切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11137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 RUk<=! U  
    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! B}q  
     
    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 t,=@hs hN  
    function z = zernfun(n,m,r,theta,nflag) 28T\@zi  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5W[3_P+  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j8[`~p b  
    %   and angular frequency M, evaluated at positions (R,THETA) on the ]cF1c90%  
    %   unit circle.  N is a vector of positive integers (including 0), and t+=12{9;f  
    %   M is a vector with the same number of elements as N.  Each element x{NNx:T1  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) U`bC>sCp  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, cg(QjH"  
    %   and THETA is a vector of angles.  R and THETA must have the same +Cn yK(V  
    %   length.  The output Z is a matrix with one column for every (N,M) <qbZG}u  
    %   pair, and one row for every (R,THETA) pair. 8!u/   
    % E8T"{ R80  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,+ns {ppn  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gdoJ4b  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral Y!++C MzU  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s{(ehP.Dd  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized H$~M`Y9I~  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. WF ?/GN  
    % -yX.Jv  
    %   The Zernike functions are an orthogonal basis on the unit circle. \6`v.B&v  
    %   They are used in disciplines such as astronomy, optics, and S2J#b"Y  
    %   optometry to describe functions on a circular domain. do:QH.q8)  
    % T&9`?QD  
    %   The following table lists the first 15 Zernike functions. ps"/}u l  
    % O" % Hprx  
    %       n    m    Zernike function           Normalization +(;8@"u  
    %       -------------------------------------------------- k~0#'I9  
    %       0    0    1                                 1 ? .c?Pu  
    %       1    1    r * cos(theta)                    2 OJMvn'y  
    %       1   -1    r * sin(theta)                    2 0zeUP {MQ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Bz~ -2#l  
    %       2    0    (2*r^2 - 1)                    sqrt(3) LQh^; ]^(  
    %       2    2    r^2 * sin(2*theta)             sqrt(6)  M*d-z  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 2Ryp@c&r^  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) jg~_'4f#  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) HA$Y1}  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) +VSZhg,Np8  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ?Wwh _TO  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rs[?v*R74  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^F>4~68d  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |n+ #1_t%  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) LWD.  
    %       -------------------------------------------------- 7< ^'DO s  
    % 0(wf{5  
    %   Example 1: pU M&"V  
    % CXBzX:T?#  
    %       % Display the Zernike function Z(n=5,m=1) OZG0AX+=#  
    %       x = -1:0.01:1; @(Z( /P;:  
    %       [X,Y] = meshgrid(x,x); ;5<P|:^  
    %       [theta,r] = cart2pol(X,Y); pp(H PKs=}  
    %       idx = r<=1; 2*+ 3Rr J  
    %       z = nan(size(X)); 6H0W`S0a  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); {5SfE$r  
    %       figure + Qt[1Xq  
    %       pcolor(x,x,z), shading interp a lrt*V|=  
    %       axis square, colorbar #-,g&)`]  
    %       title('Zernike function Z_5^1(r,\theta)') !]yQ1@)*'  
    % |-|jf  
    %   Example 2: e[s5N:IUd3  
    % ICk(z~D~  
    %       % Display the first 10 Zernike functions }qG#N  
    %       x = -1:0.01:1; |,3l`o k  
    %       [X,Y] = meshgrid(x,x); mn. `qfMh  
    %       [theta,r] = cart2pol(X,Y); ])C>\@c6Gm  
    %       idx = r<=1; h9)RJSF4  
    %       z = nan(size(X)); Po> e kz_E  
    %       n = [0  1  1  2  2  2  3  3  3  3]; LaDY`u0G%  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; `"B^{o  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ; VBpp<  
    %       y = zernfun(n,m,r(idx),theta(idx)); s,w YlVYf!  
