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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 J,:Wv`N:9~  
    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! zl<D"eP  
     
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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 9h/Hy aN  
    function z = zernfun(n,m,r,theta,nflag) 3m$ck$  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. tSe[*V4{'  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $z`l{F4eMf  
    %   and angular frequency M, evaluated at positions (R,THETA) on the C-\3,  
    %   unit circle.  N is a vector of positive integers (including 0), and  !# zO%  
    %   M is a vector with the same number of elements as N.  Each element > `mV^QD  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) h^ K]ASj  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Ahc9HA2  
    %   and THETA is a vector of angles.  R and THETA must have the same +,cd$,18  
    %   length.  The output Z is a matrix with one column for every (N,M) 6AoKuT;  
    %   pair, and one row for every (R,THETA) pair. X`J86G)  
    % 34Cnbtq^  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j#xGB]  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FmhAUe  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral $ w+.-Tr  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @1xIph<z  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized `F]  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }1%%`  
    % e ^,IZ{  
    %   The Zernike functions are an orthogonal basis on the unit circle. `sDLxgwI  
    %   They are used in disciplines such as astronomy, optics, and =dsEt\ j  
    %   optometry to describe functions on a circular domain. $N Mu  
    % F`GXho[  
    %   The following table lists the first 15 Zernike functions. )%PMDG|  
    % @|5B}%!  
    %       n    m    Zernike function           Normalization 1xu~@v 60  
    %       -------------------------------------------------- #SG.`J<%  
    %       0    0    1                                 1 Y`(~eNX^%  
    %       1    1    r * cos(theta)                    2 "0,FB4L[U5  
    %       1   -1    r * sin(theta)                    2 -+M360  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) (#Xs\IEVF  
    %       2    0    (2*r^2 - 1)                    sqrt(3) IRueq @4  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) 7XLqP  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) gVe]?Jva`  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ! ,{zDMA  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) J_fs}Y1q\  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) (z8 ;J> 7  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) JU.!<  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^d@2Y0hH  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) !v(^wqna\  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dwQ1~  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) <*WGvCh%w  
    %       -------------------------------------------------- KVh#"]<WV  
    % 9X,iQ  
    %   Example 1: KVr9kcs  
    % |a a\t  
    %       % Display the Zernike function Z(n=5,m=1) i7Cuc+ j8  
    %       x = -1:0.01:1; T?QW$cU!e:  
    %       [X,Y] = meshgrid(x,x); ,RM8D)m\  
    %       [theta,r] = cart2pol(X,Y); ];"40/X  
    %       idx = r<=1; .ZV='i()X  
    %       z = nan(size(X)); \#WWJh"W  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); em5~4;&'  
    %       figure (wuciKQ  
    %       pcolor(x,x,z), shading interp 5!cp^[rGL  
    %       axis square, colorbar y:^o ._  
    %       title('Zernike function Z_5^1(r,\theta)') r>7 +&s*yk  
    % %l14K_  
    %   Example 2: *^Ges;5 $"  
    % ,Q3OQ[Nmh  
    %       % Display the first 10 Zernike functions 97$Q?a8S@  
    %       x = -1:0.01:1; 8|<</v8i  
    %       [X,Y] = meshgrid(x,x); .@%L8_sMR  
    %       [theta,r] = cart2pol(X,Y); _x1W\#  
    %       idx = r<=1; =.&8ghJ*M  
    %       z = nan(size(X)); ?QzL#iO }h  
    %       n = [0  1  1  2  2  2  3  3  3  3]; $v[mIR  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Dr(2@ 0P  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; &M@c50&%  
    %       y = zernfun(n,m,r(idx),theta(idx)); WJu(,zM?G  
    %       figure('Units','normalized') ;6D3>Lm  
    %       for k = 1:10 9<&M~(dwT4  
    %           z(idx) = y(:,k); 9(O eH7  
    %           subplot(4,7,Nplot(k)) 'S9o!hb'@  
    %           pcolor(x,x,z), shading interp E?czolNl  
    %           set(gca,'XTick',[],'YTick',[]) eY'n S  
    %           axis square Yj*T'<e  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) </D.}ia  
    %       end sNcU>qjj6  
    % IW&*3I<K  
    %   See also ZERNPOL, ZERNFUN2. (,jsZ!sl  
    m;\nMdn  
    %   Paul Fricker 11/13/2006 WW{_D  
    o $W@@aM  
    4w=v /WDo  
    % Check and prepare the inputs: F6111Q </  
    % ----------------------------- 8a`3eM~?[  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZO cpF1y  
        error('zernfun:NMvectors','N and M must be vectors.') yYYP;N?g4k  
    end WeaT42*Q{  
    9#:fQ!3`  
    if length(n)~=length(m) nW"O+s3  
        error('zernfun:NMlength','N and M must be the same length.') O ylUuYy~j  
    end )^AZmUYZ  
    HcJ!(  
    n = n(:); 2uN3:_w  
    m = m(:); A[^#8evaK  
    if any(mod(n-m,2)) wK7w[Xt  
        error('zernfun:NMmultiplesof2', ... XHj%U  
              'All N and M must differ by multiples of 2 (including 0).') s>I]_W)Pt  
    end ^)Awjj9  
    U5wh( vi  
    if any(m>n) }2LWDQ;po  
        error('zernfun:MlessthanN', ... gaz",kK<  
              'Each M must be less than or equal to its corresponding N.') #::+# G  
    end UkpTK8>&  
    .\T!oSb4[  
    if any( r>1 | r<0 ) 7gN;9pc$  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') ][tR=Y#&y5  
    end F~fBr  
    ui,!_O .c  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) b@8z+,_  
        error('zernfun:RTHvector','R and THETA must be vectors.') %(p9AE  
    end "{qnm+G  
    ~cSXBc,+  
    r = r(:); VgIk'.  
