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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 7EXmmB~>,  
    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! 5S? yj  
     
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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 )}!'VIe^!  
    function z = zernfun(n,m,r,theta,nflag) uek3Y[n  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. \[EWxu  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n #I}!x>2  
    %   and angular frequency M, evaluated at positions (R,THETA) on the JrTBe73.]j  
    %   unit circle.  N is a vector of positive integers (including 0), and l)s+"C#  
    %   M is a vector with the same number of elements as N.  Each element *,*qv^  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 4/WCs$  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Dys"|,F  
    %   and THETA is a vector of angles.  R and THETA must have the same X)OP316yx  
    %   length.  The output Z is a matrix with one column for every (N,M) hp4(f W  
    %   pair, and one row for every (R,THETA) pair. pH%c7X/[3L  
    % -%l, Zd9  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike oJT@'{;*z  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )`7+o9&  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral 63Yu05'  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %iC63)(M  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized _ n4ma  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. {~g  
    % \#,#_  
    %   The Zernike functions are an orthogonal basis on the unit circle. {VG[m@  
    %   They are used in disciplines such as astronomy, optics, and 2z# @:Q  
    %   optometry to describe functions on a circular domain. L.[uMuUa  
    % r.^X>?  
    %   The following table lists the first 15 Zernike functions. [#'_@zZz  
    % )#~fS28j  
    %       n    m    Zernike function           Normalization m (:qZW  
    %       -------------------------------------------------- K0=E4>z,`q  
    %       0    0    1                                 1 wLe&y4  
    %       1    1    r * cos(theta)                    2 \<x_96jt!\  
    %       1   -1    r * sin(theta)                    2 R6mJFE*6T9  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) C*e[CP@u  
    %       2    0    (2*r^2 - 1)                    sqrt(3) `f+g A  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) nY-9 1q?Y  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ,ri--<  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z2V8NUn  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) p Y>-N  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) A;a(n\Sy  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) c.A/{a  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G$9|aaf`1#  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ,<r3Z$G  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n12c075  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) S&]<;N_B  
    %       -------------------------------------------------- ={@ @`yP^$  
    % qgsE7 ]  
    %   Example 1: V?dK*8s  
    % ]J=)pD rk  
    %       % Display the Zernike function Z(n=5,m=1) gs8@b5 RSb  
    %       x = -1:0.01:1; U]EuDNkO{  
    %       [X,Y] = meshgrid(x,x); `4$Qv'X*  
    %       [theta,r] = cart2pol(X,Y); A<CXdt+t  
    %       idx = r<=1; 0QH3,Ps1C  
    %       z = nan(size(X)); )u/ ^aK53^  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); `Mp7 })  
    %       figure D4]B>  
    %       pcolor(x,x,z), shading interp J K]tcP  
    %       axis square, colorbar m&~Dj#%(w  
    %       title('Zernike function Z_5^1(r,\theta)') }\L !;6oy  
    % a{Hb7&  
    %   Example 2: cPaWJ+c  
    % (Cd{#j<  
    %       % Display the first 10 Zernike functions 9`n) "r  
    %       x = -1:0.01:1; G$|;~'E  
    %       [X,Y] = meshgrid(x,x); xXxh3 k\  
    %       [theta,r] = cart2pol(X,Y); /A))"D  
    %       idx = r<=1; t:s q*d  
    %       z = nan(size(X)); =*:_swd  
    %       n = [0  1  1  2  2  2  3  3  3  3]; bKMR7&e.Ep  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; v;}`?@G  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; C9Z\G 3  
    %       y = zernfun(n,m,r(idx),theta(idx)); pH l2!{z  
    %       figure('Units','normalized') KP d C9H  
    %       for k = 1:10 pvQK6r  
    %           z(idx) = y(:,k); hd ;S>K/C  
    %           subplot(4,7,Nplot(k)) j484b2uj1  
    %           pcolor(x,x,z), shading interp A r7mH4M  
    %           set(gca,'XTick',[],'YTick',[]) $EGRaps{j>  
    %           axis square e=jT]i*cU  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [H:GKhPC`  
    %       end /< 7C[^h{-  
    % DEQE7.]3q  
    %   See also ZERNPOL, ZERNFUN2. m_Ac/ct f  
    O:^LQ  
    %   Paul Fricker 11/13/2006 3JZWhxkf[$  
    Xz .Y-5)  
    $7DcQ b9  
    % Check and prepare the inputs: K7xWE,y  
    % ----------------------------- [kuVQ$)  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d:<H?~  
        error('zernfun:NMvectors','N and M must be vectors.') "(\) &G  
    end !MJe+.  
