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

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    离线niuhelen
     
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    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 O6gI%Jdp  
    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! Er} xB~<t  
     
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    离线phility
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    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
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    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
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    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 /lDW5;d  
    function z = zernfun(n,m,r,theta,nflag) y!GjC]/  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. YFOK%7K  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N!m-gymmF  
    %   and angular frequency M, evaluated at positions (R,THETA) on the IJO`"da  
    %   unit circle.  N is a vector of positive integers (including 0), and j#y_#  
    %   M is a vector with the same number of elements as N.  Each element ' ^gF  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ~\DC )  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, |ap{+ xh  
    %   and THETA is a vector of angles.  R and THETA must have the same O:Bfbna  
    %   length.  The output Z is a matrix with one column for every (N,M) N:[m,U9a  
    %   pair, and one row for every (R,THETA) pair. `zRgP#  
    % K+Al8L?K_  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike U*, 8 ,C  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -\\}K\*MJ  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral v>.nL(VLjP  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, fG;)wQJ  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized d /&aC#'B  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JT&CJ&#[h  
    % 75wQH*  
    %   The Zernike functions are an orthogonal basis on the unit circle. -% PUY(  
    %   They are used in disciplines such as astronomy, optics, and kmNY ;b6Y$  
    %   optometry to describe functions on a circular domain. Y }'C'PR  
    % m,aJ(8G  
    %   The following table lists the first 15 Zernike functions. \bqNjlu  
    % |M  `B  
    %       n    m    Zernike function           Normalization Yi&;4vC  
    %       -------------------------------------------------- TbU\qcm]]  
    %       0    0    1                                 1 B o.x  
    %       1    1    r * cos(theta)                    2 \`jFy[(Pa'  
    %       1   -1    r * sin(theta)                    2 [yL %+I  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) #B"ki{Se*  
    %       2    0    (2*r^2 - 1)                    sqrt(3) jii2gtu'U  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) rw8O<No4.o  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) t*zve,?}  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) cQzd0X  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) jpZX5_o  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) aoz+g,1 //  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) ;gy_Qf2U  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kf_s.Dedw  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) \% !]qv  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X<K[` =I  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) kI]i,v#F  
    %       -------------------------------------------------- _a8^AG  
    % IE: x&q`3  
    %   Example 1: *58<.L|  
    % o DPs xw  
    %       % Display the Zernike function Z(n=5,m=1) %;^[WT`,  
    %       x = -1:0.01:1; zN#$eyt  
    %       [X,Y] = meshgrid(x,x); N'Ywn}!js  
    %       [theta,r] = cart2pol(X,Y); a36n}R4Q  
    %       idx = r<=1; LTS3[=AB  
    %       z = nan(size(X)); 99G/(Z}  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); fW!~*Q  
    %       figure y&t&'l/m  
    %       pcolor(x,x,z), shading interp \r^=W=  
    %       axis square, colorbar P9:7_Vc  
    %       title('Zernike function Z_5^1(r,\theta)') hUSr1jlA  
    % #p&iH9c_  
    %   Example 2: *W y0hnr;]  
    % l6Ze6X I  
    %       % Display the first 10 Zernike functions })T}e7>T  
    %       x = -1:0.01:1; ($7>\"+Tl  
    %       [X,Y] = meshgrid(x,x); 5oGnPF  
    %       [theta,r] = cart2pol(X,Y); i pjl[  
    %       idx = r<=1; [esjR`u  
    %       z = nan(size(X));  3Fo,F  
    %       n = [0  1  1  2  2  2  3  3  3  3]; H&[CSc  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; TGdD7n&Ehh  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; !-2nIY!  
