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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 @-hy:th#  
    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! ag3T[}L z  
     
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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 lx%<oC+M  
    function z = zernfun(n,m,r,theta,nflag) qF>}"m  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. 0'a.Ypf  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N b8>r UGA{  
    %   and angular frequency M, evaluated at positions (R,THETA) on the kGCd!$fsk  
    %   unit circle.  N is a vector of positive integers (including 0), and \vKMNk;kz  
    %   M is a vector with the same number of elements as N.  Each element C]{43  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,*Sj7qb#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, T'FRnC^~  
    %   and THETA is a vector of angles.  R and THETA must have the same FLi)EgZXt  
    %   length.  The output Z is a matrix with one column for every (N,M)  E{h   
    %   pair, and one row for every (R,THETA) pair. z~Gi/Ln  
    % Fz-Bd*uS  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !MoGdI-<r[  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), \  VJ3  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral -fYgTst2  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cu.f]'  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized W,}C*8{+  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. g;vG6!;E\  
    % tZwZZ0]Z  
    %   The Zernike functions are an orthogonal basis on the unit circle. `5Q0U%`W  
    %   They are used in disciplines such as astronomy, optics, and vTN$SgzfCU  
    %   optometry to describe functions on a circular domain. UZv^3_,qz  
    % nCJ)=P.d  
    %   The following table lists the first 15 Zernike functions. ,{7Z OzA  
    % v-EcJj%  
    %       n    m    Zernike function           Normalization Ee d2`~  
    %       -------------------------------------------------- JuS#p5E #  
    %       0    0    1                                 1 c V=h 8F  
    %       1    1    r * cos(theta)                    2 p Gzzv{H  
    %       1   -1    r * sin(theta)                    2 < OCy  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) b v~"_)C  
    %       2    0    (2*r^2 - 1)                    sqrt(3) cd#@"&r  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 3Jf_3c  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) YR^J7b\  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *#w+*ywVZH  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) <Zl}u:(w  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ~+7q.XL$$K  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) b+9M? k"  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D `c YQ-  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) =Z2Cg{z  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rgJKXl;@s  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) {rBS52,Z#  
    %       -------------------------------------------------- Q!iM7C!8  
    % Z~CL|=  
    %   Example 1: a"4j9cO  
    % &82Za%  
    %       % Display the Zernike function Z(n=5,m=1) Gk g)\ 3  
    %       x = -1:0.01:1; U@ Y0 z.Y  
    %       [X,Y] = meshgrid(x,x); {}y"JbXMj  
    %       [theta,r] = cart2pol(X,Y); &;DK^ta*P  
    %       idx = r<=1; <!Ed ND=  
    %       z = nan(size(X)); Tq,Kel  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); pqmtN*zV  
    %       figure &Rdg07e;>  
    %       pcolor(x,x,z), shading interp wsYvbI!  
    %       axis square, colorbar \]1qAFB5  
    %       title('Zernike function Z_5^1(r,\theta)') Q$^oIFb  
    % e3oHe1"hP  
    %   Example 2: yY_Zq\   
    % ,4M7:=gf  
    %       % Display the first 10 Zernike functions XvETys@d  
    %       x = -1:0.01:1; ' @i0~  
    %       [X,Y] = meshgrid(x,x); Q}C)az  
    %       [theta,r] = cart2pol(X,Y); FL*qV"r^n  
