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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 [Jjb<6[o  
    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! [IgB78_$  
     
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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 +frkC| .  
    function z = zernfun(n,m,r,theta,nflag) fF\s5f#:  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. )l|/lj  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8)1 k>=  
    %   and angular frequency M, evaluated at positions (R,THETA) on the z TM1 e  
    %   unit circle.  N is a vector of positive integers (including 0), and %nmD>QCe  
    %   M is a vector with the same number of elements as N.  Each element ZMI!Sl  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) S5W*,?  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, heAbxs  
    %   and THETA is a vector of angles.  R and THETA must have the same <H,q( :pM  
    %   length.  The output Z is a matrix with one column for every (N,M) <DM /"^*  
    %   pair, and one row for every (R,THETA) pair. giDe  
    % !='?+Ysxs  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |K H&,  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (eOzntp8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 5c W2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T/A[C  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized TCC([  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. qNH= W?T8.  
    % .BWCGb2bH  
    %   The Zernike functions are an orthogonal basis on the unit circle. ?/SIA9VK  
    %   They are used in disciplines such as astronomy, optics, and |BO!q9633V  
    %   optometry to describe functions on a circular domain. f*{~N!g  
    % {NS6y\,  
    %   The following table lists the first 15 Zernike functions. exn Fy-  
    % Yb~[XS |p  
    %       n    m    Zernike function           Normalization L*rND15  
    %       -------------------------------------------------- ;Tn$c70  
    %       0    0    1                                 1 |fJpX5W-l  
    %       1    1    r * cos(theta)                    2 m~LB0u$ac  
    %       1   -1    r * sin(theta)                    2 Q1?0R<jOU  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Y\Z.E ;  
    %       2    0    (2*r^2 - 1)                    sqrt(3) nO'lN<L  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) /MErS< 6  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) \5MW65  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ;{zgp  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) B ``)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) efK|)_i :  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 7V^\fh5~  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !c;Z<@  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @Qlh  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) y rSTU-5u  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 8: x{  
    %       -------------------------------------------------- * mzJ)4A  
    % wNHvYu lI  
    %   Example 1: :U,n[.$5'  
    % aCq ) hR  
    %       % Display the Zernike function Z(n=5,m=1) wRa$b  
    %       x = -1:0.01:1; yc#0c[ZQu  
    %       [X,Y] = meshgrid(x,x); ?!h jI;_&  
    %       [theta,r] = cart2pol(X,Y); O0"u-UX{  
    %       idx = r<=1; ypCarvQT  
    %       z = nan(size(X)); baD`k?](  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); x*Lm{c5+  
    %       figure K,!"5WrX*  
    %       pcolor(x,x,z), shading interp <vMdfw"(  
    %       axis square, colorbar O% 1X[  
    %       title('Zernike function Z_5^1(r,\theta)') ;^Q - 1  
    % j~|pSu.<  
    %   Example 2: N^)\+*tf1  
    % 6`ZHFem  
    %       % Display the first 10 Zernike functions zdL"PF  
    %       x = -1:0.01:1; <B }4}-}  
    %       [X,Y] = meshgrid(x,x); |>/T*zk<  
    %       [theta,r] = cart2pol(X,Y); deRnP$u0  
    %       idx = r<=1; $jpAnZR- /  
    %       z = nan(size(X)); J=%(f1X<W  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Gu3# y"a>  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; )_m#|U?Rex  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4x`.nql  
    %       y = zernfun(n,m,r(idx),theta(idx)); %Sgdhgk1  
    %       figure('Units','normalized') cc*xHv^  
    %       for k = 1:10 _{eH" ,(  
    %           z(idx) = y(:,k); F5hOKUjv  
    %           subplot(4,7,Nplot(k)) F%Xj'=  
    %           pcolor(x,x,z), shading interp R\^n2gK  
    %           set(gca,'XTick',[],'YTick',[]) 8&g`Uy/b  
    %           axis square &jg..R  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ([mC!d@a  
    %       end sQ4~oZZ  
    % i`FskEoijq  
    %   See also ZERNPOL, ZERNFUN2. NZP>aV-  
    'aW}&!H M  
    %   Paul Fricker 11/13/2006 4axc05  
    h#Z5vH  
    q ,C)AZ  
    % Check and prepare the inputs: P?.j wI  
    % ----------------------------- *0*1.>Vg  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ,%bG]5  
        error('zernfun:NMvectors','N and M must be vectors.') /2<1/[#  
    end c{qoASc?  
