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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 ./-JbW  
    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! .V'V:;BE%  
     
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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 ^+70<#Xc  
    function z = zernfun(n,m,r,theta,nflag) yYJY;".H  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. HaNboYW_K  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N YhKZ|@  
    %   and angular frequency M, evaluated at positions (R,THETA) on the y&T&1o  
    %   unit circle.  N is a vector of positive integers (including 0), and ]n1dp2aH  
    %   M is a vector with the same number of elements as N.  Each element mPZGA\  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) c$E)P$<j  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, SqPtWEq@P  
    %   and THETA is a vector of angles.  R and THETA must have the same &rq{v!=7  
    %   length.  The output Z is a matrix with one column for every (N,M) P1kB>" bR  
    %   pair, and one row for every (R,THETA) pair. A/*%J74v  
    % #~ v4caNx  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XH4d<?qu  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PK6iY7Qp)  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral KpZ:Nh$  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, <EX7WA  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized Z)< wv&K  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %FkLQ+v/<  
    % .=R lOK  
    %   The Zernike functions are an orthogonal basis on the unit circle. "l~Ci7& !a  
    %   They are used in disciplines such as astronomy, optics, and 6o&ZIYJ9k  
    %   optometry to describe functions on a circular domain. q%3<Juq~$  
    % =C7 khE  
    %   The following table lists the first 15 Zernike functions. #XIc "L)c  
    % O_,O,1  
    %       n    m    Zernike function           Normalization GY!C|7kN  
    %       -------------------------------------------------- +=@^i'  
    %       0    0    1                                 1 R'K/t|MC  
    %       1    1    r * cos(theta)                    2 &V=7D#L  
    %       1   -1    r * sin(theta)                    2 1TJ0D_,  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) `x8B n"  
    %       2    0    (2*r^2 - 1)                    sqrt(3) G$WOzY(  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) a=]W zlz  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) t1]6(@mj5  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) *7gT}O;p 5  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) )$M,Ul  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) l[h'6+o  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) r#876.JK  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d.F)9h]XHO  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |H)cuZ  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >>'C :7+Y  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) -E>)j\{PX7  
    %       -------------------------------------------------- [[L-j q.'  
    % fv'4f$U  
    %   Example 1: *:"^[Ckc  
    % >%%=0!,yX  
    %       % Display the Zernike function Z(n=5,m=1) gSi5u# }J  
    %       x = -1:0.01:1; 70gg4BS  
    %       [X,Y] = meshgrid(x,x); _9If/RD  
    %       [theta,r] = cart2pol(X,Y); |7F*MP  
    %       idx = r<=1; I."p  
    %       z = nan(size(X)); H[&@}v,L  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); t8J/\f=  
    %       figure a+z2Zd!u\x  
    %       pcolor(x,x,z), shading interp 7nE"F!d+0  
    %       axis square, colorbar 5D`26dB2  
    %       title('Zernike function Z_5^1(r,\theta)') @PkJY  
    % |m>}%{  
    %   Example 2: ;IP~Tb]&  
    % 8n)WW$  
    %       % Display the first 10 Zernike functions &y. dmW  
    %       x = -1:0.01:1; o#hI5  
    %       [X,Y] = meshgrid(x,x); <e"J4gZf&  
    %       [theta,r] = cart2pol(X,Y); a5c'V   
    %       idx = r<=1; 2W$lQ;iO  
    %       z = nan(size(X)); q?  z>  
    %       n = [0  1  1  2  2  2  3  3  3  3]; s;.=5wcvi?  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !C&%T]  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nB@UKX  
    %       y = zernfun(n,m,r(idx),theta(idx)); $[CA&Y.  
    %       figure('Units','normalized') %efGt6&  
    %       for k = 1:10 LJuW${Y  
    %           z(idx) = y(:,k); fEqC] *s  
    %           subplot(4,7,Nplot(k)) ZXXiL#^  
    %           pcolor(x,x,z), shading interp 8I#D`yVKc  
    %           set(gca,'XTick',[],'YTick',[]) W'$kZ/%[  
    %           axis square HYClm|   
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) i57( $1.  
