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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 [HF)d#A  
    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,]Q/6'>  
     
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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 P@5^`b|  
    function z = zernfun(n,m,r,theta,nflag) RM i 2Ip  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. ^k)f oD  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N U{O\  
    %   and angular frequency M, evaluated at positions (R,THETA) on the !Uj !Oy  
    %   unit circle.  N is a vector of positive integers (including 0), and rg'? ?rq  
    %   M is a vector with the same number of elements as N.  Each element #%{\59/w  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) huq6rA/i  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, b _u&%  
    %   and THETA is a vector of angles.  R and THETA must have the same jr9ZRHCU  
    %   length.  The output Z is a matrix with one column for every (N,M) M>]%Iu  
    %   pair, and one row for every (R,THETA) pair. w}(xs)`num  
    % )0GnTB;5Z  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9 /zz@  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !^LvNW\|  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral ow$#kQ&R O  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RB\WttI  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized =~arj  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ! \gRXP}  
    % \hq8/6=4s  
    %   The Zernike functions are an orthogonal basis on the unit circle. N%_~cR;  
    %   They are used in disciplines such as astronomy, optics, and ~/#?OLj(T  
    %   optometry to describe functions on a circular domain. NV91{o(-7  
    % pIrAGA;  
    %   The following table lists the first 15 Zernike functions. Bdg*XfXXk  
    % Lhc@*_2  
    %       n    m    Zernike function           Normalization 3@&H)fdp6a  
    %       -------------------------------------------------- tFSdi. |G=  
    %       0    0    1                                 1 .ClCP?HG  
    %       1    1    r * cos(theta)                    2 y-@!, @e  
    %       1   -1    r * sin(theta)                    2 ]_=HC5"  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) /~^I]D  
    %       2    0    (2*r^2 - 1)                    sqrt(3) .{;!bw  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) s7 KKH w  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) 87Uv+((H  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 0!VLPA:  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) `MwQ6%lf  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) <F3sQAe  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) 6nfkZvn  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) '-S&i{H  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c6uKK h>  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <  t (Pw  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) *0hiPj:  
    %       -------------------------------------------------- 6uXW`/lvX  
    % ~fF }  
    %   Example 1: &0eB@8{N  
    % .fsk DW  
    %       % Display the Zernike function Z(n=5,m=1) }J?fJ (  
    %       x = -1:0.01:1; `eWc p^|  
    %       [X,Y] = meshgrid(x,x); tN{t-xUgk  
    %       [theta,r] = cart2pol(X,Y); 7 h1"8#X  
    %       idx = r<=1; +.lWck  
    %       z = nan(size(X)); Tb= {g;0 @  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); $048y X 7M  
    %       figure tv OAN|+F  
    %       pcolor(x,x,z), shading interp ]k: m2$le  
    %       axis square, colorbar k2uiu  
    %       title('Zernike function Z_5^1(r,\theta)') <VU4rk^=  
    % {&#~t4  
    %   Example 2: kxW>Da<6  
    % GeaDaYh#T  
    %       % Display the first 10 Zernike functions /plUzy2Yu  
    %       x = -1:0.01:1; , imvA5  
    %       [X,Y] = meshgrid(x,x); S%X\ ,N  
    %       [theta,r] = cart2pol(X,Y); b_jZL'en  
    %       idx = r<=1; @pG lWw9*  
    %       z = nan(size(X)); p,iCM?[|  
    %       n = [0  1  1  2  2  2  3  3  3  3]; NceB'YG|  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 2-V)>98  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; H@MFj>~  
    %       y = zernfun(n,m,r(idx),theta(idx)); Px#QZZ  
    %       figure('Units','normalized') iEpq*Qj  
    %       for k = 1:10 N 2"3~  #  
    %           z(idx) = y(:,k); vA;F]epr!  
    %           subplot(4,7,Nplot(k)) aGe(vQPi9  
    %           pcolor(x,x,z), shading interp %x6Ov\s2  
    %           set(gca,'XTick',[],'YTick',[]) *?bk?*?s  
    %           axis square ^+as\  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :5S |x/  
    %       end =~hsKBt*  
    % &'(a$ S>v  
    %   See also ZERNPOL, ZERNFUN2. _@!QY   
    g. ?*F#2  
    %   Paul Fricker 11/13/2006 WBr:|F+~s  
    =zm0w~']E!  