    Vb _W&Nwd  
    if length(n)~=length(m) #o(c=  
        error('zernfun:NMlength','N and M must be the same length.') I6jDRC0<  
    end X; ~3 U 9  
    K&\3j-8^  
    n = n(:); =;) M+"  
    m = m(:); 6r|BiHP  
    if any(mod(n-m,2)) `8.Oc;*zu  
        error('zernfun:NMmultiplesof2', ... xu]>TC1  
              'All N and M must differ by multiples of 2 (including 0).') |i}5vT78  
    end Zx1I&K\Cd  
    q h+c}"4m  
    if any(m>n) qoifzEc`U  
        error('zernfun:MlessthanN', ... ,h#U<CnP#  
              'Each M must be less than or equal to its corresponding N.') f&n6;N  
    end b <1k$0J6  
    Hq>"rrVhx  
    if any( r>1 | r<0 ) )\!-n]+A  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5D~>Ed;  
    end YFGQPg  
    9b8kRz[ c  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) |%i|P)]  
        error('zernfun:RTHvector','R and THETA must be vectors.') cNd;qO0$  
    end K F:W:8  
    ^2|G0d@.:  
    r = r(:); *Dz<Pi^  
    theta = theta(:); bnm3 cR:h"  
    length_r = length(r); tH}$j  
    if length_r~=length(theta) 7jf%-X  
        error('zernfun:RTHlength', ... M_ GN3  
              'The number of R- and THETA-values must be equal.') 2E*k@  
    end m9&MTR D\  
    Dd=iYM m7  
    % Check normalization: aCwb[7N  
    % -------------------- 09r0Rb  
    if nargin==5 && ischar(nflag) SviGLv;oR  
        isnorm = strcmpi(nflag,'norm'); hPM:=@ N$  
        if ~isnorm =LUDg7P  
            error('zernfun:normalization','Unrecognized normalization flag.') dV:vM9+x  
        end DaK2P;WP  
    else r N.<S[  
        isnorm = false; ^<}>]F_  
    end r=`]L-}V  
    Gx$rk<;ZW  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I1TzPe  
    % Compute the Zernike Polynomials |.qK69  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,o_Ur.UJ  
    ]h`<E~  
    % Determine the required powers of r: 9uRs@]i  
    % ----------------------------------- ToNRY<!  
    m_abs = abs(m); J4xJGO  
    rpowers = []; 60A E~  
    for j = 1:length(n) n*HRGJ  
        rpowers = [rpowers m_abs(j):2:n(j)]; 4raKhN"  
    end On^jHqLaE  
    rpowers = unique(rpowers); Y XBU9T{r  
    Za&.sg3RG  
    % Pre-compute the values of r raised to the required powers, B F,rZZL  
    % and compile them in a matrix: +( *;F4>  
    % ----------------------------- I|{A&G}|q  
    if rpowers(1)==0 h /.^iT  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p{sbf;-x}  
        rpowern = cat(2,rpowern{:}); 9qqzCMrI0e  
        rpowern = [ones(length_r,1) rpowern]; 7n_'2qY  
    else ub#>kCL9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Qxvj`Ge  
        rpowern = cat(2,rpowern{:}); Sv E|"  
    end z@_ 9.n]  
    #]BpTpRAe<  
    % Compute the values of the polynomials: w^.^XK4v.  
