_#"4  
    if any(p)>35 UMoj9/-  
        error('zernfun2:P36', ... Q(bOar5  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... Q^(CqQo!<  
               '(P = 0 to 35).']) ^_WR) F'K  
    end 4q}+8F`0F  
    2J7|y\N,  
    % Get the order and frequency corresonding to the function number: 6"Uu;Q  
    % ---------------------------------------------------------------- t'n@yX_  
    p = p(:); vK',!1]y  
    n = ceil((-3+sqrt(9+8*p))/2); 5\+*ml  
    m = 2*p - n.*(n+2); Xs0)4U  
    *~vB6V|1  
    % Pass the inputs to the function ZERNFUN: =;Gq:mHi  
    % ---------------------------------------- _~<sb,W  
    switch nargin |oY{TQ<<d  
        case 3 ,md_eGF  
            z = zernfun(n,m,r,theta); , >LJpv  
        case 4 K./qu^+k  
            z = zernfun(n,m,r,theta,nflag); Qs&;MW4q  
        otherwise n1sYD6u<&  
            error('zernfun2:nargin','Incorrect number of inputs.') my (@~'  
    end K10G+'H^  
    7Ak<e tHD  
    % 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) k@4N7}  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. q$:1Xkl  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of TM)INo^  
    %   order N and frequency M, evaluated at R.  N is a vector of CMj =4e  
    %   positive integers (including 0), and M is a vector with the ;UQGi}?CD  
    %   same number of elements as N.  Each element k of M must be a ? i{?Q,  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) W A/dt2D|  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Hjm> I'9  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix (i~UH04r>s  
    %   with one column for every (N,M) pair, and one row for every Z& _kq|  
    %   element in R. Qit&cnO  
    % Q i18q|l8v  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Z"`w>c.  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is )U&9d  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to %e iV^>  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 C QkY6  
    %   for all [n,m]. Ac:`xk<  
    % L!}!k N:?  
    %   The radial Zernike polynomials are the radial portion of the wWl ?c  
    %   Zernike functions, which are an orthogonal basis on the unit !0;AFv`\  
    %   circle.  The series representation of the radial Zernike 8E%LhA.  
    %   polynomials is zMSwU]4I!  
    % &jDN6n3z  
    %          (n-m)/2 q/s-".%P  
    %            __ 7`|'Om?'  
    %    m      \       s                                          n-2s  u r$  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r b}! cEJY  
    %    n      s=0 ,GSiSn  
    % K9N31'  
    %   The following table shows the first 12 polynomials. l.}PxZ  
    % +7.|1x;C  
    %       n    m    Zernike polynomial    Normalization @Jd&[T27Lr  
    %       --------------------------------------------- &[G)Y D  
    %       0    0    1                        sqrt(2) ,r B(WKU  
    %       1    1    r                           2 "f91YX_)  
    %       2    0    2*r^2 - 1                sqrt(6) 9z(SOzZn  
    %       2    2    r^2                      sqrt(6) a\P:jgF  
    %       3    1    3*r^3 - 2*r              sqrt(8) .j4y0dh33  
    %       3    3    r^3                      sqrt(8) FB6Lz5:Vf  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ,Fn;*  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) pwo$qs(p  
    %       4    4    r^4                      sqrt(10) f^pBXz9&=  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) k4{!h?h  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) `l|Oj$  
    %       5    5    r^5                      sqrt(12) )1At/mr  
    %       --------------------------------------------- FGVw=G{r  
    % $}/tlA&e  
    %   Example: c.>f,vtcn  
    % aO{@.  
    %       % Display three example Zernike radial polynomials \uZpAV)5  
    %       r = 0:0.01:1; -Q? i16pM  
    %       n = [3 2 5]; =%U &$d|@G  
    %       m = [1 2 1]; vu( 5s  
    %       z = zernpol(n,m,r); @qNY"c%HV  
    %       figure WJ8i=MO67  
    %       plot(r,z) klKUX/ g  
    %       grid on ^Gk)aX  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ]xRR/S4  
    % !oH{=.w  
    %   See also ZERNFUN, ZERNFUN2. %{ BV+&  
    ip?]&5s  
    % A note on the algorithm. `l45T~`]$  
    % ------------------------ 6FDj:~  
    % The radial Zernike polynomials are computed using the series It3k#A0  
    % representation shown in the Help section above. For many special l%+ &V^:  
    % functions, direct evaluation using the series representation can ^\PRz Y  
    % produce poor numerical results (floating point errors), because L&6^(Bn   
    % the summation often involves computing small differences between NfDS6i.Fqp  
    % large successive terms in the series. (In such cases, the functions x-i,v"8  
    % are often evaluated using alternative methods such as recurrence Sh#N5kgD  
    % relations: see the Legendre functions, for example). For the Zernike ;Z*rY?v  
    % polynomials, however, this problem does not arise, because the M6wH$!zRa  
    % polynomials are evaluated over the finite domain r = (0,1), and Q0xGd(\  
    % because the coefficients for a given polynomial are generally all `)cI^!  
