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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 Jn+k$'6 %#  
    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! =Qf.  
     
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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 _ft)e3Gf  
    function z = zernfun(n,m,r,theta,nflag) KsG>,# Q  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. Fb^Ae6/i  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GQvJj4LJp  
    %   and angular frequency M, evaluated at positions (R,THETA) on the EXz{Pqz  
    %   unit circle.  N is a vector of positive integers (including 0), and G^6\OOSy  
    %   M is a vector with the same number of elements as N.  Each element `SN?4;N0  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) 8A,="YIt  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, AgU 7U/yk  
    %   and THETA is a vector of angles.  R and THETA must have the same J=OWXL!<a  
    %   length.  The output Z is a matrix with one column for every (N,M) -|/kg7IO\  
    %   pair, and one row for every (R,THETA) pair. -gzY ~a  
    % $1ZF kw  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike l0sBXs`3b  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), * SHQ[L4{  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral !vQDPLBL  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &58TX[#  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized i4p2]Nr t  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !J%m7 A  
    % bpv?$j-j  
    %   The Zernike functions are an orthogonal basis on the unit circle. NW*qw q  
    %   They are used in disciplines such as astronomy, optics, and ;A3aUN;"I  
    %   optometry to describe functions on a circular domain. Q =!f,  
    % Ze:Y"49S+>  
    %   The following table lists the first 15 Zernike functions. @?gN &Z)I  
    % ;xl_9Ht/  
    %       n    m    Zernike function           Normalization M)T{6 w  
    %       -------------------------------------------------- OQC.p,SO  
    %       0    0    1                                 1 P?Fm<s:  
    %       1    1    r * cos(theta)                    2 aN}l&4d  
    %       1   -1    r * sin(theta)                    2 FE[{*8  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) KDW=x4*p  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Ou'<9m!9  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) HXg4 T  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) ,VTX7vaH  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ROfr  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) #]_S)_Z-  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) aDreN*n  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 5%Xny8 ]|D  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IY=CTFQ8lm  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |vLlEN/S  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `;OEdeAM  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) U(t_uc5q  
    %       -------------------------------------------------- OlJkyL8|  
    % c{SD=wRt,y  
    %   Example 1: |!=KLJUA  
    % U if61)+!i  
    %       % Display the Zernike function Z(n=5,m=1) :: 2pDtMS  
    %       x = -1:0.01:1; kpU-//lk+  
    %       [X,Y] = meshgrid(x,x); u3XQ<N{Gj  
    %       [theta,r] = cart2pol(X,Y); $!-a)U,w$B  
    %       idx = r<=1; k"V@9q;*  
    %       z = nan(size(X)); V(LE4P 1  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); w' gKE'c  
    %       figure iOO1\9{@  
    %       pcolor(x,x,z), shading interp @N'0:0Nb_  
    %       axis square, colorbar ?7:?OX  
    %       title('Zernike function Z_5^1(r,\theta)') g\n0v~T+  
    % dAZh# i[  
    %   Example 2: xr<.r4  
    % fsxZQ=-PW  
    %       % Display the first 10 Zernike functions ,cqZb0VP{t  
    %       x = -1:0.01:1; U $ bLt  
    %       [X,Y] = meshgrid(x,x); g^qbd$}  
    %       [theta,r] = cart2pol(X,Y); :.k)!  
    %       idx = r<=1; |,G=k,?_p  
    %       z = nan(size(X)); '/@] V  
    %       n = [0  1  1  2  2  2  3  3  3  3]; J|Xu]fg0  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; k\J 6WT  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; >[U.P)7;  
    %       y = zernfun(n,m,r(idx),theta(idx)); =`oQcIkz  
    %       figure('Units','normalized') (6WSQqp  
    %       for k = 1:10 pJK}9p=4`  
    %           z(idx) = y(:,k); 9u->.O: p  
    %           subplot(4,7,Nplot(k)) =?, dX  
    %           pcolor(x,x,z), shading interp )ZI9n7  
    %           set(gca,'XTick',[],'YTick',[]) -}W `  
    %           axis square >cEB ,@~  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @fVCGV?'  
