切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11094阅读
    • 9回复

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

    上一主题 下一主题
    离线niuhelen
     
    发帖
    19
    光币
    28
    光券
    0
    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 WL'P)lI5  
    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! Y'ow  
     
    分享到
    离线phility
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    69
    光币
    11
    光券
    0
    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
    发帖
    19
    光币
    28
    光券
    0
    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 QC9eUYe  
    function z = zernfun(n,m,r,theta,nflag) LL3#5AA"k|  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "\3B^ e,  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N K~| 4[\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the \Z+z?K O  
    %   unit circle.  N is a vector of positive integers (including 0), and i*@< y/&'  
    %   M is a vector with the same number of elements as N.  Each element p{j.KI s7  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) Ro9tZ'N!S  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Yj6*NZ*  
    %   and THETA is a vector of angles.  R and THETA must have the same &FF"nE*  
    %   length.  The output Z is a matrix with one column for every (N,M) xo7Kn+ Kl  
    %   pair, and one row for every (R,THETA) pair. "$2 y-|  
    % "-+\R}q$  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 'LO^<  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2(#7[mgPI  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral  ~Hr}]  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -i%e!DgH  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized v%iof1 T'  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. i:OK8Q{VI  
    % <gQIq{B?  
    %   The Zernike functions are an orthogonal basis on the unit circle. i? K|TC`  
    %   They are used in disciplines such as astronomy, optics, and RT.D"WvT  
    %   optometry to describe functions on a circular domain. pQtJc*[!  
    % \cUC9/ b  
    %   The following table lists the first 15 Zernike functions. `s8{C b=}1  
    % -T[lx\}  
    %       n    m    Zernike function           Normalization p IU&^yX>  
    %       -------------------------------------------------- }wHW7SJ  
    %       0    0    1                                 1 *fn*h[pV&  
    %       1    1    r * cos(theta)                    2 k{Me[B  
    %       1   -1    r * sin(theta)                    2 <c qbUL  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Cc%LztP>  
    %       2    0    (2*r^2 - 1)                    sqrt(3) f;xkT  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) NqDHCI  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) h3z{(-~y  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) \ytJ=0r  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) enSXP~9w  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) `O[};3O&  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) eFL=G%  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o'f?YZ$.  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) e=.njMqW5  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p%*%n3bw  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) qT`k*i?  
    %       -------------------------------------------------- 5 ({t4dm  
    % |>tKq;/  
    %   Example 1: :qIXY/  
    % t 0|!(3  
    %       % Display the Zernike function Z(n=5,m=1) TTt#a6eJ  
    %       x = -1:0.01:1; b#hDHSdZ,  
    %       [X,Y] = meshgrid(x,x); @]-jl}:]  
    %       [theta,r] = cart2pol(X,Y); Ct=- 4  
    %       idx = r<=1; )Ccq4i  
    %       z = nan(size(X)); L&%s[  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); <oX7P69  
    %       figure 6T#+V37  
    %       pcolor(x,x,z), shading interp X  .5aMm  
    %       axis square, colorbar HLZ;8/|48m  
    %       title('Zernike function Z_5^1(r,\theta)') <\pfIJr$  
    % oWC@w  
    %   Example 2: j&Wl0  
    % T3pmVl  
    %       % Display the first 10 Zernike functions ,H19`;Q  
    %       x = -1:0.01:1; ?`#/ 8PN  
    %       [X,Y] = meshgrid(x,x); s8#X3Rp  
    %       [theta,r] = cart2pol(X,Y); }t%!9hr5D  
    %       idx = r<=1; HAJ7m!P  
    %       z = nan(size(X)); 2g>SHS@1>  
    %       n = [0  1  1  2  2  2  3  3  3  3]; dOoKLry  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; x`dHJq`_g  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; Ts ^"xlK  
    %       y = zernfun(n,m,r(idx),theta(idx)); PX n;C/  
    %       figure('Units','normalized') g;8jK 8 Kh  
    %       for k = 1:10 j"|=C$Kn/  
    %           z(idx) = y(:,k); Qi LEL  
    %           subplot(4,7,Nplot(k)) c(n&A~*AJ%  
    %           pcolor(x,x,z), shading interp |<.lW  
    %           set(gca,'XTick',[],'YTick',[]) (Xq)py9  
    %           axis square yLdVd P  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I9 mvt e  
    %       end q&/Yg,p\  
    % C~N/A73gF  
    %   See also ZERNPOL, ZERNFUN2. k=B] &F  
    ghX|3lI\q  
    %   Paul Fricker 11/13/2006 we;G]`@?  
