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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 0 `X%&  
    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! bTzVmqGY  
     
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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 B$7m@|p!  
    function z = zernfun(n,m,r,theta,nflag) c< gM  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ua:.97~Ym  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #;juZ*I  
    %   and angular frequency M, evaluated at positions (R,THETA) on the e#k9}n^+  
    %   unit circle.  N is a vector of positive integers (including 0), and %dZD;Vhg  
    %   M is a vector with the same number of elements as N.  Each element w;Qo9=-  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) /#$bb4  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, CTtF=\  
    %   and THETA is a vector of angles.  R and THETA must have the same h`%K \C  
    %   length.  The output Z is a matrix with one column for every (N,M) L&ws[8-  
    %   pair, and one row for every (R,THETA) pair. HH6b{f@^  
    % mU_?}}aK,  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h_]3L/  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'xb|5_D  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral &+`l $h  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FStE/2?  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized XrC{{K  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. oKt<s+r  
    % #`a-b<uz  
    %   The Zernike functions are an orthogonal basis on the unit circle. Hi|2z5=V  
    %   They are used in disciplines such as astronomy, optics, and u7j-uVG  
    %   optometry to describe functions on a circular domain. z$G?J+?J  
    % 5HG 7M&_  
    %   The following table lists the first 15 Zernike functions. qx{.`AaZW  
    % T-&CAD3 ,O  
    %       n    m    Zernike function           Normalization 0 P/A  
    %       -------------------------------------------------- B\|>i~u(  
    %       0    0    1                                 1 joDfvY*[  
    %       1    1    r * cos(theta)                    2 `P/*x[?  
    %       1   -1    r * sin(theta)                    2 j`BF k>  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) kRiWNEw  
    %       2    0    (2*r^2 - 1)                    sqrt(3) V@>?lv(\  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) `1EBnL_1  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) w^|,[G ^}H  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /N%f78 Z  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3N+P~v)T'  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) EFql g9bK  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) RU"w|Qu>pM  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *BXtE8 BU  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) &;)~bS(   
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H.idL6*G  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 9,`mH0jP  
    %       -------------------------------------------------- ?RpT_u  
    % {]<D"x ;  
    %   Example 1: qoZ*sV  
    % iZMsN*9[  
    %       % Display the Zernike function Z(n=5,m=1) 2Fx<QRz  
    %       x = -1:0.01:1; sxThz7#i)  
    %       [X,Y] = meshgrid(x,x); .yTk/x ?  
    %       [theta,r] = cart2pol(X,Y); Od&M^;BQ  
    %       idx = r<=1; mApn(&  
    %       z = nan(size(X)); 2zFdKs,  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ]nX.zE|F  
    %       figure R8'yQ#FVy  
    %       pcolor(x,x,z), shading interp k 5"3*  
    %       axis square, colorbar v9inBBC q  
    %       title('Zernike function Z_5^1(r,\theta)') <;=Y4$y[  
    % VdeK~#k  
    %   Example 2: OM4s.BLY  
    % {6%uNT>|  
    %       % Display the first 10 Zernike functions e<9nt [  
    %       x = -1:0.01:1; m/eGnv;!  
    %       [X,Y] = meshgrid(x,x); #eUfwd6.Y  
    %       [theta,r] = cart2pol(X,Y); |Y'$+[TE  
    %       idx = r<=1; ?>%u[g   
    %       z = nan(size(X)); 22BJOh   
    %       n = [0  1  1  2  2  2  3  3  3  3]; }2NH>qvY  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; U~H'c p  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 21o_9=[^  
    %       y = zernfun(n,m,r(idx),theta(idx)); G0Wd"AV+  
    %       figure('Units','normalized') )D[ypuM&  
    %       for k = 1:10 V)@MM2,  
    %           z(idx) = y(:,k); (VOKa  
    %           subplot(4,7,Nplot(k)) mSj[t   
    %           pcolor(x,x,z), shading interp ]UgA z  
    %           set(gca,'XTick',[],'YTick',[]) `|/|ej]$P  
    %           axis square 6\TstY3  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [CUJA  
    %       end .oK7E(QJ  
    % u^]yz&9V  
    %   See also ZERNPOL, ZERNFUN2. g rfF\_[:  
    ]~K&mNo  
    %   Paul Fricker 11/13/2006 rmabm\QY  
    i;xg[e8.  
