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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 =o&>fw  
    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! 9J9)AV  
     
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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 ($}`R xj1@  
    function z = zernfun(n,m,r,theta,nflag) TW[_Ko86  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. /XhIx\40 l  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )tl.s)"N  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,:Lb7bFv>  
    %   unit circle.  N is a vector of positive integers (including 0), and 1$%V{4bJ  
    %   M is a vector with the same number of elements as N.  Each element tb$LriN  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) p TeOW9  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ,ztI,1"k  
    %   and THETA is a vector of angles.  R and THETA must have the same l PK +$f$  
    %   length.  The output Z is a matrix with one column for every (N,M) V}SBuQp"  
    %   pair, and one row for every (R,THETA) pair. 3 AsT  
    % DM}YJ  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike A` AaTP  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), il \$@Bn  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Pri`K/  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %YSu8G_t  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 8'f4 Od ?  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R0L&*Bjm  
    % CC@.MA@9N  
    %   The Zernike functions are an orthogonal basis on the unit circle. H<}^'#"p  
    %   They are used in disciplines such as astronomy, optics, and DBCK2PlJ  
    %   optometry to describe functions on a circular domain. >&p0d0  
    % ^",ACWF4Sk  
    %   The following table lists the first 15 Zernike functions. Ygl%eP%Z  
    % Qbyv{/   
    %       n    m    Zernike function           Normalization yRiP{$E  
    %       -------------------------------------------------- JM\m)RH0  
    %       0    0    1                                 1 GF5^\Rf  
    %       1    1    r * cos(theta)                    2 aMvI?y {  
    %       1   -1    r * sin(theta)                    2 E[bd@[N 8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) ;Hj~n+  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ODC8D>ZYl  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) tc!wLnhG  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) Ldl 5zc  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Ns[ym>x#2  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ")!,ZD  
    %       3    3    r^3 * sin(3*theta)             sqrt(8)  R#DwF,  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) h<SQL97N  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ZG du|  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) N~NQ6:R[  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,$ ^C4I  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |)K]U  
    %       -------------------------------------------------- (>I`{9x>6  
    % ea 00\  
    %   Example 1: %0mMz.f  
    % n Ml%'[u  
    %       % Display the Zernike function Z(n=5,m=1) ;x8k[p~2  
    %       x = -1:0.01:1; "eWYv3z~-  
    %       [X,Y] = meshgrid(x,x); i6 (a@KRY  
    %       [theta,r] = cart2pol(X,Y); K%Rj8J7|u?  
    %       idx = r<=1; GR"Eas.$  
    %       z = nan(size(X)); Wf&W^Q  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); F`9ZH.  
    %       figure ;XDz)`c  
    %       pcolor(x,x,z), shading interp Zt&6Ua[Y}  
    %       axis square, colorbar D.1J_Y=9  
    %       title('Zernike function Z_5^1(r,\theta)') 8-Hsgf.*  
    % x"CZ]p&m  
    %   Example 2: }QsZ:J.  
    % ~~6^Sh60g  
    %       % Display the first 10 Zernike functions a /:@"&Y  
    %       x = -1:0.01:1; !grVR157P  
    %       [X,Y] = meshgrid(x,x); &09U@uc$  
    %       [theta,r] = cart2pol(X,Y); ,s_T pq  
    %       idx = r<=1; Zb134b'  
    %       z = nan(size(X)); x $zKzfHW  
    %       n = [0  1  1  2  2  2  3  3  3  3]; His*t1o8'O  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Kmdlf,[3d  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; vQa'S-@u  
    %       y = zernfun(n,m,r(idx),theta(idx)); !Y:0c#MPH  
    %       figure('Units','normalized') KV*xApb9y  
    %       for k = 1:10 (}5S  
    %           z(idx) = y(:,k); l?q%?v8  
    %           subplot(4,7,Nplot(k)) \J6hI\/4^  
    %           pcolor(x,x,z), shading interp f5GdZ_  
    %           set(gca,'XTick',[],'YTick',[]) >"@?ir  
    %           axis square )#}mH@  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z xb_K  