    !2'jrJGc  
    ;ml)l~~YU  
    % Check and prepare the inputs: gUpb4uN  
    % ----------------------------- " 9^j.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?F6pEt4  
        error('zernfun:NMvectors','N and M must be vectors.') w - Pk7I  
    end j|/]#@Yr  
    ;M_o)OS3  
    if length(n)~=length(m) @1 U&UH  
        error('zernfun:NMlength','N and M must be the same length.') ywb4LKD  
    end \yd s5g!:  
    YE=q:Bv  
    n = n(:); eXK o.JL  
    m = m(:); fVt9X*xK S  
    if any(mod(n-m,2)) niqN{  
        error('zernfun:NMmultiplesof2', ... 8&Oa_{1+Q  
              'All N and M must differ by multiples of 2 (including 0).') #ceaZn|@m  
    end awOd_![c'  
    Yb /i{@AJ  
    if any(m>n) qnoNT%xazo  
        error('zernfun:MlessthanN', ... 05spovO/'  
              'Each M must be less than or equal to its corresponding N.') B';6r4I-  
    end F@*+{1R  
    1XnZy5fEo  
    if any( r>1 | r<0 ) ^ mS o1?<  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') RgGyoZ  
    end Ojt`^r!V  
    Oer^Rk  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5e}A@GyC  
        error('zernfun:RTHvector','R and THETA must be vectors.') S=nP[s  
    end u&9 r2R959  
    OI?K/rn  
    r = r(:); hBFP1u/E'  
    theta = theta(:); G <uyin>  
    length_r = length(r); UH"#2< |b  
    if length_r~=length(theta) 8?i7U<CB  
        error('zernfun:RTHlength', ... ]a! xUg!S  
              'The number of R- and THETA-values must be equal.') PNA\ TXT  
    end d5>H3D{49  
    ,i0b)=!o  
    % Check normalization: g(Io/hyj  
    % -------------------- !TP@- X;  
    if nargin==5 && ischar(nflag) qBQ`~4s  
        isnorm = strcmpi(nflag,'norm'); H> '>3]G  
        if ~isnorm fsEzpUY:{W  
            error('zernfun:normalization','Unrecognized normalization flag.') Fk6x<^Q<w  
        end 1t Jg#/?  
    else shH~4<15  
        isnorm = false; s (0*  
    end gxT4PQDy  
    Hi yc#-4  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qf| U0  
    % Compute the Zernike Polynomials Maqf[ Vky  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0}:2Q#  
    x%<  
    % Determine the required powers of r: -3&G"hfK  
    % ----------------------------------- +@Ad1fJi  
    m_abs = abs(m); bC^(U`y32  
    rpowers = []; `Rd m-[&  
    for j = 1:length(n) a|BcnYN  
        rpowers = [rpowers m_abs(j):2:n(j)]; e*  
    end ur\qOX|{  
    rpowers = unique(rpowers); tj*y)28-  
    LrCk*@  
    % Pre-compute the values of r raised to the required powers, ;c -3g]  
    % and compile them in a matrix: }6-ZE9H-v  
    % ----------------------------- \@~UDP]7  
    if rpowers(1)==0 ,WQ^tI=O  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1mSaS4!"B  
        rpowern = cat(2,rpowern{:}); Y=*P 8pg  
        rpowern = [ones(length_r,1) rpowern]; O%f8I'u$  
    else Y e+Ay  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )B!d,HKt;  
        rpowern = cat(2,rpowern{:}); -#29xRPk  
    end zTD@  
    nd{R 9B  
    % Compute the values of the polynomials: 3_`szl-  
    % -------------------------------------- Y& ] 8 {  
    y = zeros(length_r,length(n)); tJ=di5&  
    for j = 1:length(n) O}#yijU3e  
        s = 0:(n(j)-m_abs(j))/2; \Xt) E[  
        pows = n(j):-2:m_abs(j); 8@M'[jT  
        for k = length(s):-1:1 (D{Ys'{q  
            p = (1-2*mod(s(k),2))* ... fMeZ]rb  
                       prod(2:(n(j)-s(k)))/              ... *mBJ? { !  