    %       end g:~+P e  
    % 3oBC   
    %   See also ZERNPOL, ZERNFUN2. sBW3{uK  
    -Zy)5NB-tZ  
    %   Paul Fricker 11/13/2006 Jq1 n0O  
    @EZ>f5IO+  
    d<T%`:s<  
    % Check and prepare the inputs: `iYc<N`  
    % ----------------------------- 0D3OE.$0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3E|;r _; 8  
        error('zernfun:NMvectors','N and M must be vectors.') UzQ$B>f  
    end W<'<'z5  
    &?<AwtNN  
    if length(n)~=length(m) 0X"\ a'M_  
        error('zernfun:NMlength','N and M must be the same length.')  )U98  
    end lo:~aJ8  
    KTmaglgp  
    n = n(:); iJnh$jo  
    m = m(:); TmP8 q  
    if any(mod(n-m,2)) i?>Hr|  
        error('zernfun:NMmultiplesof2', ... %C *^:\y  
              'All N and M must differ by multiples of 2 (including 0).') mK\aI  
    end e-6(F4  
    .ZX2^)`XD  
    if any(m>n) n2'|.y}Um:  
        error('zernfun:MlessthanN', ... Vyt E  
              'Each M must be less than or equal to its corresponding N.') n7iE8SK|k  
    end E+{5-[Zc*$  
    d]7*mzw^j  
    if any( r>1 | r<0 ) h$&rE@N|  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') l2/ @<0P  
    end P(_(w 9  
    -Enbcz(B  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,y.3Fe  
        error('zernfun:RTHvector','R and THETA must be vectors.') yQ-hnlzn~  
    end cjT[P"5$  
    /djACA  
    r = r(:); ,"H?hFQ  
    theta = theta(:); ^x3EotQ\  
    length_r = length(r); AU`OESSI  
    if length_r~=length(theta) 4*8&[b  
        error('zernfun:RTHlength', ... yWuIu>VJ  
              'The number of R- and THETA-values must be equal.') B$7[8h  
    end {PmzkT}LF  
    :uvc\|:s  
    % Check normalization: F@^N|;_2  
    % -------------------- FO^24p  
    if nargin==5 && ischar(nflag) XGk}e4;_  
        isnorm = strcmpi(nflag,'norm'); ]Zv ,  
        if ~isnorm cG(0q[  
            error('zernfun:normalization','Unrecognized normalization flag.') x!+Z{x   
        end Wa, 7P2r  
    else pn*d[M|k  
        isnorm = false; _LsYMUe  
    end 3_U\VGm  
    4k*qVOBa6R  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -x?Hj/  
    % Compute the Zernike Polynomials Hn^sW LT  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hg&u0AQ2  
    l#ygb|=x  
    % Determine the required powers of r: k(9s+0qe  
    % ----------------------------------- >3$uu+p1F  
    m_abs = abs(m); ~Jxlj(" 0(  
    rpowers = []; |VYr=hjo  
    for j = 1:length(n) K*:Im #Q  
        rpowers = [rpowers m_abs(j):2:n(j)]; H<z30r/-w  
    end Gl"wEL*  
    rpowers = unique(rpowers); QRiF!D)Nk  
    Q'C 4pn@  
    % Pre-compute the values of r raised to the required powers, ZmP1C`>  
    % and compile them in a matrix: $~ VcQ  
    % ----------------------------- D:6N9POB  
    if rpowers(1)==0 M;PlSb  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6Ok,_ !  
        rpowern = cat(2,rpowern{:}); I*9Gb$]=  
        rpowern = [ones(length_r,1) rpowern]; Tz2x9b\82  
    else *Ji9%IA  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2X^iV09  
        rpowern = cat(2,rpowern{:}); /t5g"n3  
    end YpiRF+G  
    Uv'uqt  
    % Compute the values of the polynomials: wvX"D0eVn  
    % -------------------------------------- H! #5!m&  
    y = zeros(length_r,length(n)); CP@o,v-  
    for j = 1:length(n) %Au T8  
        s = 0:(n(j)-m_abs(j))/2; +O,V6XRr  
        pows = n(j):-2:m_abs(j); yq!CWXZ2  
        for k = length(s):-1:1 i(z+a6^@|  
            p = (1-2*mod(s(k),2))* ... 35}P0+  
                       prod(2:(n(j)-s(k)))/              ... a0)vvo=bz  
                       prod(2:s(k))/                     ... EO"=\C,  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Jfe~ ,cI  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); PQQgDtiH  
            idx = (pows(k)==rpowers); Y'?Izn b  
            y(:,j) = y(:,j) + p*rpowern(:,idx); [KD}U-(Wg  
        end tI `w;e%HN  
         s^obJl3  
        if isnorm Hc>([?P%t  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E=A/4p6\$  
        end +<H !3sW  
    end z=u~]:.1O  
    % END: Compute the Zernike Polynomials gca|?tt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Z )`inw  
     I8:"h  
    % Compute the Zernike functions: 3SmqXPOw  
    % ------------------------------ IM$ 'J  
    idx_pos = m>0; z/pDOP Ku  
    idx_neg = m<0; d"z *Nb  
    h)"'YzCt  
    z = y; `Uu^I   
    if any(idx_pos) Dy98[cL  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pVdhj^n  
    end fQ^h{n  