    %       idx = r<=1; 4i|yEf  
    %       z = nan(size(X)); 8lk@ev=O&  
    %       n = [0  1  1  2  2  2  3  3  3  3]; e:D8.h+ &}  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; nLicog)!I  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; ~-ZquJ-  
    %       y = zernfun(n,m,r(idx),theta(idx)); 0A9llE  
    %       figure('Units','normalized') R~ n[g  
    %       for k = 1:10 w+($= n~  
    %           z(idx) = y(:,k); 5+Fr/C  
    %           subplot(4,7,Nplot(k)) k+b!Lw!L  
    %           pcolor(x,x,z), shading interp "NWILZwEV  
    %           set(gca,'XTick',[],'YTick',[]) KcKdhqdN-  
    %           axis square yK9:LXhf  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A:! _ &  
    %       end = Lt)15  
    % 7vZtEwC)n  
    %   See also ZERNPOL, ZERNFUN2. $FXlH;_7  
    pZHx  
    %   Paul Fricker 11/13/2006 n.is+2t  
    PgHe;^?j  
    m#w1?y)Z@X  
    % Check and prepare the inputs: NhJ]X cfP8  
    % ----------------------------- ~j3O0s<gK  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;GQCq@)-  
        error('zernfun:NMvectors','N and M must be vectors.') o_ng{SL  
    end ~P!\;S  
    =`<9N %  
    if length(n)~=length(m) s R/z)U_  
        error('zernfun:NMlength','N and M must be the same length.') !r^fX=X>'  
    end TP3KT)  
    -J &y]'  
    n = n(:); iepolO=  
    m = m(:); CZZwBt$P  
    if any(mod(n-m,2)) KEfN!6  
        error('zernfun:NMmultiplesof2', ... cPunMHD  
              'All N and M must differ by multiples of 2 (including 0).') %Yw?!GvL[  
    end z H|YVg  
    L;RHs hTy  
    if any(m>n) yK+1C68A  
        error('zernfun:MlessthanN', ... Rp^fY_  
              'Each M must be less than or equal to its corresponding N.') hufpky[&8  
    end ~cr##Ff 5  
    KpX1GrIn3  
    if any( r>1 | r<0 ) pYN.tD FO  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') A_8Xhem${  
    end P*6h $T  
    l6_dVK;s  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) x&p.-Fi  
        error('zernfun:RTHvector','R and THETA must be vectors.') Fv_B(a  
    end R1 C}S  
    {<cL@W  
    r = r(:); QJ\+u  
    theta = theta(:); H~$*R7~  
    length_r = length(r); 1VKu3  
    if length_r~=length(theta) =0t<:-?.-  
        error('zernfun:RTHlength', ... z!s1$5:"0  
              'The number of R- and THETA-values must be equal.') 0ZM#..3sI  
    end _.%U}U  
    3-/F]}0y6  
    % Check normalization: '[Zgwz;z  
    % -------------------- O\J{4EB@.  
    if nargin==5 && ischar(nflag) N?EeT}m_  
        isnorm = strcmpi(nflag,'norm'); sEymwpm9  
        if ~isnorm 6%^A6U  
            error('zernfun:normalization','Unrecognized normalization flag.') <EKTFHJ!  
        end #fx>{ vzH  
    else +R8G*2  
        isnorm = false; :y.~IQN  
    end A('o &H  
    70<{tjyc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #HDP ha  
    % Compute the Zernike Polynomials w2H^q3*  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'pnOHT  
    + mPVI  
    % Determine the required powers of r: 3y tlD'  
    % ----------------------------------- 'iWDYZ?  
    m_abs = abs(m); FLo`EE":O(  
    rpowers = []; oTJ^WePZQ  
    for j = 1:length(n) w2SN=X~#  
        rpowers = [rpowers m_abs(j):2:n(j)]; T' =6_?7K4  
    end r]0>A&,  
    rpowers = unique(rpowers); p%R+c  
    7NvnCs  
    % Pre-compute the values of r raised to the required powers, !^'6&NR#K  
    % and compile them in a matrix: Ot+Z}Z-  
    % ----------------------------- '':MhRb  
    if rpowers(1)==0 Z aYUf  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .~C%:bDnX7  
        rpowern = cat(2,rpowern{:}); a9u2Wlz  
        rpowern = [ones(length_r,1) rpowern]; =O/v]B8"  
    else :6:,s#av  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); bU\T  
        rpowern = cat(2,rpowern{:}); T65"?=<EB  
    end IISdC(5  
    Ft^X[5G4L  
    % Compute the values of the polynomials: 8VtRRtl  
    % -------------------------------------- R=<%!  