                       prod(2:s(k))/                     ... }~o ikN:  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... z]Acs  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); OK`Z@X_,bW  
            idx = (pows(k)==rpowers); rwpgBl  
            y(:,j) = y(:,j) + p*rpowern(:,idx); ., :uZyG  
        end 1]\TI7/ n  
         ?z"KnR+?Q  
        if isnorm ~F#A Pt  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zfjTQMaxh  
        end FBsn;,3<W  
    end A1*4*  
    % END: Compute the Zernike Polynomials el'j&I  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x.EgTvA&d  
    MB* u-N0v  
    % Compute the Zernike functions: 8mgQu]>  
    % ------------------------------ jNy?[ )  
    idx_pos = m>0; *=vlqpG  
    idx_neg = m<0;  q{X T  
    `)[dVfxA  
    z = y; M^ 5e~y  
    if any(idx_pos) V:\]cGA{  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0yHjrxc$  
    end KzkgWMM  
    if any(idx_neg) 55hyV{L%  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p`GWhI?  
    end 6;JP76PD  
    8D2yR#3  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) &Y=.D:z<  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. M*H< n*  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated #7\b\~5  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive jI Z+d;1  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, }Eb]9c\  
    %   and THETA is a vector of angles.  R and THETA must have the same ?C~X@sq  
    %   length.  The output Z is a matrix with one column for every P-value, -? Tz.y&  
    %   and one row for every (R,THETA) pair. {WKOJG+.  
    % 5&G 5eA  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike bHJoEYY^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) |f3U%2@  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 4$F:NW,v:)  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 u x#. :C|  
    %   for all p. `1$y(w]  
    % XW^8A 77H  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 :Dt\:`(r'  
    %   Zernike functions (order N<=7).  In some disciplines it is lc" qqt  
    %   traditional to label the first 36 functions using a single mode _l<| 1nH  
    %   number P instead of separate numbers for the order N and azimuthal }ymc5-  
    %   frequency M. =Iy/cHK  
    % Yg$@Wb6  
    %   Example: fMyE&#}z  
    % Ou? r {$(b  
    %       % Display the first 16 Zernike functions [pr 9 $Jr  
    %       x = -1:0.01:1; 6mi$.' qP  
    %       [X,Y] = meshgrid(x,x); D7M0NEY  
    %       [theta,r] = cart2pol(X,Y); E3LBPXK  
    %       idx = r<=1; 70duk:Ri0  
    %       p = 0:15; ~c!Rx'  
    %       z = nan(size(X)); !8we8)7  
    %       y = zernfun2(p,r(idx),theta(idx)); jk K#e$7  
    %       figure('Units','normalized') NoJUx['6  
    %       for k = 1:length(p) -J{Dxz  
    %           z(idx) = y(:,k); 2ve lH;  
    %           subplot(4,4,k) ^v ]UcnB0  
    %           pcolor(x,x,z), shading interp FPvuzBJ  
    %           set(gca,'XTick',[],'YTick',[]) hx*HY%\P  
    %           axis square ZGA)r0] P`  
    %           title(['Z_{' num2str(p(k)) '}']) ]bs+:  
    %       end z~BD(FDI  
    % MRjH40" 2  
    %   See also ZERNPOL, ZERNFUN. G(:s-x ig6  
    fS5GICx8R  
    %   Paul Fricker 11/13/2006 W\&WS"=~  
    >g>f;\mD7$  
    UaH26fWs  
    % Check and prepare the inputs: Jq=00fcT+  
    % ----------------------------- XyvZ&d6(d  
    if min(size(p))~=1 8$2l^  
        error('zernfun2:Pvector','Input P must be vector.') XC*uz  
    end \R6;Fef  
    ls[Ls  
    if any(p)>35 nu;} S!J  
        error('zernfun2:P36', ... c_@XQ&DC`  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... _L?v6MTj  
               '(P = 0 to 35).']) ,AdusM  
    end IUluJ.sXIf  
    tn"Y9 k|  
    % Get the order and frequency corresonding to the function number: 4$0jz'  
    % ---------------------------------------------------------------- ybD{4&ZE  
    p = p(:); A6{t%k~F  
    n = ceil((-3+sqrt(9+8*p))/2); bHhC56[M  
    m = 2*p - n.*(n+2); ML=hKwCA  
    b}ySZlmy  
    % Pass the inputs to the function ZERNFUN: *Te4U5F  
    % ---------------------------------------- c'4>D,?1  
    switch nargin mtSNl|O&{  
        case 3 1$:{{%  
            z = zernfun(n,m,r,theta); z1Bj_u{  
        case 4 k)H[XpM  
            z = zernfun(n,m,r,theta,nflag); ,H.(\p_N  
        otherwise 844tXMtPB\  
            error('zernfun2:nargin','Incorrect number of inputs.') B6tcKh9d,  
    end j[$B\H  
    ?;0nJf  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) Z]7;u>2  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. h}anTFKP  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of $L<_uqSk  
    %   order N and frequency M, evaluated at R.  N is a vector of 9Sx<tj_4P{  
    %   positive integers (including 0), and M is a vector with the _e:5XQ  
    %   same number of elements as N.  Each element k of M must be a j,|1y5f  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 4i[v ew  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is <\}Y@g8  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix .UT,lqEkv  
    %   with one column for every (N,M) pair, and one row for every E1l\~%A  
    %   element in R. `L"p)5H  
    % m]-v IUpb  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- X5L(_0?F1  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 7/^TwNsv  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 0XQ".:+h  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 r3c\;Ra7  
    %   for all [n,m]. r7Q:l ?F2  
    % o/  x5  
    %   The radial Zernike polynomials are the radial portion of the A<YZBR_  
    %   Zernike functions, which are an orthogonal basis on the unit D)O6| DiO  
    %   circle.  The series representation of the radial Zernike 7/D9n9F  
    %   polynomials is SQ^^1.V&/Y  
    % 9aF..  
