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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 #x*\dL  
    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! f\xmv|8  
     
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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 `WIZY33V  
    function z = zernfun(n,m,r,theta,nflag) 9#TD1B/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. BmKf%:l}  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ~m_{&,CA.  
    %   and angular frequency M, evaluated at positions (R,THETA) on the O}>@G  
    %   unit circle.  N is a vector of positive integers (including 0), and >"8;8Ev  
    %   M is a vector with the same number of elements as N.  Each element LN~mKoW  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) $C.a@gm  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^D<CoxG  
    %   and THETA is a vector of angles.  R and THETA must have the same dP?prT  
    %   length.  The output Z is a matrix with one column for every (N,M) fcxg6W'  
    %   pair, and one row for every (R,THETA) pair. D(l,Z  
    % 3CgID6[Sy  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike l]4=W<N  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), XwUa|"X6  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ~P#mvQE)  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &#L C'  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized R\|,GZ!`+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1aQm r=,  
    % udu<Nis4  
    %   The Zernike functions are an orthogonal basis on the unit circle. [3"F$?e5  
    %   They are used in disciplines such as astronomy, optics, and UAPd["`)y  
    %   optometry to describe functions on a circular domain. ~n- Px)  
    % eT+i &  
    %   The following table lists the first 15 Zernike functions. b3EGtC}^  
    % mFg$;F  
    %       n    m    Zernike function           Normalization <4+P37^ ~  
    %       -------------------------------------------------- 5CZyA`3V^5  
    %       0    0    1                                 1 PJiU2Y33  
    %       1    1    r * cos(theta)                    2 E/g"}yR  
    %       1   -1    r * sin(theta)                    2 K fD. J)  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) KJRAW]?{  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ;+<IWDo  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) )O"E#%  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) kL%ot<rt)w  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 9Q=VRH:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) kh9'W<tE  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) 3("C'(W  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) g35!a<JW  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nm@ h5ON_  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) gYhY1Mym  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GuO}CQs^W  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) r5DR F4,7  
    %       -------------------------------------------------- `*Yw-HL  
    % H0;Iv#S!  
    %   Example 1: EW|$qLg  
    % qS#G7~ur>y  
    %       % Display the Zernike function Z(n=5,m=1) 3Rc*vVnI  
    %       x = -1:0.01:1; N$6e KJ]  
    %       [X,Y] = meshgrid(x,x); hE|P|0U,n  
    %       [theta,r] = cart2pol(X,Y); *{3d+j/?/  
    %       idx = r<=1; IplOXD  
    %       z = nan(size(X)); g3z/yj  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); 0n{.96r0R  
    %       figure f^FFn32u  
    %       pcolor(x,x,z), shading interp -NXxxK  
    %       axis square, colorbar q7X#LYk  
    %       title('Zernike function Z_5^1(r,\theta)') ?qNU*d  
    % 6N#hN)/  
    %   Example 2: rZKfb}ANQ  
    % Q,[G?vbj  
    %       % Display the first 10 Zernike functions ^O18\a  
    %       x = -1:0.01:1; g}s$s}  
    %       [X,Y] = meshgrid(x,x); j{%;n40$  
    %       [theta,r] = cart2pol(X,Y); i)?7+<X  
    %       idx = r<=1; QselW]  