    y = zeros(length_r,length(n)); Zts1BWL[  
    for j = 1:length(n) xO^lE@a o  
        s = 0:(n(j)-m_abs(j))/2; T/FZn{I  
        pows = n(j):-2:m_abs(j); VAo`R9^D#  
        for k = length(s):-1:1 lc3N i<3v  
            p = (1-2*mod(s(k),2))* ... Gs\D`| 3=  
                       prod(2:(n(j)-s(k)))/              ... :='I>Gn  
                       prod(2:s(k))/                     ... "ql$Rz8  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^(s(4|  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 7p- RPC  
            idx = (pows(k)==rpowers); n[B[hAT  
            y(:,j) = y(:,j) + p*rpowern(:,idx); LzxO=+=9!q  
        end Ob{Tn@  
         6RG63+G  
        if isnorm vjzG H*  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);  `-JVz{z  
        end W] WH4.y  
    end yDJy'Z_F{  
    % END: Compute the Zernike Polynomials D|amKW7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v>HOz\F  
    I$R1#s  
    % Compute the Zernike functions: .4ZOm'ko{  
    % ------------------------------ rog1  
    idx_pos = m>0; [mQdc?n\  
    idx_neg = m<0; PC HKH  
    mE=Ur  
    z = y; N/'8W9#6  
    if any(idx_pos) +f%"O?  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'N^*,  
    end w+r).PS}C  
    if any(idx_neg) r\cY R}v  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); G[*z,2Kb>  
    end V;W{pd-I  
    {kBsiSvsA;  
    % EOF zernfun
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    function z = zernfun2(p,r,theta,nflag) h@Jg9AM  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. yj@k0TWT$  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 97U OH  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive $2,tT;50g  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, +q;{ %3C  
    %   and THETA is a vector of angles.  R and THETA must have the same )iM( \=1ff  
    %   length.  The output Z is a matrix with one column for every P-value, 1C<d^D_!p  
    %   and one row for every (R,THETA) pair. YU"/p|!1  
    % SO.u0!  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike -`DYDIr  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) E p;i],}  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) d:w/{m% #  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 D(;+my2  
    %   for all p. )bR0 >3/  
    % [*Ai@:F  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 'l=>H#}<B  
    %   Zernike functions (order N<=7).  In some disciplines it is J <;xkT1x  
    %   traditional to label the first 36 functions using a single mode E N%{ $  
    %   number P instead of separate numbers for the order N and azimuthal `^,E4Qy  
    %   frequency M. #g0_8>t  
    % BWQ`8  
    %   Example: qHp2;  
    % :o ~'\:/  
    %       % Display the first 16 Zernike functions C0 KFN  
    %       x = -1:0.01:1; b_a k@LYiu  
    %       [X,Y] = meshgrid(x,x); {lH'T1^m  
    %       [theta,r] = cart2pol(X,Y); mI!iSVqr  
    %       idx = r<=1; \O4s0*gw  
    %       p = 0:15; uS,?oS  
    %       z = nan(size(X)); K!Fem6R  
    %       y = zernfun2(p,r(idx),theta(idx)); ?P-O4  
    %       figure('Units','normalized') c+E\e]{  
    %       for k = 1:length(p) YPGzI]\  
    %           z(idx) = y(:,k); Q,JH/X  
    %           subplot(4,4,k) E0Q6Ryn  
    %           pcolor(x,x,z), shading interp 8b.k*,r>  
    %           set(gca,'XTick',[],'YTick',[]) }nX0h6+1  
    %           axis square #h^nvRmON  
    %           title(['Z_{' num2str(p(k)) '}']) R.*;] R>M  
    %       end |'1.a jxw  
    % <Vk}U   
    %   See also ZERNPOL, ZERNFUN. Za1mI^ L1  
    D4@?>ek6U  
    %   Paul Fricker 11/13/2006 %hN>o)  
    ?8{Os;!je  
    _hyqHvP  
    % Check and prepare the inputs: z[1uub,)1  
    % ----------------------------- $*G3'G2'iS  
    if min(size(p))~=1 >;1w-n  
        error('zernfun2:Pvector','Input P must be vector.') O5vfcX4>  
    end wkGr}  
    fo+s+Q|Y  
    if any(p)>35 2,q*8=?{6P  
        error('zernfun2:P36', ... 2F`#df  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... AC(qx:/6  
               '(P = 0 to 35).']) D{Nd2G  
    end 3))CD,|  
    lY"l6.c  
    % Get the order and frequency corresonding to the function number: B G\)B  
    % ---------------------------------------------------------------- "hI"4xSg  
    p = p(:); hBX.GFnw  
    n = ceil((-3+sqrt(9+8*p))/2); VD7-;  
    m = 2*p - n.*(n+2); X!LiekU!D  
    s=-?kcoJ2d  
    % Pass the inputs to the function ZERNFUN: 1Z)P.9c  
    % ---------------------------------------- SQ> Yf\  
    switch nargin &4|]VOf  
        case 3 FS&QF@dtgf  
            z = zernfun(n,m,r,theta); D((/fT)eD  
        case 4 1="]'!2Is  
            z = zernfun(n,m,r,theta,nflag); Qc-W2%  
        otherwise KTT!P 4  
            error('zernfun2:nargin','Incorrect number of inputs.') 0@b<?Ms9  
    end 53:u6bb;  
    AZhI~QWo  
    % EOF zernfun2
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    function z = zernpol(n,m,r,nflag) ^0>^5l'n  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 9-m_ e=jk6  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of eyDI>7W  
    %   order N and frequency M, evaluated at R.  N is a vector of {@7UfJh>  
    %   positive integers (including 0), and M is a vector with the UdkNb}L  
    %   same number of elements as N.  Each element k of M must be a  lY`WEu  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?^!J:D?  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is :/=P6b;  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix w/o8R3 F  
    %   with one column for every (N,M) pair, and one row for every A ;`[va  
    %   element in R. u"v$[8  
    % `O3#/1+  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- Q_,!(N  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is xP1D 9   
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to av(qV$2  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 K6B6@  
    %   for all [n,m]. AX@bM  
    % !$iwU3~<  
    %   The radial Zernike polynomials are the radial portion of the aRWj+[[7y  
    %   Zernike functions, which are an orthogonal basis on the unit Dy. |bUB!f  
    %   circle.  The series representation of the radial Zernike L67yL( d6a  
    %   polynomials is :+_H%4+  
    % -6F\=  
    %          (n-m)/2 :k.>H.8+~  
    %            __ u8A,f}D 3  
    %    m      \       s                                          n-2s Rb b[N#p5  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r T[},6I|!  