    %       figure('Units','normalized') J=):+F=  
    %       for k = 1:10 C(s\LI!r  
    %           z(idx) = y(:,k); \4aKLr  
    %           subplot(4,7,Nplot(k)) M2dmG<  
    %           pcolor(x,x,z), shading interp  *. 8JP  
    %           set(gca,'XTick',[],'YTick',[]) IK3qE!,&U  
    %           axis square j$+gq*I&E  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A]j}'  
    %       end g&bwtEZ  
    % e[}],W  
    %   See also ZERNPOL, ZERNFUN2. IdF$Ml#[h  
    Bq *[c=(2  
    %   Paul Fricker 11/13/2006 0vDg8i\  
    @m?{80;uQ  
    R3?:\d{  
    % Check and prepare the inputs: +lKrj\Xj  
    % ----------------------------- i *B:El1  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) l]$40 j  
        error('zernfun:NMvectors','N and M must be vectors.') }C_|gd  
    end ]/_G-2.R  
    Wk}D]o0^@  
    if length(n)~=length(m) -Un=T X  
        error('zernfun:NMlength','N and M must be the same length.') AeaPK  
    end E3f9<hm   
    P% Q@9kO>  
    n = n(:); (`pNXQ0n  
    m = m(:); ~5ubh2{  
    if any(mod(n-m,2)) QF.3c6O@  
        error('zernfun:NMmultiplesof2', ... D M}s0O$ 0  
              'All N and M must differ by multiples of 2 (including 0).') JR)/c6j  
    end 7 5|pp  
    EI\v  
    if any(m>n) XIRR Al(,  
        error('zernfun:MlessthanN', ... 2h<U  
              'Each M must be less than or equal to its corresponding N.') [fxuUmU  
    end ;R!*I%  
    gQ>2!Qc a-  
    if any( r>1 | r<0 ) lbS?/f  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6JH 56  
    end ]n5"Z,K  
    a.DX%C /5  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E=k w)<X2  
        error('zernfun:RTHvector','R and THETA must be vectors.') EE]=f=3  
    end .H2qs{N!  
    ?q!FG(  
    r = r(:); # k9 <  
    theta = theta(:); { 5-zyE  
    length_r = length(r); @!<d0_dnC  
    if length_r~=length(theta) YjLe(+ WQ  
        error('zernfun:RTHlength', ... U CRAw3=  
              'The number of R- and THETA-values must be equal.') sAYV)w3u"  
    end 7)J6/('  
    {zP#woz2Q  
    % Check normalization: |s f*hlrJ  
    % -------------------- i3PKqlp.  
    if nargin==5 && ischar(nflag) 5V@&o`!=h  
        isnorm = strcmpi(nflag,'norm'); %iJ|H(P  
        if ~isnorm vCb]%sd-U  
            error('zernfun:normalization','Unrecognized normalization flag.') W2eAhz&  
        end ] H&c'  
    else [(|v`qMv/g  
        isnorm = false; b+@D_E-RJ  
    end *d>vR1  
    `(DJs-xD  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rY,PSK/j  
    % Compute the Zernike Polynomials 8bOT*^b$H  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^PqMi:htc  
    :}9j^}"c3  
    % Determine the required powers of r: o@/xPo|  
    % ----------------------------------- SY1GR n  
    m_abs = abs(m); `c(\i$1JY)  
    rpowers = []; ?4G(N=/&  
    for j = 1:length(n) ,J(lJ,c  
        rpowers = [rpowers m_abs(j):2:n(j)]; :#$F)]y'\  
    end =Ndli>x}1  
    rpowers = unique(rpowers); .X'< D*  
    ia4k:\  
    % Pre-compute the values of r raised to the required powers, #s2B%X  
    % and compile them in a matrix: [AR>?6G-  
    % -----------------------------  AmcC:5  
    if rpowers(1)==0 .X `C^z]+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); OOB^gf}$'  
        rpowern = cat(2,rpowern{:}); =yqHC<8:  
        rpowern = [ones(length_r,1) rpowern]; 6Cc7ejt|u  
    else A-wRah.M  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); tZA:  
        rpowern = cat(2,rpowern{:}); qC@Ar)T  
    end T2weAk#J  
    XP?*=Z]  
    % Compute the values of the polynomials: l6 7KJ  
    % -------------------------------------- |RhM| i  
    y = zeros(length_r,length(n)); \[#t<dD  
    for j = 1:length(n) kus}W  J  
        s = 0:(n(j)-m_abs(j))/2; ;6m;M63z  
        pows = n(j):-2:m_abs(j); 6I|A- h  
        for k = length(s):-1:1 #?&0D>E?k  
            p = (1-2*mod(s(k),2))* ... 8h.V4/?  
