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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 !kg)84C[  
    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! BlvNBB1^  
     
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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}~l(  
    function z = zernfun(n,m,r,theta,nflag) :6z0Ep"  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y e}y_W  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N EN%Xs578  
    %   and angular frequency M, evaluated at positions (R,THETA) on the []Z| *+=Q  
    %   unit circle.  N is a vector of positive integers (including 0), and [vaG{4m  
    %   M is a vector with the same number of elements as N.  Each element IfZaK([  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) lC1X9Op  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, vN7ihe[C  
    %   and THETA is a vector of angles.  R and THETA must have the same x./jTebeO  
    %   length.  The output Z is a matrix with one column for every (N,M) 7}r!%<^  
    %   pair, and one row for every (R,THETA) pair. *3<m<<>U  
    % _+8$=k2nM  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6iFd[<.*j  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f41!+W=  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral <v('HLA  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, " I@Z:[=2  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <!zItFMD[m  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &T}v1c7)  
    % "7 )F";_(^  
    %   The Zernike functions are an orthogonal basis on the unit circle. 5.|rzk>  
    %   They are used in disciplines such as astronomy, optics, and CFZ= !s)B  
    %   optometry to describe functions on a circular domain. ;<q@>p[  
    % 't{=n[  
    %   The following table lists the first 15 Zernike functions. A}\Rms 2  
    % )}c$n  
    %       n    m    Zernike function           Normalization 0{PK]qp7  
    %       -------------------------------------------------- EW4XFP4 c  
    %       0    0    1                                 1 RkLH}`#  
    %       1    1    r * cos(theta)                    2 Ok6Y&#'P  
    %       1   -1    r * sin(theta)                    2 2.&v{gq  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) jVRd[  
    %       2    0    (2*r^2 - 1)                    sqrt(3) ^B& Z  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `bT{E.(T  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) -r-`T s  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) u(ZS sftat  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) )hQNIt3o_  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) xel&8 `  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) s !8]CV>  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~:)$~g7>b  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) I/WnF"yP  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w.l#Z} k  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 'KQu z)-  
    %       -------------------------------------------------- Y+?bo9CES!  
    % $z mES tcm  
    %   Example 1: C [2tH2*#  
    % /2HwK/RZ  
    %       % Display the Zernike function Z(n=5,m=1) Gcs+@7!b  
    %       x = -1:0.01:1; #zy,x  
    %       [X,Y] = meshgrid(x,x); RL&3 P@r  
    %       [theta,r] = cart2pol(X,Y); h'-TZXs0e1  
    %       idx = r<=1; T>uLqd{hH  
    %       z = nan(size(X)); D}"GrY 5  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~hvhT}lE  
    %       figure Wt3\&.n  
    %       pcolor(x,x,z), shading interp *h =7:*n  
    %       axis square, colorbar TVFGonVY  
    %       title('Zernike function Z_5^1(r,\theta)') ?|hzAF"U  
    % %?wuKZLnc  
    %   Example 2: p[uwG31IL`  
    % t'Q48QAb?  
    %       % Display the first 10 Zernike functions +u=xBhZ  
    %       x = -1:0.01:1; r\3In-(AT  
    %       [X,Y] = meshgrid(x,x); WJ.PPq>]F  
    %       [theta,r] = cart2pol(X,Y); 7>ODaj   
    %       idx = r<=1; zWY6D4   
    %       z = nan(size(X)); v l*RRoJ  
    %       n = [0  1  1  2  2  2  3  3  3  3]; W;-Qze\D  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; |M K-~ep  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; i5n 'f6C  
    %       y = zernfun(n,m,r(idx),theta(idx)); q$t& *O_  
    %       figure('Units','normalized') ,DE%p +q  
    %       for k = 1:10 ifgaBXT55  
    %           z(idx) = y(:,k); ^2??]R&Q  
    %           subplot(4,7,Nplot(k)) g]ihwm~  
    %           pcolor(x,x,z), shading interp .Nf*Yqs0  
    %           set(gca,'XTick',[],'YTick',[]) r=w%"3vb^  
    %           axis square MoX* e  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) TRq~n7Y7C  
    %       end 8EE7mEmLH  
    % Ci*5E$+\  
    %   See also ZERNPOL, ZERNFUN2. x9ws@=[:  
    ~T-.k 7t  
    %   Paul Fricker 11/13/2006 _N]yI0k(  
    cu"%>>,,  
    I&xRK'  
    % Check and prepare the inputs: Qxvz}r.l]  
    % ----------------------------- JIQzP?+?  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [)Ge^yI7  
        error('zernfun:NMvectors','N and M must be vectors.') vn_avYwiy  
    end an7N<-?  
