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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 (K2 p3M^  
    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! 5[@4($q8  
     
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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 T%ha2X=  
    function z = zernfun(n,m,r,theta,nflag) H p1cVs  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. fXL$CgXG\x  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N =JEnK_@?K\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !.F`8OD`u  
    %   unit circle.  N is a vector of positive integers (including 0), and id*UTY Tg  
    %   M is a vector with the same number of elements as N.  Each element n RXf\*"3  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) ,.E:mm  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, {)`5*sd  
    %   and THETA is a vector of angles.  R and THETA must have the same zf^!Zqn[8z  
    %   length.  The output Z is a matrix with one column for every (N,M) AU)Qk$c  
    %   pair, and one row for every (R,THETA) pair. Vg2s~ce{  
    % |>p\*Dl}H  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fOrqY,P'  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =:#$_qR  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral o6svSS  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, cDLS)  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized =`{!" 6a  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. h NP|  
    % siOeR@> X  
    %   The Zernike functions are an orthogonal basis on the unit circle. c?[A  
    %   They are used in disciplines such as astronomy, optics, and bu\,2t}B  
    %   optometry to describe functions on a circular domain. ]1gt|M^  
    % B9+oI c O  
    %   The following table lists the first 15 Zernike functions. Inr ~9hz  
    % "WK.sBFz4  
    %       n    m    Zernike function           Normalization jb77uH_  
    %       -------------------------------------------------- Th@L68  
    %       0    0    1                                 1 {KODwP'~  
    %       1    1    r * cos(theta)                    2 II),m8G  
    %       1   -1    r * sin(theta)                    2 ?2Bp^3ytJ  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) 3~M8.{ U#V  
    %       2    0    (2*r^2 - 1)                    sqrt(3) 3A'd7FJ0G  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) K\o!  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) jLcW;7OAC  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) %B#Ewt@[  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) dpTap<Noby  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) Nnx"b 5I}n  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) }1'C!]j  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w Gw}a[a  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) NjL,0Bp  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /&dC?bY  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) |L0s  
    %       -------------------------------------------------- ~D 5'O^  
    % b8T'DY;~  
    %   Example 1: ,]Hn*\@p[c  
    % AnIENJ  
    %       % Display the Zernike function Z(n=5,m=1) U9kt7#@FDK  
    %       x = -1:0.01:1; >b<br  
    %       [X,Y] = meshgrid(x,x); pH)V:BmJ  
    %       [theta,r] = cart2pol(X,Y); 2<U5d`  
    %       idx = r<=1; # |2w^Kn  
    %       z = nan(size(X)); 6rdm=8WFA  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); `/0X].s#o  
    %       figure .wYx_  
    %       pcolor(x,x,z), shading interp llQDZ}T  
    %       axis square, colorbar YAd.i@^  
    %       title('Zernike function Z_5^1(r,\theta)') [bE9Y;  
    % `W{Ye=|[d#  
    %   Example 2: O{LWQ"@y  
    % L +-B,466  
    %       % Display the first 10 Zernike functions O!uX:TE|Q  
    %       x = -1:0.01:1; o^_z+JFwb  
    %       [X,Y] = meshgrid(x,x); TQYud'u/  
    %       [theta,r] = cart2pol(X,Y); %vn rLt$  
    %       idx = r<=1; Hd6Qy {,*-  
    %       z = nan(size(X)); A*E$_N  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Jg |/*Or  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; q'{E $V)E  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; RIb< 7  
    %       y = zernfun(n,m,r(idx),theta(idx)); ;nSaZ$`5  
