切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11002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 Z)@[N 6\?  
    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! GCv1x->  
     
    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 raU_Z[  
    function z = zernfun(n,m,r,theta,nflag) 'm-5  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. \g)?7>M|  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R|wS*xd,  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ,Z! I^  
    %   unit circle.  N is a vector of positive integers (including 0), and p1mAoVxR  
    %   M is a vector with the same number of elements as N.  Each element h|lH`m^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) /V#? d  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Cn5;h(r  
    %   and THETA is a vector of angles.  R and THETA must have the same E0DquVrz  
    %   length.  The output Z is a matrix with one column for every (N,M) UQ@szE  
    %   pair, and one row for every (R,THETA) pair. <p2\;\?4z  
    % _g,_G  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '- #QK'p  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2:e7'}\D.  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral EV-# E  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &yOl}?u  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized 7+hc?H[&'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z/4bxO=m  
    % t3K9 |8<  
    %   The Zernike functions are an orthogonal basis on the unit circle. *Gj`1# Z$  
    %   They are used in disciplines such as astronomy, optics, and N3oa!PE  
    %   optometry to describe functions on a circular domain. ZW@cw}  
    % :2:%  
    %   The following table lists the first 15 Zernike functions. 9JJ6$cLF  
    % S?VKzVDB.S  
    %       n    m    Zernike function           Normalization ;z+}|>!  
    %       -------------------------------------------------- : Cli8#  
    %       0    0    1                                 1 Xf mN/j2  
    %       1    1    r * cos(theta)                    2 zTi 8y<}  
    %       1   -1    r * sin(theta)                    2 eW}-UeT  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) '0&HkM{ D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 7| j rk  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) SxcE@WM  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 5~E{bW$  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) N$.ls48a4-  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 3Ljj|5.q  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) !0):g/2h  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) L6ypn)l  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >enP~uW[#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Kq+vAp).  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \nL@P6X  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) IMpL+W.  
    %       -------------------------------------------------- QXEZ?gx  
    % #'RfwldD9  
    %   Example 1: lTtc#  
    % aQzmobleep  
    %       % Display the Zernike function Z(n=5,m=1) G(t&(t`[  
    %       x = -1:0.01:1; \{ C ~B;=  
    %       [X,Y] = meshgrid(x,x); */$]kE  
    %       [theta,r] = cart2pol(X,Y); Z1;+a+S=z  
    %       idx = r<=1; ]g,j  
    %       z = nan(size(X)); x`'s  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); BIg2`95F|  
    %       figure VMNdC}  
    %       pcolor(x,x,z), shading interp :?i,!0#"  
    %       axis square, colorbar RK)ikLgp  
    %       title('Zernike function Z_5^1(r,\theta)') l-Dgm  
    % 9C5F#(uY  
    %   Example 2: .t{uzDM  
    % Zu P3/d  
    %       % Display the first 10 Zernike functions zn|O)"C  
    %       x = -1:0.01:1; 8&bNI@:@  
    %       [X,Y] = meshgrid(x,x); ;$qc@)Uwp  
    %       [theta,r] = cart2pol(X,Y);  ;CV'  
    %       idx = r<=1; 2+R]q35-  