    JxLH]1b  
    % Check and prepare the inputs: 3?O| X+$p  
    % ----------------------------- <oXsn.'\  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J,D{dYLDD  
        error('zernfun:NMvectors','N and M must be vectors.') T^nX+;:|  
    end H]-W$V   
    0l:5hD,)F  
    if length(n)~=length(m) 1|nB\xgu  
        error('zernfun:NMlength','N and M must be the same length.') \ yOZ&qU  
    end 4z*_,@OA  
    X*JD  
    n = n(:); {``}TsN  
    m = m(:); Rke:*(p*n;  
    if any(mod(n-m,2)) h7y*2:l6  
        error('zernfun:NMmultiplesof2', ... _bd#C   
              'All N and M must differ by multiples of 2 (including 0).') Z|/):nVP7  
    end ZGbZu  
    ib&qH_r/  
    if any(m>n) vJCL m/}*  
        error('zernfun:MlessthanN', ... uLCU3nI  
              'Each M must be less than or equal to its corresponding N.') IRU2/Ycg  
    end m[bu(qz  
    @\h(s#sn  
    if any( r>1 | r<0 ) %nCUct@c  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3> (`Y  
    end ,9pi9\S  
    ' " tieew  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :RQ[(zD]  
        error('zernfun:RTHvector','R and THETA must be vectors.') #NE^f2  
    end sy`s$E d!  
    BdKtpje  
    r = r(:); u#,]>;  
    theta = theta(:); :$tW9*\KY  
    length_r = length(r); *]eZ Y  
    if length_r~=length(theta) 1CM1u+<iZ  
        error('zernfun:RTHlength', ... sWC"^ So  
              'The number of R- and THETA-values must be equal.') ?qbp  
    end C IDL{i8  
    KCT8Q!\  
    % Check normalization: bGJUu#  
    % -------------------- m#ie{u^  
    if nargin==5 && ischar(nflag) KwHOV$lD;  
        isnorm = strcmpi(nflag,'norm'); nGbrWu]w  
        if ~isnorm Vj]kJ,j\y  
            error('zernfun:normalization','Unrecognized normalization flag.') GVM#Xl}w9  
        end VM,ZEt3Vy  
    else GWVdNYpmr  
        isnorm = false; gQEV;hCO  
    end C|I 1 m  
    N93E;B  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pc7: hu  
    % Compute the Zernike Polynomials XZInu5(  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PsgzDhRv  
    o W[,EW+u  
    % Determine the required powers of r: ` Z/ IW  
    % ----------------------------------- 5a ~tp'  
    m_abs = abs(m); l(5-Cr  
    rpowers = []; W.|6$hRl)  
    for j = 1:length(n) JqUVGEg  
        rpowers = [rpowers m_abs(j):2:n(j)]; c6HU'%v  
    end ' XF`&3 i  
    rpowers = unique(rpowers); 4BT`|(7  
    vdm?d/0(^  
    % Pre-compute the values of r raised to the required powers, sb @hGS  
    % and compile them in a matrix: \uk#pL  
    % ----------------------------- {K:Utdu($q  
    if rpowers(1)==0 !Ia"pNDf  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pPZ/O 6  
        rpowern = cat(2,rpowern{:}); j''Iai_  
        rpowern = [ones(length_r,1) rpowern]; i .N1Cvp&  
    else 'y?|shV{]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); gDub+^ye>/  
        rpowern = cat(2,rpowern{:}); >, E$bm2  
    end swlWe}1  
    &-fx=gq=  
    % Compute the values of the polynomials: @?m8/t9 .  
    % -------------------------------------- N%f!B"NQ  
    y = zeros(length_r,length(n)); sAoM=n}!  
    for j = 1:length(n) f~FehN7  
        s = 0:(n(j)-m_abs(j))/2; =%\6}xPEl<  
        pows = n(j):-2:m_abs(j); y!gM)9vq  
        for k = length(s):-1:1 @q/1m~t  
            p = (1-2*mod(s(k),2))* ... fmJWd|  
                       prod(2:(n(j)-s(k)))/              ... X~he36-+<  
                       prod(2:s(k))/                     ... :BX{ *P  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -o_T C  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); ,)$KS*f"*z  
            idx = (pows(k)==rpowers); ;a&:r7]=  
            y(:,j) = y(:,j) + p*rpowern(:,idx); "Y]ZPFh#.  
        end #( sNk,^Ax  
         DME?kh>7  
        if isnorm {z/^X<T  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); f_!`~`04  
        end ;p 5v3<PC  
    end 66<\i ltUQ  
    % END: Compute the Zernike Polynomials Mlw9#H6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% aT!9W'uY  
    ox_h9=$-  
    % Compute the Zernike functions: NNw d;AC  
    % ------------------------------ 6b70w @P!  
    idx_pos = m>0; Ue#yDTjc  
    idx_neg = m<0; q#&#*6 )B  
    uq4s bkP  
    z = y; 4E-A@FR  
    if any(idx_pos) &}0wzcMg  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0@K:Tq-mF  
    end A dEbyL  
    if any(idx_neg) RzRvu]]8  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); )H9*NB8%  
    end iM|"H..  