    %          (n-m)/2 s!j(nUd/  
    %            __ +QXYU8bYZ  
    %    m      \       s                                          n-2s H4y1Hpa,  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r \[IdR^<YM  
    %    n      s=0 ir@N>_  
    % XftJ=  *  
    %   The following table shows the first 12 polynomials. A=qW]Im  
    % _~w V{ yp  
    %       n    m    Zernike polynomial    Normalization OO !S w  
    %       --------------------------------------------- d,oOn.n&  
    %       0    0    1                        sqrt(2) :d% -,v  
    %       1    1    r                           2 tRUsZl  
    %       2    0    2*r^2 - 1                sqrt(6) hBfzU\*0H  
    %       2    2    r^2                      sqrt(6) 8Snq75Q<   
    %       3    1    3*r^3 - 2*r              sqrt(8) ek{PA!9Sk  
    %       3    3    r^3                      sqrt(8) %8} ksl07  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) LG&Q>pt.  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) $ vw}p.  
    %       4    4    r^4                      sqrt(10) ,I2re G  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) :WfB!4%!  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) [BZ(p  
    %       5    5    r^5                      sqrt(12) ]!tYrSM!  
    %       --------------------------------------------- E!}-qbH^  
    % C>\!'^u1  
    %   Example: p=`x  
    % L1Cn  
    %       % Display three example Zernike radial polynomials !{]v='   
    %       r = 0:0.01:1; d"d)<f   
    %       n = [3 2 5]; 9Pob|UA  
    %       m = [1 2 1]; <k-@R!K~JC  
    %       z = zernpol(n,m,r); qT<qu(V:  
    %       figure $NGtxZp  
    %       plot(r,z) l LD)i J1  
    %       grid on 0p>:rU~  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ^0ZKHR(}e  
    % HyX4ob[X  
    %   See also ZERNFUN, ZERNFUN2. t!=~5YgKs  
    |7'yk__m  
    % A note on the algorithm. $L#Z?76v  
    % ------------------------ E=1/  
    % The radial Zernike polynomials are computed using the series zWmo OnK  
    % representation shown in the Help section above. For many special D 917[ <$  
    % functions, direct evaluation using the series representation can q/2K=BOh  
    % produce poor numerical results (floating point errors), because !K^kKP*l  
    % the summation often involves computing small differences between ;AL@<,8  
    % large successive terms in the series. (In such cases, the functions -Ib+/'  
    % are often evaluated using alternative methods such as recurrence Uo[5V|>X6  
    % relations: see the Legendre functions, for example). For the Zernike -TU{r_!Z(  
    % polynomials, however, this problem does not arise, because the H'h4@S  
    % polynomials are evaluated over the finite domain r = (0,1), and ]BQWA  
    % because the coefficients for a given polynomial are generally all ^1Zq0  
    % of similar magnitude. p:Ld)U*  
    % seV;f^-hR  
    % ZERNPOL has been written using a vectorized implementation: multiple C(|T/rQ-  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] ^Lv ^W  
    % values can be passed as inputs) for a vector of points R.  To achieve d>"$^${  
    % this vectorization most efficiently, the algorithm in ZERNPOL ~lalc ^  
    % involves pre-determining all the powers p of R that are required to !lN a`  
    % compute the outputs, and then compiling the {R^p} into a single g d}TTe  
    % matrix.  This avoids any redundant computation of the R^p, and 7F9g:r/^  
    % minimizes the sizes of certain intermediate variables. gS<{ekN  
    % R EH&kcn  
    %   Paul Fricker 11/13/2006 L z>{FOR  
    `~+a=Q  
    L+ETMk0  
    % Check and prepare the inputs: pQMpkAX  
    % ----------------------------- JX@6Sg<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 19-yM`O  
        error('zernpol:NMvectors','N and M must be vectors.') GoVPo'  
    end -"dy z(  
    4k2c mM$  
    if length(n)~=length(m) K#C56k q&  
        error('zernpol:NMlength','N and M must be the same length.') iN/!k.ybW}  
    end HYYx*CJ)  
    @?cXa: tX  
    n = n(:); ~Ow23N  
    m = m(:); AFB 7s z  
    length_n = length(n); *0@; kD=  
    A8Z?[,Mq!  