    %       y = zernfun(n,m,r(idx),theta(idx)); [X#bDO<t  
    %       figure('Units','normalized') +>KWY PH  
    %       for k = 1:10 g}{Rk>k  
    %           z(idx) = y(:,k); ,(N&%  
    %           subplot(4,7,Nplot(k)) 3T# zxu  
    %           pcolor(x,x,z), shading interp 8UwL%"?YB  
    %           set(gca,'XTick',[],'YTick',[]) :(} {uG  
    %           axis square ]d_Id]Qa+  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -kq=W_  
    %       end j,/OzVm9  
    % tQ5gmj  
    %   See also ZERNPOL, ZERNFUN2. .MhZ=sn  
    $V]D7kDph*  
    %   Paul Fricker 11/13/2006 9Wb9g/L  
    @NlnZfMu  
    [Rs5hO  
    % Check and prepare the inputs: } !pC}m  
    % ----------------------------- /(BQzCP9O;  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g (ZeGNV8  
        error('zernfun:NMvectors','N and M must be vectors.') W>wIcUP<<  
    end ?q7V B  
    c;Hf+n  
    if length(n)~=length(m) *^=`HE89S  
        error('zernfun:NMlength','N and M must be the same length.') 64#~p)  
    end  6?+bi\6  
    [ k^6#TQcn  
    n = n(:);  &e7yX  
    m = m(:); r|fJ~0z  
    if any(mod(n-m,2)) pJ6bX4QnDX  
        error('zernfun:NMmultiplesof2', ... 2!~ j(_TA  
              'All N and M must differ by multiples of 2 (including 0).') &1F)/$,v  
    end 09_3`K. *  
    i:&Y{iPQp  
    if any(m>n) 8n?P'iM  
        error('zernfun:MlessthanN', ... n/p M[gI  
              'Each M must be less than or equal to its corresponding N.') C;oP"K]4=  
    end r444s8Y  
    (toGU  
    if any( r>1 | r<0 ) PD|I3qv~  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ``1#^ `  
    end -/~^S]  
    qe"5&cc1  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7xVI,\qV  
        error('zernfun:RTHvector','R and THETA must be vectors.') jsf=S{^2  
    end <& 8cq@<  
    pA!+;Y!ZB<  
    r = r(:); A_{QY&%m  
    theta = theta(:); Fw!5hR`,  
    length_r = length(r); CP7Zin1S/w  
    if length_r~=length(theta) -J:](p  
        error('zernfun:RTHlength', ... {p 9y{$  
              'The number of R- and THETA-values must be equal.') /6gqpzum4  
    end b^y#.V.|k  
    5ii`!y  
    % Check normalization: :?RooJ~#  
    % -------------------- AQbbIngo  
    if nargin==5 && ischar(nflag) 4L^KR_h/  
        isnorm = strcmpi(nflag,'norm'); XsQ<ye un  
        if ~isnorm HMgZ& v  
            error('zernfun:normalization','Unrecognized normalization flag.')  3iV/7~ O  
        end Zul]ekv  
    else |42E'zH&  
        isnorm = false; .<u<!fL2  
    end gpHI)1i'H  
    6.EfM^[  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  >pv~$  
    % Compute the Zernike Polynomials j &,vju  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M7eO5  
    oE"!  
    % Determine the required powers of r: 6IPhy.8  
    % ----------------------------------- e|):%6#  
    m_abs = abs(m); +TpM7QaL  
    rpowers = []; Fu )V2[TY  
    for j = 1:length(n) T_[W=9  
        rpowers = [rpowers m_abs(j):2:n(j)]; yIXM}i:  
    end m3F.-KPO  
    rpowers = unique(rpowers); =XQ3sk6U  
    wx}\0(]Gl  
    % Pre-compute the values of r raised to the required powers, , j'=sDl  
    % and compile them in a matrix: ^^jF*)DT@  
    % ----------------------------- l"IBt:  
    if rpowers(1)==0 $Fc*^8$ryC  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p % 3B^  
        rpowern = cat(2,rpowern{:}); &I:X[=;g  
        rpowern = [ones(length_r,1) rpowern]; MZ=U} &F  
    else EK@yzJ%  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); lr+Kwve  
        rpowern = cat(2,rpowern{:}); gSZ NsiH  
    end H<}<f:  
    .3{S6#  
    % Compute the values of the polynomials: 9{70l539  
    % -------------------------------------- A. U<  
    y = zeros(length_r,length(n)); "LaNXZ9  
    for j = 1:length(n) ~< Gs<c}z  
        s = 0:(n(j)-m_abs(j))/2; gLl?e8[F  
        pows = n(j):-2:m_abs(j); 0AJ6g@ t[  
        for k = length(s):-1:1 q&jZmr  
            p = (1-2*mod(s(k),2))* ... DcSL f4A  
                       prod(2:(n(j)-s(k)))/              ... }YU#} Ip@  
                       prod(2:s(k))/                     ... +**H7: bO  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %+gze|J  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); S &s7]  
            idx = (pows(k)==rpowers); =bN[TD  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 6\4oHRJC  
        end S,G=MI"  
         8Dhq_R'r  
        if isnorm e>nRJH8pK  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Ip.5I!h[Xb  
        end L.U [eH  
    end <g>_#fz"K  
    % END: Compute the Zernike Polynomials -T4?5T_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a=p3oh?%-O  
    (G#)[0<fX  
    % Compute the Zernike functions: IJS9%m#  
    % ------------------------------ 4)JrOe&k  
    idx_pos = m>0; X]C-y,r[M  
    idx_neg = m<0; .}SW`R Pk  
    u\Fq\_  
    z = y; w gATfygr  
    if any(idx_pos) %?X~,  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [g=yuVXNZZ  
    end Va(R*38k  
    if any(idx_neg) F3H)B:  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oF]0o`U&a  
    end N(t1?R/e,  
    3t68cdFlz  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) `o/tpuI  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. (d4zNYK  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated B`"-~4YAf  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive j,EE`g&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, g B+cU  
    %   and THETA is a vector of angles.  R and THETA must have the same q/70fR7{v  
    %   length.  The output Z is a matrix with one column for every P-value, =]-!  