    U|7Qw|I7  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) 3y=<w|4F  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ]Ec[")"kT  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated StZRc\k  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive id tQXwa  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, `Kc %S^C'  
    %   and THETA is a vector of angles.  R and THETA must have the same e#6&uFce  
    %   length.  The output Z is a matrix with one column for every P-value, o`K^Wy~+k#  
    %   and one row for every (R,THETA) pair. UW/3{2  
    % R;E"Qdt  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike :gD0EqV  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) @yn^6cE  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) |/\U^AHm"h  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 JXY!c\,  
    %   for all p. nG ^M 2)(8  
    % 9j?hF$L"  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 OgXZ-<'  
    %   Zernike functions (order N<=7).  In some disciplines it is " T a9  
    %   traditional to label the first 36 functions using a single mode K"k"ml<4E  
    %   number P instead of separate numbers for the order N and azimuthal :Q r7:$S^  
    %   frequency M. o;VkoYV  
    % {|5$1v   
    %   Example: '!fFI1s  
    % 9O=05CQ  
    %       % Display the first 16 Zernike functions UJz4>JF  
    %       x = -1:0.01:1; )qyx|D  
    %       [X,Y] = meshgrid(x,x); ]\a\6&R  
    %       [theta,r] = cart2pol(X,Y); t$y&=v  
    %       idx = r<=1; BH#C<0="  
    %       p = 0:15; Q Bfhyo_  
    %       z = nan(size(X)); Xrj(,|  
    %       y = zernfun2(p,r(idx),theta(idx)); {FJX  
    %       figure('Units','normalized') *K}z@a_  
    %       for k = 1:length(p) ll(e,9.D  
    %           z(idx) = y(:,k); 7/&C;"  
    %           subplot(4,4,k) nG},v%  
    %           pcolor(x,x,z), shading interp b>bgUDq  
    %           set(gca,'XTick',[],'YTick',[]) lpfwlB'~9  
    %           axis square I[g?Ju >  
    %           title(['Z_{' num2str(p(k)) '}']) !qTpQ5Dm  
    %       end [/AdeR  
    % z<o E!1St  
    %   See also ZERNPOL, ZERNFUN. w%8ooQ|C  
    <t[Z9s$n  
    %   Paul Fricker 11/13/2006 ~Q_)>|R2  
    h{W$ fZc<  
    C`2*2Y%xkG  
    % Check and prepare the inputs: 0H OoKh  
    % ----------------------------- S om. qD  
    if min(size(p))~=1 +Q SxYV  
        error('zernfun2:Pvector','Input P must be vector.') uG=t?C6  
    end _Sly7_  
    5w,lw  
    if any(p)>35 ,#E3,bu6_4  
        error('zernfun2:P36', ... bl" (<TM  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $ ] W[y=  
               '(P = 0 to 35).']) `q f\3JT\  
    end lruF96C/Y  
    tDC0-N&6S~  
    % Get the order and frequency corresonding to the function number: BaWQ<T8p8  
    % ---------------------------------------------------------------- ]k'#g Z$  
    p = p(:); 4;BW  
    n = ceil((-3+sqrt(9+8*p))/2); =E [4H  
    m = 2*p - n.*(n+2); 'w`SBYQ5  
    .Bb$j=  
    % Pass the inputs to the function ZERNFUN: Q$xa  
    % ---------------------------------------- <[Tq7cO0  
    switch nargin Qb!!J4| !  