    %       z = nan(size(X)); .\ ;'>qy  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 6nZ]y&$G-k  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; e0TYHr)X>3  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; C(ij_>  
    %       y = zernfun(n,m,r(idx),theta(idx)); UGSZg|&6#*  
    %       figure('Units','normalized') &"^F;z/  
    %       for k = 1:10 a_RY Yj  
    %           z(idx) = y(:,k); p?i.<Z  
    %           subplot(4,7,Nplot(k)) *4}_2"[  
    %           pcolor(x,x,z), shading interp B?! L~J@p  
    %           set(gca,'XTick',[],'YTick',[]) }|.<EkA  
    %           axis square Wef%f] u  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L[x`i'0B  
    %       end M7TLQqaF  
    % 'XK 'T\m  
    %   See also ZERNPOL, ZERNFUN2. .xN<<+|_v'  
    ,U~A=bsa  
    %   Paul Fricker 11/13/2006 JT?u[p Q^  
    w:t~M[kTW  
    XwY,xg&o  
    % Check and prepare the inputs: tm+*ik=x|  
    % ----------------------------- !Y ,7%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wXIRn?z  
        error('zernfun:NMvectors','N and M must be vectors.') $G".PWc  
    end eFG/!b<17  
    {DRk{>K,  
    if length(n)~=length(m) YzESV Th  
        error('zernfun:NMlength','N and M must be the same length.') /65YHXg,  
    end <tD,Uu{P  
    gXxi; g  
    n = n(:); #L*\^ c  
    m = m(:); "`>6M&`U  
    if any(mod(n-m,2)) /eV)5`V  
        error('zernfun:NMmultiplesof2', ... !*-|!Vz  
              'All N and M must differ by multiples of 2 (including 0).') MgeC-XQM  
    end KN}#8.'>3  
    x3q^}sj%  
    if any(m>n) ?2]fE[SqY  
        error('zernfun:MlessthanN', ... '(.5!7?Qc  
              'Each M must be less than or equal to its corresponding N.') yaR>?[h  
    end y98FEG#S}  
    .C'\U[A{  
    if any( r>1 | r<0 ) "^#O7.oVi+  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') cibl j?"Wi  
    end Va8 }JD  
    S2$66xr#  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) bo\ bs1  
        error('zernfun:RTHvector','R and THETA must be vectors.') jZA1fV  
    end uj8saNu  
    Z[#8F&QV!m  
    r = r(:); G"/;Cq=t  
    theta = theta(:); eC-&.Fl  
    length_r = length(r); p:~#(/GWf  
    if length_r~=length(theta) 74([~Qs _M  
        error('zernfun:RTHlength', ... L]=]/>jQ6  
              'The number of R- and THETA-values must be equal.') cfTT7O#Dc  
    end &W\e 5X<A  
    v3DK0MW  
    % Check normalization: U1YqyG8  
    % -------------------- y!b"Cj  
    if nargin==5 && ischar(nflag) Cog}a  
        isnorm = strcmpi(nflag,'norm'); RN`TUCQL  
        if ~isnorm bJ:5pBJ3  
            error('zernfun:normalization','Unrecognized normalization flag.') 1S?~ c25=h  
        end #:?:gY<  
    else Qsbyy>o)  
        isnorm = false; [j6]!p]S$  
    end y4kn2Mw;  
    #(tdJ<HvC|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R>bg3j  
    % Compute the Zernike Polynomials A|"T8KSMB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EID-ROMO  
    sVh)Ofn  
    % Determine the required powers of r: O~5t[  
    % ----------------------------------- x// uF  
    m_abs = abs(m); WOO3z5 La  
    rpowers = []; |>ztx}\  
    for j = 1:length(n) rZgu`5 <a  
        rpowers = [rpowers m_abs(j):2:n(j)]; q]4h#?.-1v  
    end &b (*  
    rpowers = unique(rpowers); }1 O"?6  
    :q/s%`ob  
    % Pre-compute the values of r raised to the required powers, ,a>Dv@$Y  
    % and compile them in a matrix: 6w%n$tiX  
    % ----------------------------- vAM1|,U  
    if rpowers(1)==0 N:B<5l '  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g[~{iu_$d  
        rpowern = cat(2,rpowern{:}); #w''WOk@ZG  
        rpowern = [ones(length_r,1) rpowern]; "M:ui0YP  
    else Z` kVyuQ  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @x1cV_s[  
        rpowern = cat(2,rpowern{:}); 9,8/DW.K  
    end kI"9T`owR  
    y{M7kYWtHV  
    % Compute the values of the polynomials: ~C{:G;Iy0  
    % -------------------------------------- {+lU4u  
    y = zeros(length_r,length(n)); ,|*Gr"Q=  
    for j = 1:length(n) Tv#d>ZSD  
        s = 0:(n(j)-m_abs(j))/2; l$5nv5r  
        pows = n(j):-2:m_abs(j); 4V9BmVS|Th  
        for k = length(s):-1:1 m ^FKE:  