    %       end .L X8ko  
    % hR]AUH  
    %   See also ZERNPOL, ZERNFUN2. ^6Std x_  
    ]q2g[D o5  
    %   Paul Fricker 11/13/2006 oiS>:de%tc  
    PI~W6a7p  
    qU1^ K  
    % Check and prepare the inputs: k$hNibpkt  
    % ----------------------------- $2M dxw5  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) y.LJ 5K$&a  
        error('zernfun:NMvectors','N and M must be vectors.') ,3zF_y(*Y  
    end }#rdMh  
    l9 |x7GB  
    if length(n)~=length(m) $|2@of.  
        error('zernfun:NMlength','N and M must be the same length.') V`n;W6Q17  
    end y8{PAH8S  
    dX58nJ4u  
    n = n(:); jmAQ!y|W.  
    m = m(:); ~4Gs\U:!Q  
    if any(mod(n-m,2)) gyI(O>e  
        error('zernfun:NMmultiplesof2', ... _uR-Z_z  
              'All N and M must differ by multiples of 2 (including 0).') 'Gw;@[  
    end BE;J/  
    4+V+SD  
    if any(m>n) S!<1C Fh  
        error('zernfun:MlessthanN', ... 1Ugyjjlz  
              'Each M must be less than or equal to its corresponding N.') 4[S0~O{r  
    end &tULSp@J  
    ;gh#8JkI  
    if any( r>1 | r<0 ) D{](5?$`|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') M =!RJ%6f  
    end ~)zoIM\  
    ? Q`Sx  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \qrSJ=}t  
        error('zernfun:RTHvector','R and THETA must be vectors.') 9Q#eu~R  
    end @[qGoai  
    V0gk8wD  
    r = r(:); ">n38:?R  
    theta = theta(:); &~u=vuX  
    length_r = length(r); L29,Y=n@  
    if length_r~=length(theta) o|s JTY  
        error('zernfun:RTHlength', ... }@ *Me+  
              'The number of R- and THETA-values must be equal.') R|%R-J]  
    end #nE%.k|R~  
    PC| U]  
    % Check normalization: .oJs"=h:m  
    % -------------------- Sd3KY9,  
    if nargin==5 && ischar(nflag) _u`NIpXSP  
        isnorm = strcmpi(nflag,'norm'); FT1h\K|a  
        if ~isnorm 1`tE Hu.  
            error('zernfun:normalization','Unrecognized normalization flag.') hSZ0 }/  
        end ZD9UE3-  
    else &=sVq^d@qe  
        isnorm = false; x9#>0 4s  
    end 6 1= ?(Iw  
    'oZ/fUl|7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jhWNMu  
    % Compute the Zernike Polynomials _jw A_  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8+&] q#W3  
    No)v&P%  
    % Determine the required powers of r: 7L[HtwI  
    % ----------------------------------- wl{Fx+<^3  
    m_abs = abs(m); JTw'ecFev  
    rpowers = []; 62B` Z5j#  
    for j = 1:length(n) a2dlz@)J  
        rpowers = [rpowers m_abs(j):2:n(j)]; IED7v  
    end `eIX*R   
    rpowers = unique(rpowers); ZDZPJp,  
    YC:>)  
    % Pre-compute the values of r raised to the required powers, ,`/J1(\ nd  
    % and compile them in a matrix: 2&E1)^  
    % ----------------------------- qy`95^  
    if rpowers(1)==0 Mj&f7IUO  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); K?_4|  
        rpowern = cat(2,rpowern{:}); IB x?MU#.  