        case 3 7oZ@<QP'  
            z = zernfun(n,m,r,theta); BKE\SWu  
        case 4 -TzI>Fz  
            z = zernfun(n,m,r,theta,nflag); RNv{n mf  
        otherwise bGZ hUEq  
            error('zernfun2:nargin','Incorrect number of inputs.') !dfS|BA]  
    end Jh37pI  
    b$d J?%W  
    % 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) OZ(Dpx(Q  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. M0cd-Dn  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of %*$5!;  
    %   order N and frequency M, evaluated at R.  N is a vector of zWy ,Om8P  
    %   positive integers (including 0), and M is a vector with the mSU@UD|'  
    %   same number of elements as N.  Each element k of M must be a ;M}itM  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) kcLj Kp  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is AI&qU/}  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix <e;jW K  
    %   with one column for every (N,M) pair, and one row for every 8 ne/=N|,  
    %   element in R. Gx.P ]O3  
    % {I4%   
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- v2Dt3$@H6  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is "Z;~Y=hC13  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to +J"'  'cZ  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 [dl+:P:zc  
    %   for all [n,m]. Xl#Dw bx  
    % M0RRmW@f.a  
    %   The radial Zernike polynomials are the radial portion of the ]kRI}Om2  
    %   Zernike functions, which are an orthogonal basis on the unit bRWIDPh  
    %   circle.  The series representation of the radial Zernike 3Bejp+xX  
    %   polynomials is RTK}mhnV  
    % Vd,'  s  
    %          (n-m)/2 py]KTRzy  
    %            __ gh TcB  
    %    m      \       s                                          n-2s [-4KY4R  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r -M6L.gi)oJ  
    %    n      s=0 E[S? b=^  
    % ):3MYSqX  
    %   The following table shows the first 12 polynomials. B9YsA?hg  
    % cI2Fpf`2Wj  
    %       n    m    Zernike polynomial    Normalization !6M Bxg>  
    %       --------------------------------------------- uaiz*Im  
    %       0    0    1                        sqrt(2) +fM&su=wl  
    %       1    1    r                           2  #;`Oj  
    %       2    0    2*r^2 - 1                sqrt(6) {{32jU7<  
    %       2    2    r^2                      sqrt(6) ,"B?_d6  
    %       3    1    3*r^3 - 2*r              sqrt(8) fb23J|"  
    %       3    3    r^3                      sqrt(8) Gu5~ DyT`G  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) /-Wuq`P/ T  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) _l<mu?"  
    %       4    4    r^4                      sqrt(10) cA<<& C  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) rOW;yJ[  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) }g>kpa0c  
    %       5    5    r^5                      sqrt(12) {-HDkG' 8  
    %       --------------------------------------------- O16r!6=-n  
    % hd5$yU5JQ  
    %   Example: 'f*O#&?  
    % s D_G)c  
    %       % Display three example Zernike radial polynomials COSTV>s;  
    %       r = 0:0.01:1; Tp?-* K  
    %       n = [3 2 5]; #,&8&  
    %       m = [1 2 1]; lkb2?2\+  
    %       z = zernpol(n,m,r); Z ] '>  
    %       figure .Rvf/-e  
    %       plot(r,z) 34z+INkX  
    %       grid on 4@6 <  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ^2E hlK^)  
    % h$pk<<  
    %   See also ZERNFUN, ZERNFUN2. 3htq[Ren  
    fJy)STQ4  
    % A note on the algorithm. wX0l?xdI  
    % ------------------------ pA.._8(t  
    % The radial Zernike polynomials are computed using the series F}'wH-qp  
    % representation shown in the Help section above. For many special L6+C]t}>6  
    % functions, direct evaluation using the series representation can GF'f[F6oI  
    % produce poor numerical results (floating point errors), because MU1E_"Z)  
    % the summation often involves computing small differences between .xl.P7@JJ  
    % large successive terms in the series. (In such cases, the functions L+.H z&*@  
    % are often evaluated using alternative methods such as recurrence BxdX WO  
    % relations: see the Legendre functions, for example). For the Zernike (~&w-w3  
    % polynomials, however, this problem does not arise, because the 26.)Ur<F  
    % polynomials are evaluated over the finite domain r = (0,1), and s]Gd-j  
    % because the coefficients for a given polynomial are generally all zb:kanb-  
    % of similar magnitude. =IZ[_ /@  
    % @}DFp`~5|  
    % ZERNPOL has been written using a vectorized implementation: multiple k][h9'  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] PNpu*# Z`  
    % values can be passed as inputs) for a vector of points R.  To achieve uE`r/=4  
    % this vectorization most efficiently, the algorithm in ZERNPOL NEV p8)w  
    % involves pre-determining all the powers p of R that are required to =((yWn+t  
    % compute the outputs, and then compiling the {R^p} into a single [jrqzB  
    % matrix.  This avoids any redundant computation of the R^p, and ]g,lRG  
    % minimizes the sizes of certain intermediate variables. xQU"A2{}>  
    % MDHb'<o?y  
    %   Paul Fricker 11/13/2006 t'g^W  
    @h}`DNaZ^  
    <6jFKA<  
    % Check and prepare the inputs: mI"D(bx\  
    % ----------------------------- _Yq@FOu  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ORBxD"J&  
        error('zernpol:NMvectors','N and M must be vectors.') %(<(Y  
    end dJi|D  
    E'EcP4eL  
    if length(n)~=length(m) g)R1ObpZ  
        error('zernpol:NMlength','N and M must be the same length.') ((<`zx  
    end 0%<+J;'o  
    !9]d |8!  
