切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11151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 m~= ]^e  
    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! hO+O0=$}wN  
     
    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 QigoRB!z#9  
    function z = zernfun(n,m,r,theta,nflag) rr07\;  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. zP{<0o  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {d?4;Kd  
    %   and angular frequency M, evaluated at positions (R,THETA) on the TQ5MKqR$  
    %   unit circle.  N is a vector of positive integers (including 0), and !q=Q~ea  
    %   M is a vector with the same number of elements as N.  Each element ,/w852|ub  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) f@F^W YQm  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Il&"=LooZ  
    %   and THETA is a vector of angles.  R and THETA must have the same Vlp*'2VO  
    %   length.  The output Z is a matrix with one column for every (N,M) R>e3@DQ~  
    %   pair, and one row for every (R,THETA) pair. Sf4h!ly  
    % _ \v@9Q\  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike vS J<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -u3SsU)_%N  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral [:R P9r}  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]UCk_zWsn1  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized T^(n+lv  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \\R*V'e!  
    % %)6 :eIS  
    %   The Zernike functions are an orthogonal basis on the unit circle. YP\4XI  
    %   They are used in disciplines such as astronomy, optics, and j$mt*z L  
    %   optometry to describe functions on a circular domain. !s[j1=y  
    % *09\\ G  
    %   The following table lists the first 15 Zernike functions. "13 :VTs[5  
    % vRb(eg  
    %       n    m    Zernike function           Normalization 'De'(I  
    %       -------------------------------------------------- wJeqa  
    %       0    0    1                                 1 {HRxyAI!  
    %       1    1    r * cos(theta)                    2 6ImV5^l  
    %       1   -1    r * sin(theta)                    2 8|jX ~f  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) l=-d K_ I?  
    %       2    0    (2*r^2 - 1)                    sqrt(3) &PQ{e8w  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) c@o/Cv  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ;aRWJG  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) vu.S>2Wv  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ]N(zom_0d  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ">D(+ xr!)  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) aIt 0;D  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,f /IG.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) <>*''^  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1:{O RX[;  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) / =Uv  
    %       -------------------------------------------------- _qzo):G.s  
    % qYu!:xa8  
    %   Example 1: )r|zi Z{F  
    % $hE'b9qx  
    %       % Display the Zernike function Z(n=5,m=1) A$"$`)P!  
    %       x = -1:0.01:1; LWb}) #E  
    %       [X,Y] = meshgrid(x,x); Dgq[g_+l  
    %       [theta,r] = cart2pol(X,Y); ,YMdXYu`s  
    %       idx = r<=1; CIik@O*  
    %       z = nan(size(X)); !{~7)iq  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); = cI\OsV&?  
    %       figure -ZoOX"N}  
    %       pcolor(x,x,z), shading interp ah6F^Kpl{  
    %       axis square, colorbar "6NNId|Y  
    %       title('Zernike function Z_5^1(r,\theta)') l-h7ksRs  
    % q!oZ; $  
    %   Example 2: DwrCysIK  
    % )RCqsFjK  
    %       % Display the first 10 Zernike functions @Kb|  
    %       x = -1:0.01:1; k;:u| s8NS  
    %       [X,Y] = meshgrid(x,x); kFa?q} 47  
    %       [theta,r] = cart2pol(X,Y); cV!/  
    %       idx = r<=1; AO7qs:+  
    %       z = nan(size(X)); 0!'M#'m  
    %       n = [0  1  1  2  2  2  3  3  3  3]; xo_k"'f+  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 53&xTcv}x  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Pymh^i  
    %       y = zernfun(n,m,r(idx),theta(idx)); -K'84 bZ  
    %       figure('Units','normalized') n_Hn k4  
    %       for k = 1:10 3^-)gK  
    %           z(idx) = y(:,k); C<=p"pWw  
    %           subplot(4,7,Nplot(k)) &fy8,}  
    %           pcolor(x,x,z), shading interp vls> 6h  
    %           set(gca,'XTick',[],'YTick',[]) WT {Cjn  
    %           axis square fu "z%h]   
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @k #y-/~?  