    theta = theta(:); B }euIQB  
    length_r = length(r); /CO=!*7fz  
    if length_r~=length(theta) JxwKTFU'3O  
        error('zernfun:RTHlength', ... fX 1%I  
              'The number of R- and THETA-values must be equal.') O50<h O]l  
    end , +J)`+pJx  
    IB| 6\uKn  
    % Check normalization: 4gC(zJ  
    % -------------------- }Vob)r{R@  
    if nargin==5 && ischar(nflag) f~\H|E8(  
        isnorm = strcmpi(nflag,'norm'); LEP TL#WT1  
        if ~isnorm ><D2of|  
            error('zernfun:normalization','Unrecognized normalization flag.') =E]tEi  
        end tt2 S.j  
    else 7F0J*M  
        isnorm = false; 0Zwx3[bq6K  
    end /eH37H  
    HM0&%  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }(!Uq  
    % Compute the Zernike Polynomials (|Gwg\r  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `<d.I%}  
    iYvzZ7 8f  
    % Determine the required powers of r: ,M?8s2?  
    % ----------------------------------- 9uWg4U  
    m_abs = abs(m); ~mt{j7  
    rpowers = []; (>A#|N1U  
    for j = 1:length(n) [ !#Dba#  
        rpowers = [rpowers m_abs(j):2:n(j)]; u28$V]  
    end PkyX,mr#1  
    rpowers = unique(rpowers); ~Yg) 8  
    9#P~cW?  
    % Pre-compute the values of r raised to the required powers, >'q]ypA1  
    % and compile them in a matrix: ?2da6v,t  
    % ----------------------------- R|8L'H+1x  
    if rpowers(1)==0 ~K#92  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *9r(lmrfj  
        rpowern = cat(2,rpowern{:}); Uv>e :U7;  
        rpowern = [ones(length_r,1) rpowern]; us?q^>u  
    else  0LL65[  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); mxF+Fp~  
        rpowern = cat(2,rpowern{:}); 2IW!EUR  
    end +C7E]0!r  
    DFQ`(1Q  
    % Compute the values of the polynomials: Q njK<}M9  
    % -------------------------------------- ^j${#Q  
    y = zeros(length_r,length(n)); ibZ[U p?  
    for j = 1:length(n) WO9vOS>  
        s = 0:(n(j)-m_abs(j))/2; q?mpvpL G  
        pows = n(j):-2:m_abs(j); fi>.X99(G  
        for k = length(s):-1:1 :Ob^b3<t  
            p = (1-2*mod(s(k),2))* ... .wq j  
                       prod(2:(n(j)-s(k)))/              ... B,_K mHItd  
                       prod(2:s(k))/                     ... 9-{+U,3)  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... .hxFFk%5  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); VT4 >6u}  
            idx = (pows(k)==rpowers); H.XyNtJ  
            y(:,j) = y(:,j) + p*rpowern(:,idx); }]dzY(   
        end k"gm;,`  
         hy;V~J#  
        if isnorm eDP&W$s#  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +U J~/XV  
        end xLFMC?I  
    end  u? >x  
    % END: Compute the Zernike Polynomials ~E8/m_> rU  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R'tvF$3=i  
    .!L{yU,  
    % Compute the Zernike functions: !9HWx_,|Z  
    % ------------------------------ w@R"g%k-  
    idx_pos = m>0; Nb3O> &J  
    idx_neg = m<0; *a\x!c"  
    ~a2|W|?  