    %       end ,~);EC=`  
    % wV)}a5+  
    %   See also ZERNPOL, ZERNFUN2. v*qQ? S  
    W},b{NT  
    %   Paul Fricker 11/13/2006 V`-vR2(  
    & BvZF  
    PDLpNTBf  
    % Check and prepare the inputs: BnM4T~reOF  
    % ----------------------------- n 8pt\i0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hku!bJ  
        error('zernfun:NMvectors','N and M must be vectors.') {q3H5csFq  
    end SgEBh  
    tWdhDt8$&  
    if length(n)~=length(m) 0ilCS[`b  
        error('zernfun:NMlength','N and M must be the same length.') :Yj) CGl$  
    end }rdIUlVO\  
    8p!*?RRme[  
    n = n(:); :vL1}H<  
    m = m(:); }BmS )J q  
    if any(mod(n-m,2)) _NcY I  
        error('zernfun:NMmultiplesof2', ... ]O:N-Y  
              'All N and M must differ by multiples of 2 (including 0).') i0s6aAhgJ  
    end Do]*JO)(  
    "aF8l<1xn  
    if any(m>n) R <"6ojn  
        error('zernfun:MlessthanN', ... X{g%kf,D=  
              'Each M must be less than or equal to its corresponding N.') %G@5!|J  
    end }N*>QR5K  
    '?Jxt:<  
    if any( r>1 | r<0 ) TFepxF  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') {R^'=(YFy  
    end o_Si mJFK  
    2 /y}a#s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8:.nEo'  
        error('zernfun:RTHvector','R and THETA must be vectors.') M- 0i7%  
    end a? R[J==  
    i\H+X   
    r = r(:); S }>n1F_  
    theta = theta(:); Fn^C{p^  
    length_r = length(r); n tP|\E  
    if length_r~=length(theta) cW``M.d'F  
        error('zernfun:RTHlength', ... dP>w/$C}  
              'The number of R- and THETA-values must be equal.') = zl= SLe  
    end 4q$H  
    p$k\m|t  
    % Check normalization: rQP"Y[  
    % -------------------- b8f+,2Tk  
    if nargin==5 && ischar(nflag) B/"2.,  
        isnorm = strcmpi(nflag,'norm'); |8)Xc=Hz  
        if ~isnorm F8+e,x  
            error('zernfun:normalization','Unrecognized normalization flag.') p[WX'M0f  
        end > 4oY3wk8  
    else o;[bJ Z\^x  
        isnorm = false; {5%/T,  
    end $cVi;2$p  
    eu'1H@vX(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]Fb0Az  
    % Compute the Zernike Polynomials )h^NR3N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Z/k;=Sla  
    )uP[!LV[e  
    % Determine the required powers of r: L<(VG{)Z  
    % ----------------------------------- V8O.3fo`[`  
    m_abs = abs(m); 50a\e  
    rpowers = []; mo1 puU  
    for j = 1:length(n) XtBMp=7Oa  
        rpowers = [rpowers m_abs(j):2:n(j)]; iS@\ =CK  
    end 4%*hGh=  
    rpowers = unique(rpowers); FyG6 !t%  
    s%;<O:x8o  
    % Pre-compute the values of r raised to the required powers, @<_`2eW'/R  
    % and compile them in a matrix: Qrz4}0  
    % ----------------------------- J -Qh/d%]  
    if rpowers(1)==0 qvt-  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LEh)g[  
        rpowern = cat(2,rpowern{:}); #Nte^E4  
        rpowern = [ones(length_r,1) rpowern]; nj\_lL+  
    else |ZU#IQVQfn  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 'nK~'PZ,  
        rpowern = cat(2,rpowern{:}); wAbp3hX  
    end |ia@,*KD  
     d^39t4  
    % Compute the values of the polynomials: ^Y^"'"  
    % -------------------------------------- wVi%oSfM  
    y = zeros(length_r,length(n)); =hw^P%Zn  
    for j = 1:length(n) ,m07p~,V  
        s = 0:(n(j)-m_abs(j))/2; oVZ4bRl   
        pows = n(j):-2:m_abs(j); T{*^_  
        for k = length(s):-1:1 8U.$FMx :  
            p = (1-2*mod(s(k),2))* ... -Gsl[Rc0H;  
                       prod(2:(n(j)-s(k)))/              ... pH9HK  
                       prod(2:s(k))/                     ... "+iAd.qd  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @~jxG%y86  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); !=[uT+v  
            idx = (pows(k)==rpowers); ]5|z3<K^  
            y(:,j) = y(:,j) + p*rpowern(:,idx); I{dl%z73  
        end BV9*s  
         \Tq "mw9P  
        if isnorm $cK^23H/Fj  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 0->/`/xm  
        end Bt>}LLBS2  
    end vmKT F!;  
    % END: Compute the Zernike Polynomials ) YSh D  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sT<{SmBF  
    :'w?ye[e  
    % Compute the Zernike functions: J5T=!wF (  
    % ------------------------------ o`%I{?UCDJ  
    idx_pos = m>0; X Usy.l/  
    idx_neg = m<0; 9YSVK\2$  
    umDtp\  
    z = y; Js}tZ\+P75  
    if any(idx_pos) -,>:DUN2  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |t\KsW  
    end ?;8M^a/  
    if any(idx_neg) `?SGXXC  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); WzG07 2w  
    end md6*c./Z  
    y<r44a_!  
