切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 10989阅读
    • 9回复

    [求助]ansys分析后面型数据如何进行zernike多项式拟合? [复制链接]

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 #Z6'?p9  
    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! WTImRXK4  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 k?2k'2dy  
    function z = zernfun(n,m,r,theta,nflag) 7"8hC  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. MNSbtT*^  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 2(/g}  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 8T(e.I  
    %   unit circle.  N is a vector of positive integers (including 0), and LVJxn2x6  
    %   M is a vector with the same number of elements as N.  Each element /="~gq@  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) E*jP87g  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, JwJ7=P=c  
    %   and THETA is a vector of angles.  R and THETA must have the same d6W SL;$  
    %   length.  The output Z is a matrix with one column for every (N,M) <Qxh)@ N  
    %   pair, and one row for every (R,THETA) pair. F^hBtfz  
    % ?(R]9.5S  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike G#MdfKH  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =b/L?dR.-  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral =+AS/Jq  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 92^w8Z.  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized y.[Mnj  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U^Xm)lL  
    % ij?  
    %   The Zernike functions are an orthogonal basis on the unit circle. 9;veuX#(  
    %   They are used in disciplines such as astronomy, optics, and P3oI2\)*i  
    %   optometry to describe functions on a circular domain. 9Lr'YRl[W  
    % s+Q~~]HJM  
    %   The following table lists the first 15 Zernike functions. Dgy]ae(Hb3  
    % 8stwg'  
    %       n    m    Zernike function           Normalization YX` 7Hm,  
    %       -------------------------------------------------- e @IA20  
    %       0    0    1                                 1 /Ml.}7&  
    %       1    1    r * cos(theta)                    2 _U/!4A  
    %       1   -1    r * sin(theta)                    2 /tUy3myJ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `\+@Fwfx  
    %       2    0    (2*r^2 - 1)                    sqrt(3) *V+j%^91}  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) Dq)j:f#QM  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 7^g&)P  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) &B|D;|7H  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) {c (!;U  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) A,`8#-AX  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) DZ_lW  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V =-WYu  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) f aLtdQi  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -N"&/)  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2z|*xS'G  
    %       -------------------------------------------------- ?.YOI.U^  
    % v{A KEX*  
    %   Example 1: H=\3Jj(4  
    % -Y='_4s  
    %       % Display the Zernike function Z(n=5,m=1) 1 CHeufQ  
    %       x = -1:0.01:1; k2AJXw  
    %       [X,Y] = meshgrid(x,x); LGl2$#x  
    %       [theta,r] = cart2pol(X,Y); wR^R M(1  
    %       idx = r<=1; [w -l?  
    %       z = nan(size(X)); t 89!Ihk  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); q=#} yEG  
    %       figure G8;w{-{m  
    %       pcolor(x,x,z), shading interp bP^Je&nS*  
    %       axis square, colorbar ;v$4$D]L  
    %       title('Zernike function Z_5^1(r,\theta)') =dFv/F/RW  
    % [3@):8  
    %   Example 2: 1n@8Kv  
    % \.3D~2cU  
    %       % Display the first 10 Zernike functions n+PzA[  
    %       x = -1:0.01:1; DS'n  
    %       [X,Y] = meshgrid(x,x); qBCK40   
    %       [theta,r] = cart2pol(X,Y); {\(L%\sV@  
    %       idx = r<=1; ;vIrGZV<  
    %       z = nan(size(X)); d`F&aC  
    %       n = [0  1  1  2  2  2  3  3  3  3]; q5#J~n8Wr  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; et }T %~T  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ,L`$09\  
    %       y = zernfun(n,m,r(idx),theta(idx)); 1u6^z  
    %       figure('Units','normalized') ;W^o@*i{>  
    %       for k = 1:10 Oj^,m.R  
    %           z(idx) = y(:,k); ^6_Cc  
    %           subplot(4,7,Nplot(k)) 7bV{Q355P  
    %           pcolor(x,x,z), shading interp M-giR:,  
    %           set(gca,'XTick',[],'YTick',[]) 67VT\f  
    %           axis square iURk=*Z=  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fF V!)Zj  
    %       end ) lZp9O  
    % YWxc-fPZ  
    %   See also ZERNPOL, ZERNFUN2.  0gfA#|'  
    zNIsf "  
    %   Paul Fricker 11/13/2006 u,w:SM@*(  
    ivW(*c  
    o!!yd8~*r  
    % Check and prepare the inputs: iV eC=^1  
    % ----------------------------- .Fa4shNV  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (owrdPT!  