    %   and one row for every (R,THETA) pair. e3) rF5pp  
    % ~}8 3\LI}  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 78dmXOZ'_h  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (tyo4Tz1  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) i1FFf[[L  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 JS({au  
    %   for all p. ;`X-.45  
    % aJI>qk h?]  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 s) V7$D  
    %   Zernike functions (order N<=7).  In some disciplines it is kW6}57iV  
    %   traditional to label the first 36 functions using a single mode =J0FT2 d  
    %   number P instead of separate numbers for the order N and azimuthal ?#pL\1"E  
    %   frequency M. 'e;*V$+  
    % Qi6vP&  
    %   Example: YCw^u  
    % 47`{ e_YP0  
    %       % Display the first 16 Zernike functions 0)k%nIhj  
    %       x = -1:0.01:1; h-lMrI)U?h  
    %       [X,Y] = meshgrid(x,x); HmbTV(lC  
    %       [theta,r] = cart2pol(X,Y); <adu^5BI  
    %       idx = r<=1; uW Q`  
    %       p = 0:15; }-: d*YtK  
    %       z = nan(size(X)); P*I\FV  
    %       y = zernfun2(p,r(idx),theta(idx)); ( 5_oH  
    %       figure('Units','normalized') ~z32%k  
    %       for k = 1:length(p) CEqfsKrsxE  
    %           z(idx) = y(:,k); ou,W|<%  
    %           subplot(4,4,k) F_YZV)q!W  
    %           pcolor(x,x,z), shading interp aH'^`]'_=  
    %           set(gca,'XTick',[],'YTick',[]) (Clf]\_II  
    %           axis square ~NU~jmT2  
    %           title(['Z_{' num2str(p(k)) '}']) f=} u;^  
    %       end rAP+nh ans  
    % mUcHsCszH  
    %   See also ZERNPOL, ZERNFUN. !Q#u i[0q  
    0IQu6 X  
    %   Paul Fricker 11/13/2006 <pK; D  
    O=c&  
    IK~ur\3  
    % Check and prepare the inputs: ^4 es  
    % ----------------------------- =k3QymA  
    if min(size(p))~=1 HAGWA2wQ  
        error('zernfun2:Pvector','Input P must be vector.') X903;&Cim  
    end PcDPRX!@  
    z)QyQ  
    if any(p)>35 <C${1FO7If  
        error('zernfun2:P36', ... {oBVb{<  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... q.Z0Q  
               '(P = 0 to 35).']) F^A1'J  
    end (:-DuUt  
    eUF PzioW  
    % Get the order and frequency corresonding to the function number: 8b6:n1<fn  
    % ---------------------------------------------------------------- A{&Etu(K  
    p = p(:); ,Z MYCl]  
    n = ceil((-3+sqrt(9+8*p))/2); -bo0!@MK  
    m = 2*p - n.*(n+2); d{ OY  
    &W.tjqmw  
    % Pass the inputs to the function ZERNFUN: 8 hWQ  
    % ---------------------------------------- -pg7>vOq  
    switch nargin `I6)e{5t  
        case 3 B: {bmvy  
            z = zernfun(n,m,r,theta); 2<u vz<B  
        case 4 Lc<Gn y^  
            z = zernfun(n,m,r,theta,nflag); wx<5*8zP  
        otherwise `DWzp5Ax  
            error('zernfun2:nargin','Incorrect number of inputs.') Zh3]bg5  
    end MzJ5_}  
    2uiiTg>  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) .bL{fBTT~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. {&K#~[)  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of wond>m 3  
    %   order N and frequency M, evaluated at R.  N is a vector of hr]NW>;  
    %   positive integers (including 0), and M is a vector with the -qx Z3   
    %   same number of elements as N.  Each element k of M must be a b%|%Rek8  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) X)~JX}-L  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is (`>4~?|+T  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix FA4bv9:hi  
    %   with one column for every (N,M) pair, and one row for every (qB$I\  
    %   element in R. go{'mX)}u  
    % sVh!5fby&  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- RJBNY;0  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is m0=CD  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to \B2=E  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 wXNFL9F8  
    %   for all [n,m]. <niHJ*  
    % &a48DCZ  