    ~`2&'8  
    % Check and prepare the inputs: @fqV0l!GR  
    % ----------------------------- JOrELrMx  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j/H>0^  
        error('zernfun:NMvectors','N and M must be vectors.') 7G.o@p6$  
    end 9q1HSJ1)  
    .Iw ur;/\  
    if length(n)~=length(m) 2.LJp}>  
        error('zernfun:NMlength','N and M must be the same length.') 1E5a(  
    end =.36y9Mfo  
    RpO@pd m  
    n = n(:); U*3A M_w  
    m = m(:); {f+N]Oo*  
    if any(mod(n-m,2)) +0XL5( '2  
        error('zernfun:NMmultiplesof2', ... C8IkpAD  
              'All N and M must differ by multiples of 2 (including 0).') 1, "I=  
    end `$s)X$W?  
    FXP6zHsV  
    if any(m>n) DR:8oo&E  
        error('zernfun:MlessthanN', ... G2.|fp_}pG  
              'Each M must be less than or equal to its corresponding N.') b4>``n  
    end EL^8zyg%%  
    &v^!y=Bt  
    if any( r>1 | r<0 ) `|$'g^eCL  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') KH7VR^;mk  
    end XJqTmj3   
    AXwaVLEBQ  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I @ 2uF-  
        error('zernfun:RTHvector','R and THETA must be vectors.') HTmI1  
    end l)4O .*  
    lT2 4JhJ#  
    r = r(:); :Sr?6FPc  
    theta = theta(:); ~h-C&G ,v  
    length_r = length(r); =#qZ3 Qz_  
    if length_r~=length(theta) 7kKuZW@K-  
        error('zernfun:RTHlength', ... vY6oV jM  
              'The number of R- and THETA-values must be equal.') oM!xz1kVL  
    end j^flwk  
    A{# Nwd>  
    % Check normalization: 7BR8/4gcPu  
    % -------------------- nVE9^')8V  
    if nargin==5 && ischar(nflag) 1B|8ZmFJj  
        isnorm = strcmpi(nflag,'norm'); NYwR2oX  
        if ~isnorm IOL L1ar  
            error('zernfun:normalization','Unrecognized normalization flag.') oH^(qZ8W  
        end xl(@C*.sC1  
    else .%}?b~  
        isnorm = false; aTd D`h  
    end X&M4MuL  
    pd3,pQ  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mMSh2B  
    % Compute the Zernike Polynomials S4N(cn&  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Yw'NX5#)g  
    UH? p]4Nz  
    % Determine the required powers of r: xe4Oxo  
    % ----------------------------------- eg/<[ A:  
    m_abs = abs(m); hB9Ee@  
    rpowers = []; =-KMb`xT  
    for j = 1:length(n) /ASaB  
        rpowers = [rpowers m_abs(j):2:n(j)]; FOwnxYGVf  
    end 6Wj^*L!  
    rpowers = unique(rpowers); x YfD()w<I  
    ~g#r6pzN-  
    % Pre-compute the values of r raised to the required powers, ( #D*Pl  
    % and compile them in a matrix: @%/]Q<<q  
    % ----------------------------- 5| B(\wqG  
    if rpowers(1)==0 Z[Qza13lo  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);  6e,xDr  
        rpowern = cat(2,rpowern{:}); ({s6eqMhDd  
        rpowern = [ones(length_r,1) rpowern]; q:\g^_!OGA  
    else j;b42G~p  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); = ?D(g  
        rpowern = cat(2,rpowern{:}); B* kcN lW  
    end \u,}vpp z  
    Ub1hHA*)  
    % Compute the values of the polynomials: VKp*9%9  
    % -------------------------------------- +mj*o(  
    y = zeros(length_r,length(n)); IU FH:w]  
    for j = 1:length(n) , DdB^Ig<r  
        s = 0:(n(j)-m_abs(j))/2; W>_]dPBS/  
        pows = n(j):-2:m_abs(j); S$)*&46g  
        for k = length(s):-1:1 $rIoHxh. y  
            p = (1-2*mod(s(k),2))* ... N`iwC!  