    % of similar magnitude. +2MF#{ tS  
    % `,4yGgD!4  
    % ZERNPOL has been written using a vectorized implementation: multiple x<I[?GT=  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] OV{v6,>O  
    % values can be passed as inputs) for a vector of points R.  To achieve +pYgh8w@  
    % this vectorization most efficiently, the algorithm in ZERNPOL {XU!p: x  
    % involves pre-determining all the powers p of R that are required to syu/"KY^!  
    % compute the outputs, and then compiling the {R^p} into a single QtfL'su:  
    % matrix.  This avoids any redundant computation of the R^p, and T.cTL.}  
    % minimizes the sizes of certain intermediate variables. h] ho? K  
    % L9)gN.#  
    %   Paul Fricker 11/13/2006 P[fy  
    0_qr7Ui8(  
    ;BuMzG:tmZ  
    % Check and prepare the inputs: kk5&lak2V  
    % ----------------------------- 3>n&u,Xe  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Tu-lc)  
        error('zernpol:NMvectors','N and M must be vectors.') T[4xt,[a  
    end  +C\79,r  
    oI#TjF  
    if length(n)~=length(m) =x+1A)Q  
        error('zernpol:NMlength','N and M must be the same length.') }@LIb<Y  
    end mh4 VQ9  
    }V ]*FCpQ  
    n = n(:); %3dc_YPS  
    m = m(:); r.)n>  
    length_n = length(n); d"5_x]Z;  
    Q>Ct]JW&  
    if any(mod(n-m,2)) dWzf C@]  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') XR",.3LD  
    end <e#v9=}DI  
    W?SAa7+  
    if any(m<0) B^9C}QB  
        error('zernpol:Mpositive','All M must be positive.') +76'(@(1Y  
    end 'T^MaLK  
    SN)Czi#7  
    if any(m>n) DI)"F OM6  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') hpQ #`rhn  
    end wmV=GV8 d  
    Tp@Yn  
    if any( r>1 | r<0 ) #}fvjJ{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') <mP_K^9c  
    end tX% C5k  
    6Z1O:Bou  
    if ~any(size(r)==1) ,X|FyO(p  
        error('zernpol:Rvector','R must be a vector.') 8p829  
    end *CGHp8  
    #IGcQY  
    r = r(:); o_\vudXK  
    length_r = length(r); E`LaO  
    t ^>07#z  
    if nargin==4 `hY%HzV=  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); b&A/S$*  
        if ~isnorm \RDqW+,  
            error('zernpol:normalization','Unrecognized normalization flag.') -hfDf{QN  
        end hQ>$ "0K  
    else )+v5 H  
        isnorm = false; OLF6["0Rn  
    end +z9BWo!{I  
    o1 27? ^  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RF8, qz  
    % Compute the Zernike Polynomials }JOz,SQHP  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Nq9pory^  
    7#"y mE  
    % Determine the required powers of r: qm#?DSLap  
    % ----------------------------------- pqvl,G5  
    rpowers = []; sAO/yG  
    for j = 1:length(n) U(+QrC:  
        rpowers = [rpowers m(j):2:n(j)]; +?t& 7={~  
    end K~]Xx~F  
    rpowers = unique(rpowers); 5Th\wTh04  
    G~_eBy  
    % Pre-compute the values of r raised to the required powers, IecD41%  
    % and compile them in a matrix: }x{1{Bw>Y  
    % ----------------------------- Gyy4)dP  
    if rpowers(1)==0 gzK/l:  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EhD%  
        rpowern = cat(2,rpowern{:}); 0n X5Vo  
        rpowern = [ones(length_r,1) rpowern]; 7^)yo#i4  
    else SANb g&$  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _ZWU~38PM  
        rpowern = cat(2,rpowern{:}); U4a8z<l$  
    end #6y fIvap  
    %/U'Wu{*  
    % Compute the values of the polynomials: ~y Dl & S  
    % -------------------------------------- 2fayQY xD  
    z = zeros(length_r,length_n); 1mh7fZgn  
    for j = 1:length_n <Qbqxw  
        s = 0:(n(j)-m(j))/2; f[`&3+  
        pows = n(j):-2:m(j); 1A(f_ 0,.Q  
        for k = length(s):-1:1 i5WO)9Us  
            p = (1-2*mod(s(k),2))* ... -NzOX"V]3  
                       prod(2:(n(j)-s(k)))/          ... vGH]7jht  
                       prod(2:s(k))/                 ... ]We0 RD"+  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... !#5y%Bf  
                       prod(2:((n(j)+m(j))/2-s(k))); 6 P9#6mZ  
            idx = (pows(k)==rpowers); =%}(Dvjv  
            z(:,j) = z(:,j) + p*rpowern(:,idx); &B+_#V=X@  
        end Z@JTZMN_  
         0jXDjk5'<  
        if isnorm 1)xj 'n  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); b V_<5PHP  
        end ok-q9dM  
    end _=[pW2p  
    0ly6  |:  
    % 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)  'di(5  
    7a<:\F}E0  
    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。 ;[[oZ  
    +Rd;>s*.Y  
    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)