    ,WB_C\.#XN  
    if length(n)~=length(m) 9kX=99kf[  
        error('zernfun:NMlength','N and M must be the same length.') D@\;@( |  
    end %V(N U_o  
    ~l*?D7[o  
    n = n(:); H&=n:'k^  
    m = m(:); r -q3+c^+  
    if any(mod(n-m,2)) 6(J4IzZ  
        error('zernfun:NMmultiplesof2', ... (YYj3#|  
              'All N and M must differ by multiples of 2 (including 0).') G]mWaA  
    end ,s><kHJ  
    M9s43XL(&  
    if any(m>n) pgd8`$(Q  
        error('zernfun:MlessthanN', ... qQxA@kdd  
              'Each M must be less than or equal to its corresponding N.') S2 "=B&,}  
    end 3 IWLBc  
    Yb%-tv:  
    if any( r>1 | r<0 ) bKuj po6  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') p>:ef<.i  
    end K4k~r!&OU  
    y5/'!L)g  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *|k;a]HT  
        error('zernfun:RTHvector','R and THETA must be vectors.')  &(1H!  
    end ! FR%QGn1  
    {;&B^uz ]  
    r = r(:); 1O7]3&L@  
    theta = theta(:); %h"qMs S  
    length_r = length(r); R>d@tr  
    if length_r~=length(theta) C1T=O  
        error('zernfun:RTHlength', ... ,]Ro',A&  
              'The number of R- and THETA-values must be equal.') )>y k-  
    end Q'|0?nBOY  
    ^}o7*   
    % Check normalization: &@% b?~  
    % -------------------- _^ q\XPS  
    if nargin==5 && ischar(nflag) @GG(7r\/B  
        isnorm = strcmpi(nflag,'norm'); -Aa]aDAz68  
        if ~isnorm fimb]C I|x  
            error('zernfun:normalization','Unrecognized normalization flag.') ^Ue0mC7m  
        end o#xgrMB  
    else wy{\/?~c  
        isnorm = false; w{3Q( =&  
    end ,{?q^"  
    I(]BMMj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -=$% {  
    % Compute the Zernike Polynomials 20UqJM8 Ot  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #M5_em4kN  
    $s-9|Lbs`  
    % Determine the required powers of r: <t{?7_ 8  
    % ----------------------------------- k7^R,.c@  
    m_abs = abs(m); M%$ DT  
    rpowers = []; LY-lTr@A^  
    for j = 1:length(n) M[aT2A  
        rpowers = [rpowers m_abs(j):2:n(j)]; v@8S5KJ  
    end B(j02<-  
    rpowers = unique(rpowers); )Fqy%uR8  
    5M%,N-P^  
    % Pre-compute the values of r raised to the required powers, >dpbCPJ9[  
    % and compile them in a matrix: |l|_dn  
    % ----------------------------- ^ <qrM  
    if rpowers(1)==0 ! FNf>z+  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GS\%mPZ  
        rpowern = cat(2,rpowern{:}); 1GtOA3,~;-  
        rpowern = [ones(length_r,1) rpowern]; `gBD_0<T7  
    else of9q"h  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,>;!%Ui/p  
        rpowern = cat(2,rpowern{:}); 2B7h9P.NB  
    end GR,J0LT   
    fNkuX-om  
    % Compute the values of the polynomials: XQ]`&w(  
    % -------------------------------------- >']+OrQH  
    y = zeros(length_r,length(n)); BlXX:aZv  
    for j = 1:length(n) a{h%DpG  
        s = 0:(n(j)-m_abs(j))/2; I-xwJi9?,  
        pows = n(j):-2:m_abs(j); cDCJ]iDs  
        for k = length(s):-1:1  ]}Pl%.  
            p = (1-2*mod(s(k),2))* ... P7z:3o.  
                       prod(2:(n(j)-s(k)))/              ... VS?dvZ1cC  
                       prod(2:s(k))/                     ... jm[}M  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... BBcj=]"_  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); Bn\l'T  
            idx = (pows(k)==rpowers); $^t<9" t  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ?^|QiuU:n  
        end < CDA"  
         TWUUvj`.  