    n = n(:); X>zlb$  
    m = m(:); (0L=AxH  
    length_n = length(n); OJ1tV% E  
    %>zjGF<  
    if any(mod(n-m,2)) 6kR\xP]Kr  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') bd== +   
    end ^DB{qU  
    0<.R A%dj  
    if any(m<0) ,T|x)"uA`  
        error('zernpol:Mpositive','All M must be positive.') %tmp  
    end zN!j%T.e  
    V%NeZ1{ e  
    if any(m>n) EzpFOqJG  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') |PP.<ce\-  
    end 0.Vi9 7`  
    5iM[sg[y9  
    if any( r>1 | r<0 ) cb$-6ZE/  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') #;Tz[0  
    end 8-M e.2K  
    LFf`K)q  
    if ~any(size(r)==1) M%`\P\A  
        error('zernpol:Rvector','R must be a vector.') )D/ 6%]O  
    end nidr\oFUIn  
     {mTytT  
    r = r(:); i9A~<  
    length_r = length(r); )Gk`[*q ;  
    rs-,0'z,7  
    if nargin==4 I#G0, &Gv  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); C6  "  
        if ~isnorm {5j66QFoo  
            error('zernpol:normalization','Unrecognized normalization flag.') nxo+?:**  
        end t ls60h  
    else X.FGBR7=q  
        isnorm = false; BVpO#c~I  
    end ,[!LCXp  
    'V&Y[7Aeq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M;.ZM<Ga  
    % Compute the Zernike Polynomials V diJ>d[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =L]Q2V}  
    GJA`l8`SQ  
    % Determine the required powers of r: -e]7n*}H$  
    % ----------------------------------- '0Q,  
    rpowers = []; iG<Som  
    for j = 1:length(n) ytAWOt}`  
        rpowers = [rpowers m(j):2:n(j)]; 7cTk@Gq  
    end H/fUM  
    rpowers = unique(rpowers); (u1m]WYL  
    wvby?MhPY  
    % Pre-compute the values of r raised to the required powers, z=Cr7-  
    % and compile them in a matrix: l.+yn91%>  
    % ----------------------------- jZ*WN|FK?  
    if rpowers(1)==0 |j~lkzPnV  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5&!c7$K0  
        rpowern = cat(2,rpowern{:}); O XP\R  
        rpowern = [ones(length_r,1) rpowern]; H17-/|-;0!  
    else * ?Jz2[B  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /)3Lnn{W  
        rpowern = cat(2,rpowern{:}); }6u2*(TmD  
    end s^8u&y)3  
    N.do "  
    % Compute the values of the polynomials: ipE ]}0q  
    % -------------------------------------- 8V/L:h#7  
    z = zeros(length_r,length_n); >SbK.Q@ei  
    for j = 1:length_n X[yNFW}S2W  
        s = 0:(n(j)-m(j))/2; rNDrp@A>  
        pows = n(j):-2:m(j);  C})'\1O%  
        for k = length(s):-1:1 BMyzjteS+  
            p = (1-2*mod(s(k),2))* ... +-?/e-z")  
                       prod(2:(n(j)-s(k)))/          ... .ic:`1  
                       prod(2:s(k))/                 ... yV_wDeAz  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... TFQ!7'xk)  
                       prod(2:((n(j)+m(j))/2-s(k))); 4,p;Km&  
            idx = (pows(k)==rpowers); zg)sd1@  
            z(:,j) = z(:,j) + p*rpowern(:,idx); &HZmQ>!R D  
        end "tk-w{>  
         w?Ju5 5  
        if isnorm l@Z6do  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); nw:-J1kWR  
        end 7V7zGx+Z7  
    end hsRvr`#m|  
    8+Llx  
    % 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)  <Llp\XcZ  
    YSJy`  
    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。 ~U(,TjJb  
    RjF'x  
    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)