    z = y; b49h @G  
    if any(idx_pos) 8r"-3<*  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @8m%*pBg  
    end .YvIVQ  
    if any(idx_neg) ewn\'RLZ"@  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ~"\v(\Pe  
    end @|"K"j#  
    P(I%9  
    % 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) 6A =k;do  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. T YR \K  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 1'p=yHw  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive ]G8"\J4 &  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, jHE^d<=O^  
    %   and THETA is a vector of angles.  R and THETA must have the same AZik:C"Q  
    %   length.  The output Z is a matrix with one column for every P-value, ~&<vAgy,  
    %   and one row for every (R,THETA) pair. Zw{?^6;cS  
    % 3rHn?  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike |Bi7:w  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) x)eF{%QB  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ; }T+ImjA  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 A\9LJ#E  
    %   for all p. =~W=}  
    % JJg;X :p  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ylu\]pr9|C  
    %   Zernike functions (order N<=7).  In some disciplines it is nTtEv~a_n  
    %   traditional to label the first 36 functions using a single mode Ja&S_'P[  
    %   number P instead of separate numbers for the order N and azimuthal ` s+kYWg'Z  
    %   frequency M. a @3s71  
    % =E!Y f#p+q  
    %   Example: Q1yXdw  
    % .)WEg|D0Ku  
    %       % Display the first 16 Zernike functions ,UVu.RjXN  
    %       x = -1:0.01:1; $'eY-U8q  
    %       [X,Y] = meshgrid(x,x); %#&njP  
    %       [theta,r] = cart2pol(X,Y); -(lP8Y~gFY  
    %       idx = r<=1; x3U>5F@  
    %       p = 0:15; +03/A`PKrB  
    %       z = nan(size(X)); >/ A'G  
    %       y = zernfun2(p,r(idx),theta(idx)); kMLJa=]$  
    %       figure('Units','normalized') ?VRsgV'$  
    %       for k = 1:length(p) :HrFbq  
    %           z(idx) = y(:,k); \k"CtzoX  
    %           subplot(4,4,k) uF}B:53A  
    %           pcolor(x,x,z), shading interp 2,g4yXws5  
    %           set(gca,'XTick',[],'YTick',[]) h*1T3U$  
    %           axis square W)T'?b'.  
    %           title(['Z_{' num2str(p(k)) '}']) 2cIbX  
    %       end H=~7g3  
    % eGpKoq7a  
    %   See also ZERNPOL, ZERNFUN. ~9#x=nU:+V  
    )'RaMo` 4  
    %   Paul Fricker 11/13/2006 [ "3s  
    IqepR >5t  
    #XqCz>Z  
    % Check and prepare the inputs: :IJ<Mmb  
    % ----------------------------- U~?mW,iRL  
    if min(size(p))~=1 o%;ly  
        error('zernfun2:Pvector','Input P must be vector.') ,3-^EfccW  
    end K*,,j\Q.  
    kkb+qo  
    if any(p)>35 (4ZO[Ae  
        error('zernfun2:P36', ... ae@!M  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... r1vS~ 4Z  
               '(P = 0 to 35).']) @+p(%  
    end T=sAy/1oR  
    Pn.DeoHme  
    % Get the order and frequency corresonding to the function number: j[w=pF,o  
    % ---------------------------------------------------------------- 0QfDgDX  
    p = p(:); H%rNQxA2 +  
    n = ceil((-3+sqrt(9+8*p))/2); .b<W*4{j0H  
    m = 2*p - n.*(n+2); _&s pMf  
    ~&MDfpl  
    % Pass the inputs to the function ZERNFUN: J#i7'9g  
    % ---------------------------------------- ln8NcAEx  
    switch nargin 0} &/n>F  
        case 3 W1)<!nwA  
            z = zernfun(n,m,r,theta); p-EU"O  
        case 4 6~W@$SP,F  
            z = zernfun(n,m,r,theta,nflag); !plu;w  
        otherwise 9xzow,mi  
            error('zernfun2:nargin','Incorrect number of inputs.') z9OpxW@Ou  
    end `\;Z&jlpT  
    VEI ct{  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) uG6.(A1LM  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. 2?Jw0Wq5D  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of a L+>XN  
    %   order N and frequency M, evaluated at R.  N is a vector of |?g-8":H8P  
    %   positive integers (including 0), and M is a vector with the Xa?igbgAwx  
    %   same number of elements as N.  Each element k of M must be a GtZ.' ?-  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) /JC1o&z_T  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is m&xVlS  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix es#6/  
