切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11084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 d&hD[v  
    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! 8GY.){d!l  
     
    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 iOll WkF  
    function z = zernfun(n,m,r,theta,nflag) FOSbe]  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. c#  xO<  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rCE;'? Y  
    %   and angular frequency M, evaluated at positions (R,THETA) on the { UOhVJy  
    %   unit circle.  N is a vector of positive integers (including 0), and V}SyD(8~  
    %   M is a vector with the same number of elements as N.  Each element ) \ 4 |  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 6Hwxx5>r  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9Eg&CZ,9$D  
    %   and THETA is a vector of angles.  R and THETA must have the same {V0>iN:~S  
    %   length.  The output Z is a matrix with one column for every (N,M) 0V3gKd7  
    %   pair, and one row for every (R,THETA) pair. AFm,CINa  
    % \6:>{0\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike gfm;xT/y  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V!xwb:J  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral *> KHRR<N  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \B&6TeR  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized <BPRV> 0X  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. wyzOcx>M  
    % GmbIFOT~  
    %   The Zernike functions are an orthogonal basis on the unit circle. ]`d2_mu  
    %   They are used in disciplines such as astronomy, optics, and ZBJ3VK  
    %   optometry to describe functions on a circular domain. /l6\^Xf{  
    % \TUE<<?1s  
    %   The following table lists the first 15 Zernike functions. 2e.N"eLNt  
    % ~.6|dw\p!  
    %       n    m    Zernike function           Normalization +#s;yc#=2  
    %       -------------------------------------------------- [O_^MA,z  
    %       0    0    1                                 1 V&[eSVY?  
    %       1    1    r * cos(theta)                    2 -\Z `z}D  
    %       1   -1    r * sin(theta)                    2 _q)!B,y-/N  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) AK*N  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 4\6: \  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 9 mPIykAj8  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) |l7%l&!  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 2tf6GX:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) KDD@%E  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Sl>>SP  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) q}wj}t#  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~@Kf2dHes  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) C(o.Cy6  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  rN"Xz  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) 2xn<E>]  
    %       -------------------------------------------------- JUQg 'D  
    % ZPyM>XK$4  
    %   Example 1:  s4$X  
    % etyCrQ ?U  
    %       % Display the Zernike function Z(n=5,m=1) NR4Jn?l{  
    %       x = -1:0.01:1; #6W,6(#^#  
    %       [X,Y] = meshgrid(x,x); nm@']  
    %       [theta,r] = cart2pol(X,Y); >'`Sf ?+|  
    %       idx = r<=1; :<GfETIs  
    %       z = nan(size(X)); AIh*1>2Xn  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); "- eZZEl(  
    %       figure *vnXlV4L  
    %       pcolor(x,x,z), shading interp yN\e{;z`  
    %       axis square, colorbar }1U*A#aN7K  
    %       title('Zernike function Z_5^1(r,\theta)') #3 bv3m  
    % =nU/ [T.  
    %   Example 2: ZJ(rG((!  
    % a2yE:16o6  
    %       % Display the first 10 Zernike functions i8~$o:&HT  
    %       x = -1:0.01:1; } 0M{A+  
    %       [X,Y] = meshgrid(x,x); vv.PF~:  
    %       [theta,r] = cart2pol(X,Y); f^9&WT  
    %       idx = r<=1; Rri`dmH   
    %       z = nan(size(X)); Hm9<fQuM  
    %       n = [0  1  1  2  2  2  3  3  3  3]; 8!zb F<W9  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; G{b:i8}l  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; -$YJfQE6G  
    %       y = zernfun(n,m,r(idx),theta(idx)); D.*>;5:0'  
    %       figure('Units','normalized') J`oTes,  
    %       for k = 1:10 i-lKdpv  
    %           z(idx) = y(:,k); "8(U\KaX  
    %           subplot(4,7,Nplot(k)) SRL-Z&M  
    %           pcolor(x,x,z), shading interp Wx]d $_  
    %           set(gca,'XTick',[],'YTick',[]) q*8lnk  
    %           axis square >85zQ 1aL  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) wsnK3tM7-  
    %       end @6&JR<g*t  
    % s[AA7>]3  
    %   See also ZERNPOL, ZERNFUN2. }c|UX ZW  
    AhxGj+  
    %   Paul Fricker 11/13/2006 3nFt1E   
    n?E}b$6  
    f z}?*vPW  
    % Check and prepare the inputs: u7=T(4a  
    % ----------------------------- &5Y_>{,  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) - k`.j  
        error('zernfun:NMvectors','N and M must be vectors.') it1/3y =]  
    end `.^ |]|u  
    z%:&#1)  
    if length(n)~=length(m) [uR/M  
        error('zernfun:NMlength','N and M must be the same length.') AK2WN#u@Z  
    end #ia;- 3  
    1 Z[f {T)  
    n = n(:); lTz6"/  
    m = m(:); S_Z`so}  
    if any(mod(n-m,2)) <DZcra  
        error('zernfun:NMmultiplesof2', ...  >eS$  
              'All N and M must differ by multiples of 2 (including 0).') 9lspo~M  
    end ^M[P-#X_  
    ^}>/n. %  
    if any(m>n) >n$ !<  
        error('zernfun:MlessthanN', ... tcL2J.  
