切换到宽版
  • 广告投放
  • 稿件投递
  • 繁體中文
    • 11150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 )wwQv2E  
    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! c(<,qWH  
     
    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 )-9G*3  
    function z = zernfun(n,m,r,theta,nflag) V X<ZB +R  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. "9 -duDg  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +OF(CcA^  
    %   and angular frequency M, evaluated at positions (R,THETA) on the Es kh=xA {  
    %   unit circle.  N is a vector of positive integers (including 0), and %TUljX K}  
    %   M is a vector with the same number of elements as N.  Each element FG~p _[K  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) m%$z&<!  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ;C%D+"l1g  
    %   and THETA is a vector of angles.  R and THETA must have the same R.R(|!w>  
    %   length.  The output Z is a matrix with one column for every (N,M) $.}fL;BzVz  
    %   pair, and one row for every (R,THETA) pair. <v"C`cga  
    % ~u&3Ki*x  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w.cQ|_  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'f<0&Ci8  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral WIo^=?%  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, L;xc,"\3  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized QJo)  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. &G@*/2A  
    % ^6+P&MxM  
    %   The Zernike functions are an orthogonal basis on the unit circle. jz|zq\Eek  
    %   They are used in disciplines such as astronomy, optics, and 9oP8| <+  
    %   optometry to describe functions on a circular domain. vZC2F  
    % A==P?,RG  
    %   The following table lists the first 15 Zernike functions. +V&b<y;?>  
    % v'.?:S&m  
    %       n    m    Zernike function           Normalization GD|uU  
    %       -------------------------------------------------- A0M)*9 f  
    %       0    0    1                                 1 3skq%;%Wsk  
    %       1    1    r * cos(theta)                    2 (^eSm]<  
    %       1   -1    r * sin(theta)                    2 {t[j>_MYw  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) O!sZMGF$p  
    %       2    0    (2*r^2 - 1)                    sqrt(3) Rcf_31 L  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) fk P@e3  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) :9c QK]O6  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 'R~x.NM  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) >E?626*  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) C$)#s{*  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) qSMST mnQ  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $dci?7q  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) IQdiVj  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L1.<LB^4'  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ;,Sl+)@h  
    %       -------------------------------------------------- v%V$@MF  
    % g`gH]W FcG  
    %   Example 1: 8:-[wl/@  
    % Yv9(8  
    %       % Display the Zernike function Z(n=5,m=1) bR49(K$~  
    %       x = -1:0.01:1; %|o4 U0c  
    %       [X,Y] = meshgrid(x,x); 6ndt1W z  
    %       [theta,r] = cart2pol(X,Y); eUVE8pZl  
    %       idx = r<=1; +|Xx=1_?BK  
    %       z = nan(size(X)); V?HC\F-  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); _i:yI-jA  
    %       figure 3Zdkf]Gh  
    %       pcolor(x,x,z), shading interp j* g5f  
    %       axis square, colorbar SwG:?T!"}  
    %       title('Zernike function Z_5^1(r,\theta)')  HlPf   
    % s{KwO+UW  
    %   Example 2: v%= G~kF}[  
    % 0NZg[>H  
    %       % Display the first 10 Zernike functions \Q?r+VZ  
    %       x = -1:0.01:1; ?^2(|t9KU  
    %       [X,Y] = meshgrid(x,x); +l2{EiQw  
    %       [theta,r] = cart2pol(X,Y); (m=-oQ&Ro  
    %       idx = r<=1; Gu|}ax"  
    %       z = nan(size(X)); yu<sd}@  
    %       n = [0  1  1  2  2  2  3  3  3  3]; ,K6s'3O(LW  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3];  _*9eAeJ  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; A/W0O;*q  
    %       y = zernfun(n,m,r(idx),theta(idx)); mE%H5&VSI  
    %       figure('Units','normalized') {*`qL0u]^  
    %       for k = 1:10 %gJf&A  
    %           z(idx) = y(:,k); zy8W8h(?  