    %   with one column for every (N,M) pair, and one row for every wik<# ke  
    %   element in R. p0zC(v0*  
    % {y6h(@I8\  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- &V (6N%A^U  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is BU??}{  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Ma$~B0!;s  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $)o0{HsL+  
    %   for all [n,m]. ?#Z4Dg 9|  
    % 2=8PA/  
    %   The radial Zernike polynomials are the radial portion of the 5.&)hmpg  
    %   Zernike functions, which are an orthogonal basis on the unit 6<fG; :  
    %   circle.  The series representation of the radial Zernike lA/-fUA  
    %   polynomials is D^.  c:  
    % NjEi.]L*fX  
    %          (n-m)/2 ug ;Xoh5w  
    %            __ \6?a  
    %    m      \       s                                          n-2s {R@V  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ZBY2,%nAo  
    %    n      s=0 @d 7V@F0d  
    % K@cWg C  
    %   The following table shows the first 12 polynomials. *kqC^2t  
    % Gvh"3|u ?z  
    %       n    m    Zernike polynomial    Normalization +BL46 Bq  
    %       --------------------------------------------- z5M6  
    %       0    0    1                        sqrt(2) V8B4e4F  
    %       1    1    r                           2 ][?J8F  
    %       2    0    2*r^2 - 1                sqrt(6) a%b E}  
    %       2    2    r^2                      sqrt(6) ZnI15bsDx  
    %       3    1    3*r^3 - 2*r              sqrt(8) ,z/aT6M?H  
    %       3    3    r^3                      sqrt(8) *. |%uf.  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) AzXLlQ  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) kV?fie<\)  
    %       4    4    r^4                      sqrt(10) VlA]A,P}i  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) JOt(r}gU  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) !f52JQyh  
    %       5    5    r^5                      sqrt(12)  w0=  
    %       --------------------------------------------- ycc G>%>r  
    % ^ `Ozw^~  
    %   Example: ^&1O:G*"  
    % z_ycH%p  
    %       % Display three example Zernike radial polynomials \2a;z<(  
    %       r = 0:0.01:1; `}=R  
    %       n = [3 2 5]; 2m yxwA5  
    %       m = [1 2 1]; 4^2>K C_  
    %       z = zernpol(n,m,r); vZTXvdF  
    %       figure ~,1Sw7 rE  
    %       plot(r,z) b6f OHy  
    %       grid on Hh=fv~X  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') UiH7  
    % E;C{i  
    %   See also ZERNFUN, ZERNFUN2. %wOkp`1-  
    b1 w@toc  
    % A note on the algorithm. iD_y@+iz  
    % ------------------------ =cjO]  
    % The radial Zernike polynomials are computed using the series pl&nr7\  
    % representation shown in the Help section above. For many special QWfSm^ t  
    % functions, direct evaluation using the series representation can |3,WiK='  
    % produce poor numerical results (floating point errors), because H@xS<=:lM  
    % the summation often involves computing small differences between Sf*v#?  
    % large successive terms in the series. (In such cases, the functions f2IH2^)P  
    % are often evaluated using alternative methods such as recurrence nD*iSb*  
    % relations: see the Legendre functions, for example). For the Zernike Z@+nkTJ9&t  
    % polynomials, however, this problem does not arise, because the = N*Jis  
    % polynomials are evaluated over the finite domain r = (0,1), and 3[fm| aU  
    % because the coefficients for a given polynomial are generally all JXU ?'@QY  
    % of similar magnitude. YQG l8E'  
    % bGeIb-|(  
    % ZERNPOL has been written using a vectorized implementation: multiple %CiF;wJ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ?}s;,_GH  
    % values can be passed as inputs) for a vector of points R.  To achieve L>sLb(2\i  
    % this vectorization most efficiently, the algorithm in ZERNPOL -\?-  
    % involves pre-determining all the powers p of R that are required to z&d.YO_W  
    % compute the outputs, and then compiling the {R^p} into a single }BlyEcw'aN  
    % matrix.  This avoids any redundant computation of the R^p, and .@OQ$ D<  
    % minimizes the sizes of certain intermediate variables. 23^>#b7st  
    % a#r{FoU{M8  
    %   Paul Fricker 11/13/2006 VmPh''Z%-  
    T@yQOD7  