        error('zernfun:NMvectors','N and M must be vectors.') P`e!Z:  
    end &w1P\4?G  
    0JJS2oY/  
    if length(n)~=length(m) nVI! @qW  
        error('zernfun:NMlength','N and M must be the same length.') |\g5+fv9  
    end !ki.t  
    $.[#0lCI  
    n = n(:); =%> oR  
    m = m(:); 3dRr/Ilc  
    if any(mod(n-m,2)) =F;.l@:  
        error('zernfun:NMmultiplesof2', ... f`&dQ,;  
              'All N and M must differ by multiples of 2 (including 0).') d:i;z9b@to  
    end Ix(><#P  
    f0BdXsV#g  
    if any(m>n) *Otg*, \  
        error('zernfun:MlessthanN', ... S!sqbLrBn  
              'Each M must be less than or equal to its corresponding N.') Vl2XDkhq  
    end \R3H+W  
    mb!9&&2 -t  
    if any( r>1 | r<0 ) ;j)FnY=:-  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ._+J_ts  
    end PxfY&;4n!  
    w#g#8o>'  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) X 51Yfr  
        error('zernfun:RTHvector','R and THETA must be vectors.') q.()z(M 7  
    end q=9`06  
    Bdu&V*0g  
    r = r(:); Nq@+'<@p$  
    theta = theta(:); &|`C)6[C  
    length_r = length(r); '_$uW&{NI  
    if length_r~=length(theta) VV9_`myN7  
        error('zernfun:RTHlength', ... nM0[P6p  
              'The number of R- and THETA-values must be equal.') ?K3(D;5 &i  
    end leQT-l2Bk  
    `3Uj{w/Q:L  
    % Check normalization: wW%4d  
    % -------------------- Bk+{RN(w  
    if nargin==5 && ischar(nflag) @_LN3zP  
        isnorm = strcmpi(nflag,'norm'); 2~t[RY  
        if ~isnorm YXI'gn2b#  
            error('zernfun:normalization','Unrecognized normalization flag.') PClMQL#  
        end \2vg{  
    else FEJ~k1z  
        isnorm = false; nYJTKU  
    end s|NjT  
    XyOl:>%L!P  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ku..aG`  
    % Compute the Zernike Polynomials cDI [PJ9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =2 *rA'im  
    1\r|g2Z :  
    % Determine the required powers of r: yZWoN&  
    % -----------------------------------  fu9Cx  
    m_abs = abs(m); MW+b;0U`#  
    rpowers = []; ,do58i K  
    for j = 1:length(n) ?SC[G-b  
        rpowers = [rpowers m_abs(j):2:n(j)]; 41_SRh7N  
    end RAp=s  
    rpowers = unique(rpowers); EFc-foN  
    1DA1N<'  
    % Pre-compute the values of r raised to the required powers,  3S&U!  
    % and compile them in a matrix: <u=4*:QE  
    % ----------------------------- m B\C?=_  
    if rpowers(1)==0 .%82P(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bUY>st'  
        rpowern = cat(2,rpowern{:}); jU5}\oP@  
        rpowern = [ones(length_r,1) rpowern]; r lKlpl  
    else -D^}S"'  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); raQ7.7  
        rpowern = cat(2,rpowern{:}); mB0l "# F  
    end .E@|D6$D  
    10#f`OPC  
    % Compute the values of the polynomials:  ]@M5&  
    % -------------------------------------- Q*XE h  
    y = zeros(length_r,length(n)); XhPe]P  
    for j = 1:length(n) bTSL<"(]N  
        s = 0:(n(j)-m_abs(j))/2; C8L'si  
        pows = n(j):-2:m_abs(j); GAc{l=vT'  
        for k = length(s):-1:1 w2xG_q  
            p = (1-2*mod(s(k),2))* ... |0,vQv  
                       prod(2:(n(j)-s(k)))/              ... ,Hgc-7g@Y  
                       prod(2:s(k))/                     ... GTJ{h  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zY|klX})  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); M+!x}$ &v  
            idx = (pows(k)==rpowers); !(t,FYeH  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 1>Q'R  
        end p)~lL  
         ^bLRVp1  
        if isnorm p\ Lq}tk<  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); q-Qxbg[>e  
        end oW;6h.  