    %       z = nan(size(X)); !thFayq  
    %       n = [0  1  1  2  2  2  3  3  3  3]; N~S#( .}[  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; WM=)K1p0u  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; 2_Cp}Pj  
    %       y = zernfun(n,m,r(idx),theta(idx)); V gy12dE  
    %       figure('Units','normalized') +j$nbU0U  
    %       for k = 1:10 zhyf}Ta'  
    %           z(idx) = y(:,k); c]ga) A(  
    %           subplot(4,7,Nplot(k)) <YCR^?hJSi  
    %           pcolor(x,x,z), shading interp V warU(*  
    %           set(gca,'XTick',[],'YTick',[]) G,(Xz"`,  
    %           axis square <N=ow"rD  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) eq9qE^[Z&  
    %       end &iy7It  
    % +]hc!s8  
    %   See also ZERNPOL, ZERNFUN2. ^lK!tOeO  
    2t=&h|6EW  
    %   Paul Fricker 11/13/2006 qi8AK(v  
    hIa,PZ/Q  
    zXwdU5 8  
    % Check and prepare the inputs: +hlR  
    % ----------------------------- Q H>g-@  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )  FE1En  
        error('zernfun:NMvectors','N and M must be vectors.') 'p%w_VbI  
    end *$mb~k^R  
    Ie8K [ >  
    if length(n)~=length(m) u=(.}  
        error('zernfun:NMlength','N and M must be the same length.') M?['HoRo  
    end x3jjtjf  
    CwO$EL:[`  
    n = n(:); %fh-x(4v  
    m = m(:); &G3$q,`H  
    if any(mod(n-m,2)) %@C$xM"  
        error('zernfun:NMmultiplesof2', ... D{4]c)>  
              'All N and M must differ by multiples of 2 (including 0).') z34+1d  
    end w7<4D,hk  
    &Mz.i,Gh  
    if any(m>n) Prv=f@  
        error('zernfun:MlessthanN', ... }MM:qR  
              'Each M must be less than or equal to its corresponding N.') \PmM856=ms  
    end dcE(uf  
    9HlM0qE5b  
    if any( r>1 | r<0 ) *kJa$3*r  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ;*20b@  
    end Nk9w ; z&  
    J]Q-#g'Z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) u:^9ZQ+  
        error('zernfun:RTHvector','R and THETA must be vectors.') @DAaCF8  
    end 4%u\dTg/B  
    ,JJ1sf2A  
    r = r(:); AJP-7PPD  
    theta = theta(:); of`WP  
    length_r = length(r); ,awkL :  
    if length_r~=length(theta) u$^r(.EV  
        error('zernfun:RTHlength', ... ~y ?v  
              'The number of R- and THETA-values must be equal.') 8BnsYy)j  
    end sWP_fb1  
    rFto1m  
    % Check normalization: ED A6b]  
    % -------------------- ip*UujmNyR  
    if nargin==5 && ischar(nflag) !nF.whq  
        isnorm = strcmpi(nflag,'norm'); . B6mvb\  
        if ~isnorm `O?j -zR  
            error('zernfun:normalization','Unrecognized normalization flag.') pEb/yIT"  
        end !@ )JqF.  
    else >V&GL{  
        isnorm = false; LO)QEUG  
    end ;^8X(R  
    m!Aw,*m+*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h;J%Z!Rjw  
    % Compute the Zernike Polynomials $rQi$w/  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =jRC4]M})  
    QEY#U|  
    % Determine the required powers of r: YUlH5rO3  
    % ----------------------------------- biH ZyUJ  
    m_abs = abs(m); -Z:nImqzc  
    rpowers = []; LT/ *y=  
    for j = 1:length(n) Ys@\~?ym+  
        rpowers = [rpowers m_abs(j):2:n(j)]; )79F"ltz h  
    end kg$w<C@#"  
    rpowers = unique(rpowers); !LpFK0rw  
    HU-#xK  
    % Pre-compute the values of r raised to the required powers, j|y"Lcq  
    % and compile them in a matrix: 5>h# hcL  
    % ----------------------------- en16hd>^W:  
    if rpowers(1)==0 H$)otDOE  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); pA@BW:#  
        rpowern = cat(2,rpowern{:}); R^6^ {q  
        rpowern = [ones(length_r,1) rpowern]; oX6()FR  
    else D (Q=EdlO  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b*cVC^{Dy  
        rpowern = cat(2,rpowern{:}); XC[bEp$  
    end {Ytqs(`   
    %r:Uff@  
    % Compute the values of the polynomials: WL<f!   