    %   The radial Zernike polynomials are the radial portion of the 6PJ0iten  
    %   Zernike functions, which are an orthogonal basis on the unit /!7m@P|&D  
    %   circle.  The series representation of the radial Zernike ZH&%D*a&  
    %   polynomials is fyQAQZT  
    % m"+9[d_u  
    %          (n-m)/2 dVCBpCxI  
    %            __ B.&q]CA v-  
    %    m      \       s                                          n-2s ^dqyX(  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r .F 3v)  
    %    n      s=0 .&}}ro48  
    % %)q5hB  
    %   The following table shows the first 12 polynomials. N],A&}30  
    % (Ptv#LSUX  
    %       n    m    Zernike polynomial    Normalization ,ci tzh  
    %       --------------------------------------------- w6#hsRq[C  
    %       0    0    1                        sqrt(2) B8B^@   
    %       1    1    r                           2 Is?0q@  
    %       2    0    2*r^2 - 1                sqrt(6) i~l0XjQbs  
    %       2    2    r^2                      sqrt(6) skZxR5v3~L  
    %       3    1    3*r^3 - 2*r              sqrt(8) ApS/,cV  
    %       3    3    r^3                      sqrt(8) ^pZ(^  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) \7("bB=  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) ,v)@&1Wh:  
    %       4    4    r^4                      sqrt(10) 7,Z%rqf\)  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ? )0U!)tK  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) &XgB-}^:  
    %       5    5    r^5                      sqrt(12) pD`7N<F 3  
    %       --------------------------------------------- ZH~m%sA  
    % 5:56l>0  
    %   Example: =@{H7z(p&  
    % n*bbmG1  
    %       % Display three example Zernike radial polynomials G9}[g)R*  
    %       r = 0:0.01:1; fn;7Nf7{  
    %       n = [3 2 5]; PtmdUHvD  
    %       m = [1 2 1]; htMpL  
    %       z = zernpol(n,m,r); ]6$NU [  
    %       figure bl}$x/  
    %       plot(r,z) zy5@K)  
    %       grid on "C}nS=]8m  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') wf8vKl#Kfw  
    % `iQyKZS/+  
    %   See also ZERNFUN, ZERNFUN2. d!w32Y,.  
    QGLfZvTT  
    % A note on the algorithm. M,cI0i  
    % ------------------------ yUEUIPL  
    % The radial Zernike polynomials are computed using the series mn 8A%6W  
    % representation shown in the Help section above. For many special !|Vjv}UO  
    % functions, direct evaluation using the series representation can S>cT(q_&  
    % produce poor numerical results (floating point errors), because ##R]$-<4dQ  
    % the summation often involves computing small differences between m,*t}j0 7  
    % large successive terms in the series. (In such cases, the functions B8[H><)o\y  
    % are often evaluated using alternative methods such as recurrence i,* DWD+  
    % relations: see the Legendre functions, for example). For the Zernike vxbO>c   
    % polynomials, however, this problem does not arise, because the d![EnkyL;  
    % polynomials are evaluated over the finite domain r = (0,1), and %{o5 }TqD  
    % because the coefficients for a given polynomial are generally all ?^,GaZ^V  
    % of similar magnitude. <8j n_6  
    % Wq"pKI#x  
    % ZERNPOL has been written using a vectorized implementation: multiple uOm fpgO  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] 51&wH  
    % values can be passed as inputs) for a vector of points R.  To achieve y"2#bq  
    % this vectorization most efficiently, the algorithm in ZERNPOL 63F0Za}h  
    % involves pre-determining all the powers p of R that are required to 2R|2yAh  
    % compute the outputs, and then compiling the {R^p} into a single bumS>:  
    % matrix.  This avoids any redundant computation of the R^p, and Xo]FOJ 5  
    % minimizes the sizes of certain intermediate variables. MZ% P(5  
    % k]I<%  
    %   Paul Fricker 11/13/2006 S{ fNeK  
    M{hA`  
    @R`OAd y  
    % Check and prepare the inputs: 9J l9\y9  
