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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 V<ODt%  
    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! 5Ko "-  
     
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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 ed\,FWR  
    function z = zernfun(n,m,r,theta,nflag) _^&oNm1  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. frGUT#9?n  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )OjbmU!7  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]G|@F :  
    %   unit circle.  N is a vector of positive integers (including 0), and I<[(hPQUf  
    %   M is a vector with the same number of elements as N.  Each element Do2y7,jv  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) iW |]-Ba\  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, .l#Pmd!  
    %   and THETA is a vector of angles.  R and THETA must have the same D:.^]o[  
    %   length.  The output Z is a matrix with one column for every (N,M) mv30xcc  
    %   pair, and one row for every (R,THETA) pair. )NyGV!Zuu  
    % Zsf<)Vx  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G.<9K9K  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), QW~o+N~~  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral +.>O%pNj  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KZD&Ih(vC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized M5P63=1+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. uOougSBV,  
    %  hi.{  
    %   The Zernike functions are an orthogonal basis on the unit circle. N<:Ra~Ay  
    %   They are used in disciplines such as astronomy, optics, and eZg31.  
    %   optometry to describe functions on a circular domain. $g 1p!  
    % Dw.>4bA.  
    %   The following table lists the first 15 Zernike functions. $dwv1@M2  
    % ;39{iU. m  
    %       n    m    Zernike function           Normalization '# (lq5 c  
    %       -------------------------------------------------- TxxW/f9D  
    %       0    0    1                                 1 ^z)lEO  
    %       1    1    r * cos(theta)                    2 ;#f%vs>Y7i  
    %       1   -1    r * sin(theta)                    2 egP3q5~  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) jp[QA\  
    %       2    0    (2*r^2 - 1)                    sqrt(3) j- A S {w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) %81tVhg  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) aD 3$z;E  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) lXB_HDY  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .X:{s,@  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) v,>q]! |a  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) (& ~`!]  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^g~-$t<!  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) poXkH@[O  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4)XN1r:  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) jh g!K.A  
    %       -------------------------------------------------- G&3j/5V  
    % =U,;/f  
    %   Example 1: !;R{-  
    % *DG*&Me  
    %       % Display the Zernike function Z(n=5,m=1) ?BWWb   
    %       x = -1:0.01:1; lgnF\)  
    %       [X,Y] = meshgrid(x,x); pw(`+x]  
    %       [theta,r] = cart2pol(X,Y); <@zOdW|{:  
    %       idx = r<=1; ,t)mCgbcO  
    %       z = nan(size(X)); QQrvT,]  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); uO":\<1#  
    %       figure ]v9<^!  
    %       pcolor(x,x,z), shading interp 71)HxC[6vA  
    %       axis square, colorbar -Mv`|odY/  
    %       title('Zernike function Z_5^1(r,\theta)') ]k,fEn(  
    % q<;9!2py  
    %   Example 2: 9_TZ;e  
    % lezdJ  
    %       % Display the first 10 Zernike functions $s) ^zm~  
    %       x = -1:0.01:1; *$hO C%(  
    %       [X,Y] = meshgrid(x,x); uIWCVR8`Y  
    %       [theta,r] = cart2pol(X,Y); />$)o7U`+  
    %       idx = r<=1; u |f h!-  
    %       z = nan(size(X)); s';jk(i3  
    %       n = [0  1  1  2  2  2  3  3  3  3]; H M76%9!  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; bk>M4l61  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; G1P m!CM=  
    %       y = zernfun(n,m,r(idx),theta(idx)); moc_}(  
    %       figure('Units','normalized') ?=PQQx2_*u  
    %       for k = 1:10 MJ7!f+!5  
    %           z(idx) = y(:,k); xE0+3@_>>  
    %           subplot(4,7,Nplot(k)) 1p{\jCi, 2  
    %           pcolor(x,x,z), shading interp AE<AEq  
    %           set(gca,'XTick',[],'YTick',[]) YJ:CqTy  
    %           axis square [[bMYD1eO  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) J 0s8vAs  
    %       end 8, WQ}cC  
    % F<^,j7@  
    %   See also ZERNPOL, ZERNFUN2. 8`^I. tD  
    ,q:6[~n  
    %   Paul Fricker 11/13/2006 31bKgU{  
     w[VWk  
    |Yk23\!  