    %       figure('Units','normalized') .(nq"&u-*  
    %       for k = 1:10 v5 $"v?PT  
    %           z(idx) = y(:,k); L}x"U9'C  
    %           subplot(4,7,Nplot(k)) a&4>xZU #  
    %           pcolor(x,x,z), shading interp ef Ra|7!HK  
    %           set(gca,'XTick',[],'YTick',[]) naM4X@jl  
    %           axis square kLADd"C  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L 'H1\' o  
    %       end ,,b_x@y*  
    % T? _$  
    %   See also ZERNPOL, ZERNFUN2. 3|g'1X}  
    D)f hk!<  
    %   Paul Fricker 11/13/2006 q'd6\G0 }  
    f4]nz:2  
    a!xKS8-S==  
    % Check and prepare the inputs: aW$7:<A{  
    % ----------------------------- nBZqhtr  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) A0o6-M]'0  
        error('zernfun:NMvectors','N and M must be vectors.') Kx$?IxZ  
    end dt^yEapjM  
    B1J+`R3OX  
    if length(n)~=length(m) ~@MIG  
        error('zernfun:NMlength','N and M must be the same length.') Yq3(,  
    end w,9$*=k  
    p*n$iroy_{  
    n = n(:); 4|7L26,]5  
    m = m(:); 2u/(Q>#  
    if any(mod(n-m,2)) 3-~_F*%ST  
        error('zernfun:NMmultiplesof2', ... Fl^.J<Dz  
              'All N and M must differ by multiples of 2 (including 0).') s XRiUDP`  
    end ] QtGgWtC  
    +TA(crD  
    if any(m>n) UYGl  
        error('zernfun:MlessthanN', ... Xq+7l5LP  
              'Each M must be less than or equal to its corresponding N.') [t,grdw  
    end b]Oc6zR,,~  
    4- N>#  
    if any( r>1 | r<0 ) Q(E$;@   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') Su6ZO'[)  
    end hFyN|Dqhds  
    U7bG(?k)  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R~[ u|EC}  
        error('zernfun:RTHvector','R and THETA must be vectors.') Y=/HsG\W]  
    end "n:L<F,g  
    nakhepLN  
    r = r(:); D?8t'3no  
    theta = theta(:); UFC.!t-Z  
    length_r = length(r); &%C4rAd2  
    if length_r~=length(theta) >c8zMd  
        error('zernfun:RTHlength', ... yEzp+Ky  
              'The number of R- and THETA-values must be equal.') OCY7Bls4  
    end l?Bv9k.^?  
    kSoAnJ|  
    % Check normalization: _OHz6ag  
    % -------------------- g}L2\i688  
    if nargin==5 && ischar(nflag)  w~&bpCB!  
        isnorm = strcmpi(nflag,'norm'); 7Ja^d-F7  
        if ~isnorm O/iew3YF  
            error('zernfun:normalization','Unrecognized normalization flag.') L'z;*N3D  
        end *M6M'>Tin  
    else ?)5}v4b  
        isnorm = false; %ktU 51o  
    end (gs"2  
    z2wR]G5!  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nYTI\f/8v  
    % Compute the Zernike Polynomials nRb#M  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R8O<} >3a  
    RR*z3i`PP  
    % Determine the required powers of r: 'R,1Jmx  
    % ----------------------------------- w'?uJW  
    m_abs = abs(m); sW@4r/F>:D  
    rpowers = []; (*^_ wq-;  
    for j = 1:length(n) N;;!ObVHnP  
        rpowers = [rpowers m_abs(j):2:n(j)]; 2gg5:9  
    end eWW\m[k]}  
    rpowers = unique(rpowers); onHUi]yYu{  
    4}LGE>  
    % Pre-compute the values of r raised to the required powers, QJvA  
    % and compile them in a matrix: 5 S7\m5  
    % ----------------------------- x]Nq|XK  
    if rpowers(1)==0 #0hX)7(j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @1^iWM j  
        rpowern = cat(2,rpowern{:}); [[LCEw  
        rpowern = [ones(length_r,1) rpowern]; N}pE{~Y  
    else OB;AgE@  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);  UTHGjE  
        rpowern = cat(2,rpowern{:}); ^mkplp a  
    end d=Q0 /sI&  
    '~<D[](/F  
    % Compute the values of the polynomials: w3fi2B&q  
    % -------------------------------------- i *nNu-g  
    y = zeros(length_r,length(n)); 'FO^VJ;ha  
    for j = 1:length(n) V: 2|l!l*  
        s = 0:(n(j)-m_abs(j))/2; 6*tI~  
        pows = n(j):-2:m_abs(j); U3 ED3) D  
        for k = length(s):-1:1 US@ak4Y6Z  
            p = (1-2*mod(s(k),2))* ... QU8?/  
                       prod(2:(n(j)-s(k)))/              ... ^Me__Y  
                       prod(2:s(k))/                     ... $*`fn{2  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... }#a d  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); zl4Iq+5~6Q  
            idx = (pows(k)==rpowers); Ub4j3`  
            y(:,j) = y(:,j) + p*rpowern(:,idx); p@YU7_sF^!  
