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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 %)7HBj(*J  
    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! oC]|ARgQk|  
     
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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 #un#~s 7Q  
    function z = zernfun(n,m,r,theta,nflag) @Oe!*|?mS  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. JO:40V?op  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N OO.. Y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the a9 =,P  
    %   unit circle.  N is a vector of positive integers (including 0), and ;H5H7ezV  
    %   M is a vector with the same number of elements as N.  Each element _ukKzY  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) i 7:R4G(/#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, g>h5NrD N  
    %   and THETA is a vector of angles.  R and THETA must have the same `A5^D  
    %   length.  The output Z is a matrix with one column for every (N,M) z= pb<Y@X  
    %   pair, and one row for every (R,THETA) pair. ar.w'z  
    % \/C-e  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ea@N:t?(8=  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7<V(lX.{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral o/E A%q1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^7C?yC  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized cT abZc  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. xX l^\?HC  
    % DXj_\ R(}  
    %   The Zernike functions are an orthogonal basis on the unit circle. <>,V> k|  
    %   They are used in disciplines such as astronomy, optics, and 4C2JyP3  
    %   optometry to describe functions on a circular domain. <lh+mrXm  
    % 7_ g}t!b`  
    %   The following table lists the first 15 Zernike functions. \HFeEEKH  
    % WAlsh  
    %       n    m    Zernike function           Normalization M$L1!o1Xf  
    %       -------------------------------------------------- CLI!(8ZW  
    %       0    0    1                                 1 o.DT`L8  
    %       1    1    r * cos(theta)                    2 vKppXm1  
    %       1   -1    r * sin(theta)                    2 pX ]K-  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) s$e0;C!D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) U@v=q9'W  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) ` INcZr"  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 1P&XG@  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {.2A+JT,  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) t E/s|v#O  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) }YHoWYR  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }?xu/C  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) zm rQ7(y  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) q: . URl  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hk%m`|Z  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ou6|;*>d  
    %       -------------------------------------------------- uUb[Dqn  
    % AZcW f8  
    %   Example 1: b`E'MX_ m  
    % /!,>P[Vx  
    %       % Display the Zernike function Z(n=5,m=1) 'S<ebwRd=  
    %       x = -1:0.01:1; n'ft@7>%h  
    %       [X,Y] = meshgrid(x,x); 5S:#I5Wa  
    %       [theta,r] = cart2pol(X,Y); zRsG$)B  
    %       idx = r<=1; ZK4/o  
    %       z = nan(size(X)); Q}ho Y  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); weC$\st:D  
    %       figure :M(%sv</  
    %       pcolor(x,x,z), shading interp }./__gJ  
    %       axis square, colorbar Dt\F]\6sd  
    %       title('Zernike function Z_5^1(r,\theta)') I0oM\~#  
    % FQSepUl  
    %   Example 2: Kr`Cr5v  
    % *,!6#Z7  
    %       % Display the first 10 Zernike functions cMxTv4|wui  
    %       x = -1:0.01:1; 1cWUPVQ  
    %       [X,Y] = meshgrid(x,x); :N5R.@9  
    %       [theta,r] = cart2pol(X,Y); -xtj:UO  
    %       idx = r<=1; zZDa7 1>  
    %       z = nan(size(X)); lil1$K: i  
    %       n = [0  1  1  2  2  2  3  3  3  3]; g83]/s+  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; tn201TDZ]=  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; N8;/Zd;^  
    %       y = zernfun(n,m,r(idx),theta(idx)); aLTC#c%U  
    %       figure('Units','normalized') [9NzvC 9I  
    %       for k = 1:10 O#fGHI<43[  
    %           z(idx) = y(:,k); WP7*Q:5  
    %           subplot(4,7,Nplot(k)) S{aK\>>H  
    %           pcolor(x,x,z), shading interp \'6hv>W@  
    %           set(gca,'XTick',[],'YTick',[]) <<K GS  
    %           axis square <hgt{b4  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) s<LF=qGu  
    %       end hkeOe  
    % <)+9PV<w  
    %   See also ZERNPOL, ZERNFUN2. n8#iL  
    `~QS3zq  
    %   Paul Fricker 11/13/2006 sF}T9 Ue  
    8@ck" LUzD  
    !T02@e/  
    % Check and prepare the inputs: Au08k}h<G  
    % ----------------------------- !},_,J~(|  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m[,! orq  
        error('zernfun:NMvectors','N and M must be vectors.')  U=MFNp+  
    end .<j\"X(  
    {j.5!Nj]B  
    if length(n)~=length(m) !8M]n  
        error('zernfun:NMlength','N and M must be the same length.') BXy g ?  