    % 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) <t6 d)mJ%  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 2$ VTu+  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated >PH< N  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive nE<J`Wo$f  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, J_/05( 48  
    %   and THETA is a vector of angles.  R and THETA must have the same ")\ *2d  
    %   length.  The output Z is a matrix with one column for every P-value, S%V%!803!  
    %   and one row for every (R,THETA) pair. ~mcZUiP9  
    % ]1Qi=2'  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike *08+\ed"#  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 5xv,!/@  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) VLd=" ~  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^HoJ.oC/  
    %   for all p.  f }-v  
    % tAt;bYjb\  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 %f#\i#G<k  
    %   Zernike functions (order N<=7).  In some disciplines it is jhcuK:`L  
    %   traditional to label the first 36 functions using a single mode |bvGYsn_#=  
    %   number P instead of separate numbers for the order N and azimuthal %((cFQ9  
    %   frequency M. )Jz!Ut  
    % cB36p&%  
    %   Example: '7=<#Blc  
    % ?7 X3 P  
    %       % Display the first 16 Zernike functions I,z"_[^G  
    %       x = -1:0.01:1; }amE6  
    %       [X,Y] = meshgrid(x,x); dff#{  
    %       [theta,r] = cart2pol(X,Y); 'T{pdEn8u  
    %       idx = r<=1; JSUzEAKe  
    %       p = 0:15; -sD:+Te  
    %       z = nan(size(X)); rX)o3>q^?  
    %       y = zernfun2(p,r(idx),theta(idx)); P ]_Vz  
    %       figure('Units','normalized') `bi k/o=%  
    %       for k = 1:length(p) e7wKjt2fy  
    %           z(idx) = y(:,k); rOhA*_EG  
    %           subplot(4,4,k) vy:6_  
    %           pcolor(x,x,z), shading interp !?Tzk&'  
    %           set(gca,'XTick',[],'YTick',[]) `;T? 9n  
    %           axis square 3?]S,~!F  
    %           title(['Z_{' num2str(p(k)) '}']) t>-XT|lV  
    %       end 0Mq6yu^  
    % "vvFq ,c  
    %   See also ZERNPOL, ZERNFUN. tl2Lq0  
    vkh;qPD  
    %   Paul Fricker 11/13/2006 L7[X|zmy*x  
     6.vNe  
    5M]6'X6I  
    % Check and prepare the inputs: Q9nu"x %  
    % ----------------------------- q w"e0q%)  
    if min(size(p))~=1 6l=M;B7:i  
        error('zernfun2:Pvector','Input P must be vector.') OHQ3+WJ  
    end )8\Z=uC  
    M!{Rq1M  
    if any(p)>35 a#>t+.dd  
        error('zernfun2:P36', ... AZ}%MA; q  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... rjt O`Mt`  
               '(P = 0 to 35).']) 6pS Rum  
    end oF` -cyj"  
    pq@$&G  
    % Get the order and frequency corresonding to the function number: ;Ce 2d+K  
    % ---------------------------------------------------------------- >hh"IfIZ4  
    p = p(:); C[^a/P`i  
    n = ceil((-3+sqrt(9+8*p))/2); Q9SPb6O2  
    m = 2*p - n.*(n+2); a'c9XG}  
    s; ~J2h[  
    % Pass the inputs to the function ZERNFUN: x Xl$Mp7  
    % ---------------------------------------- &Qz"nCvJ  
    switch nargin F&-5&'6G+  
        case 3 G`&'Bt{Z*  
            z = zernfun(n,m,r,theta); I]s:Ev[~  
        case 4 ,2Sv1v$  
            z = zernfun(n,m,r,theta,nflag); AJdlqbd'+  
        otherwise h%4 ~0  
            error('zernfun2:nargin','Incorrect number of inputs.') <%T%NjNPQ  
    end Nj"_sA p  
    s#4))yUR6Z  
    % 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) 4?* `:  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. C]{V%jU  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of u:p:*u_^I  
    %   order N and frequency M, evaluated at R.  N is a vector of kY0g}o'<  
    %   positive integers (including 0), and M is a vector with the Bil;@,Z#  
    %   same number of elements as N.  Each element k of M must be a K[Ws/yc^a  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 6-Vl#Lyb  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is `-E.n'+  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Fb $5&~d  