    % -------------------------------------- bm(.(0MI  
    y = zeros(length_r,length(n)); ZJ |&t  
    for j = 1:length(n) b!z=:  
        s = 0:(n(j)-m_abs(j))/2; h.aXW]]}(P  
        pows = n(j):-2:m_abs(j); ]hY4 MS  
        for k = length(s):-1:1 JE[J}-2  
            p = (1-2*mod(s(k),2))* ... ,<=_t{^  
                       prod(2:(n(j)-s(k)))/              ... 3}i(i0+  
                       prod(2:s(k))/                     ... 3x E^EXV  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... gg :{Xf*`  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); v`~egE17  
            idx = (pows(k)==rpowers); qk!,:T  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 8Y~\:3&1<  
        end WL1$LLzN  
         :n$?wp  
        if isnorm A55F* d  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); !F# ^Peb  
        end #(r1b'jfP  
    end [J43]  
    % END: Compute the Zernike Polynomials pt9fOih[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ROr|  <  
    EZ)GW%Bm2  
    % Compute the Zernike functions: vOBXAF  
    % ------------------------------ F ss@/-  
    idx_pos = m>0; v'u}%FC  
    idx_neg = m<0; wWB^m@:4  
    h2ou ]  
    z = y; )|L#i2?:  
    if any(idx_pos) Y q-7!  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Q Pp>%iE@  
    end Cg%}=  
    if any(idx_neg) rJc=&'{&)N  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); lgh+\pj  
    end RJA#cv~f  
    Ip;;@o&D  
    % 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) u#v];6N  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. aUEnQ%YU"  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated H4g1@[{|0O  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive yI_MY L[  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, <5sP%Fs)  
    %   and THETA is a vector of angles.  R and THETA must have the same >Mk#19j[/  
    %   length.  The output Z is a matrix with one column for every P-value,  -bQi4  
    %   and one row for every (R,THETA) pair. Y EhPAQNj  
    % g 9:V00^<  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike IyGW>g6_.  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Rln@9muXA  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) :V:siIDn  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 @!2vS@f  
    %   for all p. a #Pr)H  
    % I8{ohFFo  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 QF9$SCmv  
    %   Zernike functions (order N<=7).  In some disciplines it is ,(&5y:o  
    %   traditional to label the first 36 functions using a single mode 8WMGuv  
    %   number P instead of separate numbers for the order N and azimuthal  '' Pfs<!  
    %   frequency M. :N ]H"u9X  
    % K.::P84m;  
    %   Example: 1|xo4fmV  
    % 8Hh= Sp^  
    %       % Display the first 16 Zernike functions nSeb?|$D6  
    %       x = -1:0.01:1; Kma-W{vGD  
    %       [X,Y] = meshgrid(x,x);  eAbp5}B  
    %       [theta,r] = cart2pol(X,Y); G;v3kGn  
    %       idx = r<=1; Q@? {|7:  
    %       p = 0:15; q OX=M  
    %       z = nan(size(X)); RS /*Dp^  
    %       y = zernfun2(p,r(idx),theta(idx)); n% ={!WD  
    %       figure('Units','normalized') TWT h!  
    %       for k = 1:length(p) "y$s`n4Mj  
    %           z(idx) = y(:,k); 9:]|TIPi  
    %           subplot(4,4,k) 3pI)  
    %           pcolor(x,x,z), shading interp +]jJ:V  
    %           set(gca,'XTick',[],'YTick',[]) 8Xk,Nbcqt  
    %           axis square pJPP6Be<  
    %           title(['Z_{' num2str(p(k)) '}']) ;0]s:0WD0P  
    %       end k{w  
    % ]:F?k#c  
    %   See also ZERNPOL, ZERNFUN. :ej`]yK |  
    * 4RL  
    %   Paul Fricker 11/13/2006 ^fxS=Qs+  
    <+)B8I^  
    f l*O)r  
    % Check and prepare the inputs: G.sf>.[  
    % ----------------------------- MFO1v%m  
    if min(size(p))~=1 x] j&Knli  
        error('zernfun2:Pvector','Input P must be vector.') Qvhz$W[P>  
    end N2e]S8-  
    B*T;DE   
    if any(p)>35 OIdoe0JR:O  
        error('zernfun2:P36', ... X UcM~U-  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... |!.VpN&  
               '(P = 0 to 35).']) v.Zr,Z=eV  
    end (GXFPEH8  
    ]O.Z4+6w  
    % Get the order and frequency corresonding to the function number: R\,qL-Br  
    % ---------------------------------------------------------------- t6a$ZN;  
    p = p(:); E.+BqWZ!  
    n = ceil((-3+sqrt(9+8*p))/2); h$rk]UM/Q  
    m = 2*p - n.*(n+2); o1]ZeF  
    {BS`v5*  
    % Pass the inputs to the function ZERNFUN: 8u4FagQ,  
    % ---------------------------------------- sRDxa5<MD  
    switch nargin #>\%7b59>  
        case 3 I18<brZJ  
            z = zernfun(n,m,r,theta); Vf] ;hm  
        case 4 XD$;K$_7  
            z = zernfun(n,m,r,theta,nflag); 2r>I,TNHl  
        otherwise h;S?  
            error('zernfun2:nargin','Incorrect number of inputs.') !+Us)'L  
    end nh? ~S`  
    8$C?j\J|*  
    % 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) q_[V9  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. c&;Xjy  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of @w{"6xc%a  
    %   order N and frequency M, evaluated at R.  N is a vector of 8KyF0r?  