    end J@w Q3#5a  
    s,O:l0  
    n = n(:); \&|)?'8rS  
    m = m(:); ntE;*F yH  
    if any(mod(n-m,2)) 3G|n`dj  
        error('zernfun:NMmultiplesof2', ... Vr0-evwfo  
              'All N and M must differ by multiples of 2 (including 0).') EOWLGleD1  
    end 0\84~t'[  
    > f,G3Ay  
    if any(m>n) VeidB!GyP  
        error('zernfun:MlessthanN', ... -bT1Qh X  
              'Each M must be less than or equal to its corresponding N.') )*ocX)AE  
    end G4][`C]8c  
    ;HRIB)wF  
    if any( r>1 | r<0 ) 'Y{fah  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') HM ;9%rtO  
    end ).e_iE[&  
    'H- : >'k  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) u*0Ck*pZ  
        error('zernfun:RTHvector','R and THETA must be vectors.') 6tBL?'pG  
    end S,,,D+4  
    `+cc{k  
    r = r(:); raRb K8CQ  
    theta = theta(:); 9q;n@q:29  
    length_r = length(r); ;@ xSJqT  
    if length_r~=length(theta) cX u"-/  
        error('zernfun:RTHlength', ...  oZTKG'  
              'The number of R- and THETA-values must be equal.') (;-< @~2  
    end &|'k)6Rx  
    !2>MaV1,  
    % Check normalization: O+hN?/>v  
    % -------------------- QQ^P IQj  
    if nargin==5 && ischar(nflag) ibo{!>m  
        isnorm = strcmpi(nflag,'norm'); *^+8_%;1  
        if ~isnorm swEE >=  
            error('zernfun:normalization','Unrecognized normalization flag.') +Zgh[a  
        end CU'$JF  
    else <]#'6'  
        isnorm = false; 60?/Z2w5  
    end WBdC}S }3t  
    7kJ =C  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Obwj=_+upd  
    % Compute the Zernike Polynomials w3oh8NRs_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d*;wHA,}F  
    R+Q..9 P  
    % Determine the required powers of r: AdV&w: ^yf  
    % ----------------------------------- 4,kdP)Md$  
    m_abs = abs(m); #1c%3KaZ I  
    rpowers = []; d 2f   
    for j = 1:length(n) jinDKJ,n;  
        rpowers = [rpowers m_abs(j):2:n(j)]; {z:aZ]QhKc  
    end ]Q-*xho  
    rpowers = unique(rpowers); X}Heaqn  
    ^)|8N44O  
    % Pre-compute the values of r raised to the required powers, ##Jg>HL'  
    % and compile them in a matrix: ^p3"_;p)h  
    % ----------------------------- }cUq1r-bW  
    if rpowers(1)==0 @AM;58.  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $e>(M&9,  
        rpowern = cat(2,rpowern{:}); {akSK  
        rpowern = [ones(length_r,1) rpowern]; )xKZ)SxV  
    else %dA6vHI,  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >6xZF'4  
        rpowern = cat(2,rpowern{:}); ;la sk4|  
    end );X &J:-l+  
    vhe[:`=a  
    % Compute the values of the polynomials: :5`=9 _|  
    % -------------------------------------- !>gi9z,  
    y = zeros(length_r,length(n)); <7-Qn(m,  
    for j = 1:length(n) ;A^Ii>`  
        s = 0:(n(j)-m_abs(j))/2; (.Q.S[<Y  
        pows = n(j):-2:m_abs(j); :Y/>] tS4  
        for k = length(s):-1:1 \C<|yD  
            p = (1-2*mod(s(k),2))* ... }.bhsy  
                       prod(2:(n(j)-s(k)))/              ... wB%:RI,  
                       prod(2:s(k))/                     ... Vu6$84>-,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !kAjne8]d  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); "'Bx<FA  
            idx = (pows(k)==rpowers); [=f(u wY>g  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 4KH8dau.fF  
        end <UI^~Azc#  
         -nM=^ i4)  
        if isnorm ,|:TML  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); XEK%\o}  
        end U7GgGMw  
    end `[.b>ztqgJ  
    % END: Compute the Zernike Polynomials v[-.]b*5A$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fjD/<`}v  
    mar6/*`I#+  
    % Compute the Zernike functions: Tvdg:[V<  
    % ------------------------------ `XT8}9z!  