    5Ci}w|c/>  
    if length(n)~=length(m) WIGb7}egR  
        error('zernfun:NMlength','N and M must be the same length.') .U3p~M+  
    end )5t_tPv  
    L9kP8&&KK  
    n = n(:); W#wM PsB  
    m = m(:); + mcN6/  
    if any(mod(n-m,2)) ZRHTvxf  
        error('zernfun:NMmultiplesof2', ... NWpRzh8$u  
              'All N and M must differ by multiples of 2 (including 0).') a@a1/ 3  
    end "L)pH@)  
    ?~K2&eo  
    if any(m>n) 1)R)+`y  
        error('zernfun:MlessthanN', ... D[r  
              'Each M must be less than or equal to its corresponding N.') MQ+ek4  
    end t}tKm  
    v\ox:C  
    if any( r>1 | r<0 ) 6:!fyia  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') <#Lw.;(U;k  
    end 7h<K)aT  
    !+6l.`2WI  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 8tL61x{]  
        error('zernfun:RTHvector','R and THETA must be vectors.') /LD*8 a  
    end yR!>80$j  
    4_Jdh48-d  
    r = r(:); &zp5do;m  
    theta = theta(:); QD<4(@c5|  
    length_r = length(r); } :mI6zsNj  
    if length_r~=length(theta) ^ \?9W  
        error('zernfun:RTHlength', ... B<R-|-#  
              'The number of R- and THETA-values must be equal.') t5k&xV=~ #  
    end YZ>cE#  
    v(^rq  
    % Check normalization: LZVO9e]  
    % -------------------- P Cf|^X#B  
    if nargin==5 && ischar(nflag) m&q;.|W  
        isnorm = strcmpi(nflag,'norm'); #r:`bQ0;  
        if ~isnorm wj^I1;lO  
            error('zernfun:normalization','Unrecognized normalization flag.') .T|NB8 rS  
        end O2G+ '  
    else {P-PH$ E-  
        isnorm = false; Kq$Zyf=E  
    end A E711l-  
    nf4 P2<L!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s]iOC6v  
    % Compute the Zernike Polynomials XbC8t &Q],  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3(:mRb}  
    + LwoBn>6  
    % Determine the required powers of r: >D<=9G(a  
    % ----------------------------------- x\rZoF.NQ  
    m_abs = abs(m); *eP4dGe&  
    rpowers = []; @nP}q!y  
    for j = 1:length(n) }WbN)  
        rpowers = [rpowers m_abs(j):2:n(j)]; + joE  
    end !iVFzG @m  
    rpowers = unique(rpowers); dX*>?a  
    h+UscdU l  
    % Pre-compute the values of r raised to the required powers, :RsPGj6   
    % and compile them in a matrix: 1l_}O1  
    % ----------------------------- 2M?lgh4"  
    if rpowers(1)==0 p L@zZK0  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); v+#j>   
        rpowern = cat(2,rpowern{:}); ib_Gy77Os  
        rpowern = [ones(length_r,1) rpowern]; VK;x6*Y  
    else \6hL W_q1  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,NEs{! T  
        rpowern = cat(2,rpowern{:}); !5j3gr ~  
    end \z9?rvT:  
    (NdgF+'=  
    % Compute the values of the polynomials: >!1f`  
    % -------------------------------------- G)hH?_U#T  
    y = zeros(length_r,length(n)); +ca296^  
    for j = 1:length(n) :dN35Y]a  
        s = 0:(n(j)-m_abs(j))/2; \&5@yh  
        pows = n(j):-2:m_abs(j); Wp}9%Mq~Jy  
        for k = length(s):-1:1 >k}/$R+  
            p = (1-2*mod(s(k),2))* ... UD2<!a'T  
                       prod(2:(n(j)-s(k)))/              ... rfRo*u2"  
                       prod(2:s(k))/                     ... cJEz>Z6[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... C..2y4bA}  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 0:'jU  
            idx = (pows(k)==rpowers); ?d<:V.1U@  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 51qIo4$  
        end ok s=|'&  
         !rg0U<bO!  