    % Check and prepare the inputs: ^K;,,s;0  
    % ----------------------------- 0?sIod  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1nvs51?H  
        error('zernfun:NMvectors','N and M must be vectors.') =Qz 8"rt#  
    end u`("x5sa  
    >j$f$*x  
    if length(n)~=length(m) <rCl  
        error('zernfun:NMlength','N and M must be the same length.') ff{ESFtD  
    end i5)trSM|  
    ;vd%=vR  
    n = n(:); ^@tn+'.  
    m = m(:); }~A-ELe:  
    if any(mod(n-m,2)) 0"<g g5  
        error('zernfun:NMmultiplesof2', ... *emUQ/uvf  
              'All N and M must differ by multiples of 2 (including 0).') ,ciNoP*-~%  
    end t#<q O6&B  
    F1/f:<}  
    if any(m>n) O?{pln  
        error('zernfun:MlessthanN', ... os#j;C]l  
              'Each M must be less than or equal to its corresponding N.') ZPMX19  
    end m_St"`6 .  
    j)J4[j  
    if any( r>1 | r<0 ) qOk4qbl[  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Nf$Y-v?i  
    end JQ.ZAhv  
    pX!S*(Q{  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) rl6vt*g  
        error('zernfun:RTHvector','R and THETA must be vectors.')  snN1  
    end w0Us8JNGz  
    a+J :1'  
    r = r(:); p 6jR,m8S  
    theta = theta(:); VS 8|lgQ  
    length_r = length(r); )iEK7d^-  
    if length_r~=length(theta) A$^}zP'u0<  
        error('zernfun:RTHlength', ... .Yh-m  
              'The number of R- and THETA-values must be equal.') YDDwvk H  
    end VQLo vt"  
    \8<bb<`  
    % Check normalization: LkNfcBa_  
    % -------------------- Imv kB~8N  
    if nargin==5 && ischar(nflag) "qwRcuHY  
        isnorm = strcmpi(nflag,'norm'); fzw6VGTf  
        if ~isnorm nY(jN D  
            error('zernfun:normalization','Unrecognized normalization flag.') tCA |sN  
        end *d(wO l5[  
    else u8o!ncy  
        isnorm = false; 0w(<pNA  
    end _|~2i1 Ms,  
    CZ1 tqAk-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^t#]E#  
    % Compute the Zernike Polynomials 2t[inzn=E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A0&~U0*(~  
    (VC_vz-  
    % Determine the required powers of r: o5zth^p[  
    % ----------------------------------- o F @{&  
    m_abs = abs(m); :Z`4ea"w  
    rpowers = []; NUm3E4  
    for j = 1:length(n) W.H_G.C%  
        rpowers = [rpowers m_abs(j):2:n(j)]; ts)0+x  
    end t6 js@Ih  
    rpowers = unique(rpowers); E{lq@it32p  
    Lw-j#}&6E  
    % Pre-compute the values of r raised to the required powers, oYOf<J  
    % and compile them in a matrix: (|bht0  
    % ----------------------------- r;S%BFMJS  
    if rpowers(1)==0 [[TB.'k  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Sgr<z d'b  
        rpowern = cat(2,rpowern{:}); x\t>|DB  
        rpowern = [ones(length_r,1) rpowern]; 7b Gzun&  
    else e2Xx7*vS  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); xG<S2R2VQh  
        rpowern = cat(2,rpowern{:}); f'r/Q2{n  
    end %yptML9  
    W%Um:C\I  
    % Compute the values of the polynomials: )5]z[sE  
    % -------------------------------------- 3)GXu>) t  
    y = zeros(length_r,length(n)); ?J)%.~!  