    %       end ]<_!@J6k  
    % hE#8_34%s  
    %   See also ZERNPOL, ZERNFUN2. Z!i'Tbfn  
    PaeafL65=  
    %   Paul Fricker 11/13/2006 -bu. *=  
    ~t3?er& R  
    3Co>3d_  
    % Check and prepare the inputs: ]~M {@h!<  
    % ----------------------------- _,?HrL9  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 0m!ZJHe  
        error('zernfun:NMvectors','N and M must be vectors.') v;qL? _:=c  
    end hgr ,v"  
    4_:e+ ql  
    if length(n)~=length(m) p[VCt" j  
        error('zernfun:NMlength','N and M must be the same length.') l YA+k5  
    end s ;Nu2aOp7  
    ~9;mZi1-  
    n = n(:); *ik)>c_  
    m = m(:); 3:Egqw  
    if any(mod(n-m,2)) 5e8-?w% e  
        error('zernfun:NMmultiplesof2', ... M6Z`Pwv];  
              'All N and M must differ by multiples of 2 (including 0).') GeT CN  
    end 7IW7'klkvD  
    4i&!V9@:  
    if any(m>n) CMjPp`rA  
        error('zernfun:MlessthanN', ... Y tj>U  
              'Each M must be less than or equal to its corresponding N.') {cHTg04  
    end l>P~M50D?{  
    Jpnp'  
    if any( r>1 | r<0 ) DYk->)   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') TEyPlSGG  
    end \/%Q PE8  
    (8F?yBu  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )  cJ{P,K  
        error('zernfun:RTHvector','R and THETA must be vectors.') -* j;  
    end a2)*tbM 9\  
    m,fr?d/;  
    r = r(:); m,_oX1h  
    theta = theta(:); . k DCcnm  
    length_r = length(r); X KeK;+  
    if length_r~=length(theta) gz:c_HJ  
        error('zernfun:RTHlength', ... yG_.|%e  
              'The number of R- and THETA-values must be equal.') ;G&O"S><]c  
    end LYKm2C*d  
    Du4?n8 o  
    % Check normalization: ~%q e,  
    % -------------------- u-cC}DP  
    if nargin==5 && ischar(nflag) [qo* ,CRz  
        isnorm = strcmpi(nflag,'norm'); ~$ Yuxo  
        if ~isnorm p{u}t!`!d  
            error('zernfun:normalization','Unrecognized normalization flag.') ~_6rD`2cJ  
        end #jR?C9&!(  
    else ld0WZj  
        isnorm = false; /;[')RO`  
    end h<jIg$rA  
    I!%@|[ Ow  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8;bOw  
    % Compute the Zernike Polynomials hD=D5LYAZ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |LhuZ_;1xo  
    {<Zqw]  
    % Determine the required powers of r: |1$X`|S  
    % ----------------------------------- d@~)Wlje  
    m_abs = abs(m); z#ET-[ I  
    rpowers = []; eLWzd_ln  
    for j = 1:length(n) R``qQ;cc  
        rpowers = [rpowers m_abs(j):2:n(j)]; }\*|b@)]  
    end 8A=(,)`}9  
    rpowers = unique(rpowers); @bE?WXY  
    JaTW/~ TU  
    % Pre-compute the values of r raised to the required powers, /$Jh5Bv  
    % and compile them in a matrix: ~Y$1OA8  
    % ----------------------------- Q0A1N[  
    if rpowers(1)==0 e;v2`2z2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uDUSR+E>  
        rpowern = cat(2,rpowern{:}); <aS1bQgaU  
        rpowern = [ones(length_r,1) rpowern]; $~l :l[Zs  
    else -A~<IyPt  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F.6SX (x  
        rpowern = cat(2,rpowern{:}); #YV;Gp(2h  
    end P^r8JhDJ  
    36z{TWF  
    % Compute the values of the polynomials: LNW p$"  
    % -------------------------------------- (nG  
    y = zeros(length_r,length(n)); \wP$"Z}j  
    for j = 1:length(n) -8: @xG2  
        s = 0:(n(j)-m_abs(j))/2; w\a#Bfcv  
        pows = n(j):-2:m_abs(j); 0Oq1ay^  
        for k = length(s):-1:1 xC]/i(+bA  
            p = (1-2*mod(s(k),2))* ... auU{I y   
                       prod(2:(n(j)-s(k)))/              ... nfEk,(:  