    %           subplot(4,7,Nplot(k)) ^4O1:_|G  
    %           pcolor(x,x,z), shading interp L/"XIMI*Xg  
    %           set(gca,'XTick',[],'YTick',[]) y0M^oLx  
    %           axis square d5\w'@Di  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) K]oFV   
    %       end @.a[2,o_  
    % O]~cv^  
    %   See also ZERNPOL, ZERNFUN2. w=s:e M@  
    {XC# -3O  
    %   Paul Fricker 11/13/2006 60*2k  
    n87B[R  
    Nqk*3Q"f  
    % Check and prepare the inputs: cc*A/lD  
    % ----------------------------- 4H]Go~<  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VjBV2x  
        error('zernfun:NMvectors','N and M must be vectors.') >jME == U0  
    end OSK 3X Qc  
    s|dcO  
    if length(n)~=length(m) >>Z.]  
        error('zernfun:NMlength','N and M must be the same length.') LS+ _y <v=  
    end &#F>%~<or  
    .v9#|d d+  
    n = n(:); G}&B{Ir  
    m = m(:); 4)!aYvaER  
    if any(mod(n-m,2)) 0g,;Yzm  
        error('zernfun:NMmultiplesof2', ... [DC8X P5 <  
              'All N and M must differ by multiples of 2 (including 0).') HbX>::J8  
    end yJ c#y   
    t Q385en  
    if any(m>n) 1\=)b< y  
        error('zernfun:MlessthanN', ... <[@AMdS  
              'Each M must be less than or equal to its corresponding N.') 3J32W@}.K  
    end IKMkpX!]  
    7](,/MeGG  
    if any( r>1 | r<0 ) 7;jwKA;k  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') *8M 0h9S$  
    end `|P fa  
    T ]hVO'z  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) g'ha7~w(p  
        error('zernfun:RTHvector','R and THETA must be vectors.') T@GT=1E)  
    end c3W9"  
    [fiB!G ]?  
    r = r(:); V##=-KZ  
    theta = theta(:); pwtB{6)VH{  
    length_r = length(r); Aw~ =U!  
    if length_r~=length(theta) o|YY,G=C  
        error('zernfun:RTHlength', ... ig5 d-A  
              'The number of R- and THETA-values must be equal.') c>#T\AEkF  
    end ?`bi8 Ck  
    ~[l6;bn  
    % Check normalization: zePVB -@u  
    % -------------------- HT0VdvLw  
    if nargin==5 && ischar(nflag) 4$#nciAe  
        isnorm = strcmpi(nflag,'norm'); S.pL^Ru  
        if ~isnorm +!h~T5Ck  
            error('zernfun:normalization','Unrecognized normalization flag.')  S {oW  
        end XP'<\  
    else r"sK@  
        isnorm = false; ?f f!(U  
    end NF8'O  
    M3P\1  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r6S-G{o  
    % Compute the Zernike Polynomials }K':tX?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]CHO5'%,$  
    ySAkj-< /P  
    % Determine the required powers of r: (k/[/`3ST  
    % ----------------------------------- N3O3V5':!  
    m_abs = abs(m); UKX9C"-5v  
    rpowers = []; d5Hp&tm  
    for j = 1:length(n) sA$x2[*O  
        rpowers = [rpowers m_abs(j):2:n(j)]; TgMa! Vz  
    end HHVCw7r0  
    rpowers = unique(rpowers); :0@R(ct;>  
    ko<u0SjF)u  
    % Pre-compute the values of r raised to the required powers, KmS$CFsGL  
    % and compile them in a matrix: ^/@Z4(E  
    % ----------------------------- j3>0oe!  
    if rpowers(1)==0 .TZ0F xW  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &O8vI ,M  
        rpowern = cat(2,rpowern{:}); )aSj!X'`;  
        rpowern = [ones(length_r,1) rpowern]; >f+qImH  
    else  dpG l  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &! i'Q;q  
        rpowern = cat(2,rpowern{:}); 1g_p`(  
    end (5CdA1|  
    :&SvjJR  
    % Compute the values of the polynomials: h0**[LDH  
    % -------------------------------------- Ao?y2 [sE  
    y = zeros(length_r,length(n)); QAGR\~  
    for j = 1:length(n) /B"FGa04p(  
        s = 0:(n(j)-m_abs(j))/2; @}9*rWJIE  
        pows = n(j):-2:m_abs(j); c{.y9P6  
        for k = length(s):-1:1 cft/;A u{  
            p = (1-2*mod(s(k),2))* ... D+4oV6}~  
                       prod(2:(n(j)-s(k)))/              ... P+ ejyl,  
                       prod(2:s(k))/                     ... . -ihxEbzr  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M2Q*#U>6r  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); CE,0@%6F*  
            idx = (pows(k)==rpowers); CgT5sk}  
            y(:,j) = y(:,j) + p*rpowern(:,idx); LV}Z[\?   
        end i ZU 1w7Z  
         %u=b_4K"j  
        if isnorm T-MC|>pv  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); aI.5w9  
        end zX]4DLl,  
    end gvzBV +3'  
    % END: Compute the Zernike Polynomials oS>VN<  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~ eNKu  
    d.e_\]o<@  
    % Compute the Zernike functions: y26?>.!  