              'Each M must be less than or equal to its corresponding N.') ~V+l_ :  
    end .zC*Z&e,.[  
    ai^|N.!  
    if any( r>1 | r<0 ) )^/0cQcJ  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]J@/p:S>  
    end ngUHkpYS5  
    |y1;&<  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K2ewucn  
        error('zernfun:RTHvector','R and THETA must be vectors.') 1;wb(DN*c  
    end !'W-6f  
    9UD @MA  
    r = r(:); +jV_Wz  
    theta = theta(:); bd \=h1  
    length_r = length(r); lG"H4Aa>  
    if length_r~=length(theta) LwdV3vb#  
        error('zernfun:RTHlength', ... -cfx2;68  
              'The number of R- and THETA-values must be equal.') +nU.p/cK+\  
    end ]P1YHw9  
    ` }8&E(<  
    % Check normalization: E%3TP_B3  
    % -------------------- 3,6Ox45  
    if nargin==5 && ischar(nflag) 8cdsToF(e.  
        isnorm = strcmpi(nflag,'norm'); Ijedo/  
        if ~isnorm U[||~FW'  
            error('zernfun:normalization','Unrecognized normalization flag.') `ROG~0lN(  
        end `X8@/wf#  
    else LWmB, Zf/  
        isnorm = false; &<1 `O  
    end FPv" N'/  
    Y25uU%6t_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )[&zCq Dc  
    % Compute the Zernike Polynomials #`ejU&!6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \:/Lc{*}MD  
    |wp ,f%WK  
    % Determine the required powers of r: Lj 8<' "U#  
    % ----------------------------------- k-jahm4  
    m_abs = abs(m); o`?zF+M0  
    rpowers = []; EzT`,#b  
    for j = 1:length(n) jP=Hf=:$  
        rpowers = [rpowers m_abs(j):2:n(j)]; g22gIj]  
    end 9&  
    rpowers = unique(rpowers); I%;Jpe  
    ZYMw}]#((E  
    % Pre-compute the values of r raised to the required powers, qL 5>o>J  
    % and compile them in a matrix: 3V;gW%>  
    % ----------------------------- /q1s;I  
    if rpowers(1)==0 G+WM`:v8%  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); HEY4$Lf(I  
        rpowern = cat(2,rpowern{:}); x;#zs64f  
        rpowern = [ones(length_r,1) rpowern]; ~`cwG` 'N  
    else .<&s%{EW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wAF,H8 -DK  
        rpowern = cat(2,rpowern{:}); |jG~,{  
    end K* vU5S  
    1>pe&n/  
    % Compute the values of the polynomials: f )NHM'  
    % -------------------------------------- bcz-$?]  
    y = zeros(length_r,length(n)); c:\shAM&  
    for j = 1:length(n) JUt7En;XE  
        s = 0:(n(j)-m_abs(j))/2; 0A[esWmP  
        pows = n(j):-2:m_abs(j); :tj-gDa\Y  
        for k = length(s):-1:1 SvuTc!$?  