                       prod(2:s(k))/                     ... s4\2lBU?  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... GWsFW[T?~  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 9lwg`UWl,  
            idx = (pows(k)==rpowers); :nn'>  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 2TO1i0  
        end Y-9F*8<  
         Ex{]<6UAu  
        if isnorm K,Vl.-4?  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _`_$U MK;  
        end y+_U6rv[  
    end A}o1I1+  
    % END: Compute the Zernike Polynomials \hVFK6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z(cgI5Pu  
    ! `o =2b=N  
    % Compute the Zernike functions: ~PHG5?X  
    % ------------------------------ f3O'lc3  
    idx_pos = m>0; {[eY/)6H  
    idx_neg = m<0; CS  
    x :s-\>RcA  
    z = y; )deuB5kz  
    if any(idx_pos) OmW|\d PU  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {Ffr l(*  
    end uQ}kq7gd  
    if any(idx_neg) (<t)5?@%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); DuaOi1Gw  
    end wS*UXF&f  
    S!uyplYKF  
    % 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) dCBJV  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. o Vs&r?\Z  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated |1EM )zh6  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive uKplPze?  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 8AK#bna~-  
    %   and THETA is a vector of angles.  R and THETA must have the same &WIPz\  
    %   length.  The output Z is a matrix with one column for every P-value, -Rmz`yOq}  
    %   and one row for every (R,THETA) pair. K=;p^dE  
    % giv cq'L  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike #'8E%4  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) dPS}\&1  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) dS-l2 $n  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 qzXch["So  
    %   for all p. %',. K)IR  
    % F2#^5s(  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 n/?_]  
    %   Zernike functions (order N<=7).  In some disciplines it is B<W{kEY  
    %   traditional to label the first 36 functions using a single mode |zSkQ_?54  
    %   number P instead of separate numbers for the order N and azimuthal { Uh/ ~zu  
    %   frequency M. r__uPyIMG/  
    % [/I4Pe1Yj%  
    %   Example: N( Cfv3{  
    % ,+f'%)s_x  
    %       % Display the first 16 Zernike functions |<OZa;c+  
    %       x = -1:0.01:1; zg|]Ic  
    %       [X,Y] = meshgrid(x,x); *9G;n!t  
    %       [theta,r] = cart2pol(X,Y); /.vB /{2  
    %       idx = r<=1; V7.EDE2A3  
    %       p = 0:15; cxVnlgq1  
    %       z = nan(size(X)); (sZ B-  
    %       y = zernfun2(p,r(idx),theta(idx)); 4e* rBTl  
    %       figure('Units','normalized') 0q81H./3  
    %       for k = 1:length(p) `CO?} rW  
    %           z(idx) = y(:,k); b}Jcj  
    %           subplot(4,4,k) 2x0[@cT i?  
    %           pcolor(x,x,z), shading interp Rc @p!Xi  
    %           set(gca,'XTick',[],'YTick',[]) 4Xlq Ym  
    %           axis square (2 X`imJ  
    %           title(['Z_{' num2str(p(k)) '}']) XB2[{XH,  
    %       end ?EdF&^[3rD  
    % ^+I{*0{/[  
    %   See also ZERNPOL, ZERNFUN. , j7&(V~  
    b:oB $E  
    %   Paul Fricker 11/13/2006 v#F .FK  
    PXWBc\  
    sdrALl;w|  
    % Check and prepare the inputs: %kUIIH V}  
    % ----------------------------- nF]lSg&]X  
    if min(size(p))~=1 b8(94t|;U  
        error('zernfun2:Pvector','Input P must be vector.') ki39$A'8  
    end jCkYzQUPz  
    du  Pzt  
    if any(p)>35 qh{hpX)\D  
        error('zernfun2:P36', ... ZJ1 %  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... wP29 xV"5  
               '(P = 0 to 35).']) )V7bi^r  
    end Bg {"{poy  
    O1V s!  