        if isnorm <}d/v_+pnh  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9Uk(0A  
        end sltk@  
    end \M9 h&I\7  
    % END: Compute the Zernike Polynomials B={/nC}G~  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uJgI<l'|e3  
    pA<eTlH  
    % Compute the Zernike functions: Q uB+vL  
    % ------------------------------ h1# S+k  
    idx_pos = m>0; Gz ?2b#7v  
    idx_neg = m<0; RU6KIg{H  
    [g#s&bF  
    z = y; [OzzL\)3l  
    if any(idx_pos) U IfH*6X  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 2}w#3K  
    end < kz[:n:  
    if any(idx_neg) +P|2m"UA  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); <;%0T xK|U  
    end iNQ0p:<k  
    W2`.RF^  
    % 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) X8b|]Nr  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. MLV:U  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated >Q=Ukn;k  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive nLj&Uf&  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, $o.Kn9\  
    %   and THETA is a vector of angles.  R and THETA must have the same ! RPb|1Y}+  
    %   length.  The output Z is a matrix with one column for every P-value, P? (vW&B  
    %   and one row for every (R,THETA) pair. H8f]}  
    % %H& ].47  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike \0ov[T N.>  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ^P?vkO"pB?  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) EY=FDlV  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 QL97WK\$  
    %   for all p. MS*G-C  
    % ` H XEZ|  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ly7!R$X  
    %   Zernike functions (order N<=7).  In some disciplines it is ]CX[7Q+'  
    %   traditional to label the first 36 functions using a single mode 3?.1n Gu  
    %   number P instead of separate numbers for the order N and azimuthal oq$w4D0Z  
    %   frequency M. Km!nM$=k  
    % M4KWN'  
    %   Example: /syVGmS'M  
    % ka/XK[/'  
    %       % Display the first 16 Zernike functions 'e@=^FC  
    %       x = -1:0.01:1; Qf xH9_  
    %       [X,Y] = meshgrid(x,x); ;Lo&}U3F,!  
    %       [theta,r] = cart2pol(X,Y); BAV>o|-K  
    %       idx = r<=1; U>P|X=)  
    %       p = 0:15; >$H|:{D  
    %       z = nan(size(X));  9S1)U$  
    %       y = zernfun2(p,r(idx),theta(idx)); !VP %v&jKm  
    %       figure('Units','normalized') {q3:Z{#>7  
    %       for k = 1:length(p) 7NL% $Vf  
    %           z(idx) = y(:,k); 8;8c"'Mn  
    %           subplot(4,4,k) _w(ln9   
    %           pcolor(x,x,z), shading interp [ohBPQO  
    %           set(gca,'XTick',[],'YTick',[]) 33K*qaRAD  
    %           axis square fP[& a9l  
    %           title(['Z_{' num2str(p(k)) '}']) <7XT\?%F  
    %       end p!zJ;rh)  
    % WR#0<cz(  
    %   See also ZERNPOL, ZERNFUN. ys'T~Cs  
    A}5fCx.{  
    %   Paul Fricker 11/13/2006 XiQkrZ  
    Tycq1i^  
    FT89*C)oD  
    % Check and prepare the inputs: q; ji w#_  
    % ----------------------------- 6CCbBA  
    if min(size(p))~=1 wO.iKX;  
        error('zernfun2:Pvector','Input P must be vector.') qAY%nA>jO  
    end 'dJ/RJ~  
    {gsdG-  
    if any(p)>35 MCc$TttaVz  
        error('zernfun2:P36', ... %Dig)<yx  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... nSx]QREL!  