            p = (1-2*mod(s(k),2))* ... ,sQ93(Vo  
                       prod(2:(n(j)-s(k)))/              ... <$i4?)f(  
                       prod(2:s(k))/                     ... ^[q /Mw  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... b"CAKl  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); w{,4rk;Hr  
            idx = (pows(k)==rpowers); 8]"(!i_;)  
            y(:,j) = y(:,j) + p*rpowern(:,idx); )K]pnH|  
        end Q*ju sm  
         :td ~g;w  
        if isnorm SW 8x]B  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); U ?b".hJ2  
        end WeJ@x L  
    end 1mgLX_U9  
    % END: Compute the Zernike Polynomials {aOkV::  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d8x%SQ!V  
    |m* .LTO  
    % Compute the Zernike functions: <"tDAx  
    % ------------------------------ ,.mBJ SE3  
    idx_pos = m>0; 8l+H"M&|  
    idx_neg = m<0; p,!$/Q+l  
    >fs2kha  
    z = y; lK(Fg  
    if any(idx_pos) H3KTir"on  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); lj[, |[X7`  
    end c:hK$C)T  
    if any(idx_neg) ]k%PG-9  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M]rO;^;6?  
    end M {a #  
    _GA$6#]  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) {7"0,2 Hb?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. CboLH0Fa  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Oe!6){OG)  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive @!%n$>p/V  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, :1wrVU-?h  
    %   and THETA is a vector of angles.  R and THETA must have the same -j2 (R?a  
    %   length.  The output Z is a matrix with one column for every P-value, `dkV_ O0  
    %   and one row for every (R,THETA) pair. yi6N-7  
    % +s[\g>i  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike @4GA^h  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) u?H 2%hD  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 5 t{ja  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 qRB7Ec_  
    %   for all p. 6^F '|Wh  
    % 5Jk<xWKj  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 t;q7t!sC]  
    %   Zernike functions (order N<=7).  In some disciplines it is wa09$4>_w  
    %   traditional to label the first 36 functions using a single mode %&blJ6b  
    %   number P instead of separate numbers for the order N and azimuthal iz^qR={bW  
    %   frequency M. XEH}4;C'{  
    % kI\tqNJi  
    %   Example: x~DLW1I  
    % PGn);Baq  
    %       % Display the first 16 Zernike functions nHOr AD|&  
    %       x = -1:0.01:1; =t0tK}Y+4  
    %       [X,Y] = meshgrid(x,x); y-aRXF=W  
    %       [theta,r] = cart2pol(X,Y); ?A*Kg;IU  
    %       idx = r<=1; oOU1{[  
    %       p = 0:15; J ++v@4Z  
    %       z = nan(size(X)); ^rAa"p9  
    %       y = zernfun2(p,r(idx),theta(idx)); Ty4S~ClO#'  
    %       figure('Units','normalized') _F(P*[[&  
    %       for k = 1:length(p) c-1q2y  
    %           z(idx) = y(:,k); N3A<:%s  
    %           subplot(4,4,k) cu9Qwm  
    %           pcolor(x,x,z), shading interp M4f;/`w  
    %           set(gca,'XTick',[],'YTick',[]) |i %2%V#  
    %           axis square E#%}ZY  
    %           title(['Z_{' num2str(p(k)) '}']) PR7f(NC  
    %       end ,XKCz ]8V  
    % G-um`/<%  
    %   See also ZERNPOL, ZERNFUN. hUpnI@  
    b'p4wE>  
    %   Paul Fricker 11/13/2006 ^q[gxuL_  
    rxZi8w>}  
    o+O}Te  
    % Check and prepare the inputs: +g*k*e>l  
    % ----------------------------- K`%tGVY  
    if min(size(p))~=1 Zk-~a r  
        error('zernfun2:Pvector','Input P must be vector.') [3/VCYje  
    end %Q"(/jm?  
    v1G"3fy9  
    if any(p)>35 W#F Q,+0)  
        error('zernfun2:P36', ... XFwLz  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... S7iDTG_@t  
               '(P = 0 to 35).']) ~eh0[mF^]  
    end <O~WB  
    x34f9! 't  
    % Get the order and frequency corresonding to the function number: K|S:{9Q  
    % ---------------------------------------------------------------- VU.@R,  
    p = p(:); Do7=#|bAM  
    n = ceil((-3+sqrt(9+8*p))/2); a|j%n  
    m = 2*p - n.*(n+2); "eAy^,  
    P 1>AOH2yG  
    % Pass the inputs to the function ZERNFUN: ]c)_&{:V  
    % ---------------------------------------- b{M7w  
    switch nargin zU5Hb2a  
        case 3 O'*@ Ytn  
            z = zernfun(n,m,r,theta); B}?IEpYp  
        case 4 \Q$HXK  
            z = zernfun(n,m,r,theta,nflag); dE`-\J  
        otherwise |AhF7Mj*  
            error('zernfun2:nargin','Incorrect number of inputs.') {jKI^aC<[  
    end QfjN"25_  
    N!&:rK  
    % 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) 5II(mSg8  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. XMN:]!1J  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of &BE  g  
    %   order N and frequency M, evaluated at R.  N is a vector of M\<w#wZ  
    %   positive integers (including 0), and M is a vector with the lK7m=[ j  
    %   same number of elements as N.  Each element k of M must be a a `Q ot  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) | tQiFC  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is o|pT;1a"  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Z"-L[2E/{!  