    % Get the order and frequency corresonding to the function number: s:CsUl|  
    % ---------------------------------------------------------------- (V 5_q,2  
    p = p(:); M, f6UYo=  
    n = ceil((-3+sqrt(9+8*p))/2); .}C pX  
    m = 2*p - n.*(n+2); l!5fuB8  
    r Z%l?(  
    % Pass the inputs to the function ZERNFUN: Yv\>\?865  
    % ---------------------------------------- eh`n?C  
    switch nargin Tc$Jvy-G4A  
        case 3 \b6H4aQii  
            z = zernfun(n,m,r,theta); k "7l\;N  
        case 4 `ZhDoLpH<  
            z = zernfun(n,m,r,theta,nflag); as6a)t.^  
        otherwise ):-\TVz~  
            error('zernfun2:nargin','Incorrect number of inputs.') [z=KHk  
    end jOV,q%)^,:  
    <2^XKaS`  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 4TX~]tEyky  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ?9W2wqN>o  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of HFlMx  
    %   order N and frequency M, evaluated at R.  N is a vector of `?P)RS30  
    %   positive integers (including 0), and M is a vector with the 4}&$s  
    %   same number of elements as N.  Each element k of M must be a n3x< L:)  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) N #C,q&;  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is .A%*AlX  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix P~xP@? I%  
    %   with one column for every (N,M) pair, and one row for every ~IQ3B $4H&  
    %   element in R. :$ qa  
    % A-S!Z2m\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- so=Ux2  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is N./l\NtZ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to | 6/ # H*  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 85gdmla@9  
    %   for all [n,m]. dkUh[yo"H  
    % O> .gcLA  
    %   The radial Zernike polynomials are the radial portion of the >/J!:Htk+K  
    %   Zernike functions, which are an orthogonal basis on the unit #"&<^  
    %   circle.  The series representation of the radial Zernike _98 %?0  
    %   polynomials is Le,e,#hiY  
    % ?xX9o  
    %          (n-m)/2 ^p7g[E&  
    %            __ ais@|s;  
    %    m      \       s                                          n-2s ;;f&aujSHD  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @ vHj>N  
    %    n      s=0 xryXO(  
    % E'KKR1t  
    %   The following table shows the first 12 polynomials. OUM^ u*  
    % ^ s.necg0  
    %       n    m    Zernike polynomial    Normalization }[FP"#  
    %       --------------------------------------------- u*w'.5l  
    %       0    0    1                        sqrt(2) FV~ENpncP  
    %       1    1    r                           2 d$ f3 Cre  
    %       2    0    2*r^2 - 1                sqrt(6) 0cF +4,5  
    %       2    2    r^2                      sqrt(6) .G/>X%X  
    %       3    1    3*r^3 - 2*r              sqrt(8) <0sT  
    %       3    3    r^3                      sqrt(8) !TwH;#U w  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) D:Rr|m0Tk  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) <13').F  
    %       4    4    r^4                      sqrt(10) a]]>(Txc  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :@oy5zib  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) Ul{{g$  
    %       5    5    r^5                      sqrt(12) .).}ffhOL  
    %       --------------------------------------------- G?$0OU  
    % Q |1-j  
    %   Example: I%?ia5]H  
    % GeydVT-  
    %       % Display three example Zernike radial polynomials Or:a\qQ1  
    %       r = 0:0.01:1; h+d  \u  
    %       n = [3 2 5]; I7C*P~32{n  
    %       m = [1 2 1]; Kf>]M|G c  
    %       z = zernpol(n,m,r); J{w[vcf  
    %       figure @a]O(S>Ub  
    %       plot(r,z) oY(q(W0ze  
    %       grid on {3H)c^Q  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <$+Cd=71\  
    % N3U.62  
    %   See also ZERNFUN, ZERNFUN2. q-<t'uhs[  
    ^1 U<,<  
    % A note on the algorithm. j-}WA"  
    % ------------------------ >Y>>lE! k  
    % The radial Zernike polynomials are computed using the series #)7THx/=  
    % representation shown in the Help section above. For many special ]IQ`.:g=9  
    % functions, direct evaluation using the series representation can &l-1.muQ  
    % produce poor numerical results (floating point errors), because {9_}i#,vR  
    % the summation often involves computing small differences between o?]N2e&(  
    % large successive terms in the series. (In such cases, the functions [$iKx6\  
    % are often evaluated using alternative methods such as recurrence _&U.DMt2 C  
    % relations: see the Legendre functions, for example). For the Zernike 4Rv.m* ^B  
    % polynomials, however, this problem does not arise, because the 9snc *<  
    % polynomials are evaluated over the finite domain r = (0,1), and bd & /B&a  
    % because the coefficients for a given polynomial are generally all sgxD5xj}4  
    % of similar magnitude.  G9qN1q~  
    % yKb+bm&5:'  
    % ZERNPOL has been written using a vectorized implementation: multiple  HQ0fY  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ,e93I6  
    % values can be passed as inputs) for a vector of points R.  To achieve ~u^MRe|`  
    % this vectorization most efficiently, the algorithm in ZERNPOL a 9H^e<g  
    % involves pre-determining all the powers p of R that are required to (fUpj^E)p  
    % compute the outputs, and then compiling the {R^p} into a single =F 9!)r  
    % matrix.  This avoids any redundant computation of the R^p, and !M*$p Qi}  
    % minimizes the sizes of certain intermediate variables. sngM4ikhs  
    % .W*"C  
    %   Paul Fricker 11/13/2006 y(92Th$  
    8x/]H(J  
    '2=$pw  
    % Check and prepare the inputs: h,!`2_&UQ  
    % ----------------------------- Rv=rO|&]  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) YQ _]Jv k  
        error('zernpol:NMvectors','N and M must be vectors.') I|eYeJ3  
    end XhEJF !  