            p = (1-2*mod(s(k),2))* ... 6D| F1UFU  
                       prod(2:(n(j)-s(k)))/              ... &Sg]P  
                       prod(2:s(k))/                     ... 29=ob("  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <zpxodM@T  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); fln[Q2zl  
            idx = (pows(k)==rpowers); 6D ]fDeH\  
            y(:,j) = y(:,j) + p*rpowern(:,idx); B9,39rG/7+  
        end TFOx=_.%i  
         [.&JQ  
        if isnorm vVMoCG"f  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); qMEd R;o  
        end ^W sgAyCB  
    end  j=pg5T  
    % END: Compute the Zernike Polynomials ]-t>F  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J#Q>dC7  
    Jt}`oFQ5l  
    % Compute the Zernike functions: ktPM66`b  
    % ------------------------------ ~0+<-T  
    idx_pos = m>0; )*_G/<N) |  
    idx_neg = m<0; g}R#0gkdk}  
    'Ev[G6vo  
    z = y; 7(D)U)9h  
    if any(idx_pos) /*;a6S8q  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [ PN2^  
    end T}{zh  
    if any(idx_neg) >!qtue7B  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); BEax[=&W  
    end xyo~p,(~t  
    (Zx--2lc  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) )-"<19eu  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. /pkN=OBR  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated >VZxDJ$R  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive v6DjNyg<x  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, uO]|YF  
    %   and THETA is a vector of angles.  R and THETA must have the same Id^q!4Th9  
    %   length.  The output Z is a matrix with one column for every P-value, $aEv*{$y  
    %   and one row for every (R,THETA) pair. wZ0bD&B  
    % U:99w  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike x]`F#5j  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) L8n?F#q  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) cQxUEY('+  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 66-\}8f8a  
    %   for all p. 94O\M RQ*  
    % Wm"q8-<<  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 vN v'%;L  
    %   Zernike functions (order N<=7).  In some disciplines it is FO(QsR=\s  
    %   traditional to label the first 36 functions using a single mode "5dke^yk0  
    %   number P instead of separate numbers for the order N and azimuthal 4Th?q{X  
    %   frequency M. _'Jjt9@S  
    % @i> r(X  
    %   Example: 1P"{TMd?  
    % Fs~*-R$  
    %       % Display the first 16 Zernike functions FZ%h7Oe  
    %       x = -1:0.01:1; &Jb$YKt  
    %       [X,Y] = meshgrid(x,x); g]JJ!$*1  
    %       [theta,r] = cart2pol(X,Y); Zgp9Uu}"  
    %       idx = r<=1; Lp"OXJ*es  
    %       p = 0:15; \VEnP=*:W  
    %       z = nan(size(X)); >Vx_Xv`Jwb  
    %       y = zernfun2(p,r(idx),theta(idx)); %Iflf]l  
    %       figure('Units','normalized') w%TrL+v  
    %       for k = 1:length(p) ;X]B0KFe7  
    %           z(idx) = y(:,k); <sm"3qs"_  
    %           subplot(4,4,k) hC8WRxEGq  
    %           pcolor(x,x,z), shading interp 'Q=)-  
    %           set(gca,'XTick',[],'YTick',[]) "9^b1UH<  
    %           axis square ^HR8.9^[1u  
    %           title(['Z_{' num2str(p(k)) '}']) b{-"GqMO  
    %       end ( ./MFf  
    % # "c'eG0  
    %   See also ZERNPOL, ZERNFUN. Q jXJo$I6  
    :4)x  
    %   Paul Fricker 11/13/2006 KwMt@1Z  
    XM+.Hel  
    >WZbb d-  
    % Check and prepare the inputs: @=AQr4&  
    % ----------------------------- LKI\(%ba#  
    if min(size(p))~=1 R=a4zVQ  
        error('zernfun2:Pvector','Input P must be vector.') e<{ d{  
    end *7Y#G8 s  
    (x/:j*`K  
    if any(p)>35 -0q|AB<  
        error('zernfun2:P36', ... 68bvbig  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... gCsN\z  
               '(P = 0 to 35).']) Q[#8ErUY  
    end T#!% Uzz  
    l=T;hk  
    % Get the order and frequency corresonding to the function number: {Mb<on W  
    % ---------------------------------------------------------------- XP!m]\E&I  
    p = p(:); B_[I/ ?  