               '(P = 0 to 35).']) ^2JpWY:|7  
    end Q_p[k KH  
    . +  
    % Get the order and frequency corresonding to the function number: RJ4. kt  
    % ---------------------------------------------------------------- $uj(G7_  
    p = p(:); WYrI|^[>  
    n = ceil((-3+sqrt(9+8*p))/2); Dyg?F )6  
    m = 2*p - n.*(n+2); #VVr"*7$  
    o)Z=m:t,lK  
    % Pass the inputs to the function ZERNFUN: v~|?3/{Q  
    % ----------------------------------------  dy>!KO  
    switch nargin )G1P^WV4  
        case 3 !>z:m!MlQ  
            z = zernfun(n,m,r,theta); `ln1$  
        case 4 hk>;pU(  
            z = zernfun(n,m,r,theta,nflag); nBItO~l  
        otherwise $s5a G)?7  
            error('zernfun2:nargin','Incorrect number of inputs.') i38[hQR9a  
    end Q.U$nph\%d  
    >~nF=   
    % 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) kgdT7  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. c]-*P7W  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of LwcIGhy  
    %   order N and frequency M, evaluated at R.  N is a vector of DL'iS  
    %   positive integers (including 0), and M is a vector with the A]0:8@k5  
    %   same number of elements as N.  Each element k of M must be a 3r+.N  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) =ONHK F[UJ  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is RoGwK*j0+  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix t"072a  
    %   with one column for every (N,M) pair, and one row for every )0 W`  
    %   element in R. D]G)j  
    % VZ& A%UFC  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 7JJ/D4uT  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is &fa5laJb  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to .iMN,+qP  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $j}OB6^I  
    %   for all [n,m]. j^tW Iz  
    % C)'q QvA  
    %   The radial Zernike polynomials are the radial portion of the :r#)z4d5  
    %   Zernike functions, which are an orthogonal basis on the unit 7{@l%jx][  
    %   circle.  The series representation of the radial Zernike uDw.|B2ui  
    %   polynomials is jHXwOJq %  
    % 1923N]b  
    %          (n-m)/2 \s"U{N-  
    %            __ 'H5M|c$s  
    %    m      \       s                                          n-2s ]?O2:X  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r j>uj=B@  
    %    n      s=0 7>XDNI  
    % tGA :[SP  
    %   The following table shows the first 12 polynomials. Yim<>. !  
    % OU5*9_7.  
    %       n    m    Zernike polynomial    Normalization tE6!+c<7  
    %       --------------------------------------------- E%oY7.~-  
    %       0    0    1                        sqrt(2) g_5QA)4x  
    %       1    1    r                           2 H{J'# 9H  
    %       2    0    2*r^2 - 1                sqrt(6) tCxF~L@  
    %       2    2    r^2                      sqrt(6) HK\~Qnq  
    %       3    1    3*r^3 - 2*r              sqrt(8) ~qe%Yq  
    %       3    3    r^3                      sqrt(8) F  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) H]4Hj  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) sGh(#A0Pt  
    %       4    4    r^4                      sqrt(10) 3 rLTF\  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) rc&%m  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) su*Pk|6%  
    %       5    5    r^5                      sqrt(12) ~{sG| ;/!*  
    %       --------------------------------------------- `.s({/|[  
    % u:0aM}9A  
    %   Example: ]Q]W5WDe:  
    % 8-<:i  
    %       % Display three example Zernike radial polynomials s3 7'&K  
    %       r = 0:0.01:1; AJ#Nenmj  
    %       n = [3 2 5]; {*{Ox[Nh{  
    %       m = [1 2 1]; @(r /dZc  
    %       z = zernpol(n,m,r); 6aM*:>C"  
    %       figure )95f*wte  
    %       plot(r,z) WA dCF-S  
    %       grid on  V#VN %{  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') rE@T79"  
    % ca+5=+X7  
    %   See also ZERNFUN, ZERNFUN2. df7wN#kO+  
    y;/VB,4V  
    % A note on the algorithm. %|s+jeUDn|  
    % ------------------------ 3R+|5Uq8~  
    % The radial Zernike polynomials are computed using the series 2eMTxwt*S  
    % representation shown in the Help section above. For many special %^RN#_ro(3  
    % functions, direct evaluation using the series representation can (5]}5W*  
    % produce poor numerical results (floating point errors), because .^B*e6DAD  
    % the summation often involves computing small differences between /SYw;<=  
    % large successive terms in the series. (In such cases, the functions $DG?M6   
    % are often evaluated using alternative methods such as recurrence 8WnwQ%;m?  