    %   positive integers (including 0), and M is a vector with the ;/=6~%  
    %   same number of elements as N.  Each element k of M must be a i*2l4  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ]0@ 06G(y  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Bl!R bh\  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix QDpzIjJj  
    %   with one column for every (N,M) pair, and one row for every J'#R9NO<  
    %   element in R. UTph(U#  
    % XYdr~/[HPy  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- X>kW)c4{b  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is *>8Y/3Y\B  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to a0=>@?  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 &&K"3"um  
    %   for all [n,m]. v!77dj 6I  
    % +L\bg| ;  
    %   The radial Zernike polynomials are the radial portion of the Y4)v>&H  
    %   Zernike functions, which are an orthogonal basis on the unit cL yed3uU  
    %   circle.  The series representation of the radial Zernike wS}Rl}#Oh?  
    %   polynomials is kQQhZ8Ch  
    % tX7TP(  
    %          (n-m)/2 'ghwc:Og|%  
    %            __ cNvh2JI  
    %    m      \       s                                          n-2s #) bqn|0l  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `6QQS3fk!  
    %    n      s=0 "pW@[2Dkx/  
    % /o]j  
    %   The following table shows the first 12 polynomials. m. DC  
    % L$4nbOu\~  
    %       n    m    Zernike polynomial    Normalization qbu5aK}+  
    %       --------------------------------------------- #,PB(  
    %       0    0    1                        sqrt(2) $G@^!(  
    %       1    1    r                           2 7R5!(g  
    %       2    0    2*r^2 - 1                sqrt(6) #23m_w^L  
    %       2    2    r^2                      sqrt(6) 0f3C; u-q-  
    %       3    1    3*r^3 - 2*r              sqrt(8) A.@Af+  
    %       3    3    r^3                      sqrt(8) QLum=YB  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) (D <o=Q  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 7UA|G2Zr  
    %       4    4    r^4                      sqrt(10) gt{$G|bi  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) #7yy7Y5  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) hD! 9[Gb  
    %       5    5    r^5                      sqrt(12) 4,P!D3SH  
    %       --------------------------------------------- \B1<fF2  
    % ^":Dk5gl  
    %   Example: FST}:*dOe5  
    % !-Br?  
    %       % Display three example Zernike radial polynomials 9 &p;2/H  
    %       r = 0:0.01:1; Xsit4Ma  
    %       n = [3 2 5]; ?)V}_%fVv  
    %       m = [1 2 1]; r(uf yC&  
    %       z = zernpol(n,m,r); "Ir.1FN  
    %       figure I!uGI  
    %       plot(r,z) P|{Et=R`1  
    %       grid on oYNp0Hc  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') <=7N2t)s4  
    % k>;a5'S  
    %   See also ZERNFUN, ZERNFUN2. PquATAzQA  
    <"rckPv_H  
    % A note on the algorithm. UOtrq=y  
    % ------------------------ .e8S^lSl  
    % The radial Zernike polynomials are computed using the series dgsD~.((A  
    % representation shown in the Help section above. For many special jYi{[* *  
    % functions, direct evaluation using the series representation can :U$U:e  
    % produce poor numerical results (floating point errors), because cgvD>VUw  
    % the summation often involves computing small differences between ?B`Yq\L)  
    % large successive terms in the series. (In such cases, the functions zgR@-OtFZ  
    % are often evaluated using alternative methods such as recurrence m"RE[dQ  
    % relations: see the Legendre functions, for example). For the Zernike 4hh=z>$|l)  
    % polynomials, however, this problem does not arise, because the OP}8u"\Z  
    % polynomials are evaluated over the finite domain r = (0,1), and BL Q&VI4  
    % because the coefficients for a given polynomial are generally all BpQ/$?5E"  
    % of similar magnitude. b$Ch2Qz0q  
    % ^&-H"jF  
    % ZERNPOL has been written using a vectorized implementation: multiple z`Cq,Sz/  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] Ul?92  
    % values can be passed as inputs) for a vector of points R.  To achieve q|fZdTw  
    % this vectorization most efficiently, the algorithm in ZERNPOL sBfPhBT|  
    % involves pre-determining all the powers p of R that are required to `*hrU{b  
    % compute the outputs, and then compiling the {R^p} into a single m&X6a C'[  
    % matrix.  This avoids any redundant computation of the R^p, and ' y9yx[P  
    % minimizes the sizes of certain intermediate variables. <DjFMTCN  
    % H D95>%  
    %   Paul Fricker 11/13/2006 r=3knCEWK  
    j/9Uf|z-_  
    "p{cz(  
    % Check and prepare the inputs: &^W91C?<6  
    % ----------------------------- t%f6P  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) cjel6 nj  
        error('zernpol:NMvectors','N and M must be vectors.') Xu8_<%  
    end 8Qg,UX  
    c#6g[TE@  
    if length(n)~=length(m) bkgJz+u  
        error('zernpol:NMlength','N and M must be the same length.') =1}Umn|ZLS  
    end :W\xZ  
     MXj7Z3  
    n = n(:); Ka"Z,\T   
    m = m(:); G`HL^/Z*  
    length_n = length(n); """gV)Y  
    01nbR+e  
    if any(mod(n-m,2)) :z!N_]t  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') UHEn+Tc>  
    end '`*{ig  
    YIQm;E EG  
    if any(m<0) AGN5=K*D  
        error('zernpol:Mpositive','All M must be positive.') 9w=GB?/  
    end ByK!r~>Z1Q  
    6O>GVJbw  
    if any(m>n) i: ZL0nH-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') <6s?M1J  
    end a3<.F&c+c  
    9p#Laei].  