    [!'fE #"a  
    if length(n)~=length(m) ,)beK*Iw  
        error('zernpol:NMlength','N and M must be the same length.') )&pcRFl  
    end /t=Fx94  
    D\CjR6DE  
    n = n(:); G.l ~!;  
    m = m(:); l'm\ *=3  
    length_n = length(n); *^=zQ~  
    Z6\H4,k&  
    if any(mod(n-m,2)) q1_iV.G<  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') appWq}db  
    end M:/)|fk  
    57~/QEdy  
    if any(m<0) L,7+26XV"B  
        error('zernpol:Mpositive','All M must be positive.') (5\VOCT>4%  
    end LEn+0^hX  
    >Y 8\I  
    if any(m>n) o4F?Rx,L  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') U ,7O{YM  
    end 2, R5mL$  
    3-6Lbe9H  
    if any( r>1 | r<0 ) q>5 K:5  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') H:5- S  
    end ,SB5"  
    =b_/_b$q  
    if ~any(size(r)==1) ~M Mv+d88  
        error('zernpol:Rvector','R must be a vector.') Wfp>BC  
    end ;'i>^zX`  
    <l!{j?Kx  
    r = r(:); ~Kiu " g  
    length_r = length(r); UY~N4IR8  
    yO*HJpc   
    if nargin==4 `2l j{N  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); OGWZq(c"6  
        if ~isnorm &|Cd1z#?  
            error('zernpol:normalization','Unrecognized normalization flag.') ,C;%AS/  
        end 5!jU i9  
    else 0hv}*NYd  
        isnorm = false; a,`f`;\7N%  
    end f3oGB*5>  
    8D:0Vhx\I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?&_u$Nn  
    % Compute the Zernike Polynomials R^k)^!/$f  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ra)AQ n  
    0 qp Pz|h  
    % Determine the required powers of r: p{0NKyOvU  
    % ----------------------------------- d]r?mnN W  
    rpowers = []; pz0Q@n/X  
    for j = 1:length(n) P+<4w  
        rpowers = [rpowers m(j):2:n(j)]; @|6#]&v`  
    end q .s'z}  
    rpowers = unique(rpowers); M"!{Dx~  
    w:HRzU>  
    % Pre-compute the values of r raised to the required powers, AQm#a;  
    % and compile them in a matrix: F1GFn|OA  
    % ----------------------------- vlVHoF;&  
    if rpowers(1)==0 6.6;oa4j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w0&|8y  
        rpowern = cat(2,rpowern{:}); K*9~ g('  
        rpowern = [ones(length_r,1) rpowern]; 6^NL>|?  
    else # ~(lY}  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  sDl @  
        rpowern = cat(2,rpowern{:}); 3{H&{@Q  
    end S(#v<C,hd  
    sa w  
    % Compute the values of the polynomials: j^6,V\;l  
    % -------------------------------------- (MzThGJK_  
    z = zeros(length_r,length_n); /C6$B)w_*{  
    for j = 1:length_n 6(8zt"E  
        s = 0:(n(j)-m(j))/2; {&uN q^Ch  
        pows = n(j):-2:m(j); 9+"ISXS  
        for k = length(s):-1:1 B+4WnR1%T  
            p = (1-2*mod(s(k),2))* ... Nz;f| 2h  
                       prod(2:(n(j)-s(k)))/          ... w[]\%`69}Z  
                       prod(2:s(k))/                 ... 1qm _Qs&  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... MHQM'  
                       prod(2:((n(j)+m(j))/2-s(k))); h pKrP  
            idx = (pows(k)==rpowers); &6&$vF65c  
            z(:,j) = z(:,j) + p*rpowern(:,idx); o-+H-  
        end V\*J"ZP&  
         Y$0K}`{  
        if isnorm fRK=y+gl@  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); [Wn6d:  
        end 4Ul*`/d  
    end nj=nSD  
    B4t,@,\O  
    % 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)  VB@M=ShKK  
    mW_ N-z  
    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。 QxeK-x^  
    gBJM|"_A?  
    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)