                       prod(2:(n(j)-s(k)))/              ... <+MyZM(z>  
                       prod(2:s(k))/                     ... %^sTU4D5  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [(X y.L7x  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); |-sPLU&s%  
            idx = (pows(k)==rpowers); ^;!0j9"* :  
            y(:,j) = y(:,j) + p*rpowern(:,idx); OsBo+fwT  
        end 3LDS Z1f  
         ;g{qYj_  
        if isnorm +$4(zP s@  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sn7AR88M;  
        end Lg8nj< TF  
    end w=b)({`M  
    % END: Compute the Zernike Polynomials  _zlqtO  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oFOnjK"|F  
    g^*<f8 ~d  
    % Compute the Zernike functions: zJe#m|Z  
    % ------------------------------ YK|bXSA[  
    idx_pos = m>0; OL4z%mDZi  
    idx_neg = m<0; h-iJlm  
    V_plq6z  
    z = y; o7IxJCL=Q  
    if any(idx_pos) xsWur(>]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); pr%nbl  
    end 2]%h$f+  
    if any(idx_neg) [M+f-kl  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); AwQ?l(iZ"p  
    end R|i/lEq  
     i2~  
    % EOF zernfun
    离线niuhelen
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    只看该作者 4楼 发表于: 2011-03-12
    function z = zernfun2(p,r,theta,nflag) y\&>Z yOY  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. "s\L~R.&  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated o/ui)U_   
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive {[+2n]f_G  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ,^S@EDq  
    %   and THETA is a vector of angles.  R and THETA must have the same '= l[;Q^Q  
    %   length.  The output Z is a matrix with one column for every P-value, s: 3z'4oX  
    %   and one row for every (R,THETA) pair. +iI&c s  
    % Q,80Hor#J  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike j2 !3rI  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 1T:Y0  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 3"rzb]=R  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 -j&Tc` j_  
    %   for all p. O@YTAT&d#  
    % .; &# )l  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 s`#(   
    %   Zernike functions (order N<=7).  In some disciplines it is 7#wn<HDY%  
    %   traditional to label the first 36 functions using a single mode 1Z,[|wJ  
    %   number P instead of separate numbers for the order N and azimuthal Wa?; ^T  
    %   frequency M. RxQh2<?  
    % JsotOic%  
    %   Example: itzyCw2|#  
    % !~h}8'a?  
    %       % Display the first 16 Zernike functions j^`hzh3S  
    %       x = -1:0.01:1; BATG FS&  
    %       [X,Y] = meshgrid(x,x); pC_O:f>vJ  
    %       [theta,r] = cart2pol(X,Y); 'TA UE{{  
    %       idx = r<=1; ?-Vjha@BO  
    %       p = 0:15; "]-Xmdk09  
    %       z = nan(size(X)); b=/curl&  
    %       y = zernfun2(p,r(idx),theta(idx)); gkHNRAL  
    %       figure('Units','normalized') \cCV6A[  
    %       for k = 1:length(p) G}9=)  
    %           z(idx) = y(:,k); c5mZG7-  
    %           subplot(4,4,k) xzx$TUL  
    %           pcolor(x,x,z), shading interp w;l<[q?_  
    %           set(gca,'XTick',[],'YTick',[]) C{d7J'Avk  
    %           axis square ^gFqRbuS  
    %           title(['Z_{' num2str(p(k)) '}']) $U/YR&vcw  
    %       end :\=CRaA  
    % QFIL)'K  
    %   See also ZERNPOL, ZERNFUN. !\ g+8>  
    rLX4jT^  
    %   Paul Fricker 11/13/2006 Y \:0Ev  
    Ve 4u +0  
    a/< Csad  
    % Check and prepare the inputs: 9+keX{/c  
    % ----------------------------- -@ZiS^l  
    if min(size(p))~=1 @'=Uq  
        error('zernfun2:Pvector','Input P must be vector.') K!KMQr`  
    end @}:uu$OH  
    F0690v0mB[  
    if any(p)>35 AdWq Q  
        error('zernfun2:P36', ... `ImE% r!  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 1J' 3g  
               '(P = 0 to 35).']) 5#QXR+ T  
    end FW.$5*f='  
    `N5|Ho*C  
    % Get the order and frequency corresonding to the function number: Sv;_HZ  
    % ---------------------------------------------------------------- l (3bW1{n  
    p = p(:); ./$cMaDJ  
    n = ceil((-3+sqrt(9+8*p))/2); q=lAb\i  
    m = 2*p - n.*(n+2); 4GB7A]^E  
    TW^/sx  
    % Pass the inputs to the function ZERNFUN: Y\0}R,]a-  
    % ---------------------------------------- 03j]d&P%d  
    switch nargin wK}\_2?  