        end Nq9@^ E-{M  
         @]gP"Pp  
        if isnorm %h2U(=/:  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); pN1W|Wv2  
        end FgKDk!ci  
    end %dhnp9'  
    % END: Compute the Zernike Polynomials AdKv!Ta5b  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4B^f"6'  
    S^a")U4  
    % Compute the Zernike functions: Aum&U){yY  
    % ------------------------------ [;83 IoU}  
    idx_pos = m>0; bTb|@  
    idx_neg = m<0; &,3.V+Sz  
    gR?=z}`@p  
    z = y; 9p9:nx\  
    if any(idx_pos) D)K/zh)  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #zZQ@+5zw  
    end H+;>>|+:~  
    if any(idx_neg) yAW%y  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3K_J"B*7  
    end m!tB;:6  
    C8e{9CF  
    % 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) =j#uH`jgW  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 6H53FMqr  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ,KdD owc  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 0h _9  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?<%GY dus  
    %   and THETA is a vector of angles.  R and THETA must have the same 6`J*{%mP  
    %   length.  The output Z is a matrix with one column for every P-value, -2{NI.-Xd  
    %   and one row for every (R,THETA) pair. XBh0=E?qiS  
    % Uz} #.  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike b>er'U  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) .-awl1 W  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .!/DM-C  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 &'TZU"_  
    %   for all p. h.l^f>, /  
    % 6k|o<`~,  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 /q^)thJ~  
    %   Zernike functions (order N<=7).  In some disciplines it is g=XvqD<  
    %   traditional to label the first 36 functions using a single mode LPc)-t|p"  
    %   number P instead of separate numbers for the order N and azimuthal wqkD  
    %   frequency M. {^a"T'+  
    % TwH%P2)x  
    %   Example: A,Wwt [Qw  
    % !ow:P8K?  
    %       % Display the first 16 Zernike functions >B!E 6ah  
    %       x = -1:0.01:1; |-a5|3  
    %       [X,Y] = meshgrid(x,x); HIsIW%B  
    %       [theta,r] = cart2pol(X,Y); jhgS@g=@ZC  
    %       idx = r<=1; MxQhkY-=  
    %       p = 0:15; HkVnTC  
    %       z = nan(size(X)); U5\^[~vW  
    %       y = zernfun2(p,r(idx),theta(idx)); ^~9fQJNs  
    %       figure('Units','normalized') eyUguA<lK\  
    %       for k = 1:length(p) @a$_F3W  
    %           z(idx) = y(:,k); w$[&ejFb  
    %           subplot(4,4,k) YcOPqvQ  
    %           pcolor(x,x,z), shading interp 2FU+o\1 %  
    %           set(gca,'XTick',[],'YTick',[]) [% \>FT[  
    %           axis square RtO3!dGT.  
    %           title(['Z_{' num2str(p(k)) '}']) lR5[UKr  
    %       end 3`.*~qW  
    % IO3p&sJ/  
    %   See also ZERNPOL, ZERNFUN. }Z#KPI8\Q  
    b+Sq[  
    %   Paul Fricker 11/13/2006 v3PtiKS  
    js;p7wi  
    Zgy~Y0Di  
    % Check and prepare the inputs: MdXOH$ ps  
    % ----------------------------- VRU"2mQ.P6  
    if min(size(p))~=1 S52'!WTq  
        error('zernfun2:Pvector','Input P must be vector.') .8e]-^Z  
    end QOiPDu=8z  
    {*mf Is  
    if any(p)>35 KCP$i@Pjv  
        error('zernfun2:P36', ... G5*"P!@6  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ]8@s+ N  
               '(P = 0 to 35).']) VaP9&tWXj  
    end D+#OB|&Dn  
    I]Ev6>=;  
    % Get the order and frequency corresonding to the function number: + OKk~GYf  
    % ---------------------------------------------------------------- #O 2g]YH  
    p = p(:); yX 9 .yq  
    n = ceil((-3+sqrt(9+8*p))/2); I\e/ Bv^  
    m = 2*p - n.*(n+2); ^Gi9&fS,  
    q8A;%.ZLG  
    % Pass the inputs to the function ZERNFUN: *$e1Bv6 $  
    % ---------------------------------------- Db4(E*/pj!  
    switch nargin O|K-UTWH%  
        case 3 BOt1J_;(rO  
            z = zernfun(n,m,r,theta); `A\,$(q+  
        case 4 ]2E#P.-!b  
            z = zernfun(n,m,r,theta,nflag); %#t*3[  
        otherwise ?vt#M^Q   
            error('zernfun2:nargin','Incorrect number of inputs.') f/xQy}4+~E  
    end u00w'=pe)  
    M>qqe!c*  
    % 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) QmgO00{  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. ~ =GwNo_  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of [KQ#b  
    %   order N and frequency M, evaluated at R.  N is a vector of +Y?) ?  