    end ~xIj F1Z  
    % END: Compute the Zernike Polynomials 1R. 4:Dn_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9Ok9bC'?8@  
    9*:gr#(5  
    % Compute the Zernike functions: WGAXIQ  
    % ------------------------------ T,_(?YJW  
    idx_pos = m>0; X1vNF|o~  
    idx_neg = m<0; 1JEnnqu  
    5#E |R  
    z = y; 5%}wV,Y  
    if any(idx_pos) 6yy;JQAke  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }!i` 0p  
    end qSx(X!YS  
    if any(idx_neg) pZZf[p^s|  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p*l$Wj  
    end <*EZ@XoN>  
    4m-I5!=O  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) (.3'=n|kE  
    %ZERNFUN2 Single-index Zernike functions on the unit circle.  H@uE>  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated [/RM=4Nh5  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive mceG!@t  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, @$eT~ C  
    %   and THETA is a vector of angles.  R and THETA must have the same [hRU&z;W  
    %   length.  The output Z is a matrix with one column for every P-value, xdy^ ^3"  
    %   and one row for every (R,THETA) pair. +2C?9:bH  
    % s:y ^_W)d  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike F&;   
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ;o<m}bGaT  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) K^t?gt@k}  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 uENdI2EY8y  
    %   for all p. 2yo cu!4l  
    % insY(.N  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 |vFj*XU  
    %   Zernike functions (order N<=7).  In some disciplines it is ;pRcVL_4  
    %   traditional to label the first 36 functions using a single mode /\Q*MLwD  
    %   number P instead of separate numbers for the order N and azimuthal lnbmoHv  
    %   frequency M. ] q~<=   
    % qO`qJ/  
    %   Example: )fU(AXSP  
    % @oe\"vz  
    %       % Display the first 16 Zernike functions f*xpE`&  
    %       x = -1:0.01:1; (!?K7<Jv  
    %       [X,Y] = meshgrid(x,x); >P. 'CU  
    %       [theta,r] = cart2pol(X,Y); dv N<5~  
    %       idx = r<=1; 5c -N0@\  
    %       p = 0:15; Ps R>V)L  
    %       z = nan(size(X)); sP$Ks#/  
    %       y = zernfun2(p,r(idx),theta(idx)); T,JA#Rk|1N  
    %       figure('Units','normalized') #NRh\Wj|  
    %       for k = 1:length(p) ")lw9t`  
    %           z(idx) = y(:,k); b*,3< 9  
    %           subplot(4,4,k) oYM,8 K  
    %           pcolor(x,x,z), shading interp RM*f|j  
    %           set(gca,'XTick',[],'YTick',[]) v+1i= s2$  
    %           axis square 'qv;sB.  
    %           title(['Z_{' num2str(p(k)) '}']) 1x >iz `A  
    %       end -g`IH-B  
    % .gYt0raSY  
    %   See also ZERNPOL, ZERNFUN. X,v4d~>]  
    2RppP?M!  
    %   Paul Fricker 11/13/2006 8TZENRzx-|  
    p/]s)uYp$  
    Jfg7\&|  
    % Check and prepare the inputs: M1u{A^d.Z  
    % ----------------------------- <`g3(?   
    if min(size(p))~=1 i</J@0}y  
        error('zernfun2:Pvector','Input P must be vector.') @Z\~  
    end mrZ`Lm#>pS  
    &$ p[  
    if any(p)>35 IjZ@U%g@;  
        error('zernfun2:P36', ... r[HT9  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... E20 :uZ7\  
               '(P = 0 to 35).']) !0fI"3P@r  
    end KAb(NZK  
    E`@43Nz  
    % Get the order and frequency corresonding to the function number: V,LVB_6  
    % ---------------------------------------------------------------- u3ds QU  
    p = p(:); if~rp-\P  
    n = ceil((-3+sqrt(9+8*p))/2); %<}=xJf>1  
    m = 2*p - n.*(n+2); [BXyi  
    ^9ng)  
    % Pass the inputs to the function ZERNFUN: l C\E  
    % ---------------------------------------- y(8d?]4:_  
    switch nargin Zg $Tf  
        case 3 =,Ttw>   
            z = zernfun(n,m,r,theta); D@vMAW  
        case 4 zk>h u<_  
            z = zernfun(n,m,r,theta,nflag); SFO&=P:U  
        otherwise _bI+QC#   
            error('zernfun2:nargin','Incorrect number of inputs.') 4 iH&:Al  
    end En5!"w|j  
    %ejeyc  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) ^K*-G@B  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. #`j][F@N  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of S OI)/u  
    %   order N and frequency M, evaluated at R.  N is a vector of aQh?}=da  