    for j = 1:length(n) G::6?+S  
        s = 0:(n(j)-m_abs(j))/2; 9E (>mN  
        pows = n(j):-2:m_abs(j); R?X9U.AcW  
        for k = length(s):-1:1 V+D "_  
            p = (1-2*mod(s(k),2))* ... 4 (Y5n?/  
                       prod(2:(n(j)-s(k)))/              ... H&%=>hyX  
                       prod(2:s(k))/                     ... 9>zN 27  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =U@*adgw  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); eIg2m <9u  
            idx = (pows(k)==rpowers);  )?4m}  
            y(:,j) = y(:,j) + p*rpowern(:,idx); sU{+.k{  
        end M2V.FYV{j>  
         xaS kn  
        if isnorm u,oxUySeG  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 21cIWvy  
        end tkJ/ h<  
    end v~@Y_ `l  
    % END: Compute the Zernike Polynomials b^A&K@[W#,  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iY( hGlV  
    Y*"%;e$tg  
    % Compute the Zernike functions: +mxsjcq0  
    % ------------------------------ -=g`7^qa>  
    idx_pos = m>0; Jl5<9x  
    idx_neg = m<0; rJNf&x%6  
    hefV0)4K  
    z = y; 8uCd|dJ  
    if any(idx_pos) OFUN hbg  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ',O@0L]L  
    end Mzb_o2^(  
    if any(idx_neg) ZJw9 2Sb  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <{cPa\  
    end J qU%$[w  
    2TAy'BB;)  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 4CN8>J'-  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. (/hF~A  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated />n0&~k[h  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive DO %YOv  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, &1Az`[zKGW  
    %   and THETA is a vector of angles.  R and THETA must have the same 1L$u8P^<  
    %   length.  The output Z is a matrix with one column for every P-value, }!.7QpA$  
    %   and one row for every (R,THETA) pair. g@j:TQM_0  
    % Mz"kaO  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike J4co@=AJ  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 7IIM8/BI  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :z"Uw*  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 n_c0=YH  
    %   for all p. bxyU[`  
    % ^Xb!dnT.*a  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 [PQG]"  
    %   Zernike functions (order N<=7).  In some disciplines it is '.^JN@  
    %   traditional to label the first 36 functions using a single mode | _S9U|  
    %   number P instead of separate numbers for the order N and azimuthal a~{St v  
    %   frequency M. S"_vD<q  
    % L9AfLw5&X  
    %   Example: ! q1Ql18n  
    % $/d~bk@=l  
    %       % Display the first 16 Zernike functions ||_F /AD  
    %       x = -1:0.01:1; X0=R @_KY  
    %       [X,Y] = meshgrid(x,x); wTTQIo 60  
    %       [theta,r] = cart2pol(X,Y); d3 N %V.w  
    %       idx = r<=1; =l_eliM/  
    %       p = 0:15; |a"(Ds2U  
    %       z = nan(size(X)); ( NiuAy  
    %       y = zernfun2(p,r(idx),theta(idx)); 9(B)  
    %       figure('Units','normalized') 89 lPeFQ`  
    %       for k = 1:length(p) D@W m-  
    %           z(idx) = y(:,k); RhDa`kV%t  
    %           subplot(4,4,k) "<.b=mN-  
    %           pcolor(x,x,z), shading interp S63L>p|ml  
    %           set(gca,'XTick',[],'YTick',[]) ](0A/,#q6  
    %           axis square XM_S"  
    %           title(['Z_{' num2str(p(k)) '}']) Dk7"#q@kx  
    %       end Gge"`AT  
    % L ~;_R*Th  
    %   See also ZERNPOL, ZERNFUN. VSSiuo'5w  
    bRIb'%=+GA  
    %   Paul Fricker 11/13/2006 Z`:V~8=l  
    }k ,Si9O  
    \ tQi7yj4  
    % Check and prepare the inputs: N.2rF  
    % ----------------------------- ^7_<rs   
    if min(size(p))~=1 fh2Pn!h+  
        error('zernfun2:Pvector','Input P must be vector.') 1`)R#$h  
    end  T  
    kPxrI=  
    if any(p)>35 4frZ .r;V  
        error('zernfun2:P36', ... ?c43cYb  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... p~e6ah?1  
               '(P = 0 to 35).']) _R8-Hj E  
    end ?pVODnP k  
    D/giM#"  
    % Get the order and frequency corresonding to the function number: ,*bxNs'/  
    % ---------------------------------------------------------------- jwE<}y I  
    p = p(:); ,qr)}s-  
    n = ceil((-3+sqrt(9+8*p))/2); Cf10 ud   
    m = 2*p - n.*(n+2); |e pe;/  
    T8RQM1D_s  
    % Pass the inputs to the function ZERNFUN: B)c.`cfr*\  
    % ---------------------------------------- Nd( $s[  
    switch nargin >o[T#U  
        case 3 ]IoS-)$Z/  
            z = zernfun(n,m,r,theta); g:*yjj  
        case 4 /Ia#udkNMp  
            z = zernfun(n,m,r,theta,nflag); *F9uv)[kz  
        otherwise U}{r.MryFG  
            error('zernfun2:nargin','Incorrect number of inputs.') .rMGI "  
    end -MU^%t;-  
    fY6&PuDf.  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) ;1o"Oij  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 0/*z]2  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 0phGn+"R  
    %   order N and frequency M, evaluated at R.  N is a vector of 19lx;^b  
    %   positive integers (including 0), and M is a vector with the a{{([uZ  
    %   same number of elements as N.  Each element k of M must be a .E@yB`AR  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) )v'DQAL  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is "rX`h  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix iveWau292  
    %   with one column for every (N,M) pair, and one row for every dM$]OAT  
    %   element in R. 5jbd!t@L  
    % he!e~5<@y  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- .m4K ]^m  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 0BBWuNF.  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _p90Zm-3X  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 g#H#i~E^  
    %   for all [n,m]. nGg>lRL  
    % x2 tx{Z  
    %   The radial Zernike polynomials are the radial portion of the WJhI6lu  
    %   Zernike functions, which are an orthogonal basis on the unit 4sG^ bZ,  
    %   circle.  The series representation of the radial Zernike qf'uXH  
    %   polynomials is O!;!amvz  
    % ]ErAa"?  