    %   positive integers (including 0), and M is a vector with the ^F)t>K$0m  
    %   same number of elements as N.  Each element k of M must be a M ^ZEAZi  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) `cQAO1-5  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is S>Z07d6&  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix Q~R%|Q{&  
    %   with one column for every (N,M) pair, and one row for every %1mIngW=g  
    %   element in R. _V`F_C\\#  
    % Ec9%RAxl  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 9dVHh?E  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _-|/$ jZ  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to mzf~qV^T  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 hbdB67,  
    %   for all [n,m]. R ;k1(p  
    % #V{!|Y'  
    %   The radial Zernike polynomials are the radial portion of the 6E@TcN~ ,!  
    %   Zernike functions, which are an orthogonal basis on the unit R XN0v@V  
    %   circle.  The series representation of the radial Zernike buldA5*!o  
    %   polynomials is :F8h}\a*  
    % '4Drs}j5  
    %          (n-m)/2 G %A!yV  
    %            __ frokl5L@  
    %    m      \       s                                          n-2s `]`S"W7&  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r r^7eK)XA_  
    %    n      s=0 D&o ~4Qvc]  
    % HvR5-?qQ  
    %   The following table shows the first 12 polynomials. ?o1QjDG  
    % A vww @$  
    %       n    m    Zernike polynomial    Normalization Cxd^i  
    %       --------------------------------------------- uZM%F)  
    %       0    0    1                        sqrt(2) <a&w$Zc/  
    %       1    1    r                           2 5O(U1 *  
    %       2    0    2*r^2 - 1                sqrt(6) C{:U<q  
    %       2    2    r^2                      sqrt(6) 5dX /<  
    %       3    1    3*r^3 - 2*r              sqrt(8) EfB.K}b^  
    %       3    3    r^3                      sqrt(8) y@@h)P#  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) XRmE  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) : HM~!7e  
    %       4    4    r^4                      sqrt(10) >Hu3Guik]  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) C#V_Gb  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) \[G"/]J  
    %       5    5    r^5                      sqrt(12) ?bH`  
    %       --------------------------------------------- g"~`\ xhx  
    % AJ>$`=  
    %   Example: O5MV&Zb(  
    % )<%CI#s#  
    %       % Display three example Zernike radial polynomials ef7BG(  
    %       r = 0:0.01:1; ;VzdlCZ@  
    %       n = [3 2 5]; jM-7  
    %       m = [1 2 1]; foUBMl  
    %       z = zernpol(n,m,r); O1@3V/.Wu  
    %       figure 4k9$' k  
    %       plot(r,z) HVdB*QEH  
    %       grid on t33\f<e  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') PM i.)%++  
    % # 0d7  
    %   See also ZERNFUN, ZERNFUN2.  HzL~B#  
    u+y3( 0  
    % A note on the algorithm. ;?q-]J?  
    % ------------------------ l, 9r d[  
    % The radial Zernike polynomials are computed using the series 2Lytk OMf  
    % representation shown in the Help section above. For many special KRX\<@  
    % functions, direct evaluation using the series representation can Y[|9 +T  
    % produce poor numerical results (floating point errors), because Aj]/A  
    % the summation often involves computing small differences between k0&FUO  
    % large successive terms in the series. (In such cases, the functions  od$$g(  
    % are often evaluated using alternative methods such as recurrence p-m\0tQ  
    % relations: see the Legendre functions, for example). For the Zernike  Ci 'V  
    % polynomials, however, this problem does not arise, because the o=RxQk1N  
    % polynomials are evaluated over the finite domain r = (0,1), and QL @SE@"  
    % because the coefficients for a given polynomial are generally all ^F qs,^~W  
    % of similar magnitude. aTfc>A;  
    %  #]QS   
    % ZERNPOL has been written using a vectorized implementation: multiple /eT9W[a  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] )6BySk  
    % values can be passed as inputs) for a vector of points R.  To achieve sOVpDtZ]LR  