    % relations: see the Legendre functions, for example). For the Zernike J2:y6kGj>  
    % polynomials, however, this problem does not arise, because the ]U"94S U:)  
    % polynomials are evaluated over the finite domain r = (0,1), and vVOh3{e|  
    % because the coefficients for a given polynomial are generally all 7uzk p&+:  
    % of similar magnitude. <[w=TdCPs  
    % Xh56T^,2  
    % ZERNPOL has been written using a vectorized implementation: multiple / m=HG^!  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] x7O-Y~[2  
    % values can be passed as inputs) for a vector of points R.  To achieve 21"1NJzP  
    % this vectorization most efficiently, the algorithm in ZERNPOL <)1qt 9  
    % involves pre-determining all the powers p of R that are required to 3Z1CWzq(  
    % compute the outputs, and then compiling the {R^p} into a single j]`PSl+w  
    % matrix.  This avoids any redundant computation of the R^p, and l\i)$=d&g  
    % minimizes the sizes of certain intermediate variables. TYW&!sm  
    % EFz&N\2  
    %   Paul Fricker 11/13/2006 ]\|VpIg  
    'inFKy'H  
    5{g?,/(  
    % Check and prepare the inputs: !r<pmr3f@7  
    % ----------------------------- 50X([hIr  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $< JaLS  
        error('zernpol:NMvectors','N and M must be vectors.') WlU0:(d  
    end 7 qS""f7  
    =i[\-  
    if length(n)~=length(m) a |X a3E  
        error('zernpol:NMlength','N and M must be the same length.') Hj}K{20  
    end @{2 5xTt  
    B6={&7U2  
    n = n(:); $` ""  
    m = m(:); nR*ryv  
    length_n = length(n); W)bLSL]`E  
    gw!vlwC&T  
    if any(mod(n-m,2)) 7<*yS310  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [@.!~E)P  
    end ~A\GT$  
    6e |*E`I  
    if any(m<0) {z{bY\  
        error('zernpol:Mpositive','All M must be positive.') o4Om}]Ti  
    end tS6qWtE  
    %%[LKSTb  
    if any(m>n) a^I\ /&aw'  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #pnI\  
    end BI%$c~wS  
    lN Yt`xp  
    if any( r>1 | r<0 ) #AJM6* G9  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') t7aefV&_,  
    end XwJ7|cB  
    EFM5,gB.m  
    if ~any(size(r)==1) ;{N!Eb`S  
        error('zernpol:Rvector','R must be a vector.') %WjXg:R  
    end te-jfmu2  
    \XZ/v*d0  
    r = r(:); Yo6*C  
    length_r = length(r); 9dx/hFA  
    1G^`-ri6  
    if nargin==4 asppRL||  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); Li4zTR|U  
        if ~isnorm b0Ps5G\ u  
            error('zernpol:normalization','Unrecognized normalization flag.') ;~m8;8)  
        end k5'Vy8q  
    else vg32y /l]S  
        isnorm = false; M/"I2m   
    end rX2.i7i,  
    u. F9g #  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% z7fp#>uw  
    % Compute the Zernike Polynomials N 5lDS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *Q "wwpl?  
    $Nhs1st*8  
    % Determine the required powers of r: p8Qk 'F=h  
    % ----------------------------------- *RJG!t*t  
    rpowers = []; n{ar gI8wF  
    for j = 1:length(n) @niHl  
        rpowers = [rpowers m(j):2:n(j)]; t.i 8 2Q  
    end &w_j/nW^'  
    rpowers = unique(rpowers); u04kF^  
    iP ->S\  
    % Pre-compute the values of r raised to the required powers, Yg||{  
    % and compile them in a matrix: 4V)kx[j  
    % ----------------------------- "R;U/+  
    if rpowers(1)==0 )q8pk2  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Yi%;|]  
        rpowern = cat(2,rpowern{:}); &5B'nk"  
        rpowern = [ones(length_r,1) rpowern]; 65JF`]  
    else }C"%p8=HM  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s<<ooycBrQ  
        rpowern = cat(2,rpowern{:}); z]_wjYn Z  
    end ^@s1Z7  
    *av<E  
    % Compute the values of the polynomials: iL-(O;n  
    % -------------------------------------- *&^Pj%DX  
    z = zeros(length_r,length_n); R'as0 u\  
    for j = 1:length_n BYL)nCc  
        s = 0:(n(j)-m(j))/2; /~%&vpF-L  
        pows = n(j):-2:m(j); FrGgga$  
        for k = length(s):-1:1 @k,#L`3^  
            p = (1-2*mod(s(k),2))* ... 2*;~S4 4  
                       prod(2:(n(j)-s(k)))/          ... HdUQCugxx:  
                       prod(2:s(k))/                 ... gwuI-d^  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... q376m-+  
                       prod(2:((n(j)+m(j))/2-s(k))); pP&7rRhw  
            idx = (pows(k)==rpowers); [ )Iv^ U9  
            z(:,j) = z(:,j) + p*rpowern(:,idx); /K@XzwM  
        end %rL.|q9  
         -A^_{4X  
        if isnorm c<B/V0]  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1));  dVtG/0  
        end %} SrL*  
    end dd%6t  
    8Z8gRcv{p  
    % 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)  E ~<JC"]  
    G_,jgg7  
    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。 )jP1or  
    jQB9j  
    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)