    %   with one column for every (N,M) pair, and one row for every u"xJjS  
    %   element in R. bvBHYf:^  
    % KW^<,qt5w  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- R<ND=[}s  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is $ <8~k^  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to SO\/-]9#  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 07g':QU@  
    %   for all [n,m]. zvc`3  
    % FyoEQ%.bI  
    %   The radial Zernike polynomials are the radial portion of the qml2XJ>  
    %   Zernike functions, which are an orthogonal basis on the unit ![6EUMx  
    %   circle.  The series representation of the radial Zernike RkEN ,xWE  
    %   polynomials is pv!oz2w1  
    % ,|?CU r9Y  
    %          (n-m)/2 Flxvhl)L  
    %            __ 3 voT^o  
    %    m      \       s                                          n-2s fU3`v\X  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r lq:}0<k  
    %    n      s=0 V&]DzjT/  
    % ikBYd }5  
    %   The following table shows the first 12 polynomials.  =SOe}!  
    %  _?vo U  
    %       n    m    Zernike polynomial    Normalization F1%vtk;2?  
    %       --------------------------------------------- uQb!=]  
    %       0    0    1                        sqrt(2) <+#o BN  
    %       1    1    r                           2 c?2MBtnu  
    %       2    0    2*r^2 - 1                sqrt(6) o_M.EZO  
    %       2    2    r^2                      sqrt(6) ?jQ](i&  
    %       3    1    3*r^3 - 2*r              sqrt(8) X.F^$  
    %       3    3    r^3                      sqrt(8) <Peebv&v  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) /.Nov  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ?YM4b5!3T  
    %       4    4    r^4                      sqrt(10) nP~({ :l8X  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) RR;AJ8wd  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ~rr 4ok  
    %       5    5    r^5                      sqrt(12) 5qUTMT['T  
    %       --------------------------------------------- )+")Sz3zx  
    % ?Ucu#UO  
    %   Example: 8N%Bn&   
    % }V;+l8  
    %       % Display three example Zernike radial polynomials :1q 4"tv|  
    %       r = 0:0.01:1; c)md  
    %       n = [3 2 5]; sAJ7R(p  
    %       m = [1 2 1]; -tsDMji~V  
    %       z = zernpol(n,m,r); e,_-Je  
    %       figure Fk;o E'"D  
    %       plot(r,z) ow=UtA-^O  
    %       grid on fEE /-}d  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') g @I6$Z  
    % tc r//  
    %   See also ZERNFUN, ZERNFUN2. %Pqk63QF  
    ^taBG3P  
    % A note on the algorithm. *Oc.9 F88"  
    % ------------------------ ZR v"h/~  
    % The radial Zernike polynomials are computed using the series e pCLM_yA  
    % representation shown in the Help section above. For many special Z|9u]xL  
    % functions, direct evaluation using the series representation can f~OU*P>V@  
    % produce poor numerical results (floating point errors), because Ioy  
    % the summation often involves computing small differences between wv QMnE8\  
    % large successive terms in the series. (In such cases, the functions {j{+0V  
    % are often evaluated using alternative methods such as recurrence ik|-L8  
    % relations: see the Legendre functions, for example). For the Zernike -7uwOr  
    % polynomials, however, this problem does not arise, because the H2xeP%;$  
    % polynomials are evaluated over the finite domain r = (0,1), and $uui:wU%Q  
    % because the coefficients for a given polynomial are generally all R`";Z$~{  
    % of similar magnitude. kc'pN&]r:  
    % LWsP ya  
    % ZERNPOL has been written using a vectorized implementation: multiple $P7iRM]  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] plu$h-$d  
    % values can be passed as inputs) for a vector of points R.  To achieve m\>a,oZH  
    % this vectorization most efficiently, the algorithm in ZERNPOL iGDLZE+?  