    %   with one column for every (N,M) pair, and one row for every A X^3uRQJ  
    %   element in R. c9/ 'i  
    % A@lhm`Aa  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ?Ix'2v  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is :ok!,QN  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +$an*k9  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 @/Wty@PU  
    %   for all [n,m]. X.W#=$;$:  
    % 8*Nt&`@  
    %   The radial Zernike polynomials are the radial portion of the {&Gk.ODI7  
    %   Zernike functions, which are an orthogonal basis on the unit ! S$oaCxM  
    %   circle.  The series representation of the radial Zernike s}p GJ&C  
    %   polynomials is *]DJAF]  
    % ~P_d0A~T  
    %          (n-m)/2 |M0,%~Kt  
    %            __ '44nk(hM69  
    %    m      \       s                                          n-2s @O*ev| o@x  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r eo}S01bt  
    %    n      s=0 RF\1.HJG  
    % ML9T (th6v  
    %   The following table shows the first 12 polynomials. 4YB7og%P  
    %  Cq~ah  
    %       n    m    Zernike polynomial    Normalization kcZ;SYosj  
    %       --------------------------------------------- Rqd%#v  
    %       0    0    1                        sqrt(2) 8u~\]1 (  
    %       1    1    r                           2 'KIi!pA.  
    %       2    0    2*r^2 - 1                sqrt(6) xN":2qy#T  
    %       2    2    r^2                      sqrt(6) T(fR/~:z?  
    %       3    1    3*r^3 - 2*r              sqrt(8) M,v@G$pW  
    %       3    3    r^3                      sqrt(8) &t=>:C$1Y  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3K;b~xg`nw  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Duo#WtC  
    %       4    4    r^4                      sqrt(10) D2wgSrY  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)  C.TCDl  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) %<#$:Qb.  
    %       5    5    r^5                      sqrt(12) ]$`s}BN  
    %       --------------------------------------------- NiQc2\4%  
    % \]d*h]Hms  
    %   Example: R4J>M@-0v  
    %  PtVNG  
    %       % Display three example Zernike radial polynomials |jCE9Ve#  
    %       r = 0:0.01:1; ]mGsNQ ].H  
    %       n = [3 2 5]; 8aIf{(/k  
    %       m = [1 2 1]; |@n{tog+-  
    %       z = zernpol(n,m,r); {Z{NH:^  
    %       figure Qak@~b  
    %       plot(r,z) dXcMysRc%&  
    %       grid on 8T1DcA*  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') KzH}5:qI  
    % 1Mhc1MU  
    %   See also ZERNFUN, ZERNFUN2. 4~D>oNx4  
    MBTt'6M  
    % A note on the algorithm. jU9zCMyNF  
    % ------------------------ laRKt"A  
    % The radial Zernike polynomials are computed using the series {XUfxNDf  
    % representation shown in the Help section above. For many special 0 Vgn N  
    % functions, direct evaluation using the series representation can SJuf`  
    % produce poor numerical results (floating point errors), because So]FDd  
    % the summation often involves computing small differences between <OO/Tn'a  
    % large successive terms in the series. (In such cases, the functions D8Waf  
    % are often evaluated using alternative methods such as recurrence {&j{V-}f  
    % relations: see the Legendre functions, for example). For the Zernike g!|E!\p  
    % polynomials, however, this problem does not arise, because the fkUH]CdaB  
    % polynomials are evaluated over the finite domain r = (0,1), and ~v,KI["o  
    % because the coefficients for a given polynomial are generally all 4R^j"x 5  
    % of similar magnitude. rL+n$p X-  
    % JFk|Uqs(  
    % ZERNPOL has been written using a vectorized implementation: multiple KUqS(u  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] RxeRO2  
    % values can be passed as inputs) for a vector of points R.  To achieve z# ?w/NE  
    % this vectorization most efficiently, the algorithm in ZERNPOL _!g NF=  
    % involves pre-determining all the powers p of R that are required to D]`B;aE>A*  