    n = ceil((-3+sqrt(9+8*p))/2); \reVA$M [  
    m = 2*p - n.*(n+2); zOMxg00  
    _IOUhMo  
    % Pass the inputs to the function ZERNFUN: G Wa6FX:/  
    % ---------------------------------------- uUx7>algF  
    switch nargin 1}c /l<d  
        case 3 y2?9pVLa\y  
            z = zernfun(n,m,r,theta); hR0a5   
        case 4 GTfM *b  
            z = zernfun(n,m,r,theta,nflag); Wk3-J&QbS  
        otherwise k kD#Bb  
            error('zernfun2:nargin','Incorrect number of inputs.')  ? .SiT5  
    end S9$,.aq  
    '3^qW  
    % 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) r6Vw!^]8u8  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 1V[ZklS  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >{~xO 6H  
    %   order N and frequency M, evaluated at R.  N is a vector of }oG6XI9  
    %   positive integers (including 0), and M is a vector with the Ca?w"m~h  
    %   same number of elements as N.  Each element k of M must be a 2P`./1L  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) _nzq(m1@  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [#\OCdb*3  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix D?S|]]Y!q  
    %   with one column for every (N,M) pair, and one row for every &,PA+#  
    %   element in R. M^HYkXn[  
    % fk?!0M6d  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 7=X6_AD  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 5M'cOJ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to cf>lY  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 2<n 18-|OQ  
    %   for all [n,m].  )$f?v22  
    %  Jn|<G  
    %   The radial Zernike polynomials are the radial portion of the >m%TUQ#%  
    %   Zernike functions, which are an orthogonal basis on the unit 0)h.[O8@>  
    %   circle.  The series representation of the radial Zernike ^fd*KM  
    %   polynomials is E>*b,^J7g  
    % `g(#~0R  
    %          (n-m)/2 _a"| :kX  
    %            __ CiHx.5TiC  
    %    m      \       s                                          n-2s B/lIn' =  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r xA:;wV  
    %    n      s=0 @|ZUyat  
    % q0>9T  
    %   The following table shows the first 12 polynomials. ,mCf{V]#  
    % /#: *hn  
    %       n    m    Zernike polynomial    Normalization B3[X{n$px  
    %       --------------------------------------------- T$r/XAs  
    %       0    0    1                        sqrt(2) xZ2 1i QeN  
    %       1    1    r                           2 N@k' s   
    %       2    0    2*r^2 - 1                sqrt(6) d72 yu3  
    %       2    2    r^2                      sqrt(6) RDQ]_wsyKG  
    %       3    1    3*r^3 - 2*r              sqrt(8) Dn#5H{D-d  
    %       3    3    r^3                      sqrt(8) x7l}u`N4  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) q2*)e/}H  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) SV ~QH&0'  
    %       4    4    r^4                      sqrt(10) }mZCQJ#`  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) a8[%-eW,  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) "tk1W>liIN  
    %       5    5    r^5                      sqrt(12) }*-fh$QJ  
    %       --------------------------------------------- f]Aa$\@b  
    % I hSXU<]  
    %   Example: P*?2+.  
    % ;2fzA<RkK  
    %       % Display three example Zernike radial polynomials L!/{Z  
    %       r = 0:0.01:1; $ <[r3  
    %       n = [3 2 5]; "k [$euV  
    %       m = [1 2 1]; *Y53b Z  
    %       z = zernpol(n,m,r); ZZ!6O/M  
    %       figure Eqny'44  
    %       plot(r,z) {t0!N]'  
    %       grid on Oa@SyroF=  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') #|:q"l9  
    % yl' IL#n]r  
    %   See also ZERNFUN, ZERNFUN2. 066\zAPdH  
    1T~`$zS7  
    % A note on the algorithm. J$jLGy&'  
    % ------------------------ }\N ~%?6D  
    % The radial Zernike polynomials are computed using the series g;o5m}  
    % representation shown in the Help section above. For many special PDgZb  
    % functions, direct evaluation using the series representation can 4T)`%Oo<}  
    % produce poor numerical results (floating point errors), because <Z]j89wzDZ  
    % the summation often involves computing small differences between $'*{&/@  
    % large successive terms in the series. (In such cases, the functions ^eRbp?H*T  
    % are often evaluated using alternative methods such as recurrence 2Z^p)  
    % relations: see the Legendre functions, for example). For the Zernike XNvlx4  
    % polynomials, however, this problem does not arise, because the \Z~@/OVc  
    % polynomials are evaluated over the finite domain r = (0,1), and #f=41d%  
    % because the coefficients for a given polynomial are generally all M M @&QaK  
    % of similar magnitude. V%M@zd?u.  