    % -------------------------------------- TDMyZ!d  
    y = zeros(length_r,length(n)); xz#.3|_('  
    for j = 1:length(n) Ke_ & dgsq  
        s = 0:(n(j)-m_abs(j))/2; X~H ~k1  
        pows = n(j):-2:m_abs(j); RZV8{  
        for k = length(s):-1:1 @`</Z)  
            p = (1-2*mod(s(k),2))* ... $oJ)W@>  
                       prod(2:(n(j)-s(k)))/              ... \w]c<gM K  
                       prod(2:s(k))/                     ... )bM #s">Y  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...  F%}0q&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); \mBH6GS  
            idx = (pows(k)==rpowers); Sb9In_* 0  
            y(:,j) = y(:,j) + p*rpowern(:,idx); e>Z F? (a0  
        end N1O& fMz  
         u_5O<UP5  
        if isnorm LB)sk$)  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pO~VI$7  
        end )ZU=`!4  
    end xSQ0]vE  
    % END: Compute the Zernike Polynomials f +#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hKT]M[Pv  
    Mohy;#8Wk  
    % Compute the Zernike functions: m-~eCFc  
    % ------------------------------ ()E:gq Q  
    idx_pos = m>0; 7jb{E+DrG  
    idx_neg = m<0; h%hE$2  
    Tz=YSQy$9  
    z = y; /R_*u4}iD  
    if any(idx_pos) $rZ:$d.C  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); `f@VX :aL}  
    end Y'.WO[dgf  
    if any(idx_neg) O$><E8q  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); )6S;w7  
    end .CrrjS w  
    2Qoj>Wy{  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) qM`XF32A$  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 2u[:3K-@,  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ^6?NYHMr=  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive cx]O#b6B.  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, Tl#Jf3XY}  
    %   and THETA is a vector of angles.  R and THETA must have the same +s6 wF{  
    %   length.  The output Z is a matrix with one column for every P-value, 1MtvnPY  
    %   and one row for every (R,THETA) pair. -DO*,Eecv  
    % 7k<4/|CQ{  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike dVDQ^O&  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) kT(}>=]g  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) TtL2}Wdd.%  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 x M1>kbo|  
    %   for all p. \WM*2&  
    % :!a9|Fh~  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 {&Kq/sRz  
    %   Zernike functions (order N<=7).  In some disciplines it is ~Od4( }/G  
    %   traditional to label the first 36 functions using a single mode wHW";3w2~  
    %   number P instead of separate numbers for the order N and azimuthal GHHErXT\a  
    %   frequency M. e75 k-  
    % IJ^KYho  
    %   Example: @<]xbWhuw  
    % $'f<4  
    %       % Display the first 16 Zernike functions U.oxLbJ`  
    %       x = -1:0.01:1; 8#%p[TLj  
    %       [X,Y] = meshgrid(x,x); -jB1tba  
    %       [theta,r] = cart2pol(X,Y); v>3)^l:=Y*  
    %       idx = r<=1; xMsos?5}  
    %       p = 0:15; ;Ef:mr"Nu  
    %       z = nan(size(X)); PXGS5,  
    %       y = zernfun2(p,r(idx),theta(idx)); ;lST@>  
    %       figure('Units','normalized') 4z-sR/d  
    %       for k = 1:length(p) P'#m1ntxQ  
    %           z(idx) = y(:,k); @GGzah#  
    %           subplot(4,4,k) 7N^9D H{`  
    %           pcolor(x,x,z), shading interp Vw*;xek?  
    %           set(gca,'XTick',[],'YTick',[]) lrjlkgSN  
    %           axis square %S8e:kc6  
    %           title(['Z_{' num2str(p(k)) '}'])  :GC <U|p  
    %       end <^,w,A  
    % ,ZcW+!  
    %   See also ZERNPOL, ZERNFUN. W[o~AbU  
    BRP9j y  
    %   Paul Fricker 11/13/2006 ;T2)nSAqt  
    v]g/ 5qI&  
    |U?5% L  
    % Check and prepare the inputs: Lj"~6l`)  
    % ----------------------------- WYTeu "  
    if min(size(p))~=1  C~vU  
        error('zernfun2:Pvector','Input P must be vector.') oC>QJ(o,8  
    end [ADr _  
    A)En25,X  
    if any(p)>35 lTPo2-j/eK  
        error('zernfun2:P36', ... E%\iNU!  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... .+7GecYz  
               '(P = 0 to 35).']) i^(_Gk  
    end (veGztt  
    BVxg=7%St  
    % Get the order and frequency corresonding to the function number: lIO.LF3  
    % ---------------------------------------------------------------- $}<+~JpGfP  
    p = p(:); DO(-)i zC  
    n = ceil((-3+sqrt(9+8*p))/2); ~=HN30  
    m = 2*p - n.*(n+2); H,qIHQW#  
    gZgb-$b  
    % Pass the inputs to the function ZERNFUN: QthHQA  
    % ---------------------------------------- ;Jt*s  
    switch nargin PYqx&om  