    idx_pos = m>0; V5ve  
    idx_neg = m<0; ^ud-N;]MKs  
    <]{$XcNm  
    z = y; K+2sq+ 3q  
    if any(idx_pos) #kho[`9  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); k :KN32%  
    end Q7V*~{  
    if any(idx_neg) d2cslD d  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); DjzHEqiH  
    end |AgdD  
    L$T23*9XY  
    % 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) 2=?:(e9  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. *?l-:bc]  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated U"SH fI:  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive roiUVisq*  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ]x;*Z&  
    %   and THETA is a vector of angles.  R and THETA must have the same J #ukH`|-  
    %   length.  The output Z is a matrix with one column for every P-value, 1$+-?:i C  
    %   and one row for every (R,THETA) pair. [.ya&E)x  
    % |{STkV]  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike E2hy%y9Tp  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) jUtFDw  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) "#ctT-g`6  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 s2IjZF{  
    %   for all p. seNJ6p=`  
    % /Soc,PjZ  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 %1\MW+  
    %   Zernike functions (order N<=7).  In some disciplines it is lMn1e6~K  
    %   traditional to label the first 36 functions using a single mode S)n+E\c  
    %   number P instead of separate numbers for the order N and azimuthal `jI$>{oa  
    %   frequency M. cN{(XmX5n  
    % k'(d$;Jgr  
    %   Example: 1o Z!Up0  
    % vJ&g3ky  
    %       % Display the first 16 Zernike functions VuU{7:  
    %       x = -1:0.01:1; o+}>E31a  
    %       [X,Y] = meshgrid(x,x); sYMgi D  
    %       [theta,r] = cart2pol(X,Y); G C#s;X  
    %       idx = r<=1; Exu5|0AAE  
    %       p = 0:15; YA?46[:  
    %       z = nan(size(X)); ktEdbALK  
    %       y = zernfun2(p,r(idx),theta(idx)); t_Q\uo}  
    %       figure('Units','normalized') !e<D2><^  
    %       for k = 1:length(p) 6g6BE^o\  
    %           z(idx) = y(:,k); "ZHA.M]`  
    %           subplot(4,4,k) N(7UlS,u'  
    %           pcolor(x,x,z), shading interp {S$]I)tV  
    %           set(gca,'XTick',[],'YTick',[]) j0X Jf<  
    %           axis square 6Fe$'TP  
    %           title(['Z_{' num2str(p(k)) '}']) Dr_ (u<[  
    %       end 3D2\#6yo  
    % Y8s.Q  
    %   See also ZERNPOL, ZERNFUN. A.8[FkiNmD  
    #a$k3C  
    %   Paul Fricker 11/13/2006 3hD\6,@  
    fzk^QrB  
    Y!1x,"O'H  
    % Check and prepare the inputs: rBLcj;,  
    % ----------------------------- uE;bNs'  
    if min(size(p))~=1 F\Z|JCA  
        error('zernfun2:Pvector','Input P must be vector.') nS` :)#;  
    end T {![a{  
    mne?r3d  
    if any(p)>35 >Ohh) $  
        error('zernfun2:P36', ... 5ltrr(MeD  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |[3%^!f\  
               '(P = 0 to 35).']) a3wk#mH  
    end Y9w^F_relL  
    UG,<\k&  
    % Get the order and frequency corresonding to the function number: U;q GUqI  
    % ---------------------------------------------------------------- tD j/!L`  
    p = p(:); !zW22M  
    n = ceil((-3+sqrt(9+8*p))/2); Vx#n0z  
    m = 2*p - n.*(n+2); VUQx"R9-  
    X$0&tmum  
    % Pass the inputs to the function ZERNFUN: 4?c4GT9(6S  
    % ---------------------------------------- ; `Vbl_"L  
    switch nargin )J>-;EYb8  
        case 3 _@/nc:)H  
            z = zernfun(n,m,r,theta); nX>HRdC  
        case 4 p_:bt7 B  
            z = zernfun(n,m,r,theta,nflag); fgW>~m.W  
        otherwise IwZe2$f  
            error('zernfun2:nargin','Incorrect number of inputs.') U(~d^9/#  
    end >S[NI<=8S  
    5<RZ ht$i  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) [:"7B&&A  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. _@_w6Rh  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Wu|ANc  
    %   order N and frequency M, evaluated at R.  N is a vector of ,+U,(P5>s  
    %   positive integers (including 0), and M is a vector with the YkJnZ_k/P  
    %   same number of elements as N.  Each element k of M must be a 6 66f;h  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ,5Nf9z!hk(  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Y| 2Gj(*8  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix # M18&ld,r  
    %   with one column for every (N,M) pair, and one row for every Ef"M e(  
    %   element in R. 56l1&hp8In  
    % gBPYGci2F  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- o4Hp|iK&0  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is xL#UMvZ>;h  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /xh/M@G3  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Re]7G.y  
    %   for all [n,m]. qf?X:9Wt  
    % 3?Tk[m1b  
    %   The radial Zernike polynomials are the radial portion of the ?_BK(kL_  
    %   Zernike functions, which are an orthogonal basis on the unit Jd-u ?  