    % this vectorization most efficiently, the algorithm in ZERNPOL qo}yEl1  
    % involves pre-determining all the powers p of R that are required to p I.~j]*:{  
    % compute the outputs, and then compiling the {R^p} into a single :`K2?;DC8  
    % matrix.  This avoids any redundant computation of the R^p, and M1]w0~G  
    % minimizes the sizes of certain intermediate variables. i03=Af3  
    % ~;-2eKw  
    %   Paul Fricker 11/13/2006 nltOX@P-  
    j >`FZKxp  
    8QMMKO ui\  
    % Check and prepare the inputs: A~ v[6*~>  
    % ----------------------------- 8#R%jjr%T  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) t<"`gM^|  
        error('zernpol:NMvectors','N and M must be vectors.') A k~|r#@  
    end qY!LzKM0  
    ;dtA-EfOZ  
    if length(n)~=length(m) ]<ay_w;  
        error('zernpol:NMlength','N and M must be the same length.') 3UF^Ff<wo  
    end Q-A_8  
    m<LzgX  
    n = n(:); xnJ#}-.7  
    m = m(:); &xvNR=K[`  
    length_n = length(n); Pqj\vdzx  
    p.<d+S<  
    if any(mod(n-m,2)) e A3 NyL  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') bMsThoePT  
    end T24$lhM  
    ' R2*3<  
    if any(m<0) <IBUl}|\  
        error('zernpol:Mpositive','All M must be positive.') Dw 5Ze  
    end <WbO&;%  
    i-#Dc (9  
    if any(m>n) VZe'6?#  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.')  %{UW!/  
    end Q/J<$W*,  
    qG2P?DR  
    if any( r>1 | r<0 )  ;\f0II3  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') JaR!9GVN7  
    end KZ1m 2R}'  
    o.Bbb=*rZ  
    if ~any(size(r)==1) 0'Qvis[kt  
        error('zernpol:Rvector','R must be a vector.') ~eS/gF?  
    end zG c ]*R  
    !HtW~8|:  
    r = r(:); ]zj&U#{  
    length_r = length(r); GO*D4<#u  
    :T>OJ"p  
    if nargin==4 &LG|YvMY6  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); >]A#_p  
        if ~isnorm xX0 wn?,~  
            error('zernpol:normalization','Unrecognized normalization flag.') Jh36NE8r  
        end HX,i{aWWy  
    else {%RwZ'  
        isnorm = false; |9BX  ~`{  
    end -G#m'W&  
    K@oyvJ$  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ] yWywa\  
    % Compute the Zernike Polynomials G8MLg#  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7AqbfLO  
    /n:Q>8^n'W  
    % Determine the required powers of r: g&Uu~;jq]  
    % ----------------------------------- e2>AL  
    rpowers = []; yigq#h^  
    for j = 1:length(n) ^ 4p$@5zH  
        rpowers = [rpowers m(j):2:n(j)]; !w0=&/Y{R  
    end ] r%fAm j  
    rpowers = unique(rpowers); dB QCr{7  
    6\v4#  
    % Pre-compute the values of r raised to the required powers, bj_/  
    % and compile them in a matrix: +D[C.is>]}  
    % ----------------------------- Mhb~wDQl  
    if rpowers(1)==0 ,Ij/ ^EC}  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); gR_Exs'K  
        rpowern = cat(2,rpowern{:}); *+00  
        rpowern = [ones(length_r,1) rpowern]; W59xe&l  
    else l<(jm{q?u  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); rt?*eC1b+Z  
        rpowern = cat(2,rpowern{:}); CL?=j| Ea  
    end T[g(S0dz  
    h&!$ `)   
    % Compute the values of the polynomials: U'Y,T$Q  
    % -------------------------------------- 7 9k+R9m  
    z = zeros(length_r,length_n); /)dyAX(  
    for j = 1:length_n m,6[;  
        s = 0:(n(j)-m(j))/2; -D1 A  
        pows = n(j):-2:m(j); b&d4(dk  
        for k = length(s):-1:1 ^gY'^2bzxu  
            p = (1-2*mod(s(k),2))* ... Df]*S  
                       prod(2:(n(j)-s(k)))/          ... 0,8RA_Ca}  
                       prod(2:s(k))/                 ... Qw"%Xk  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... _fHj8- s/  
                       prod(2:((n(j)+m(j))/2-s(k))); & IsPqO  
            idx = (pows(k)==rpowers); gO@LJ  
            z(:,j) = z(:,j) + p*rpowern(:,idx); M6V^ur 1  
        end x cZF_elt7  
         q$`>[&I~)  
        if isnorm 3;!!`R>e  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); w6-<HPW<S  
        end [L ' >  
    end WD*z..`  
    W A*1_  
    % 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)  2( m#WK7>F  
    N,3iSH=cN[  
    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。 [+!&iN  
    8)ng> l  
    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)