    %    n      s=0 2g)q (  
    % r7U[QTM%  
    %   The following table shows the first 12 polynomials. iuEe#B;!  
    % -y&>&D  
    %       n    m    Zernike polynomial    Normalization x51p'bNy  
    %       --------------------------------------------- <Vyl*a{%  
    %       0    0    1                        sqrt(2)  :5^5l  
    %       1    1    r                           2 =}0>S3a.7  
    %       2    0    2*r^2 - 1                sqrt(6) 3WkrG.$[b  
    %       2    2    r^2                      sqrt(6) !C' Y 7  
    %       3    1    3*r^3 - 2*r              sqrt(8) pDr/8HEh  
    %       3    3    r^3                      sqrt(8) >> -{AR0  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) y>UQm|o<W  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) %kBrxf  
    %       4    4    r^4                      sqrt(10) yq[CA`zVN  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) O'~^wu.  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) QE;,mC>  
    %       5    5    r^5                      sqrt(12) i2O$oHd  
    %       --------------------------------------------- i"!j:YEo  
    % czo*_q%  
    %   Example: V,tYqhQ3  
    % XHuHbriI  
    %       % Display three example Zernike radial polynomials A:sP%c;  
    %       r = 0:0.01:1; v>Kv!OY:c  
    %       n = [3 2 5]; $*0XWrE  
    %       m = [1 2 1]; ap|V}j C  
    %       z = zernpol(n,m,r); [DSzhi]  
    %       figure F JxH{N6a  
    %       plot(r,z) Wc##.qU  
    %       grid on [QT H~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') P'5Q}7  
    % \xZ6+xZd1  
    %   See also ZERNFUN, ZERNFUN2. bb`DyUy ^+  
    9qJ:h-?M  
    % A note on the algorithm. gAGcbepX  
    % ------------------------ pvqbk2BO  
    % The radial Zernike polynomials are computed using the series 5=_))v<Tp  
    % representation shown in the Help section above. For many special t+WUz#i"  
    % functions, direct evaluation using the series representation can +^aM(4K\  
    % produce poor numerical results (floating point errors), because >RmL0d#B  
    % the summation often involves computing small differences between 0 s%{m<  
    % large successive terms in the series. (In such cases, the functions g>f_'7F&  
    % are often evaluated using alternative methods such as recurrence \9.@T g8`  
    % relations: see the Legendre functions, for example). For the Zernike 9[\$\l  
    % polynomials, however, this problem does not arise, because the SC0_ h(zb,  
    % polynomials are evaluated over the finite domain r = (0,1), and K&vqk/JW1  
    % because the coefficients for a given polynomial are generally all qZ+^ND(I  
    % of similar magnitude. :"oUnBY%  
    % n<7R6)j6  
    % ZERNPOL has been written using a vectorized implementation: multiple O#D N3yu?  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] +@C|u'  
    % values can be passed as inputs) for a vector of points R.  To achieve  A,|lDsvM  
    % this vectorization most efficiently, the algorithm in ZERNPOL ivl_=  
    % involves pre-determining all the powers p of R that are required to M}F) P&Y  
    % compute the outputs, and then compiling the {R^p} into a single =g^JJpS  
    % matrix.  This avoids any redundant computation of the R^p, and bcprhb  
    % minimizes the sizes of certain intermediate variables. c f1GA  
    % ApAHa]Ccp  
    %   Paul Fricker 11/13/2006 <NX6m|DD  
    e~BUAz  
    %MUwd@,  
    % Check and prepare the inputs: ji|tc9#6  
    % ----------------------------- 3HmJixy  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }#f~"-O  