    % ------------------------------ ~K$dQb])  
    idx_pos = m>0; ]g] ]\hS  
    idx_neg = m<0; \9t/*%:  
    k'6x_ G  
    z = y; hqDnmzG  
    if any(idx_pos) {!0f.nv  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i<\WRzVT  
    end $I0&I[_LzK  
    if any(idx_neg) :,Zs {\oI3  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); z:1"d R   
    end (e"\%p`  
    )L+>^cJI<  
    % EOF zernfun
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) =D3K})&  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. ;$Pjl8\  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated FZBdQhYF  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive JZup} {a  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, vqhu%ZyP  
    %   and THETA is a vector of angles.  R and THETA must have the same <Z j>}  
    %   length.  The output Z is a matrix with one column for every P-value, u>\u}c  
    %   and one row for every (R,THETA) pair. (jI_Dk;  
    % 230ijq3Y G  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike a{ p1Yy-]  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) wbQs>pc  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) G2 0   
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 f0@4 >\g  
    %   for all p. Uz_OUTFM  
    % [;Y*f,UG_-  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ' e:rL.  
    %   Zernike functions (order N<=7).  In some disciplines it is Y52TC@'  
    %   traditional to label the first 36 functions using a single mode s}lp^Uh=  
    %   number P instead of separate numbers for the order N and azimuthal HVz|*?&6  
    %   frequency M. g/+|gHq^  
    % YH,u*.I^/  
    %   Example: 2TccIv  
    % e_+SBN1`P&  
    %       % Display the first 16 Zernike functions jZgCDA8Mr!  
    %       x = -1:0.01:1; *@eZt*_  
    %       [X,Y] = meshgrid(x,x); VQ7A"&hh  
    %       [theta,r] = cart2pol(X,Y); Yln[ZmK9g  
    %       idx = r<=1; -uei nd]  
    %       p = 0:15; K3^2;j1F Q  
    %       z = nan(size(X)); #_kV o3  
    %       y = zernfun2(p,r(idx),theta(idx)); 3~EPX`#[W  
    %       figure('Units','normalized') LI9 Uc\  
    %       for k = 1:length(p) 8 3Tv-X  
    %           z(idx) = y(:,k); VmON}bb[zz  
    %           subplot(4,4,k) ,5}")T["u  
    %           pcolor(x,x,z), shading interp )}to7r7 `  
    %           set(gca,'XTick',[],'YTick',[]) ==npFjB  
    %           axis square U>hpYqf_  
    %           title(['Z_{' num2str(p(k)) '}']) LMRq.wxbbB  
    %       end UM}MK  
    % F;8Uvj  
    %   See also ZERNPOL, ZERNFUN. ]sD lZJX<M  
    f {j`d&|  
    %   Paul Fricker 11/13/2006 (R]b'3,E$  
    m't8\fo^w  
    c7@[RG !  
    % Check and prepare the inputs: dO!B=/  
    % ----------------------------- ~COd(,ul  
    if min(size(p))~=1 8,L)=3m-  
        error('zernfun2:Pvector','Input P must be vector.') -Mzm~@_s]  
    end E`b<^l`  
    ,56objaE  
    if any(p)>35 hQ}y(2A.XI  
        error('zernfun2:P36', ... 'MQJt2QU9{  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... /Jc54d  
               '(P = 0 to 35).']) d`3>@*NR<  
    end )E c /5=A  
    | $D`*  
    % Get the order and frequency corresonding to the function number: c JOT{  
    % ---------------------------------------------------------------- %I#[k4,N  
    p = p(:); }K|40oO5  
    n = ceil((-3+sqrt(9+8*p))/2); |3C5"R3ZGO  
    m = 2*p - n.*(n+2); 'wjL7P I  
    fl+2 '~  
    % Pass the inputs to the function ZERNFUN:  zt2#6v  
    % ---------------------------------------- >k8FUf(c  
    switch nargin jg3T1ROL  
        case 3 -] `OaL!  
            z = zernfun(n,m,r,theta); Vi>`g{\  
        case 4 Cznp(z  
            z = zernfun(n,m,r,theta,nflag); 78kT}kgW  
        otherwise ]5+<Rqdbg  
            error('zernfun2:nargin','Incorrect number of inputs.') #\&64  
    end &5n0J  
    i\C~]K~O!  