    %   positive integers (including 0), and M is a vector with the sV'v* 1|  
    %   same number of elements as N.  Each element k of M must be a VR v02m5  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) @ta?&Qf)  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 2f`xHI/@fj  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix ji##$xC  
    %   with one column for every (N,M) pair, and one row for every 3M$X:$b  
    %   element in R. 0Bu*g LY  
    % )G4rJ~#@  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- oeGS  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is qT 0_L  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to irmwc'n]  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 .Qk{5=l6P  
    %   for all [n,m]. jZ/+~{<  
    % lE a W7j  
    %   The radial Zernike polynomials are the radial portion of the HPTHF  
    %   Zernike functions, which are an orthogonal basis on the unit uWrFunh%  
    %   circle.  The series representation of the radial Zernike 2H>aC wfX  
    %   polynomials is {jhcZ"#>\  
    % Z~R dFC  
    %          (n-m)/2 D1! {S7  
    %            __ #4q1{)=  
    %    m      \       s                                          n-2s Q;@X2 JSp  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .$^wy3:F"  
    %    n      s=0 <O bHf`Q  
    % %/md"S  
    %   The following table shows the first 12 polynomials. .m!s". ?[  
    % r?afv.@L2  
    %       n    m    Zernike polynomial    Normalization IrUi E q  
    %       --------------------------------------------- b.,$# D{p  
    %       0    0    1                        sqrt(2) NlMQHma  
    %       1    1    r                           2 `rq<jtf+  
    %       2    0    2*r^2 - 1                sqrt(6) !*8#jy  
    %       2    2    r^2                      sqrt(6) @92gb$xT  
    %       3    1    3*r^3 - 2*r              sqrt(8) #!Ze\fOC  
    %       3    3    r^3                      sqrt(8) FSVS4mtiX\  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) -7,vtd[h  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) !`Xt8q\r  
    %       4    4    r^4                      sqrt(10) 4UazD_`'  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Yd.027  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) F\Y,JUn[G  
    %       5    5    r^5                      sqrt(12) #a.\P.{L  
    %       --------------------------------------------- CHg]Ul  
    % &R0OeRToUb  
    %   Example: *<?XTs<  
    % rQ &S<  
    %       % Display three example Zernike radial polynomials GA7u5D"0  
    %       r = 0:0.01:1; f{f_g8f[  
    %       n = [3 2 5]; QWKs[yfdo  
    %       m = [1 2 1]; :(+]b  
    %       z = zernpol(n,m,r);  .?70=8{  
    %       figure &1 oaZY w  
    %       plot(r,z) :"y0oCu7`W  
    %       grid on FE>3 D1\  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') .r(^h/IF  
    % |zT%$  
    %   See also ZERNFUN, ZERNFUN2. R&#[6 r(h  
    BZ2nDW*%  
    % A note on the algorithm. /5jKX 5r  
    % ------------------------ jjYM3LQcdP  
    % The radial Zernike polynomials are computed using the series G^ K*+  
    % representation shown in the Help section above. For many special 8>2&h  
    % functions, direct evaluation using the series representation can xp~YIeSg  
    % produce poor numerical results (floating point errors), because m\1VF\  
    % the summation often involves computing small differences between l#p }{  
    % large successive terms in the series. (In such cases, the functions HUK" OH  
    % are often evaluated using alternative methods such as recurrence 8g-P_[>  
    % relations: see the Legendre functions, for example). For the Zernike TS/Cp{  
    % polynomials, however, this problem does not arise, because the n#)PvV~  
    % polynomials are evaluated over the finite domain r = (0,1), and 7:#  
    % because the coefficients for a given polynomial are generally all 5FZ47m ~{Z  
    % of similar magnitude. lGl[^ 0  
    % (21']x  
    % ZERNPOL has been written using a vectorized implementation: multiple `:V}1ioX5  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r(pwOOx  
    % values can be passed as inputs) for a vector of points R.  To achieve :EYu 4Y  
    % this vectorization most efficiently, the algorithm in ZERNPOL H\ {E%7^h-  