                       prod(2:(n(j)-s(k)))/              ... {TAw)!R~  
                       prod(2:s(k))/                     ... M{G xjmdx  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y=2Un).&  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); C1QV[bJK  
            idx = (pows(k)==rpowers); EJm4xkYLj1  
            y(:,j) = y(:,j) + p*rpowern(:,idx); c Zvf"cIs  
        end uGCp#>+  
         YaL]>.;Z:"  
        if isnorm Hwu4:^OL|  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -9o{vmB{  
        end C_->u4 -  
    end <KQ(c`KW7  
    % END: Compute the Zernike Polynomials MzTW8  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s".HEP~]=  
    j*zD0I]  
    % Compute the Zernike functions: 9%!dNnUk  
    % ------------------------------ Mqv[XHfB  
    idx_pos = m>0; nPA@h  
    idx_neg = m<0; Q_O*oT(0  
    nvyB/  
    z = y; T20VX 8gX  
    if any(idx_pos) r:9gf?(&  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); $j*Qo/x d  
    end g1|w?pI1  
    if any(idx_neg) N.hzKq][  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Zdn!qyR`  
    end YYUe)j{T  
    3&*'6D Tg  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) FFzH!=7T?  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. {#hVD4$b  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated t9u|iTY f!  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive PRr*]$\&Mj  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 5w<A;f  
    %   and THETA is a vector of angles.  R and THETA must have the same .j?kEN?w  
    %   length.  The output Z is a matrix with one column for every P-value, DTY<0Q.  
    %   and one row for every (R,THETA) pair. >D _F!_  
    % Pil;/t)"  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike lLiQ;@  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) '%7 Bxof  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) BD*G1k_q  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 )=_ycf^MC  
    %   for all p. LmL Gki$w  
    % ]gP5f@`  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 "H+,E_&(  
    %   Zernike functions (order N<=7).  In some disciplines it is e7k%6'@  
    %   traditional to label the first 36 functions using a single mode *g$i5!yM'  
    %   number P instead of separate numbers for the order N and azimuthal `W5-.Tv  
    %   frequency M. O\Eqr?%L)  
    % wNDbHR  
    %   Example: E1*QdCV2  
    % 7%d8D>uw8  
    %       % Display the first 16 Zernike functions h9CIZU[Nh  
    %       x = -1:0.01:1; OW5t[~y]  
    %       [X,Y] = meshgrid(x,x); V|FrN*m  
    %       [theta,r] = cart2pol(X,Y); 3V;gW%>  
    %       idx = r<=1; /q1s;I  
    %       p = 0:15; f_\_9o"l  
    %       z = nan(size(X)); {eHAg<+  
    %       y = zernfun2(p,r(idx),theta(idx)); @;)PSp*j  
    %       figure('Units','normalized') S# we3  
    %       for k = 1:length(p) %SA!p;  
    %           z(idx) = y(:,k); Z4q~@|+%  
    %           subplot(4,4,k) HW6.O|3  
    %           pcolor(x,x,z), shading interp j1U 5~%^  
    %           set(gca,'XTick',[],'YTick',[]) r"wtZ]69  
    %           axis square mP^SS Je  
    %           title(['Z_{' num2str(p(k)) '}']) p3]Q^KFS  
    %       end ]<trA$ 0  
    % q=5l4|1  
    %   See also ZERNPOL, ZERNFUN. Mi 0sC24b|  
    C/tr$.2H=  
    %   Paul Fricker 11/13/2006 b2) \ MNH  
    ,YLF+^w-  
    :qj<p3w~}  
    % Check and prepare the inputs: ,+1m`9}  
    % ----------------------------- j8$Zv%Ca%  
    if min(size(p))~=1 Poy^RpnX  
        error('zernfun2:Pvector','Input P must be vector.') &m'kI  
    end |g&ym Fc  
    w*!wQ,o  
    if any(p)>35 C"eXs#A  
        error('zernfun2:P36', ... s] au/T6b  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... {"}V&X160o  
               '(P = 0 to 35).']) ;hDa@3|]34  
    end Q!'qC*Gyfn  
    kfrY1  
    % Get the order and frequency corresonding to the function number: `8g7q 5  
    % ---------------------------------------------------------------- WxUxc75  
    p = p(:); WlVl[/qt  
    n = ceil((-3+sqrt(9+8*p))/2); u$*>`Xe6  
    m = 2*p - n.*(n+2); #@f[bP}a  
    v#oi0-9o[  
    % Pass the inputs to the function ZERNFUN: #1/}3+=5B  
    % ---------------------------------------- SoQR#(73HK  
    switch nargin i*[n{=*l@  
        case 3 |+f-h,  
            z = zernfun(n,m,r,theta); P~ 0Jg# V  
        case 4 t~p y=\  
            z = zernfun(n,m,r,theta,nflag); 1|| nR4yK  
        otherwise A o/vp-e  
            error('zernfun2:nargin','Incorrect number of inputs.') m VxO$A,  
    end {|<"C?  