    zG ='U  
    % Check and prepare the inputs: 4DCh+|r  
    % ----------------------------- ;Y*K!iFWH  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;(`bP  
        error('zernpol:NMvectors','N and M must be vectors.') *GE6zGdN  
    end d[3me{Rs  
    mv8H:T  
    if length(n)~=length(m) '1b 1N5~  
        error('zernpol:NMlength','N and M must be the same length.') xc}[q`vK  
    end N { oVz],  
    a`w=0]1&*  
    n = n(:); =figat  
    m = m(:); oa"Bpi9i  
    length_n = length(n); )Ac,F6w  
    WkA47+DsV  
    if any(mod(n-m,2)) h76j|1gI  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') NzQvciJ@"  
    end 6P _+:Mf  
    _j , Tc*T  
    if any(m<0) Ljz)%y[s  
        error('zernpol:Mpositive','All M must be positive.') Y0,{fw<  
    end pwfQqPC#_  
    ,Lp"Ia  
    if any(m>n) $mGzJ4&  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') C<^i`[&P$  
    end tB>!1}v  
    ^;PjO|mD Z  
    if any( r>1 | r<0 ) Q*#Lr4cm{  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') )m7%cyfC  
    end _ $>);qIP4  
    ! I?C8)  
    if ~any(size(r)==1) #1J ,!seJ  
        error('zernpol:Rvector','R must be a vector.') PxrT@.T$  
    end )4uWB2ZRoi  
    wd*B3  
    r = r(:); [IOI&`?D  
    length_r = length(r); o4FHR+u<M  
    p+=zl`\=|  
    if nargin==4 =H;n$ -P  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); wGLMLbj5  
        if ~isnorm G>vK$W$f N  
            error('zernpol:normalization','Unrecognized normalization flag.') R&W%E%uj  
        end 5"nq h}5  
    else akbB=:M,x  
        isnorm = false; Fc`IRPW<  
    end p4z4[=-:  
    >WLX5i&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Xf&YcHo  
    % Compute the Zernike Polynomials g$GGo[_0  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2Ty]s~  
    Wd;t(5Xl  
    % Determine the required powers of r: 1<m`38'  
    % ----------------------------------- s&nat4{B  
    rpowers = []; tdRvg7v,N%  
    for j = 1:length(n) QYo04`Rl  
        rpowers = [rpowers m(j):2:n(j)]; ||X3g"2W9  
    end ~|=D.}#$  
    rpowers = unique(rpowers); p}pRf@(`\  
    h!SsIy(  
    % Pre-compute the values of r raised to the required powers, F;#$Q  
    % and compile them in a matrix: e+BZoK ^  
    % ----------------------------- } F{s\qUt  
    if rpowers(1)==0 ?=Ceo#Er  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); afX|R  
        rpowern = cat(2,rpowern{:}); `$jun  
        rpowern = [ones(length_r,1) rpowern]; yrEh5v:  
    else P,] ./m\J  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CJN~p]\  
        rpowern = cat(2,rpowern{:}); o A2oX  
    end }6a}8EyFP  
    k#pO+[ x  
    % Compute the values of the polynomials: o7E|wS  
    % -------------------------------------- -51LF=(!L  
    z = zeros(length_r,length_n); LI.WcI3uS  
    for j = 1:length_n 5XX)8gAo  
        s = 0:(n(j)-m(j))/2; m BvO<?ec  
        pows = n(j):-2:m(j); .:&`PaMt  
        for k = length(s):-1:1 J(}PvkA  
            p = (1-2*mod(s(k),2))* ... rGNa[1{kRs  
                       prod(2:(n(j)-s(k)))/          ... V. i{IW  
                       prod(2:s(k))/                 ... yC$m(Y12FN  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... [(Z(8{3i  
                       prod(2:((n(j)+m(j))/2-s(k))); }y*D(`  
            idx = (pows(k)==rpowers); 39O rY  
            z(:,j) = z(:,j) + p*rpowern(:,idx); 4Lg ,J9  
        end "[7-1}l  
         99*k&mb  
        if isnorm py\:u5QS  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); $)NS]wJ]3  
        end sm'_0EUg  
    end #%QHb,lhl  
    %`k [xz  
    % EOF zernpol
    离线niuhelen
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    只看该作者 6楼 发表于: 2011-03-12
    这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
    离线li_xin_feng
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    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
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    只看该作者 8楼 发表于: 2014-03-04
    我也一直想了解这个多项式的应用,还没用过呢
    离线phoenixzqy
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    只看该作者 9楼 发表于: 2014-04-22
    回 guapiqlh 的帖子
    guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  Ke@Bf  
    JfJ ln[  
    数值分析方法看一下就行了。其实就是正交多项式的应用。zernike也只不过是正交多项式的一种。 ;(,GS@sP  
    PJKY$s.  
    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)