    % involves pre-determining all the powers p of R that are required to ;HR 6X  
    % compute the outputs, and then compiling the {R^p} into a single |X,$?ZDap  
    % matrix.  This avoids any redundant computation of the R^p, and +SO2M|ru&  
    % minimizes the sizes of certain intermediate variables. r[6#G2  
    % 2%`^(\y  
    %   Paul Fricker 11/13/2006 OiYNH~hv  
    mJSK; @w<O  
    m*\B2\2gJ  
    % Check and prepare the inputs: q;CayN'I  
    % ----------------------------- ]d[Rf$>vu0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :U!'U;uQ  
        error('zernpol:NMvectors','N and M must be vectors.') xi;/^)r  
    end KuIBYaK, g  
    PbbXi  
    if length(n)~=length(m) +Gk! t]dy  
        error('zernpol:NMlength','N and M must be the same length.') \8=e |a5`  
    end q-A`/9  
    -08&&H  
    n = n(:); VfQMFb',o  
    m = m(:); N>Vacc_[  
    length_n = length(n); e$ThSh\+(  
    dCa}ITg  
    if any(mod(n-m,2)) S`ax*`  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 3Ne9% "  
    end TS\9<L9S  
    (~q#\  
    if any(m<0) ^2C0oX  
        error('zernpol:Mpositive','All M must be positive.') Y1#-^,qg  
    end !w @1!Xpn1  
    z\xiACIc  
    if any(m>n) + 9F^F>mu  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') j l;kcGE  
    end HiQoRk  
    %bCcsdK  
    if any( r>1 | r<0 ) Es.toOH$S  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 6V.awg,  
    end +io;K]C  
    +A]&AkTw  
    if ~any(size(r)==1) &>&dhdTQ  
        error('zernpol:Rvector','R must be a vector.') 8O"x;3I9  
    end 3g?MEM~  
    [z W_%O kP  
    r = r(:); >P<k[vF  
    length_r = length(r); v< 65(I>  
    LFk5rv'sM0  
    if nargin==4 bs<WH`P  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); C ffTv  
        if ~isnorm %Nwyx;>9^K  
            error('zernpol:normalization','Unrecognized normalization flag.') *%ed;>6:Q  
        end ^2&O3s  
    else Y|hzF:ll  
        isnorm = false; 9f@#SB_H  
    end ",MK'\E  
    +Fu@I{"A  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S(g<<Te  
    % Compute the Zernike Polynomials G=r(SJq  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  IgzCh  
    %f_)<NP9=  
    % Determine the required powers of r: .fio<mqi  
    % ----------------------------------- m NUN6qVP~  
    rpowers = []; BxSk%$J  
    for j = 1:length(n) 377j3dP  
        rpowers = [rpowers m(j):2:n(j)]; 1Y H4a|bc  
    end kr/1Dsr4  
    rpowers = unique(rpowers); ?=/}Ft  
    [oQ`HX1g  
    % Pre-compute the values of r raised to the required powers, #U ",,*2  
    % and compile them in a matrix: j6&zRFX  
    % ----------------------------- )z ?&" I  
    if rpowers(1)==0 *@-q@5r}!  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); TS\A`{^T  
        rpowern = cat(2,rpowern{:}); EWu iaw.  
        rpowern = [ones(length_r,1) rpowern]; .LeF|EQU\@  
    else pO-s@"j]  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); oW yN:Qh  
        rpowern = cat(2,rpowern{:}); H3p4,Y}'#  
    end N=O+X~  
    H#1*'e>  
    % Compute the values of the polynomials: f^[{k {t  
    % -------------------------------------- ;JPbBwm  
    z = zeros(length_r,length_n); %S(#cf!HP  
    for j = 1:length_n \,@Yl.,+  
        s = 0:(n(j)-m(j))/2; 9a"Y,1  
        pows = n(j):-2:m(j); ;y?D1o^r8W  
        for k = length(s):-1:1 C$AIP\j- )  
            p = (1-2*mod(s(k),2))* ... a0V8L+v(  
                       prod(2:(n(j)-s(k)))/          ... ijZydn  
                       prod(2:s(k))/                 ... i(&6ys5  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... T>%uRK$  
                       prod(2:((n(j)+m(j))/2-s(k))); Ru  vG1"  
            idx = (pows(k)==rpowers); _Cv[`e.  
            z(:,j) = z(:,j) + p*rpowern(:,idx); U&Sbm~Qi  
        end NE; (..  
         a.Rp#}f  
        if isnorm ZZ]OR;8  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); yVmtsQ-}a  
        end Mu_mm/U_  
    end SBN_>;$c5}  
    % L %1g  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
    发帖
    4352
    光币
    5478
    光券
    1
    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  3{L vKe  
    i K[8At"Xo  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 ;o@`l$O   
    98}vbl31j  
    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)