        case 3 WO$PW`k  
            z = zernfun(n,m,r,theta); `pF|bZ?v  
        case 4 IC+Z C   
            z = zernfun(n,m,r,theta,nflag); w!)B\l^+c  
        otherwise 9iWDEk  
            error('zernfun2:nargin','Incorrect number of inputs.') ^.,pq?_  
    end eX 9{wb(  
    I!.o& dk  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) st ( l85  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. DDn@M|*$  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of KDgJ~T  
    %   order N and frequency M, evaluated at R.  N is a vector of /j./  
    %   positive integers (including 0), and M is a vector with the Gvv~P3Dm  
    %   same number of elements as N.  Each element k of M must be a aM?Xi6 U5  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) bLGgu#  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is [=9-AG~}  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Oi&.pY:X-  
    %   with one column for every (N,M) pair, and one row for every /K2VSj3\  
    %   element in R. cu(2BDfiL  
    % 2}>jq8Y47  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ,xB&{ J  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is >>=lh  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 5Fm.] /  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 =2pGbD;*  
    %   for all [n,m]. G>&=rmK"  
    % szMh}q"u  
    %   The radial Zernike polynomials are the radial portion of the E~_2Jf\U  
    %   Zernike functions, which are an orthogonal basis on the unit |{+D65R  
    %   circle.  The series representation of the radial Zernike ?`Qw=8]`  
    %   polynomials is 8s pGDg\g  
    % !!4_x  
    %          (n-m)/2 VdQ}G!d  
    %            __ AU}e^1h  
    %    m      \       s                                          n-2s r9 'lFj  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r EcrM`E#kaZ  
    %    n      s=0 ,x!P|\w.G{  
    % mf6?8!O}>  
    %   The following table shows the first 12 polynomials. Kvv&# eO\  
    % -fuSCj  
    %       n    m    Zernike polynomial    Normalization ~T>_}Q[M2p  
    %       --------------------------------------------- T[B@7$Dp*  
    %       0    0    1                        sqrt(2) -X5rGp++  
    %       1    1    r                           2 ct=|y(_  
    %       2    0    2*r^2 - 1                sqrt(6) ~"!F&  
    %       2    2    r^2                      sqrt(6) lBh|+K N  
    %       3    1    3*r^3 - 2*r              sqrt(8) bwUsE U 0  
    %       3    3    r^3                      sqrt(8) 7$WO@yOsh  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \ }>1$kH;  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) &K2[>5 mG  
    %       4    4    r^4                      sqrt(10) Q*Per;%J  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 23@e?A=C  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) DtG><g}[]  
    %       5    5    r^5                      sqrt(12) T!eeMsI  
    %       --------------------------------------------- rc1EJ(c  
    % 0*YLFqN  
    %   Example: YUJlQ2e(  
    % QgO@oV*S  
    %       % Display three example Zernike radial polynomials YOwo\'|=  
    %       r = 0:0.01:1; "12.Bi.O"[  
    %       n = [3 2 5]; )Xg,;^  
    %       m = [1 2 1]; /lkIbmV  
    %       z = zernpol(n,m,r); )VQ:L:1t(  
    %       figure &N GYV  
    %       plot(r,z) YFOSv]w  
    %       grid on 2;r(?ebw  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') K/79Tb-  
    % }legh:/*?O  
    %   See also ZERNFUN, ZERNFUN2. 55MsF}p  
    x+l.04a@  
    % A note on the algorithm. KL,=Z&.<=  
    % ------------------------ >|WNsjkU%  
    % The radial Zernike polynomials are computed using the series RoSh|$JF  
    % representation shown in the Help section above. For many special v>YdPQky  
    % functions, direct evaluation using the series representation can zM"OateA  
    % produce poor numerical results (floating point errors), because JDfkm+}uY  
    % the summation often involves computing small differences between )Y}t~ Zfx  
    % large successive terms in the series. (In such cases, the functions bPEf2Z G4  
    % are often evaluated using alternative methods such as recurrence )+RTA y[k  
    % relations: see the Legendre functions, for example). For the Zernike qEPvV  
    % polynomials, however, this problem does not arise, because the /1ooOq]  
    % polynomials are evaluated over the finite domain r = (0,1), and lHv;C*(_=  
    % because the coefficients for a given polynomial are generally all db$Th=s[  