    ]\c,BWC@e  
    % 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) /@oLe[Mz$  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. yi6N-7  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of +s[\g>i  
    %   order N and frequency M, evaluated at R.  N is a vector of l* dV\ B  
    %   positive integers (including 0), and M is a vector with the On_@HQ/FI  
    %   same number of elements as N.  Each element k of M must be a blt'={Z?.x  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) vfc[p ^  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is kuMKX`_  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix xTV{^=\rS  
    %   with one column for every (N,M) pair, and one row for every onei4c>@  
    %   element in R. )}lRd#V  
    % "MOpsb,  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- "M H6fF  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is HIc a nk  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to JAb6zpP  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 7=}F{U  
    %   for all [n,m]. -_A$DM!^=w  
    % lFG9=Wf  
    %   The radial Zernike polynomials are the radial portion of the PX O!t]*  
    %   Zernike functions, which are an orthogonal basis on the unit Qd`T5[b\  
    %   circle.  The series representation of the radial Zernike &)9{HRP  
    %   polynomials is D{7w!z  
    % '0aG N<c  
    %          (n-m)/2 Ty4S~ClO#'  
    %            __ _F(P*[[&  
    %    m      \       s                                          n-2s ODw`E9  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r N3A<:%s  
    %    n      s=0 ~2 *9{  
    % j]4,<ppWSH  
    %   The following table shows the first 12 polynomials. |i %2%V#  
    % Sb&lhgW]c  
    %       n    m    Zernike polynomial    Normalization k[|~NLB8  
    %       --------------------------------------------- m?CZQq,  
    %       0    0    1                        sqrt(2) PRu&3BP  
    %       1    1    r                           2 -yH,5vD  
    %       2    0    2*r^2 - 1                sqrt(6) @tUoD>f  
    %       2    2    r^2                      sqrt(6) m\u26`M  
    %       3    1    3*r^3 - 2*r              sqrt(8) 'xK.U I  
    %       3    3    r^3                      sqrt(8) kyYLP"oB=  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Yc Q=vt{  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) 5p"BD'^:  
    %       4    4    r^4                      sqrt(10) uXZg1 F)  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) X"asfA[6K  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) #(;<-7M2  
    %       5    5    r^5                      sqrt(12) cD}Sf>  
    %       --------------------------------------------- 'o4p#`R:8  
    % | }&RXD  
    %   Example: C}}/)BYi  
    % VDF)zA1V  
    %       % Display three example Zernike radial polynomials ,I.WX,OR  
    %       r = 0:0.01:1; X$?3U!  
    %       n = [3 2 5]; Zl/< w(f_  
    %       m = [1 2 1]; X* eW#|$\  
    %       z = zernpol(n,m,r); BzJ;%ywS  
    %       figure qi[Z,&  
    %       plot(r,z) kQ>^->w  
    %       grid on GRqT-/n"  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') pV[''  
    % Td\o9  
    %   See also ZERNFUN, ZERNFUN2. k\)Cw  
    W m&  
    % A note on the algorithm. T<b+s#n4  
    % ------------------------ A#h/B+  
    % The radial Zernike polynomials are computed using the series dR^"X3$  
    % representation shown in the Help section above. For many special QfjN"25_  
    % functions, direct evaluation using the series representation can R=j% S!  
    % produce poor numerical results (floating point errors), because F'm(8/A$  
    % the summation often involves computing small differences between yl&UM qI(  
    % large successive terms in the series. (In such cases, the functions TX8<J>x  
    % are often evaluated using alternative methods such as recurrence 8P' ana  
    % relations: see the Legendre functions, for example). For the Zernike gN6rp(?y  
    % polynomials, however, this problem does not arise, because the wNJzwC&iQ  
    % polynomials are evaluated over the finite domain r = (0,1), and s,]%dG!  