    if any(mod(n-m,2)) E?h2e~ ,]  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ,, #rv-*  
    end lGHu@(n<  
    V #\ZS{'J  
    if any(m<0) [W\atmd"  
        error('zernpol:Mpositive','All M must be positive.') d8 Nh0!  
    end pW^ ?g|_}  
    DoB3_=yJ+  
    if any(m>n) :!YJ3:\  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #\ S$$gP  
    end {,C8}8 a W  
    +P)[|y +e  
    if any( r>1 | r<0 ) QZa#i L  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') Y {|~A  
    end [W;dguh  
    Jas|P}{=fT  
    if ~any(size(r)==1) z,x"vK(  
        error('zernpol:Rvector','R must be a vector.') Rf0\CEc  
    end #5:A?aj  
    lJY=*KB(6  
    r = r(:); =RE_Urt:  
    length_r = length(r); R$&&kmJ  
    #|1QA3KzO  
    if nargin==4 =X5&au o  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ~ 2oP,  
        if ~isnorm @ZPTf>J}  
            error('zernpol:normalization','Unrecognized normalization flag.') D!T4k]^  
        end Zy3&Zt  
    else Y[]+C8"O  
        isnorm = false; .%b_3s".  
    end ~#km0<r?  
    i[^lJ)[>N  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U5$DJ5>8  
    % Compute the Zernike Polynomials GJ_)Cl+5E  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RGuHXf  
    [ .uaO  
    % Determine the required powers of r: >MY.Fr#.m  
    % ----------------------------------- +5|nCp6||j  
    rpowers = []; D2 cIVx3:(  
    for j = 1:length(n) 2(J tD  
        rpowers = [rpowers m(j):2:n(j)]; Jl4XE%0  
    end 4 Wd5Goe:  
    rpowers = unique(rpowers); Q~!hr0 ZR  
    T`{MQ:s  
    % Pre-compute the values of r raised to the required powers, |(v=1#i  
    % and compile them in a matrix: pyJOEL]1F  
    % ----------------------------- FS+^r\)  
    if rpowers(1)==0 vK7,O%!S  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); LVl0:!>~  
        rpowern = cat(2,rpowern{:}); yzR=:0J  
        rpowern = [ones(length_r,1) rpowern]; Hf!4(\yN  
    else Zw\V}uXI?  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D\*_ulc]  
        rpowern = cat(2,rpowern{:}); R:^?6f<Z}  
    end <FT\u{9$  
    B^Mtj5Oc  
    % Compute the values of the polynomials: wSF#;lqd  
    % -------------------------------------- R+hS;F nh%  
    z = zeros(length_r,length_n); lfeWtzOf  
    for j = 1:length_n oySM?ZE  
        s = 0:(n(j)-m(j))/2; Z9~Wlt'?  
        pows = n(j):-2:m(j); )nxIxr0d-  
        for k = length(s):-1:1 P]{.e UB@c  
            p = (1-2*mod(s(k),2))* ... j|dzd<kE6  
                       prod(2:(n(j)-s(k)))/          ... ^uEl QI  
                       prod(2:s(k))/                 ... gc[J.[  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... tvxcd*{  
                       prod(2:((n(j)+m(j))/2-s(k))); 6YGr"Kj &  
            idx = (pows(k)==rpowers); ;*H~Yb0  
            z(:,j) = z(:,j) + p*rpowern(:,idx); E'6P>6l5  
        end >&Q. .`q  
         Ao0PFY  
        if isnorm &YKzK)@  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Q9zpX{JT  
        end _cN)q  
    end :"IH*7xp  
    k T>}(G||  
    % EOF zernpol
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    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
    发帖
    59
    光币
    0
    光券
    0
    只看该作者 7楼 发表于: 2012-09-28
    我也正在找啊
    离线guapiqlh
    发帖
    856
    光币
    846
    光券
    0
    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
    发帖
    4352
    光币
    5478
    光券
    1
    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)  /D~:Ufw  
    yl*S|= 8;k  
    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。 zuOIos  
    &c'unKH  
    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)