    %   circle.  The series representation of the radial Zernike  X0&[cyP!  
    %   polynomials is P;DGs]PF  
    %  WgayH  
    %          (n-m)/2 3{FUFx  
    %            __ :J~sz)n4  
    %    m      \       s                                          n-2s >[&Zs3>  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `44 }kkBT  
    %    n      s=0 pe@j`Sm:Ej  
    % =ec"G2$?"  
    %   The following table shows the first 12 polynomials. |x$2- RUP  
    % pdEUDuX  
    %       n    m    Zernike polynomial    Normalization J{qpGRQNa  
    %       --------------------------------------------- 2nkUvb%=  
    %       0    0    1                        sqrt(2) Ce-D^9kC  
    %       1    1    r                           2 (I0QwB  
    %       2    0    2*r^2 - 1                sqrt(6) /j(3 ~%]o4  
    %       2    2    r^2                      sqrt(6) p0bMgP  
    %       3    1    3*r^3 - 2*r              sqrt(8) xa]e9u%  
    %       3    3    r^3                      sqrt(8) $5v:z   
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) {wy#HYhv  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) /^^wHW:  
    %       4    4    r^4                      sqrt(10) i-E/#zni  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) r?:xD(}Q  
    %       5    3    5*r^5 - 4*r^3            sqrt(12)  Unc_e  
    %       5    5    r^5                      sqrt(12) _=ziw|zI  
    %       --------------------------------------------- [_w;=l0 ;  
    % x-Ug(/!^  
    %   Example: A2{s ?L,  
    % n"dYN3dE  
    %       % Display three example Zernike radial polynomials p8_ CY[U  
    %       r = 0:0.01:1; ;:#g\|(<+  
    %       n = [3 2 5]; 45H(.}&f  
    %       m = [1 2 1]; k%2Rv4)hU  
    %       z = zernpol(n,m,r); = k7}[!T  
    %       figure EI!e0 V1!  
    %       plot(r,z)  OK\F  
    %       grid on ,d~6LXr<fM  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 6>R|B?I%  
    % d^W1;0  
    %   See also ZERNFUN, ZERNFUN2. o{I]c#W  
    H%^j yGS  
    % A note on the algorithm. `S+B-I0  
    % ------------------------ lVR a{._m  
    % The radial Zernike polynomials are computed using the series gK+/wTQ%  
    % representation shown in the Help section above. For many special '%\FT-{  
    % functions, direct evaluation using the series representation can w</qUOx  
    % produce poor numerical results (floating point errors), because *p=a-s5-  
    % the summation often involves computing small differences between lJ$j[Y  
    % large successive terms in the series. (In such cases, the functions  *CS2ndp  
    % are often evaluated using alternative methods such as recurrence ]46#u=y~3  
    % relations: see the Legendre functions, for example). For the Zernike %IhUQ6  
    % polynomials, however, this problem does not arise, because the 8DO3L "  
    % polynomials are evaluated over the finite domain r = (0,1), and s?pd&_kOv3  
    % because the coefficients for a given polynomial are generally all K%iA-h  
    % of similar magnitude.  HLsG<#  
    % W v4o:_}  
    % ZERNPOL has been written using a vectorized implementation: multiple d&ap u{  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] h.WvPZ2U  
    % values can be passed as inputs) for a vector of points R.  To achieve /hksESiU  
    % this vectorization most efficiently, the algorithm in ZERNPOL ro`2IE>  
    % involves pre-determining all the powers p of R that are required to 3X#)PX9b){  
    % compute the outputs, and then compiling the {R^p} into a single wOU\&u|  
    % matrix.  This avoids any redundant computation of the R^p, and iwS55o  
    % minimizes the sizes of certain intermediate variables. ,"U_oa3  
    % 0oiz V;B5%  
    %   Paul Fricker 11/13/2006 I {&8iUN  
    }=T=Z#OgH  
    1xsIM'&  