        rpowern = [ones(length_r,1) rpowern]; \ A\a=A[  
    else U9;C#9E  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _wWh7'u~G  
        rpowern = cat(2,rpowern{:}); ui4H(A'}  
    end 0@rrY  
    R1z\b~@"  
    % Compute the values of the polynomials: 9$,?Grw~  
    % -------------------------------------- Eb`U^*A  
    y = zeros(length_r,length(n)); 30Nya$$A=  
    for j = 1:length(n) 5=g{%X  
        s = 0:(n(j)-m_abs(j))/2; 4 uv'l3  
        pows = n(j):-2:m_abs(j); (=${@=!z  
        for k = length(s):-1:1 im^G{3z  
            p = (1-2*mod(s(k),2))* ... tr2@{xb  
                       prod(2:(n(j)-s(k)))/              ... #F5O>9hA  
                       prod(2:s(k))/                     ... jxL5L[  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &oevgG  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); $4`RJ{ZJw]  
            idx = (pows(k)==rpowers); WlVC0&  
            y(:,j) = y(:,j) + p*rpowern(:,idx); `j088<?j  
        end rMqWXGl`(  
         S%b7NK  
        if isnorm !!ZNemXct$  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -OZRSjmY  
        end b0]y$*{j  
    end 2.Ym  
    % END: Compute the Zernike Polynomials R|h9ilc  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >Qc0g(w  
    t &u,Od  
    % Compute the Zernike functions: {4&G\2<^^  
    % ------------------------------ F]3iL^v  
    idx_pos = m>0; |jW82L+!N%  
    idx_neg = m<0; pB{QO4q n  
    ,,oiL  
    z = y; m~\BkE/[l  
    if any(idx_pos) :|oH11 y  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i\G@kJNnF  
    end 7|3Z+#|T  
    if any(idx_neg) ecA[  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); KYVB=14  
    end 5aw#!K=J'  
    ;Gxp'y  
    % EOF zernfun
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) qev1bBW  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. D0=D8P}H:  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated A\lnH5A  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive +Tde#T&[  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, L.lmbxn  
    %   and THETA is a vector of angles.  R and THETA must have the same ; P I=jp  
    %   length.  The output Z is a matrix with one column for every P-value, |h(!CFR  
    %   and one row for every (R,THETA) pair. #ldNWwvRGj  
    % w``t"v4  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Zs e3e  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) S0 M-$  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) Jg3}U j2By  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 5NbI Vz  
    %   for all p. j/wG0~<kz  
    % x$I~y D  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Z T95g  
    %   Zernike functions (order N<=7).  In some disciplines it is Qs8iu`'  
    %   traditional to label the first 36 functions using a single mode R>BI;IcX  
    %   number P instead of separate numbers for the order N and azimuthal VR2BdfKU,  
    %   frequency M. F4>}mIA  
    % #Qc[W +%  
    %   Example: ,g@U *06  
    % vLJ<_&6  
    %       % Display the first 16 Zernike functions 8vz9o <I  
    %       x = -1:0.01:1; <^|8\<J  
    %       [X,Y] = meshgrid(x,x); C78YHjy  
    %       [theta,r] = cart2pol(X,Y); `,tv&siSA  
    %       idx = r<=1; ()v[@"J  
    %       p = 0:15; ${ad[hs  
    %       z = nan(size(X)); Pe!uk4}w  
    %       y = zernfun2(p,r(idx),theta(idx)); K&<bn22  
    %       figure('Units','normalized') [Nr6 qxWg  
    %       for k = 1:length(p) T/MbEqAf  
    %           z(idx) = y(:,k); :,y V?E6]  
    %           subplot(4,4,k) ' wvZnb  
    %           pcolor(x,x,z), shading interp 2sjV*\Udf  
    %           set(gca,'XTick',[],'YTick',[]) :)t1>y>3  
    %           axis square 1[D~Ee p  
    %           title(['Z_{' num2str(p(k)) '}']) 52/^>=t  
    %       end 8fTuae$^  
    % 0wB ?U~  
    %   See also ZERNPOL, ZERNFUN. Vd4x!Vk  
    FgrOZI;_  
    %   Paul Fricker 11/13/2006 f8+($Ys  
    ~EW (2B{u  
    %A`f>v.7 c  
    % Check and prepare the inputs: x&+/da-E/5  
    % ----------------------------- 0^*4LM|z  
    if min(size(p))~=1 3X89mIDr  
        error('zernfun2:Pvector','Input P must be vector.') Uc!} D  
    end fBS;~;l  
    $dFEC}1t  
    if any(p)>35 %L}9nc%~eP  
        error('zernfun2:P36', ... 25t2tj@S  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... X_Is#&6;  
               '(P = 0 to 35).']) ? Sj,HLo@U  
    end BC%t[H} >R  
    f}Eoc>n  
    % Get the order and frequency corresonding to the function number: acdaDY  
    % ---------------------------------------------------------------- ;t:B:4r(j  
    p = p(:); 8k2prv^  
    n = ceil((-3+sqrt(9+8*p))/2); ox{)O/aj  
    m = 2*p - n.*(n+2); i*#Gq6qZq  
    zK(9k0+s  
    % Pass the inputs to the function ZERNFUN: iyc}a6g  
    % ---------------------------------------- M,8a$Mdqh  
    switch nargin tcSn`+Bu_`  
        case 3 Z 3m5DK  
            z = zernfun(n,m,r,theta); \'&:6\-fw  
        case 4 A/lxXy}D  
            z = zernfun(n,m,r,theta,nflag); {kD|8["Ie'  
        otherwise `8\ _ ]w0  
            error('zernfun2:nargin','Incorrect number of inputs.') <QQgOaS`2  
    end ~#h@.yW^JN  
    A?ma5h  
    % 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) ?9mFI(r~  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. &"BmCDOq  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of R{NmWj['Mg  
    %   order N and frequency M, evaluated at R.  N is a vector of 4};iL)  
    %   positive integers (including 0), and M is a vector with the {oy(08 `6  
    %   same number of elements as N.  Each element k of M must be a F6dm_Oq&  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) QxwZ$?w%  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 1 9$ufod  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix sycN  
    %   with one column for every (N,M) pair, and one row for every z'O$[6m6  
    %   element in R. sEt5!&  
    % lj/ ?P9  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- %0YwaxXPn7  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Wy.2*+5FX0  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to HTao)`.  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 Q!7Er  
    %   for all [n,m]. dB ?+-aE  
    % 9`f]Rf"  
    %   The radial Zernike polynomials are the radial portion of the bU/5ug.  