    % compute the outputs, and then compiling the {R^p} into a single HG 6{`i  
    % matrix.  This avoids any redundant computation of the R^p, and u2\qg;dP  
    % minimizes the sizes of certain intermediate variables. |JQP7z6j]  
    % <"Cwy0V kp  
    %   Paul Fricker 11/13/2006 3jdB8a]T_  
    ?GfA;O  
    JfINAaboi  
    % Check and prepare the inputs: $0C/S5b  
    % ----------------------------- *A9{H>Vq  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3#ZKuGg=  
        error('zernpol:NMvectors','N and M must be vectors.') n&78~@H  
    end _89G2)U=C  
    $u.T1v  
    if length(n)~=length(m) : MmXH&yR  
        error('zernpol:NMlength','N and M must be the same length.') t-'GRme  
    end m%(JRh  
    nMvIL2:3  
    n = n(:); v#2qwd3x  
    m = m(:); 9wJmX<Rm  
    length_n = length(n); |]3);^0  
    4< >:]  
    if any(mod(n-m,2)) K}(n;6\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') j1CD;9i)%  
    end D>^ix[:J  
    G[-jZ  
    if any(m<0) "J:NW_U  
        error('zernpol:Mpositive','All M must be positive.') % +"AF+c3r  
    end )g<qEyJR  
    RSeezP6#  
    if any(m>n) ojqX#>0K  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') %,q#f#  
    end >A "aOV>K  
    S^n4aBm\+  
    if any( r>1 | r<0 ) VQx-gm8}!  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') htSk2N/  
    end -dN;\x  
    A2bV[+Q  
    if ~any(size(r)==1) .7rsbZzs  
        error('zernpol:Rvector','R must be a vector.') ?0&>?-?  
    end >c>ar>4xF  
    sjkl? _  
    r = r(:); P[oB'  
    length_r = length(r); 3A1kH` X^q  
    e(5R8ud  
    if nargin==4 PS]X Lz  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); <W^~Y31:0  
        if ~isnorm uCr  
            error('zernpol:normalization','Unrecognized normalization flag.') En_8H[<%  
        end tqf-,BLh  
    else "n-xsAG  
        isnorm = false; "t`e68{Ls  
    end /qze  
    b#^D8_9h  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t\0JNi$2  
    % Compute the Zernike Polynomials >R-$JrU.=  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J>v>6OC6i  
    ]`m5!V_Y  
    % Determine the required powers of r: |(g2fByDf  
    % ----------------------------------- Bqgw%_  
    rpowers = []; cIkLdh   
    for j = 1:length(n) UG$i5PV%i  
        rpowers = [rpowers m(j):2:n(j)]; ]F#kM211  
    end T^>cT"ux_  
    rpowers = unique(rpowers); >s~`K^zS  
    gE(03SX  
    % Pre-compute the values of r raised to the required powers, A 76yz`D  
    % and compile them in a matrix: $OuA<-  
    % ----------------------------- pDfF'jt9  
    if rpowers(1)==0 ^PszZ10T  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?2c:|FD  
        rpowern = cat(2,rpowern{:}); d|lzkY~  
        rpowern = [ones(length_r,1) rpowern]; 8t; nU;E*  
    else Yuck]?#0  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); *i90[3l  
        rpowern = cat(2,rpowern{:}); ? ~8V;Qn  
    end W;W\L? r  
    T;7|d5][  
    % Compute the values of the polynomials: 8a1{x(\z.  
    % -------------------------------------- [c~zO+x  
    z = zeros(length_r,length_n); 35et+9  
    for j = 1:length_n 9m>_q Wa A  
        s = 0:(n(j)-m(j))/2; s3S73fNOk  
        pows = n(j):-2:m(j); ymu#u   
        for k = length(s):-1:1 SY.V_O$l }  
            p = (1-2*mod(s(k),2))* ... y6\#{   
                       prod(2:(n(j)-s(k)))/          ... I(|{/{P,  
                       prod(2:s(k))/                 ... 7="V7  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... [;+YO)  
                       prod(2:((n(j)+m(j))/2-s(k))); wu3ZSLY  
            idx = (pows(k)==rpowers); &nn":  
            z(:,j) = z(:,j) + p*rpowern(:,idx); eP8wTStC  
        end T%F'4_~No  
         Wit1WI;18  
        if isnorm PT=%]o]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Hr'#0fW  
        end IAQ=d4V&  
    end eyOAG4QTV  
    pK%'S  
    % 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)  {C 7=  
    &:Q""e!  
    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。 a"~W1|JC"  
    l*h6 JgU  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。