    % because the coefficients for a given polynomial are generally all x*XH]&V  
    % of similar magnitude. t ~7V { xk  
    % Zi\['2CG  
    % ZERNPOL has been written using a vectorized implementation: multiple Q4*-wF-P  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] L5YnG_M&  
    % values can be passed as inputs) for a vector of points R.  To achieve /'.=sH  
    % this vectorization most efficiently, the algorithm in ZERNPOL 2;3f=$3  
    % involves pre-determining all the powers p of R that are required to G bP!9I  
    % compute the outputs, and then compiling the {R^p} into a single bru/AZ#de  
    % matrix.  This avoids any redundant computation of the R^p, and arK_oh0B  
    % minimizes the sizes of certain intermediate variables. Lv[OUW#S  
    % Y5q3T`x E  
    %   Paul Fricker 11/13/2006 0IkM  
    vxQ8t!-u  
    u"xJjS  
    % Check and prepare the inputs: bvBHYf:^  
    % ----------------------------- K4Dp:2/K%  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) BP[|nL  
        error('zernpol:NMvectors','N and M must be vectors.') revF;l6->C  
    end SO\/-]9#  
    I751 t  
    if length(n)~=length(m) V%0I%\0Y  
        error('zernpol:NMlength','N and M must be the same length.') az;Q"V'6  
    end bizTd  
    ,Tagj`@bHc  
    n = n(:); <+j)P4O4  
    m = m(:); U5HKRO  
    length_n = length(n); \!50UVzm)  
    oPKr* `'  
    if any(mod(n-m,2)) /Dt d#OAdr  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') &QX`NO 6  
    end NSA F4e  
    )jrT6x^IB  
    if any(m<0) {Rq1HH  
        error('zernpol:Mpositive','All M must be positive.') Uggw-sRU  
    end HL3XyP7  
    1k%k`[VC  
    if any(m>n) 0H_!Kg  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') CI:^\-z  
    end r\6"5cQ=  
    6 9y;`15  
    if any( r>1 | r<0 ) :p&!RI(l  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') %#L]]-%  
    end _96~rel_P  
    ,tH5e&=U01  
    if ~any(size(r)==1) /Ss7"*JLe  
        error('zernpol:Rvector','R must be a vector.')  6Si-u  
    end iZ3W"Vd`b  
    UM*jKi2]"  
    r = r(:); |wE3UWsy  
    length_r = length(r); -m= 8&B  
    sd#|3  
    if nargin==4 PYRd] %X  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); p}b/XnV$~  
        if ~isnorm 'uDjFQX  
            error('zernpol:normalization','Unrecognized normalization flag.') f&,{XZ  
        end lOwS&4UT  
    else nnb8Gcr  
        isnorm = false; m4E)qCvy  
    end L(>=BK*  
    +|Hioq* ,t  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'D1A}X  
    % Compute the Zernike Polynomials ;< )~Y-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gkBdR +  
    w6dFb6~R  
    % Determine the required powers of r: [ =x s4=  
    % ----------------------------------- v4miU;|\  
    rpowers = []; C${ S^v  
    for j = 1:length(n) E@05e  
        rpowers = [rpowers m(j):2:n(j)]; mV73 \P6K  
    end tj]9~eJ-  
    rpowers = unique(rpowers); Y!q!5Crfi  
    SQ,?N XZ  
    % Pre-compute the values of r raised to the required powers, :4)Qt  
    % and compile them in a matrix: H2xeP%;$  
    % ----------------------------- $uui:wU%Q  
    if rpowers(1)==0 R`";Z$~{  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kc'pN&]r:  
        rpowern = cat(2,rpowern{:}); LWsP ya  
        rpowern = [ones(length_r,1) rpowern]; CxhY$%C (L  
    else X7UuwIIP  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); J?JeU/:+  
        rpowern = cat(2,rpowern{:}); Fhz*&JC#  
    end ]{ BE r*  
    Uq_j\A;c  
    % Compute the values of the polynomials: 6<,dRn  
    % -------------------------------------- aKUS5jDu  
    z = zeros(length_r,length_n); a9zw)A  
    for j = 1:length_n {k.MS-q  
        s = 0:(n(j)-m(j))/2; Ed0IWPx  
        pows = n(j):-2:m(j); \7MHaQvS   
        for k = length(s):-1:1 ^[Ua46/"m  
            p = (1-2*mod(s(k),2))* ... *?+V65~dW  
                       prod(2:(n(j)-s(k)))/          ... Dlo xrdOY&  
                       prod(2:s(k))/                 ... FEPXuCb  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... fW`&'!  
                       prod(2:((n(j)+m(j))/2-s(k))); JxLf?ad.  
            idx = (pows(k)==rpowers); yq_LW>|Z  
            z(:,j) = z(:,j) + p*rpowern(:,idx); MC 0TaP  
        end f"7M^1)h2%  
         w#JJXXQI  
        if isnorm @ DZD  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); =Cv/Y%DN  
        end :0K8h  
    end hqk}akXt  
    }pkj:NT  
    % 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)  m(D]qYwh  
    @mw "W{  
    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。 K5 3MMH[q#  
    pvWau1ArNq  
    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)