        if isnorm cqY.^f.  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6 ]PM!6  
        end XDk o{jEJ  
    end sBtG}Mo)  
    % END: Compute the Zernike Polynomials Y@H,Lk  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% } Tr83B|  
    B" m:<@ "  
    % Compute the Zernike functions: ~f10ZB_k>'  
    % ------------------------------ :.o=F`W  
    idx_pos = m>0; 9c{%m4  
    idx_neg = m<0; sNfb %r  
    qTHg[sME  
    z = y; ZBR^[OXO  
    if any(idx_pos) J(0=~Z[  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pq?[wp"  
    end _8li4;F  
    if any(idx_neg) LnTe_Q7_  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); cm@oun  
    end 'Z2N{65  
    1mn$Rh&dO  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) t6bWSz0  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Gj7QG IKx  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 2gL[\/s  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive *T>#zR{  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, t66f 7AR  
    %   and THETA is a vector of angles.  R and THETA must have the same I6hhU;)C  
    %   length.  The output Z is a matrix with one column for every P-value, !v5sWVVR  
    %   and one row for every (R,THETA) pair. h"BhTx7E}  
    % 2>MP:yY;K  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 0$"Q&5Y  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Sa[EnC  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) j |'# 5H`  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 7o965h  
    %   for all p. ZaRr2Z:!  
    % |,a%z-l  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36  zfjDb  
    %   Zernike functions (order N<=7).  In some disciplines it is vN0L( B  
    %   traditional to label the first 36 functions using a single mode \9>g;qPg}  
    %   number P instead of separate numbers for the order N and azimuthal neB.Wu~WH  
    %   frequency M. Ql#W /x,e  
    % UYcyk $da  
    %   Example: ]m/@wW9  
    % \2gvp6  
    %       % Display the first 16 Zernike functions nz&b5Xb2  
    %       x = -1:0.01:1; [I++>4  
    %       [X,Y] = meshgrid(x,x); "]SJbuzh  
    %       [theta,r] = cart2pol(X,Y); f>s#Ngvc  
    %       idx = r<=1; 0i`v:Lq%  
    %       p = 0:15; >uyeI&z  
    %       z = nan(size(X)); 5&n988g C8  
    %       y = zernfun2(p,r(idx),theta(idx)); AF*ni~  
    %       figure('Units','normalized') GFQG(7G9  
    %       for k = 1:length(p) 4 [5lX C  
    %           z(idx) = y(:,k); A{i][1N  
    %           subplot(4,4,k) nj~$%vmA  
    %           pcolor(x,x,z), shading interp iJCY /*C}  
    %           set(gca,'XTick',[],'YTick',[]) q*F~~J!P  
    %           axis square Ypn%[sSOp  
    %           title(['Z_{' num2str(p(k)) '}']) I*+LJy;j  
    %       end taWirq d9  
    % -~( 0O  
    %   See also ZERNPOL, ZERNFUN. .fLiXx  
    r{R[[]p  
    %   Paul Fricker 11/13/2006 c]%;^)  
    ,`%k'ecN  
    D% v:PYf  
    % Check and prepare the inputs: =A0"0D{\  
    % ----------------------------- uGuc._}=  
    if min(size(p))~=1 Z9J =vzsHE  
        error('zernfun2:Pvector','Input P must be vector.') i_[ HcgT-  
    end DJ1XN pm  
    nJldz;  
    if any(p)>35 H7z>S G0  
        error('zernfun2:P36', ... YZ"+c&V"  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... @b ::6n/u  
               '(P = 0 to 35).']) a2c x  
    end =RW* %8C  
    5(iSOsb  
    % Get the order and frequency corresonding to the function number: bK_0NrXP  
    % ---------------------------------------------------------------- gD,YQ%aq  
    p = p(:); v{mv*`~nA\  
    n = ceil((-3+sqrt(9+8*p))/2); Q-! i$#-  
    m = 2*p - n.*(n+2); i$`|Y*  
    Dh\S`nfFq  
    % Pass the inputs to the function ZERNFUN: G zJ9N`  
    % ---------------------------------------- ;-3h~k  
    switch nargin p<of<YU)  
        case 3 8~&F/C*  
            z = zernfun(n,m,r,theta); $?]@_=  
        case 4 _qC+'RE3  
            z = zernfun(n,m,r,theta,nflag); W;3 R;  
        otherwise _%A/ )  
            error('zernfun2:nargin','Incorrect number of inputs.') ZfFIX5Qd\  
    end )^jQkfL  
    5z9r S<  
    % 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) [=XZza.z  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 4u3 \xR?w6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of v t^r1j  
    %   order N and frequency M, evaluated at R.  N is a vector of ,3wI~ j=  
    %   positive integers (including 0), and M is a vector with the $?H]S]#|}.  