    if any(idx_neg) Ua}g  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -=@K %\\~5  
    end "sC$%D<oc  
    _P>1`IR  
    % 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) ER0TY,  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. Z#H@BWN7  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated AEBw#v!,o  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive #Lu4OSM+  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, e,PQ)1  
    %   and THETA is a vector of angles.  R and THETA must have the same b=6ZdN1  
    %   length.  The output Z is a matrix with one column for every P-value, c }g$1of87  
    %   and one row for every (R,THETA) pair. A{xSbbDk  
    % Rt*-#`I $  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike :/n ?4K^  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) LX&=uv%-^  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) qg/Y;tGSx  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 gEX:S(1 QP  
    %   for all p. 8Xt=eL/P  
    % W+fkWq7`Xx  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 }s8*QfK>  
    %   Zernike functions (order N<=7).  In some disciplines it is Z3&XTsq  
    %   traditional to label the first 36 functions using a single mode M)bC%(xJ  
    %   number P instead of separate numbers for the order N and azimuthal ',v0vyO8  
    %   frequency M. s2,`eV  
    % #l8K8GLuf  
    %   Example: i[V,IP +  
    % lk5_s@V l  
    %       % Display the first 16 Zernike functions 0~LnnD N  
    %       x = -1:0.01:1; bYRQI=gW':  
    %       [X,Y] = meshgrid(x,x); r-Xjy*T  
    %       [theta,r] = cart2pol(X,Y); oW` *FD  
    %       idx = r<=1; nJnO/~|  
    %       p = 0:15; -M(58/y  
    %       z = nan(size(X)); @5uyUSt]  
    %       y = zernfun2(p,r(idx),theta(idx)); GLoL4el  
    %       figure('Units','normalized') i1kh@s~8UC  
    %       for k = 1:length(p) 0:nt#n~_  
    %           z(idx) = y(:,k); DY%T`}  
    %           subplot(4,4,k) [m@e^6F0U  
    %           pcolor(x,x,z), shading interp Fp* &os  
    %           set(gca,'XTick',[],'YTick',[]) la6e`  
    %           axis square WoN]eO  
    %           title(['Z_{' num2str(p(k)) '}']) eFeCS{LV+  
    %       end ]Y/pSwnV  
    % dRarNW  
    %   See also ZERNPOL, ZERNFUN. M\x7=*\  
    l7FZ;%&  
    %   Paul Fricker 11/13/2006 *HfW(C$  
    G/^5P5y%@  
    <{P^W;N7  
    % Check and prepare the inputs: c'VtRE# z~  
    % ----------------------------- s;=J'x)~%  
    if min(size(p))~=1 9=sMKc%!-  
        error('zernfun2:Pvector','Input P must be vector.') Y?q*hS0!H  
    end ce-m)o/  
    ESXU, qK]v  
    if any(p)>35 1[,#@!k@  
        error('zernfun2:P36', ... c"_H%x<[  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... aF_ZV bS  
               '(P = 0 to 35).']) KfN`ZZ<  
    end R&d_ WB4w  
    s`7 _J9  
    % Get the order and frequency corresonding to the function number: tue%L]hc  
    % ---------------------------------------------------------------- -t706(#k  
    p = p(:); g#bfY=C  
    n = ceil((-3+sqrt(9+8*p))/2); ]f+D& qZ B  
    m = 2*p - n.*(n+2); Xj&~N;Ysb  
    .>\>F{#~  
    % Pass the inputs to the function ZERNFUN: =FC;d[U  
    % ---------------------------------------- 0DP%44Cv9  
    switch nargin q=L* 99S  
        case 3 c&iK+qvh{  
            z = zernfun(n,m,r,theta); wr6xuoH  
        case 4 `'{%szmD  
            z = zernfun(n,m,r,theta,nflag); 5d>YE  
        otherwise .$T:n[@  
            error('zernfun2:nargin','Incorrect number of inputs.') u{dN>}{  
    end =<27qj  
    kA9 X!)2w  
    % EOF zernfun2
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) 88L bO(q\d  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. fbbl92p  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of )-:f;#xJ  
    %   order N and frequency M, evaluated at R.  N is a vector of XQoT},C  
    %   positive integers (including 0), and M is a vector with the M:q ;z(  
    %   same number of elements as N.  Each element k of M must be a f0`rJ?us  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) y5tAp  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is oL/^[TXjH  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix aGkVC*T  
    %   with one column for every (N,M) pair, and one row for every Ac(Vw%  
    %   element in R. Kgio}y  
    % HC`3AQ12!&  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- \EfwS% P  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is >Ti2E+}[M  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9^h%}>  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 K@u."eaD  