    %   Zernike functions, which are an orthogonal basis on the unit 12gcma}  
    %   circle.  The series representation of the radial Zernike bLUn>ch  
    %   polynomials is ~e@ QJ=r  
    % n,hHh=.Fu  
    %          (n-m)/2 oZHsCQ%  
    %            __ @aN<nd`q)  
    %    m      \       s                                          n-2s .n[!3X|d  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r X3tpW`alo  
    %    n      s=0 [?r`8K2!,  
    % $xNM^O  
    %   The following table shows the first 12 polynomials. \CM/KrCR  
    % qXR>Z=K<  
    %       n    m    Zernike polynomial    Normalization fKY6stJE  
    %       --------------------------------------------- dms R>Q  
    %       0    0    1                        sqrt(2) C|5eV=f)P  
    %       1    1    r                           2 ` :eXXE  
    %       2    0    2*r^2 - 1                sqrt(6) dy-m9fc6%  
    %       2    2    r^2                      sqrt(6) "1$OPt5  
    %       3    1    3*r^3 - 2*r              sqrt(8) Q9tBHz  
    %       3    3    r^3                      sqrt(8) 51W\%aB  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) }i!hzkK#  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) YQ}Rg5 o  
    %       4    4    r^4                      sqrt(10) $`(}ygmP  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) "X4OUk  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) L (XGD  
    %       5    5    r^5                      sqrt(12) 'e_^s+l)a  
    %       --------------------------------------------- biKom|<nm  
    % lZ.x@hDS  
    %   Example: OE]z C  
    % ?wt%e;  
    %       % Display three example Zernike radial polynomials } 1^/[?  
    %       r = 0:0.01:1; \-a^8{.^E  
    %       n = [3 2 5]; jq yqOhb4  
    %       m = [1 2 1]; =hxj B*")  
    %       z = zernpol(n,m,r); N`1:U 4}  
    %       figure 8ME_O~,N  
    %       plot(r,z) *&9_+F8ly  
    %       grid on vQ>x5\r5O_  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 89*CoQ  
    % wkOo8@J\  
    %   See also ZERNFUN, ZERNFUN2. 8u Tq0d6(  
    }J73{  
    % A note on the algorithm. OJPx V~y  
    % ------------------------ U+>!DtOYK  
    % The radial Zernike polynomials are computed using the series CMB:%  
    % representation shown in the Help section above. For many special rCR?]1*Z  
    % functions, direct evaluation using the series representation can j9)P3=s  
    % produce poor numerical results (floating point errors), because ,V'+16xW  
    % the summation often involves computing small differences between hNgbHzW  
    % large successive terms in the series. (In such cases, the functions )8VrGg?  
    % are often evaluated using alternative methods such as recurrence EtvZk9d6h*  
    % relations: see the Legendre functions, for example). For the Zernike u&yAMWl  
    % polynomials, however, this problem does not arise, because the };6[Byf  
    % polynomials are evaluated over the finite domain r = (0,1), and [* ,k  
    % because the coefficients for a given polynomial are generally all f2ygN6(>  
    % of similar magnitude. ~Mbo`:>(4v  
    % :@x24wN/  
    % ZERNPOL has been written using a vectorized implementation: multiple = Ryh@X&  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] s\y+ xa:  
    % values can be passed as inputs) for a vector of points R.  To achieve 3>Y G  
    % this vectorization most efficiently, the algorithm in ZERNPOL " 2A`M~  
    % involves pre-determining all the powers p of R that are required to A]L;LkEM  
    % compute the outputs, and then compiling the {R^p} into a single Fka&\9i  
    % matrix.  This avoids any redundant computation of the R^p, and RAYDl=}  
    % minimizes the sizes of certain intermediate variables. JX8Hn |  
    % ;U=IbK*  
    %   Paul Fricker 11/13/2006 ts%XjCN[  
    9Jd{HI=  
    Sm-gi|A  
    % Check and prepare the inputs: 1Z%^U ?  