    %   same number of elements as N.  Each element k of M must be a JiKImz  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) #D!$~ h&i  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Y;fuh[#  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix #[no~&E  
    %   with one column for every (N,M) pair, and one row for every X?KGb{  
    %   element in R. &E.OyqGZV  
    % %3]3r*e&5  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 'b LP ~  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is kA1RfSS  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to z `\# $  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ,3G$`  
    %   for all [n,m]. i0ILb/LS  
    % X tJswxw`K  
    %   The radial Zernike polynomials are the radial portion of the "F&Tnhh4  
    %   Zernike functions, which are an orthogonal basis on the unit 6tOP}X  
    %   circle.  The series representation of the radial Zernike <2n'}&F  
    %   polynomials is Rb{+Ki  
    % qsI{ b<n  
    %          (n-m)/2 FpP\-+Sl  
    %            __ V^j3y`K  
    %    m      \       s                                          n-2s k%"$$uo  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ' " Bex`  
    %    n      s=0 =ft9T&ciD  
    % }j& O/ Up  
    %   The following table shows the first 12 polynomials. 'g. :MQ8  
    % Bfbl#ZkyL  
    %       n    m    Zernike polynomial    Normalization g;$E1U=R-E  
    %       --------------------------------------------- w+Ad$4Pf"  
    %       0    0    1                        sqrt(2) gs$3)t  
    %       1    1    r                           2 !.9l4@z#  
    %       2    0    2*r^2 - 1                sqrt(6) RI?NB6U  
    %       2    2    r^2                      sqrt(6) J09*v )L  
    %       3    1    3*r^3 - 2*r              sqrt(8) Vz%"9`r  
    %       3    3    r^3                      sqrt(8) <MRC%!.  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) %(fL?  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) rU],J!LF  
    %       4    4    r^4                      sqrt(10) 1Pu ,:Jt  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) C,[ L/!  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) X d!Cp  
    %       5    5    r^5                      sqrt(12) baqn7k"  
    %       --------------------------------------------- IoQr+:_R  
    % Z&TD+fT<  
    %   Example: 8a7YHUL<3i  
    % r i,2clp  
    %       % Display three example Zernike radial polynomials 5@K\c6   
    %       r = 0:0.01:1; IT,"8 s  
    %       n = [3 2 5]; g .3f2w  
    %       m = [1 2 1]; XnD0eua#  
    %       z = zernpol(n,m,r); nZe\5`  
    %       figure $$42pb.  
    %       plot(r,z) m^ z,,t9  
    %       grid on e "_&z# 2_  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') V!mWn|lf  
    % ma3Qi/  
    %   See also ZERNFUN, ZERNFUN2. ~M*7N@D  
    Ks|gL#)*Ku  
    % A note on the algorithm. 'HCnB]1  
    % ------------------------ .a {QA  
    % The radial Zernike polynomials are computed using the series 8:~b &>   
    % representation shown in the Help section above. For many special anLbl#UV  
    % functions, direct evaluation using the series representation can !TGr.R  
    % produce poor numerical results (floating point errors), because {798=pC<.  