    %          (n-m)/2 A}W&=m8!  
    %            __ |)0kvf?  
    %    m      \       s                                          n-2s ?}O\'Fa8  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r o^lKM?t  
    %    n      s=0 i)eub`uMy  
    % S=o Ab&  
    %   The following table shows the first 12 polynomials. F_@PSA+  
    % sl`\g1<{`  
    %       n    m    Zernike polynomial    Normalization aP2  
    %       --------------------------------------------- I]zCsT.  
    %       0    0    1                        sqrt(2) 0Y[mh@(  
    %       1    1    r                           2 b}axw+  
    %       2    0    2*r^2 - 1                sqrt(6) yht_*7.lM  
    %       2    2    r^2                      sqrt(6) z}kD:A)a  
    %       3    1    3*r^3 - 2*r              sqrt(8) qy.Mi{=~:  
    %       3    3    r^3                      sqrt(8) T?Hs_u{  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \-c70v63X  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) o-49o5:1  
    %       4    4    r^4                      sqrt(10) 5a_1x|Fhi  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ,US]  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) F~&bgl[YZ  
    %       5    5    r^5                      sqrt(12) m+,a=sR  
    %       --------------------------------------------- ;}1*M !  
    % 29}(l#S}m  
    %   Example: h_fA  
    % # M%-q8  
    %       % Display three example Zernike radial polynomials qm] k (/w  
    %       r = 0:0.01:1; >e'6RZRLA  
    %       n = [3 2 5]; W}XDzR'<  
    %       m = [1 2 1]; 1dHN<xy  
    %       z = zernpol(n,m,r); vz/.*u  
    %       figure MejM(o_kk  
    %       plot(r,z) T =_Hd  
    %       grid on 1iig0l6\m  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 3)p#}_u{  
    % Y!"LrkC  
    %   See also ZERNFUN, ZERNFUN2. t4~Bn<=  
    |3QKxS0  
    % A note on the algorithm. eM{,B  
    % ------------------------ ~X)Aw 3}F  
    % The radial Zernike polynomials are computed using the series 'z>|N{-xG  
    % representation shown in the Help section above. For many special e@w-4G(;  
    % functions, direct evaluation using the series representation can !S$LRm\ '  
    % produce poor numerical results (floating point errors), because Jvgx+{Xu  
    % the summation often involves computing small differences between aF]4%E  
    % large successive terms in the series. (In such cases, the functions .\".}4qQ  
    % are often evaluated using alternative methods such as recurrence *FmY4w  
    % relations: see the Legendre functions, for example). For the Zernike ?45bvkCT  
    % polynomials, however, this problem does not arise, because the H0LEK(K  
    % polynomials are evaluated over the finite domain r = (0,1), and .T#h5[S2x  
    % because the coefficients for a given polynomial are generally all ko2?q  
    % of similar magnitude. zU}Ru&T9  
    % |@!4BA  
    % ZERNPOL has been written using a vectorized implementation: multiple Lzm9Kh;  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Mj2`p#5wKh  
    % values can be passed as inputs) for a vector of points R.  To achieve N7=lSBm  
    % this vectorization most efficiently, the algorithm in ZERNPOL tHgu#k0  
    % involves pre-determining all the powers p of R that are required to  _xjw:  
    % compute the outputs, and then compiling the {R^p} into a single (_Ph{IN  
    % matrix.  This avoids any redundant computation of the R^p, and A]c'`Nf  
    % minimizes the sizes of certain intermediate variables. wxS.!9K  
    % }%x2Z{VF  
    %   Paul Fricker 11/13/2006 5%Hw,h   
    14Y_ oH9  
    KP,#x$Bg  
    % Check and prepare the inputs: CC"}aV5  
    % ----------------------------- R6eKI,y\"  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +-"uJIwMD  
        error('zernpol:NMvectors','N and M must be vectors.') % 8u97f W  
    end  0@7%  
    QM wrt  
    if length(n)~=length(m) e98f+,E/  
        error('zernpol:NMlength','N and M must be the same length.') pL 2P .  