    %   for all [n,m]. |ssIUJ  
    % QZq9$;>dW  
    %   The radial Zernike polynomials are the radial portion of the v\tbf  
    %   Zernike functions, which are an orthogonal basis on the unit uNGxz*e  
    %   circle.  The series representation of the radial Zernike vrldRn'*9  
    %   polynomials is $Lbe5d?\  
    % Br$PL&e~  
    %          (n-m)/2 CO+jB  
    %            __ |MR?8A^"  
    %    m      \       s                                          n-2s !rqR]nd  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 8 =Lv7G%  
    %    n      s=0 sN) .Jo  
    % U@AfRUF&  
    %   The following table shows the first 12 polynomials. e9~4wt  
    % !>);}J!e]  
    %       n    m    Zernike polynomial    Normalization _o"3gfH&sJ  
    %       --------------------------------------------- @BW8`Ky1  
    %       0    0    1                        sqrt(2) 1EPOYvf%U  
    %       1    1    r                           2 E"yf!*  
    %       2    0    2*r^2 - 1                sqrt(6) ~,65/O  
    %       2    2    r^2                      sqrt(6) {!?RG\EYN  
    %       3    1    3*r^3 - 2*r              sqrt(8) `GWq3c5  
    %       3    3    r^3                      sqrt(8) .v&h>@'m  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) T/6=A$4 #  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) D;n%sRq(Z  
    %       4    4    r^4                      sqrt(10) ">&:(<  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 1@dx(_  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) ~ J{{n_G{  
    %       5    5    r^5                      sqrt(12) TRok4uc  
    %       --------------------------------------------- XFe7qt;%  
    % 6EWB3.x19  
    %   Example: A>2p/iMc  
    % E,:pIw  
    %       % Display three example Zernike radial polynomials JB_`lefW,'  
    %       r = 0:0.01:1; 4N0nU  
    %       n = [3 2 5]; wdV?& W+  
    %       m = [1 2 1]; 'vIkA=  
    %       z = zernpol(n,m,r); 0l@+xS;  
    %       figure i)3\jO0&GU  
    %       plot(r,z) oA%[x  
    %       grid on X1dG'PQ  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') x7@HPf  
    % * v]UgPk  
    %   See also ZERNFUN, ZERNFUN2. Y\|J1I,Z4  
    "A+F&C>  
    % A note on the algorithm. w8ld* z  
    % ------------------------ 9?c^~77  
    % The radial Zernike polynomials are computed using the series 93DBZqN  
    % representation shown in the Help section above. For many special T$;XJx  
    % functions, direct evaluation using the series representation can -Lb^O/  
    % produce poor numerical results (floating point errors), because EQ $9IaY.  
    % the summation often involves computing small differences between L:Eb(z/D  
    % large successive terms in the series. (In such cases, the functions rv%[?Ml  
    % are often evaluated using alternative methods such as recurrence *b_54X%3  
    % relations: see the Legendre functions, for example). For the Zernike ;BVhkW A  
    % polynomials, however, this problem does not arise, because the +}/!yQtH  
    % polynomials are evaluated over the finite domain r = (0,1), and `0D1Nh"%k  
    % because the coefficients for a given polynomial are generally all /vMyf),2  
    % of similar magnitude. fC|u  
    % GA}^Rh`T-  
    % ZERNPOL has been written using a vectorized implementation: multiple _AbEQ\P{  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] $ '*BS  
    % values can be passed as inputs) for a vector of points R.  To achieve +cH(nZ*f  
    % this vectorization most efficiently, the algorithm in ZERNPOL 2GzpWV(  
    % involves pre-determining all the powers p of R that are required to j@!}r|-T  
    % compute the outputs, and then compiling the {R^p} into a single <z)G& h@  
    % matrix.  This avoids any redundant computation of the R^p, and G'f"w5%qZv  
    % minimizes the sizes of certain intermediate variables. e8bJ]  
    % 3>Snd9Q  
    %   Paul Fricker 11/13/2006 @~3c;9LkY  
    I!D*(>  
    n#cN[C9  
    % Check and prepare the inputs: [+z:^a1?V  
    % -----------------------------  0 XzO`*  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) KK$A 4`YoR  
        error('zernpol:NMvectors','N and M must be vectors.') ): C4}&l  
    end yRkMR$5&  
    0g-ESf``{n  
    if length(n)~=length(m) J3;KQ}F.I  
        error('zernpol:NMlength','N and M must be the same length.') e`F|sz]k"H  
    end FG:BRS<m~  
    |lV9?#!  