    % EOF zernfun2
    离线niuhelen
    发帖
    19
    光币
    28
    光券
    0
    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) G$E+qk nJL  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. Q6hWHfS  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )BmO[AiOM  
    %   order N and frequency M, evaluated at R.  N is a vector of jbTsrj"g  
    %   positive integers (including 0), and M is a vector with the ^ ^k]2oG  
    %   same number of elements as N.  Each element k of M must be a ~JTp8E9kw  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) a1g aB:w5n  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is en-HX3'  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix cc:,,T /i  
    %   with one column for every (N,M) pair, and one row for every lH"4"r  
    %   element in R. c3C<P  
    % d7qYz7=d  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- *V?p&/>MT  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is R_B`dP<"~Y  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to DgKe!w$  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 h5}:>yc  
    %   for all [n,m]. >.r> aH  
    % ab^>_xD<  
    %   The radial Zernike polynomials are the radial portion of the NH;.!x q:  
    %   Zernike functions, which are an orthogonal basis on the unit ':DLv{R  
    %   circle.  The series representation of the radial Zernike 9;tY'32/  
    %   polynomials is A3Oe=rB  
    % /s "Lsbe  
    %          (n-m)/2 Q3 yW#eD  
    %            __ >M^4p   
    %    m      \       s                                          n-2s */y (~O6  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r p #Y2v  
    %    n      s=0 |6GDIoZ  
    % x#}{z1op9  
    %   The following table shows the first 12 polynomials.  _!E)a  
    % um*!+Q  
    %       n    m    Zernike polynomial    Normalization ' ;3#t(J;  
    %       --------------------------------------------- o>Q=V 0?  
    %       0    0    1                        sqrt(2) EJ=ud9  
    %       1    1    r                           2 agj_l}=gO  
    %       2    0    2*r^2 - 1                sqrt(6) Mh8s@g  
    %       2    2    r^2                      sqrt(6) < m/@_"  
    %       3    1    3*r^3 - 2*r              sqrt(8) h+j{;evN  
    %       3    3    r^3                      sqrt(8) t>JPK_b0  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) >wf.C%  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) cc{^0JT  
    %       4    4    r^4                      sqrt(10) `} S; _g!  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Lb}$)AcC  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) pd}Cg'}X  
    %       5    5    r^5                      sqrt(12) XxLauJP K  
    %       --------------------------------------------- N^%7  
    % \JbOT%1  
    %   Example: 1`)e}p&  
    % ][1 iKT  
    %       % Display three example Zernike radial polynomials R_2T"  
    %       r = 0:0.01:1;  |: ,i  
    %       n = [3 2 5];  &sg~owz  
    %       m = [1 2 1]; 0YO/G1O&  
    %       z = zernpol(n,m,r); %JPBD]&M  
    %       figure ~zhP[qA})  
    %       plot(r,z) >8.o  
    %       grid on {3eg4j.Z  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') }fh<LCwTi  
    % ,dXJCX8so  
    %   See also ZERNFUN, ZERNFUN2. &9F(C R  
    2k M;7:  
    % A note on the algorithm. D8f4X w}=  
    % ------------------------ T,/:5L9  
    % The radial Zernike polynomials are computed using the series kV:T2}]|H  
    % representation shown in the Help section above. For many special S(PU"}vZy  
    % functions, direct evaluation using the series representation can lw=!v%L  
    % produce poor numerical results (floating point errors), because tA{h x -  
    % the summation often involves computing small differences between .> 5[;  
    % large successive terms in the series. (In such cases, the functions 'nN'bVl/  
    % are often evaluated using alternative methods such as recurrence nR8r$2B+t  
    % relations: see the Legendre functions, for example). For the Zernike U5ME`lN*`  
    % polynomials, however, this problem does not arise, because the QE+HL8c^s  
    % polynomials are evaluated over the finite domain r = (0,1), and SaFNPnk=  
    % because the coefficients for a given polynomial are generally all 1"f)\FPGe  
    % of similar magnitude. 3gs!ojG  
    % y&6 pc   
    % ZERNPOL has been written using a vectorized implementation: multiple `rb}"V+  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] #AF.1;(k  
    % values can be passed as inputs) for a vector of points R.  To achieve #j4RX:T*[  
    % this vectorization most efficiently, the algorithm in ZERNPOL +*Zjo&pc  
    % involves pre-determining all the powers p of R that are required to Iad&Z8E  
    % compute the outputs, and then compiling the {R^p} into a single  w\y)  
    % matrix.  This avoids any redundant computation of the R^p, and "=,IbC  
    % minimizes the sizes of certain intermediate variables. X.eocy  
    % Y|cj&<o  
    %   Paul Fricker 11/13/2006 FK~*X3'  
    Y(F>;/AA  
    Ogu";p(  
    % Check and prepare the inputs: ffR<G&"n~b  
    % ----------------------------- ?H!QV;ku  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2?@Ozr2Uh  
        error('zernpol:NMvectors','N and M must be vectors.') L~E|c/  
    end _*++xF1  
    ou=33}uO  
    if length(n)~=length(m) a/xnf<(H  
        error('zernpol:NMlength','N and M must be the same length.') ''H;/&nDX  
    end /kAbGjp0  
    x9\]C' *sO  
    n = n(:); =F09@C,  
    m = m(:); _b9>ZF~  
    length_n = length(n); ;s?,QvE{r#  
    yOO@v6jO)  
    if any(mod(n-m,2)) "'~&D/7  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') 7)*q@  
    end )yUSuK(Vu  
    v+C%t!dx  
    if any(m<0) RV),E:?  