    end ( y0  
    Kg?(Ax4  
    n = n(:); 5e1;m6  
    m = m(:); v,, .2UR4  
    length_n = length(n); icS% ])3LF  
    !p #m?|Km  
    if any(mod(n-m,2)) \USl 9*E  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 2 8>  
    end `X)y5*##wq  
    S*PcK>  
    if any(m<0) O2N7qV3 U,  
        error('zernpol:Mpositive','All M must be positive.') w eX%S&#?  
    end {+Zj}3o  
    <UsFBF  
    if any(m>n) &"l Sq2  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') c6-~PKJL  
    end Q%>6u@'  
    ZQ1,6<^9i[  
    if any( r>1 | r<0 ) x_L5NsO:  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]8 vsr$E#  
    end [Z]%jABR  
    y$$|_ l@  
    if ~any(size(r)==1) 8SGqDaRt  
        error('zernpol:Rvector','R must be a vector.')  /dI8o  
    end 7! sR%h5p  
    u0;k_6N  
    r = r(:); \gCh'3  
    length_r = length(r); @V}!elV  
    KwAc Ga}J  
    if nargin==4 t4d^DZDh!  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); F% < ZEVm  
        if ~isnorm .RW&=1D6  
            error('zernpol:normalization','Unrecognized normalization flag.') dp}s]`x+  
        end )gVz?-u+D  
    else u7nTk'#r  
        isnorm = false; .~ O- <P#  
    end *q-VY[2  
    7?@ -|{  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n:"0mWnL$y  
    % Compute the Zernike Polynomials PRal>s&f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lJvfgP-j  
    "W^+NeLc  
    % Determine the required powers of r: q:cCk#ra  
    % ----------------------------------- 8hV>Q  
    rpowers = []; 9 ;Qgby  
    for j = 1:length(n) J7pF*2  
        rpowers = [rpowers m(j):2:n(j)];  MFyi#nq  
    end Nr"gj$v  
    rpowers = unique(rpowers); +aQM %~  
    2WUl8?f2Y  
    % Pre-compute the values of r raised to the required powers, oM^VtH=>  
    % and compile them in a matrix: .^xQtnq  
    % ----------------------------- f = 'AI  
    if rpowers(1)==0 RF[Uy?es  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +[Izz~ _p  
        rpowern = cat(2,rpowern{:}); ~K@p`CRbV  
        rpowern = [ones(length_r,1) rpowern]; :z-?L0C=0  
    else 0" F\ V  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MK.TBv  
        rpowern = cat(2,rpowern{:}); b5)1\ANq  
    end SFjRSMi  
    >H5_,A}f  
    % Compute the values of the polynomials:  G){A&F  
    % -------------------------------------- o&$Of  
    z = zeros(length_r,length_n); 14`S9SL{V  
    for j = 1:length_n 5SEGV|%  
        s = 0:(n(j)-m(j))/2; 8I~*9MUp  
        pows = n(j):-2:m(j); B{K_?ae!  
        for k = length(s):-1:1 6!@p$ pm)a  
            p = (1-2*mod(s(k),2))* ... ]+5Y\~I  
                       prod(2:(n(j)-s(k)))/          ... LfvNO/:,  
                       prod(2:s(k))/                 ... u p zBd]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... e$`;z%6y  
                       prod(2:((n(j)+m(j))/2-s(k))); ^|%N _ s  
            idx = (pows(k)==rpowers); q{}U5(,{0  
            z(:,j) = z(:,j) + p*rpowern(:,idx); s54AM]a{j  
        end 8/@*6J  
         F?Fxm*Wa/  
        if isnorm )}J}d)  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); iU|X/>k?  
        end p^C$(}Yh  
    end 79uAsI2-Y  
    ZEB,Q~  
    % 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)  Zk n1@a  
    ,a?$F1Z-  
    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。 )6Qk|gIu(  
    #[ hJm'G  
    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)