    if any( r>1 | r<0 ) 1px8af]  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ZO2$Aan  
    end `KgWaf-  
    L.uX  
    if ~any(size(r)==1) &O'6va  
        error('zernpol:Rvector','R must be a vector.') lKd+,<  
    end X&[S.$_U  
    E2s lpo  
    r = r(:); DUk&`BSJ  
    length_r = length(r); PSO9{!  
    +%'S>g0W=  
    if nargin==4 <J`",h  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); \tj7Jy  
        if ~isnorm #S/]=D  
            error('zernpol:normalization','Unrecognized normalization flag.') U9[ &ci  
        end t} zffe-  
    else :K ^T@F5n  
        isnorm = false; TU-4+o%;  
    end =""z!%j  
    3TRzDE(J  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K 9ytot  
    % Compute the Zernike Polynomials Ny@CP}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @hlT7C)xK  
    JM-spi o  
    % Determine the required powers of r: hlpi-oW`  
    % ----------------------------------- 9 wun$!>&  
    rpowers = []; NW'rqgG  
    for j = 1:length(n) GHaOFLY  
        rpowers = [rpowers m(j):2:n(j)]; (cX;a/BR  
    end fb7Gy  
    rpowers = unique(rpowers); gAA2S5th  
    v2e*mNK5  
    % Pre-compute the values of r raised to the required powers, qn VxP&  
    % and compile them in a matrix: %T hY6y(  
    % ----------------------------- >~-8RM  
    if rpowers(1)==0 2NHkK_B1P  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); wo($7'.@  
        rpowern = cat(2,rpowern{:}); e6C;A]T2E  
        rpowern = [ones(length_r,1) rpowern]; $niJw@zC  
    else RKu'WD?sdH  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )~ {T  
        rpowern = cat(2,rpowern{:}); "K3"s Ec%  
    end B? $9M9  
    gOpi>  
    % Compute the values of the polynomials: "<3F[[;~  
    % -------------------------------------- zo,`Vibx<  
    z = zeros(length_r,length_n); d NQ?8P-&  
    for j = 1:length_n UEZnd8  
        s = 0:(n(j)-m(j))/2; cFcn61x-  
        pows = n(j):-2:m(j); G%{J.J41F  
        for k = length(s):-1:1 p^|IN'lx,  
            p = (1-2*mod(s(k),2))* ... Mu,}?%  
                       prod(2:(n(j)-s(k)))/          ... hk =nXv2M  
                       prod(2:s(k))/                 ... dzIc X*"  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... SV2DvrIR  
                       prod(2:((n(j)+m(j))/2-s(k))); `nAR/Ye  
            idx = (pows(k)==rpowers); .+|HJ(  
            z(:,j) = z(:,j) + p*rpowern(:,idx); _l`d+ \#  
        end >K }j}M%  
         ^I=W<  
        if isnorm BI\ )vr$  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Y$3 &?LA  
        end d5zv8?|X+  
    end +Bg$]~ T  
    v6Vieo=  
    % 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)  s7Z+--I)L  
    Pb] EpyAW  
    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。 n>XfXt =  
    )c<X.4  
    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)