        error('zernpol:NMvectors','N and M must be vectors.') .3 T#:Hl  
    end GCA?sFwo>  
    j%s:d(H`  
    if length(n)~=length(m) };;6706a  
        error('zernpol:NMlength','N and M must be the same length.') A@lY{e  
    end ?qjlWCV|e  
    W[tX%B  
    n = n(:); l+8G6?@]>  
    m = m(:); , 8F(R%v  
    length_n = length(n); `~3y[j]kO  
    )y}W=Q>T  
    if any(mod(n-m,2)) >NN&j#;x~  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') |nj,]pA  
    end )[hQK_e]  
    8L*P!j9`EY  
    if any(m<0) dg]: JU  
        error('zernpol:Mpositive','All M must be positive.') RBzBR)@5   
    end 4!KUPgg  
    f"G?#dW/1  
    if any(m>n) (>Tq  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') v= I 'rx  
    end n$T'gX#5  
    =m?x|Zc_v  
    if any( r>1 | r<0 ) ta 66AEc9  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') ;4 ON  
    end 2aUy1*aM  
    RK`C31Ws  
    if ~any(size(r)==1) S2 0L@e"U  
        error('zernpol:Rvector','R must be a vector.') k9<P]%  
    end Xy(o0/7F9  
    eO:wx.PW  
    r = r(:); n++L =&Wd  
    length_r = length(r); dLMKfh/4Q  
    qEoa%O  
    if nargin==4 Sc zYL?w^  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); l4sFT)}-J  
        if ~isnorm +5+?)8Ls  
            error('zernpol:normalization','Unrecognized normalization flag.') 4>&%N\$*  
        end 80M"`6  
    else "NamP\hj  
        isnorm = false; ]%PQ3MT.  
    end P+~{q.|._c  
    }t[?g)"M#-  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _JjR= m  
    % Compute the Zernike Polynomials SGREpOlJ+  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p=65L  
    ,NQucp  
    % Determine the required powers of r: ?p8(Uc#73  
    % ----------------------------------- M 8BN'% S  
    rpowers = []; 5@6%/='I q  
    for j = 1:length(n) !V6O~#  
        rpowers = [rpowers m(j):2:n(j)]; ]HK|xO(  
    end <Yk#MeiEp  
    rpowers = unique(rpowers); xmEmdOoD  
    n%>c4*t  
    % Pre-compute the values of r raised to the required powers, c.u$NnDU6  
    % and compile them in a matrix: Ck|3DiRQ  
    % ----------------------------- ~EVD NnHEr  
    if rpowers(1)==0 L)cy&"L|  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  4_E{  
        rpowern = cat(2,rpowern{:}); _BV'J92.  
        rpowern = [ones(length_r,1) rpowern]; e.<$G'  
    else ,[lS)`G  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (C RY$+d  
        rpowern = cat(2,rpowern{:}); MHv2r  
    end JwNG`M Gc  
    ?]'Rz\70  
    % Compute the values of the polynomials: q#$4Kt;  
    % -------------------------------------- pFLR!/J  
    z = zeros(length_r,length_n); I&oHVFY+  
    for j = 1:length_n %8)GuxG*  
        s = 0:(n(j)-m(j))/2; J2BW>T!tuw  
        pows = n(j):-2:m(j); 4Lo8Eue  
        for k = length(s):-1:1 =#AeOqs( q  
            p = (1-2*mod(s(k),2))* ... G] -$fz  
                       prod(2:(n(j)-s(k)))/          ... AS34yM(h  
                       prod(2:s(k))/                 ... ;hz"`{(JY  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... R$<LEwjSw  
                       prod(2:((n(j)+m(j))/2-s(k))); #-ioLt%  
            idx = (pows(k)==rpowers); xHv<pza:  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 0Dj<-n{9  
        end 2},|RQETy  
         QfuKpcT &  
        if isnorm -0 [^w  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 7-"ml\z  
        end $d4^e&s  
    end K<O1PrC  
    k#8,:B2  
    % 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)  icW?a9b&  
    ld58R  
    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。 TCAtb('D  
    5:(uD3]  
    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)