        error('zernpol:Mpositive','All M must be positive.') LerRrN}~  
    end C(n_*8{  
    O% 8>siU  
    if any(m>n) RG:_:%@%}  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') TGGbO:s3  
    end g`dAj4B  
    auAwZi/  
    if any( r>1 | r<0 ) UN|S!&C$  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') F#a'N c9  
    end c~u91h?  
    dg#w!etB  
    if ~any(size(r)==1) ]v#T9QQN  
        error('zernpol:Rvector','R must be a vector.') k,61Va  
    end 465?,EpS  
    4e?MthJ>  
    r = r(:); .V@3zzv\  
    length_r = length(r); P52qtN<  
    _8e0vi!~2  
    if nargin==4 A().1h1_k  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); V W(+sSQ  
        if ~isnorm O+t'E9Fa  
            error('zernpol:normalization','Unrecognized normalization flag.') 7n7UL0Oc1  
        end 2E0oLl[  
    else uOPLJ?%  
        isnorm = false; uQg&]bSv  
    end yT[)V[}  
    @b{$s  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0+NGFX \p  
    % Compute the Zernike Polynomials cUTG! P\R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {T3~js   
    {dwlW`{  
    % Determine the required powers of r: .9q`Tf  
    % ----------------------------------- B?9"Ztb  
    rpowers = []; )H+p6<  
    for j = 1:length(n) =g?k`v p  
        rpowers = [rpowers m(j):2:n(j)]; aa}U87]k  
    end Z<#h$XUA  
    rpowers = unique(rpowers); VYkUUp  
    bT`et*]  
    % Pre-compute the values of r raised to the required powers, DJ`xCs!R  
    % and compile them in a matrix: #!t6'*  
    % ----------------------------- EAoq2_(`a  
    if rpowers(1)==0 2:&L|;  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); :A5h<=[  
        rpowern = cat(2,rpowern{:}); cC9haxW  
        rpowern = [ones(length_r,1) rpowern]; Gm=e;X;r  
    else 8Fv4\dr  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !UHX? <3r  
        rpowern = cat(2,rpowern{:}); Tj6kCB  
    end XQZiJ %'  
    Y^eF(  
    % Compute the values of the polynomials: p MR4]G  
    % -------------------------------------- C)ic;!$Qhb  
    z = zeros(length_r,length_n); X?Or.  
    for j = 1:length_n lD$\t/8B  
        s = 0:(n(j)-m(j))/2; 8d(l)[GZt  
        pows = n(j):-2:m(j); );{76  
        for k = length(s):-1:1 czH# ~  
            p = (1-2*mod(s(k),2))* ... Px&)kEQ  
                       prod(2:(n(j)-s(k)))/          ... fzUG1|$e  
                       prod(2:s(k))/                 ... u56F;y  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... qUk-BG8^  
                       prod(2:((n(j)+m(j))/2-s(k))); UQjYWXvi  
            idx = (pows(k)==rpowers); G-'CjiMu  
            z(:,j) = z(:,j) + p*rpowern(:,idx); @#yl_r%  
        end 63kZ#5g(Dw  
         PoD/i@  
        if isnorm ;f /2u  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); xW58B  
        end v.c.5@%%o  
    end 9r ](/"=f  
    gps.  
    % 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)  -] G=Q1 1  
    ,o}CBB! 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。 #q5tG\gnM  
    V;hO1xfR3&  
    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)