    % Check and prepare the inputs: #cdrobJ  
    % ----------------------------- )Z qY`by!  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2?./S)x)  
        error('zernpol:NMvectors','N and M must be vectors.') yhzZ[vw7k  
    end x-%4-)  
    5?M d  
    if length(n)~=length(m) '#Pg:v_  
        error('zernpol:NMlength','N and M must be the same length.') U2TR>0l  
    end H\!p%Y  
    M*n@djL$\~  
    n = n(:); k/]4L!/ T  
    m = m(:); Vq>$ZlvS  
    length_n = length(n); 5wgeA^HE2y  
    Rt=zqfJ  
    if any(mod(n-m,2)) }?KfL$@$  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') + 2?=W1`  
    end BfOQ/k))  
    }X?M6;$)  
    if any(m<0) |ryV7VJ8  
        error('zernpol:Mpositive','All M must be positive.') @})]4H  
    end !eE;MaS>  
    ~8l(,N0  
    if any(m>n) (]gd$BgD  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') q"C(`S.@  
    end {>rGe#Vu  
    Al-;-t#Dc  
    if any( r>1 | r<0 ) pS6p}S=1]  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') o +KDK{MD  
    end n4%ZR~9WH  
    Ae[Na:G+  
    if ~any(size(r)==1) K0xka[x=(  
        error('zernpol:Rvector','R must be a vector.') 9?J 3G,&  
    end Y5,[udF:O  
    H.=S08c3kA  
    r = r(:); |0N6]%r  
    length_r = length(r); 8urX]#  
    oQ:.pq{T  
    if nargin==4 ]qpLaBD  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); lNRGlTD%  
        if ~isnorm 2*)2c[/0F  
            error('zernpol:normalization','Unrecognized normalization flag.') Svqj@@_f  
        end Ql8s7%  
    else kVeR{i<*(  
        isnorm = false; J<;@RK,c_  
    end cM 5V%w  
    bpgvLZb>s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% egXbe)ld  
    % Compute the Zernike Polynomials hI>vz"J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `WRM7  
    /9@[gv A  
    % Determine the required powers of r: w 8B SY  
    % ----------------------------------- /?*GJN#  
    rpowers = []; =`pH2SJT  
    for j = 1:length(n) I6-.;)McO  
        rpowers = [rpowers m(j):2:n(j)]; 9Rd& Jq^  
    end $~c wB  
    rpowers = unique(rpowers); 6 @A'N(I=O  
    B9(@ .  
    % Pre-compute the values of r raised to the required powers, oT0TbZu%  
    % and compile them in a matrix: ?[VL 2dP0  
    % ----------------------------- OUFx M  
    if rpowers(1)==0 @ Cd#\D|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q"9 2][}  
        rpowern = cat(2,rpowern{:}); X 7R&>Pf  
        rpowern = [ones(length_r,1) rpowern]; %iR"eEE  
    else +oevNM  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); $H8B%rT]  
        rpowern = cat(2,rpowern{:}); N> jQe  
    end GTuxMg`  
    Q&]f9j_  
    % Compute the values of the polynomials: |5TzRz  
    % -------------------------------------- U-U"RC>  
    z = zeros(length_r,length_n); [JVEKc ym  
    for j = 1:length_n 3ZUME\U  
        s = 0:(n(j)-m(j))/2; ISHzlEY  
        pows = n(j):-2:m(j); v2l*n  
        for k = length(s):-1:1 \)`OEGdOR\  
            p = (1-2*mod(s(k),2))* ... urhOvC$a  
                       prod(2:(n(j)-s(k)))/          ... =n<Lbl(7  
                       prod(2:s(k))/                 ... YN}vAFR`  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Rn$[P.||  
                       prod(2:((n(j)+m(j))/2-s(k))); \"pp-str  
            idx = (pows(k)==rpowers); ?>s[B7wMp  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 6^e}^~|  
        end `n?Rxhkwp  
         pf$gvL  
        if isnorm #IR,KX3]A  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); K1$   
        end %k'>bmJ  
    end aqEmF  
    5=_bK^Am  
    % 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)  F>5b[q6~4  
    aK]AhOG   
    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。 x8x-b>|$&<  
    x6$3 KDQm  
    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)