    % involves pre-determining all the powers p of R that are required to kL7#W9  
    % compute the outputs, and then compiling the {R^p} into a single @=]~\[e\  
    % matrix.  This avoids any redundant computation of the R^p, and {*ZY(6^  
    % minimizes the sizes of certain intermediate variables. Ogt]_  
    % 1QZ&Mj^^  
    %   Paul Fricker 11/13/2006 XS0xLt=  
     HBys  
    V]c;^  
    % Check and prepare the inputs: @\oz4^  
    % ----------------------------- cWGDee(  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }),w1/#5u8  
        error('zernpol:NMvectors','N and M must be vectors.') b96%")  
    end cr?7O;,  
    &~UJf4b|A  
    if length(n)~=length(m) i`/+,<  
        error('zernpol:NMlength','N and M must be the same length.') rV({4cIe9R  
    end ]`g <w#  
    3Y)PU=  
    n = n(:); @cRZk`|1n  
    m = m(:); xR"M*%{@0  
    length_n = length(n); +5.t. d  
    z|?R/Gf8  
    if any(mod(n-m,2)) qjJBcu_C'S  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') #>Y'sd5'A  
    end 7f<EoSK  
    q'oMAMf}  
    if any(m<0) gef6pfV  
        error('zernpol:Mpositive','All M must be positive.') ?6c-7QV  
    end ODc9r }  
    sC00un%  
    if any(m>n) O=)  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') _8}QlT  
    end p\C%%  
    '`Bm'Dd  
    if any( r>1 | r<0 ) d_S*#/k  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ~9F,%  
    end 4> ^K:/y  
    'tN25$=V&W  
    if ~any(size(r)==1) M,j(=hRJ/E  
        error('zernpol:Rvector','R must be a vector.') =5D nR  
    end =S[yE]v^  
    sfr(/mp(  
    r = r(:); w(L>#?  
    length_r = length(r); q;5 i4|  
    e98lhu"|H  
    if nargin==4 =H0vE7{*  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); !KKT[28v  
        if ~isnorm in<Rq"L  
            error('zernpol:normalization','Unrecognized normalization flag.') wn Y$fT9  
        end g u)=wu0  
    else , "jbq~  
        isnorm = false; *?QE2&S:  
    end lcON+j  
    : "6q,W  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rQ4*k'lA:  
    % Compute the Zernike Polynomials _u"nvgVz9  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% It_M@  
    {)- .xG  
    % Determine the required powers of r: g#NZ ,~  
    % ----------------------------------- UH@a s  
    rpowers = []; sGY_{CZ:  
    for j = 1:length(n) %I!:ITa  
        rpowers = [rpowers m(j):2:n(j)]; ;E~4)^  
    end NRnRMY-  
    rpowers = unique(rpowers); rdJm{<  
    =1h9rlFj"D  
    % Pre-compute the values of r raised to the required powers,  g]*  
    % and compile them in a matrix: v]2S`ffP  
    % ----------------------------- oq-<ob  
    if rpowers(1)==0 s/"&9F3  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); bLz*A-  
        rpowern = cat(2,rpowern{:}); ;;5Uwd'-  
        rpowern = [ones(length_r,1) rpowern]; JXiZB 8}  
    else aYL|@R5;e  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Fhq9D{TeY,  
        rpowern = cat(2,rpowern{:}); {v aaFs  
    end R8*Q$rH<  
    OYM@szM  
    % Compute the values of the polynomials: +c:3o*  
    % -------------------------------------- @Un/c:n  
    z = zeros(length_r,length_n); +&tgJ07A  
    for j = 1:length_n n?#!VN3  
        s = 0:(n(j)-m(j))/2; (VyNvB  
        pows = n(j):-2:m(j); 2^~<("+w  
        for k = length(s):-1:1 : Ud[f`t  
            p = (1-2*mod(s(k),2))* ... YF#H Sf7  
                       prod(2:(n(j)-s(k)))/          ... 1rw0sAuGy  
                       prod(2:s(k))/                 ... 3[p_!eoW  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... +ww^ev%  
                       prod(2:((n(j)+m(j))/2-s(k))); kI*(V [i  
            idx = (pows(k)==rpowers); >,C4rC+:XN  
            z(:,j) = z(:,j) + p*rpowern(:,idx); G DSfT{kK\  
        end .F&9.#>  
         h*0S$p<[1  
        if isnorm `|1MlRM9  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); I4H`YOD%  
        end F9c`({6k  
    end fnzy5+9"  
    VvByHcLv  
    % 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)  qc8Ta"  
    SE`l(-tL  
    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。 *3Nn +T  
    Wc'Ehyi;  
    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)