    n = n(:); C<XDQ>?  
    m = m(:); "cM5=;  
    length_n = length(n); I1O?)x~  
    yw!`1#3.  
    if any(mod(n-m,2)) LP /4e`  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') & jvG]>CS'  
    end \ 6 a  
    P.DWC'IBN  
    if any(m<0) v19`7qgR(  
        error('zernpol:Mpositive','All M must be positive.') F9w&!yW:  
    end ArK9E!`^  
    !yrHVc  
    if any(m>n) cU|jT8Q4H  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') #jiqRhm  
    end )A=g# D#  
    +9CUnRv  
    if any( r>1 | r<0 ) q(^J7M)  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') qCN7i&k,  
    end o#/iR]3  
    c F]3gM  
    if ~any(size(r)==1) Dj"=kL0  
        error('zernpol:Rvector','R must be a vector.') [)n}!5fE  
    end |3ETF|)?  
    _B FX5ifK  
    r = r(:); l&xD3u^G  
    length_r = length(r); k]pD3.QJ  
    x`i`]6q  
    if nargin==4 XtdLKYET  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); z-gG(  
        if ~isnorm #SNI dc>9\  
            error('zernpol:normalization','Unrecognized normalization flag.') C/ VYu-p%  
        end >]8.xkQq  
    else REBDr;tv  
        isnorm = false; #)}bUNc'  
    end JZxF)] ^  
    tuV?:g?  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (`? snMc  
    % Compute the Zernike Polynomials 5zna?(#}  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "e<. n  
    SJ^?D8  
    % Determine the required powers of r: 7#qL9+G  
    % ----------------------------------- b)^ZiRW``  
    rpowers = []; { BL1j  
    for j = 1:length(n) n3j h\  
        rpowers = [rpowers m(j):2:n(j)]; ? 4Juw?  
    end X>8?p'*  
    rpowers = unique(rpowers); 9yw/-nA  
    h ]$?~YE  
    % Pre-compute the values of r raised to the required powers, })vr*[  
    % and compile them in a matrix: D-:<]D:  
    % ----------------------------- x]cZm^  
    if rpowers(1)==0 1t=X: ]0j  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); q*TKs#3  
        rpowern = cat(2,rpowern{:}); G11.6]?Gg  
        rpowern = [ones(length_r,1) rpowern]; -8 =u{n  
    else h!CX`pBM  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )Hm[j)YI  
        rpowern = cat(2,rpowern{:}); : ";D.{||  
    end b7sE  
    rGGepd  
    % Compute the values of the polynomials: e4%*I8 ^e  
    % -------------------------------------- ey\{C`(__y  
    z = zeros(length_r,length_n); 3N0X?* (x|  
    for j = 1:length_n ruA+1-<f  
        s = 0:(n(j)-m(j))/2; ai  _fN  
        pows = n(j):-2:m(j); T&"dBoUq>G  
        for k = length(s):-1:1 e - ]c  
            p = (1-2*mod(s(k),2))* ... `R52{B#&/  
                       prod(2:(n(j)-s(k)))/          ... Mq lo:7 ^F  
                       prod(2:s(k))/                 ... 5po' (r|U  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... :_,]?n  
                       prod(2:((n(j)+m(j))/2-s(k))); aX'g9E  
            idx = (pows(k)==rpowers); zQ %z "tQ  
            z(:,j) = z(:,j) + p*rpowern(:,idx); ;=\5$J9  
        end 'qF3,Rw  
         3]OP9!\6  
        if isnorm nk|N.%E  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); }>X\"  
        end ^~<Rzq!  
    end W%9K5(e  
    K0RYI69_  
    % 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)  f._l105.  
    B{7Kzwh;  
    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。 x9&-(kBU  
    N\9}\Rk@  
    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)