    % of similar magnitude. z]^&^VFu  
    % /c'3I  
    % ZERNPOL has been written using a vectorized implementation: multiple =z'- B~  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] jhrmQS  
    % values can be passed as inputs) for a vector of points R.  To achieve =i  }  
    % this vectorization most efficiently, the algorithm in ZERNPOL \~d|MP}"F:  
    % involves pre-determining all the powers p of R that are required to {[hH: \  
    % compute the outputs, and then compiling the {R^p} into a single 5:/ zbt\C  
    % matrix.  This avoids any redundant computation of the R^p, and s$css{(ek  
    % minimizes the sizes of certain intermediate variables. z(d@!Cd  
    % (xpj?zlmM  
    %   Paul Fricker 11/13/2006 6js94ko[  
    ]3wg-p+  
    /"+YE&>\  
    % Check and prepare the inputs: f9u^/QVS&  
    % ----------------------------- <uDEDb1|l  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) h 1G`z  
        error('zernpol:NMvectors','N and M must be vectors.') ewg&DBbN"  
    end r/'9@oM  
    )$Xd#bzD|  
    if length(n)~=length(m) jnsV'@v8Nj  
        error('zernpol:NMlength','N and M must be the same length.') ce th)Xm  
    end _"_ W KlN  
    ^ }Rqe  
    n = n(:); }Iz'#I Xx  
    m = m(:); y`wTw/5N  
    length_n = length(n); ]J+ }WR  
    z=:<]j#=  
    if any(mod(n-m,2)) 3?SofPtc/  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') i71 ,  
    end uN20sD}  
    l_GvdD  
    if any(m<0) RB.&,1  
        error('zernpol:Mpositive','All M must be positive.') l|z 'Lwwm5  
    end 7yo/ sb9h  
    l?v`kAMR  
    if any(m>n) :L#t?~  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') (G $nN*rlu  
    end {Ak{ ct\t  
     {I+   
    if any( r>1 | r<0 ) n_\V G[f  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') rq?:I:0  
    end i <0H W  
    |#8u:rguy  
    if ~any(size(r)==1) 8"/5Lh(  
        error('zernpol:Rvector','R must be a vector.') YYU Di@K  
    end ~L(=-B`Ow  
    nlJ~Q_E(  
    r = r(:); oV utHt  
    length_r = length(r); ?;@xAj  
    X{zg-k(@  
    if nargin==4 wU,{ 5w  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); k<RJSK8  
        if ~isnorm <6^MVaD  
            error('zernpol:normalization','Unrecognized normalization flag.') j_S///  
        end EM]~yn!+  
    else ?#?[6t  
        isnorm = false; Dz/I"bZLC  
    end HPl!r0 h  
    Z*S 9pkWcF  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | n5F_RL  
    % Compute the Zernike Polynomials m<)0 XE6w  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l<5O\?Vo]  
    N|hNh$J[  
    % Determine the required powers of r: v(D{_  
    % ----------------------------------- Qb}7lm{r  
    rpowers = []; OrP-+eg  
    for j = 1:length(n) n ^P=a'+  
        rpowers = [rpowers m(j):2:n(j)]; BE. v+'c"  
    end )R$+dPu>  
    rpowers = unique(rpowers); RK?b/9y  
    54~`8f  
    % Pre-compute the values of r raised to the required powers,  ^ZnlWZ@r  
    % and compile them in a matrix: &09z`* ,  
    % ----------------------------- &os9K)  
    if rpowers(1)==0 # D"TY-$.=  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |S48xsFvq  
        rpowern = cat(2,rpowern{:}); wHm{4  
        rpowern = [ones(length_r,1) rpowern]; !9=hUpRN  
    else 3Tze`Q 9  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^|y6oj  
        rpowern = cat(2,rpowern{:}); 2?YN8 n9n  
    end *O-1zIlp  
    pOP`n3m0  
    % Compute the values of the polynomials: Q4e*Z9YJ  
    % -------------------------------------- <>$`vuU  
    z = zeros(length_r,length_n); W5,e;4/hL  
    for j = 1:length_n ULc oti=,  
        s = 0:(n(j)-m(j))/2; jn vJ`7zFP  
        pows = n(j):-2:m(j); v#*9rNEj0  
        for k = length(s):-1:1 NIufL }6\  
            p = (1-2*mod(s(k),2))* ... &ywAzGV{s  
                       prod(2:(n(j)-s(k)))/          ... P5s'cPX  
                       prod(2:s(k))/                 ... z =1 J{]  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... %T@3-V_  
                       prod(2:((n(j)+m(j))/2-s(k))); hJY= )  
            idx = (pows(k)==rpowers); +c4]}9f!  
            z(:,j) = z(:,j) + p*rpowern(:,idx); *y[i~{7:  
        end hZ NS$  
         vQB;a?)o  
        if isnorm 0[ MQp"z  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ucP}( $  
        end K{)N:|y%!$  
    end .),ql_sXr  
    z%cq%P8g  
    % 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)  ;CbQ}k  
    c.5?Q >!+  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 <]f ru1  
    Az +}[t  
    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)