    % 3dtL[aVwY  
    % ZERNPOL has been written using a vectorized implementation: multiple wz:,gpH  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] !14v Ovj4{  
    % values can be passed as inputs) for a vector of points R.  To achieve l0',B*og  
    % this vectorization most efficiently, the algorithm in ZERNPOL }.=wQ_  
    % involves pre-determining all the powers p of R that are required to )T(1oK(g  
    % compute the outputs, and then compiling the {R^p} into a single g6 SZ4WV  
    % matrix.  This avoids any redundant computation of the R^p, and G&6`?1k  
    % minimizes the sizes of certain intermediate variables. fE>JoQs38  
    % ?6MUyH]a  
    %   Paul Fricker 11/13/2006 *'n=LB8R  
    yWH!v]S  
    4>HQ2S{t  
    % Check and prepare the inputs: YZ->ep}  
    % ----------------------------- ?FZ) LZM  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <\ ".6=E#W  
        error('zernpol:NMvectors','N and M must be vectors.') ?%Pi#%P  
    end +- hfl/$  
    <{eJbNp  
    if length(n)~=length(m) a63Ud<_a7  
        error('zernpol:NMlength','N and M must be the same length.') z=rSb4"W  
    end mLHl]xs4  
    7$L*nf  
    n = n(:); 8*]dA ft  
    m = m(:); [F27i#'I]  
    length_n = length(n); 7<5=fYb r  
    u$"Ew^C  
    if any(mod(n-m,2)) CT=5V@_u\  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') r,GgMk  
    end Qz<i{r-z  
    ]6WP;.[  
    if any(m<0) j  W -K  
        error('zernpol:Mpositive','All M must be positive.') J@q!N;eh|  
    end ]#FQde4]5  
    3HndE~_C&  
    if any(m>n)  zK:2.4  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') bg*@N  
    end =d JRBl  
    V67<Ky>  
    if any( r>1 | r<0 ) =A/$[POr  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') +L=a\8Ep  
    end 2cv=7!K4Uv  
    l27J  
    if ~any(size(r)==1) d]^\qeG^p  
        error('zernpol:Rvector','R must be a vector.') QLH6Nmk  
    end HhUk9 >7  
    )dRB I)P  
    r = r(:); 6"o@d8>v  
    length_r = length(r); 6[]O3Aa  
    >td\PW~X  
    if nargin==4 G>+iisb%  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ZX40-6#O  
        if ~isnorm 4~0 @(3  
            error('zernpol:normalization','Unrecognized normalization flag.') cq 1)b\|  
        end 4AN(4"$N  
    else a +`;:tX,  
        isnorm = false; D^H4]7wG@  
    end R lmeZy4.  
    V_H0z  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b5UIX Kim  
    % Compute the Zernike Polynomials -+ Mh( 'K  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9Qc=D"'  
    24d{ol)  
    % Determine the required powers of r: |\h<!xR  
    % ----------------------------------- !u%XvxJwDb  
    rpowers = []; !MD uj  
    for j = 1:length(n) P<R'S  
        rpowers = [rpowers m(j):2:n(j)]; E"t79dD  
    end R"{oj]d;$F  
    rpowers = unique(rpowers); C,dRdEB>  
    GuRJ  
    % Pre-compute the values of r raised to the required powers, YR0.m%U,  
    % and compile them in a matrix: fzjZiBK@  
    % ----------------------------- x@,B))WlGr  
    if rpowers(1)==0 SHUn<+/e  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _!E/ em  
        rpowern = cat(2,rpowern{:}); {'q(a4  
        rpowern = [ones(length_r,1) rpowern]; h[j(@P  
    else w?A6S-z  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,gn**E  
        rpowern = cat(2,rpowern{:}); uBxs`'C  
    end <FU1|  
    K}Rq<z W  
    % Compute the values of the polynomials: ;cW9NS3:  
    % -------------------------------------- 5^GrG|~  
    z = zeros(length_r,length_n); Gbc2\A\  
    for j = 1:length_n "P8cgj C  
        s = 0:(n(j)-m(j))/2; JReJlDu  
        pows = n(j):-2:m(j); C4t@;U=x  
        for k = length(s):-1:1 ](sT,'  
            p = (1-2*mod(s(k),2))* ... V=5v7Y3( j  
                       prod(2:(n(j)-s(k)))/          ... sn:wLc/GAd  
                       prod(2:s(k))/                 ... 0^zp*u  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... bRK[u\,  
                       prod(2:((n(j)+m(j))/2-s(k))); eR:!1z_h  
            idx = (pows(k)==rpowers); Nmu=p~f}3`  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 3^p;'7x  
        end mpDQhD[n  
         #(Ezt% ^  
        if isnorm >iFi~)i_4y  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =&v&qn e9  
        end jmh$6 N% F  
    end .V\: )\<|  
    . I#dR*  
    % 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)  {-fhp@;  
    ?Dr K2;q  
    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。 L~PBD?l  
    6yN8 (&`  
    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)