    % ----------------------------- )RA7Y}e|m  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =o+t_.)N  
        error('zernpol:NMvectors','N and M must be vectors.') {Ivu"<`L3  
    end ^H&6'A`  
    nA%-<  
    if length(n)~=length(m) 5r`g6@  
        error('zernpol:NMlength','N and M must be the same length.') p?6w/n  
    end gqG l>=.m  
    Z\LW<**b  
    n = n(:); ^Z\1z!{R  
    m = m(:); kO/dZ%vj  
    length_n = length(n); *-` /A  
    VI37  
    if any(mod(n-m,2)) mxDy!:@=  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Xj|j\2$ 0  
    end O:k@'&  
    Nu|?s-   
    if any(m<0) qRB&R$  
        error('zernpol:Mpositive','All M must be positive.') qj=12;  
    end IvH0sS`F  
    IsnC_"f  
    if any(m>n) >&Bg F*mm  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') % sT=>\  
    end B#sc!eLmU&  
    [R& P.E7w'  
    if any( r>1 | r<0 ) [.|tD  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 4sROMk=l  
    end /5zzzaj {  
    rK)%n!Z  
    if ~any(size(r)==1) =C5 [75z#+  
        error('zernpol:Rvector','R must be a vector.') 5E}0 <&  
    end MqXA8D  
    .>h|e_E  
    r = r(:); [=..#y!U  
    length_r = length(r); rZGA9duy  
    !4-NbtT  
    if nargin==4 PvKe|In(  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); =d( 6 )  
        if ~isnorm e|]g ?!  
            error('zernpol:normalization','Unrecognized normalization flag.') P_Po g^  
        end eN,m8A`/S  
    else D`,@EW].  
        isnorm = false; g/JAr<  
    end scN}eg:5  
    &X +@,!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24|:VxO  
    % Compute the Zernike Polynomials !tX14O~B-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PP$Ig2Q  
    sHh2>f@x$  
    % Determine the required powers of r: AE^&hH0^  
    % ----------------------------------- qdUlT*fw  
    rpowers = []; BOfO$J}  
    for j = 1:length(n) .hZ =8y9  
        rpowers = [rpowers m(j):2:n(j)]; a?Q~C<k  
    end .{)b^gE  
    rpowers = unique(rpowers); `| R8WM  
    Dt.OZ4w5  
    % Pre-compute the values of r raised to the required powers, d|DIq T~{W  
    % and compile them in a matrix: Zw"6-h4  
    % ----------------------------- /rJvw   
    if rpowers(1)==0 :tR%y"  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]:]2f 9y  
        rpowern = cat(2,rpowern{:}); ~+^,o_hT  
        rpowern = [ones(length_r,1) rpowern]; h7(twct  
    else !A!zG)Ue<  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]P]lG-  
        rpowern = cat(2,rpowern{:}); h'T\gF E%  
    end iHQFieZ.E  
    0qJ 3@d  
    % Compute the values of the polynomials: mX,#|qLf  
    % -------------------------------------- K\n %&w  
    z = zeros(length_r,length_n); ~x>IN1Vci  
    for j = 1:length_n t41\nTZr  
        s = 0:(n(j)-m(j))/2; x-Xb4?{  
        pows = n(j):-2:m(j); Na3tK}x  
        for k = length(s):-1:1 Rp.@  
            p = (1-2*mod(s(k),2))* ... ;|9VPv/  
                       prod(2:(n(j)-s(k)))/          ... lWnV{/q\X  
                       prod(2:s(k))/                 ... Fd|:7NRA<  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... w )DO"Z7  
                       prod(2:((n(j)+m(j))/2-s(k))); nb?bx{M  
            idx = (pows(k)==rpowers); n>Zkx+jLj<  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /j3oHi$  
        end ~V5k  
         <J`_Qc8C  
        if isnorm V@cRJ3ZF  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); V 9=y@`;  
        end ?V*>4A  
    end I+u=H2][2  
    x2|DI)J1'  
    % EOF zernpol
    离线niuhelen
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    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
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    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
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    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)  [{Y$]3?}  
    <-lz_  
    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。  gu"Agct4  
    Q |l93Rb`  
    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)