    % ----------------------------- d/5i4g[q  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) z +,l"#Vv  
        error('zernpol:NMvectors','N and M must be vectors.') 12qX[39/  
    end Gx /sJ(  
    `>KB8SY:qK  
    if length(n)~=length(m) cdiDfiE  
        error('zernpol:NMlength','N and M must be the same length.') .|W0B+Z8  
    end CeQL8yJ;  
    Ks'msSMC  
    n = n(:); GcN[bH(@  
    m = m(:); ,l/~epx4v)  
    length_n = length(n); 8 g0By;h;  
    WO$9Svh8  
    if any(mod(n-m,2)) ~2u~}v5m7  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [&12`!;j  
    end s |B  
    7i^7sT8t  
    if any(m<0) Ua0fs|t1v  
        error('zernpol:Mpositive','All M must be positive.') [ u7p:?WDW  
    end Wy1#K)LRb  
    _~~:@fy  
    if any(m>n) =nPIGI72VO  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 7Nx5n<  
    end >%Rb}Ki4  
    mHCp^g4Q  
    if any( r>1 | r<0 ) Mj&`Y gW5a  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') );EW(7KeL  
    end BkywYCWZ )  
    t8SvU  
    if ~any(size(r)==1) @CGci lS=  
        error('zernpol:Rvector','R must be a vector.') ab"6]%_  
    end 7|$cM7_r  
    ' cM2]<  
    r = r(:); tFlLKziU  
    length_r = length(r); '"u>;Bq  
    K @x4>9 3n  
    if nargin==4 iA*^`NMaT  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); /$ a>f>EJ  
        if ~isnorm bu[PQsT  
            error('zernpol:normalization','Unrecognized normalization flag.') _cPGS=Ew  
        end st~ 1[in  
    else \l)Jb*t  
        isnorm = false; abog\0  
    end Iw@ou  
    R(YhVW_l  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /mS|Byx  
    % Compute the Zernike Polynomials '+?L/|'  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GD*rTtDWn  
    3B*b d  
    % Determine the required powers of r: nE"##2X  
    % ----------------------------------- iYbp^iVg  
    rpowers = []; &lbZTY}  
    for j = 1:length(n) rq#8}T>  
        rpowers = [rpowers m(j):2:n(j)]; $y%X#:eLJ  
    end yg}zK>j^vC  
    rpowers = unique(rpowers); BhAWIH8@C  
    h?t#ABsVK  
    % Pre-compute the values of r raised to the required powers, R#"LP7\  
    % and compile them in a matrix: g2?kC^=z=  
    % ----------------------------- Ih Yso7g  
    if rpowers(1)==0 !4;A"B(  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0%x"Va~"z  
        rpowern = cat(2,rpowern{:}); U`)\|\NY  
        rpowern = [ones(length_r,1) rpowern]; qDSZ:36  
    else ,<Ag&*YE4  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P:lmQHls+  
        rpowern = cat(2,rpowern{:}); L@mNfLK  
    end l]g /rs  
    4o/}KUu(*  
    % Compute the values of the polynomials: IBP3  
    % -------------------------------------- JAt$WW{  
    z = zeros(length_r,length_n); 8x)&4o@  
    for j = 1:length_n tW^oa  
        s = 0:(n(j)-m(j))/2; xi+bBqg<.K  
        pows = n(j):-2:m(j); I, 7~D!4G  
        for k = length(s):-1:1 &^q!,7.J  
            p = (1-2*mod(s(k),2))* ... \,n|V3#G  
                       prod(2:(n(j)-s(k)))/          ... /z=xEnU#  
                       prod(2:s(k))/                 ... k4n 4 BL  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... 3/?^d;=  
                       prod(2:((n(j)+m(j))/2-s(k))); W;Pdbf"  
            idx = (pows(k)==rpowers); !O*'mX  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ~mSW.jy}=-  
        end 3t4i2]  
         X mmb^2I  
        if isnorm H[WsHq;T+9  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); <w,NMu"  
        end 95XQ?%  
    end o"kVA;5<G  
    {th=MldJ?  
    % 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)  XzB3Xs?W2  
    s{:Thgv,9  
    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。 a/n~#5-  
    sitgz)Ki^  
    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)