    % the summation often involves computing small differences between @ozm;  
    % large successive terms in the series. (In such cases, the functions wtq,`'B  
    % are often evaluated using alternative methods such as recurrence ]XY0c6 <  
    % relations: see the Legendre functions, for example). For the Zernike @?m+Z"o|z  
    % polynomials, however, this problem does not arise, because the CDJ$hu  
    % polynomials are evaluated over the finite domain r = (0,1), and ^mAJ[^%  
    % because the coefficients for a given polynomial are generally all )q^(T1  
    % of similar magnitude. "8FSA`>=  
    % |l$ u<3  
    % ZERNPOL has been written using a vectorized implementation: multiple  -W9gH  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )mj<{Td`  
    % values can be passed as inputs) for a vector of points R.  To achieve L,6MF,vx  
    % this vectorization most efficiently, the algorithm in ZERNPOL Ny]lvgu9X  
    % involves pre-determining all the powers p of R that are required to a"k'm}hVY$  
    % compute the outputs, and then compiling the {R^p} into a single Trpgx  
    % matrix.  This avoids any redundant computation of the R^p, and N_0pO<<cs  
    % minimizes the sizes of certain intermediate variables. s~=g*99H  
    % z[*zuo  
    %   Paul Fricker 11/13/2006 gbJG`zC>U  
    RTZ:U@  
    (,shiK[5f  
    % Check and prepare the inputs: %Or2iuO%-,  
    % ----------------------------- JfSdUWxT  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I-TlrW=t  
        error('zernpol:NMvectors','N and M must be vectors.') FQ1arUOFW,  
    end  Ll?g.z"  
    @bE~@4mOu  
    if length(n)~=length(m) $ND90my  
        error('zernpol:NMlength','N and M must be the same length.') p x0Sy|  
    end @FU~1u3d  
    A4}#U=3tI  
    n = n(:); j|k @MfA  
    m = m(:); h>| g2h  
    length_n = length(n); db'K!M)  
    IEc>.J|T&  
    if any(mod(n-m,2)) 1b8c67j[  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ,b4g.CV  
    end ;KL9oV!<f  
    ;sCU [4  
    if any(m<0) Hl/7(FJqc>  
        error('zernpol:Mpositive','All M must be positive.') {79qtq%W{  
    end 1!d)PK>1$  
    sSz%V[X WL  
    if any(m>n) 4 ]sCr+   
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') brfKd]i  
    end {!MVc<G.  
    Vli3>K&  
    if any( r>1 | r<0 ) y)tYSTJK  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') V'C-'Ythwf  
    end i_NJ -K  
    fy`+Efuj  
    if ~any(size(r)==1) VcrVaBw  
        error('zernpol:Rvector','R must be a vector.') 6Etss!_  
    end }s(C^0x  
    >IBTBh_ka  
    r = r(:); Ww=O=c5uOu  
    length_r = length(r); >gnF]<  
    X% X$Y6  
    if nargin==4 i+1Qf  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); j3{HkcjJG  
        if ~isnorm Hsgy'X%om  
            error('zernpol:normalization','Unrecognized normalization flag.') 3(C :X1  
        end dHq#  
    else bs BZ E  
        isnorm = false; bQ"N ;d)e  
    end K?[)E3  
    6{8/P'@/Zz  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "9ue76  
    % Compute the Zernike Polynomials ,z G(u 1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %E  aE,  
    d@Q][7  
    % Determine the required powers of r: S+iP^*L,c  
    % ----------------------------------- <XvYa{t]{  
    rpowers = []; ol<lCp  
    for j = 1:length(n) iE=P'"I  
        rpowers = [rpowers m(j):2:n(j)]; ZtR&wk  
    end ||XIWKF<n2  
    rpowers = unique(rpowers); vf N#NY6  
    `R0Y+#$8h  
    % Pre-compute the values of r raised to the required powers, VAs ( .y  
    % and compile them in a matrix: L1{T ?aII  
    % ----------------------------- rn H}#u+  
    if rpowers(1)==0 FQ!Oxlq,Q  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); k2.G%]j  
        rpowern = cat(2,rpowern{:}); r@yD8D \  
        rpowern = [ones(length_r,1) rpowern]; m:3J!1  
    else *i@T!O(1)M  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); drIK(u\_  
        rpowern = cat(2,rpowern{:}); +sRP<as  
    end r :NH6tAL  
    vd(dNu&,<  
    % Compute the values of the polynomials: hbfsHT  
    % -------------------------------------- lV)G@l[1  
    z = zeros(length_r,length_n); hlC%HA  
    for j = 1:length_n ]4o?BkL  
        s = 0:(n(j)-m(j))/2; {xToz]YA  
        pows = n(j):-2:m(j); 5 VKcV&D  
        for k = length(s):-1:1 sUbF Rq  
            p = (1-2*mod(s(k),2))* ... np=kTJ  
                       prod(2:(n(j)-s(k)))/          ... m8HYW zN  
                       prod(2:s(k))/                 ... YZ**;"<G  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... ~#Aa Ldq  
                       prod(2:((n(j)+m(j))/2-s(k))); OXCQfT@\  
            idx = (pows(k)==rpowers); ^K;hn,R=  
            z(:,j) = z(:,j) + p*rpowern(:,idx); +Vy_9I(4Z  
        end ${>DhfF  
         4.'JLArw  
        if isnorm |Euus5[  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); n_9x"m$  
        end r.<JDdj  
    end *KJ7nRKx(w  
    sOz sY7z3Z  
    % 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)  T`Ro)ORC#  
    Yw[{beo  
    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。 5%& ]  
    3SFg#  
    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)