        case 3 $Q*<96M  
            z = zernfun(n,m,r,theta); CwJDmz\tk  
        case 4 JBnK K  
            z = zernfun(n,m,r,theta,nflag); AO UL^$&  
        otherwise ] 7 _`]7p  
            error('zernfun2:nargin','Incorrect number of inputs.') 5 Qoew9rA  
    end |_ G )qp;  
    i{I~mrm/'\  
    % 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) "YB** Y  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. C.kxQ<  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of [le)P$#z  
    %   order N and frequency M, evaluated at R.  N is a vector of uw},`4`  
    %   positive integers (including 0), and M is a vector with the Tz9`uW~Mf  
    %   same number of elements as N.  Each element k of M must be a Qeu\&%C!<  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 0 P[RyQI  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ]QuM<ms  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix 9h0X&1u  
    %   with one column for every (N,M) pair, and one row for every TT9z_Q5~  
    %   element in R. nhN);R~o"1  
    % -rKO )}  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- )z8!f}:De=  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is Pf F=m'  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to pMs AyCAk  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ~8m=1)A{(  
    %   for all [n,m]. Cg616hyut  
    % r",]Voibd  
    %   The radial Zernike polynomials are the radial portion of the 6DZ),F,M  
    %   Zernike functions, which are an orthogonal basis on the unit d(:3   
    %   circle.  The series representation of the radial Zernike -8N|xQ378  
    %   polynomials is r_YIpnJ  
    % Yhp]x   
    %          (n-m)/2 vzn{h)D  
    %            __ y ?G_y  
    %    m      \       s                                          n-2s >q7BVF6V |  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r `nO71mo  
    %    n      s=0 dCu'>G\bP  
    % I!wX[4p eg  
    %   The following table shows the first 12 polynomials. [&*6_q"V  
    % C%~a`e|/Y  
    %       n    m    Zernike polynomial    Normalization >E,U>@+  
    %       --------------------------------------------- kcDyuM`  
    %       0    0    1                        sqrt(2) Ys8SDlMo  
    %       1    1    r                           2  9dzdrT  
    %       2    0    2*r^2 - 1                sqrt(6) 7E!7"2e a  
    %       2    2    r^2                      sqrt(6) 0[<~?`:)  
    %       3    1    3*r^3 - 2*r              sqrt(8) )+H[kiN  
    %       3    3    r^3                      sqrt(8) H[b}kZW:a  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) U}$DhA"r"  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) r ]>\~&?^F  
    %       4    4    r^4                      sqrt(10) )wVIb)`R>Y  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 0hZ1rqq8C  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) IcIOC8WC  
    %       5    5    r^5                      sqrt(12) !,Zp? g)  
    %       --------------------------------------------- { BEo &  
    % u>pBB@  
    %   Example: B cj/y4"  
    % dO7;}>F$n  
    %       % Display three example Zernike radial polynomials #Dfo#]k(  
    %       r = 0:0.01:1; -A-tuyIsh"  
    %       n = [3 2 5]; E| :!Q8"%w  
    %       m = [1 2 1]; ZX~ _g@  
    %       z = zernpol(n,m,r); 6x=YQwn~  
    %       figure LEECW_:  
    %       plot(r,z) vs6,  
    %       grid on x7T +>  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') O--7<Q\  
    % c<#<k}y  
    %   See also ZERNFUN, ZERNFUN2. wve=.n  
    o/o:2p.  
    % A note on the algorithm. H6aM&r9}  
    % ------------------------ n-QJ;37\  
    % The radial Zernike polynomials are computed using the series 8[ry |J  
    % representation shown in the Help section above. For many special D@X+{  
    % functions, direct evaluation using the series representation can -RJE6~>'\  
    % produce poor numerical results (floating point errors), because CVXytS?@x  
    % the summation often involves computing small differences between M`D$!BJr  
    % large successive terms in the series. (In such cases, the functions  uIMe  
    % are often evaluated using alternative methods such as recurrence <Q<+4Y{R  
    % relations: see the Legendre functions, for example). For the Zernike `:M^8SYrL  
    % polynomials, however, this problem does not arise, because the nU`Lhh8y  
    % polynomials are evaluated over the finite domain r = (0,1), and &@3m -Z  
    % because the coefficients for a given polynomial are generally all }jSj+*  
    % of similar magnitude. W 4YE~  
    % j[6Raf/(n  
    % ZERNPOL has been written using a vectorized implementation: multiple l0tYG[  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] r+<{S\ Q  
    % values can be passed as inputs) for a vector of points R.  To achieve rsa&Oo D>  
    % this vectorization most efficiently, the algorithm in ZERNPOL #t!}K_  
    % involves pre-determining all the powers p of R that are required to .]Mn^2#j  
    % compute the outputs, and then compiling the {R^p} into a single /)uM[ dnai  
    % matrix.  This avoids any redundant computation of the R^p, and vuz4qCQ  
    % minimizes the sizes of certain intermediate variables. /,|CrNwY*  
    % !p 8psi0  
    %   Paul Fricker 11/13/2006 `"k9wC1  
    \Btk;ivg  
    !PUp>(  
    % Check and prepare the inputs: A[UP"P~u/  
    % ----------------------------- =FW5Tkw0  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :46h+?   
        error('zernpol:NMvectors','N and M must be vectors.') |sgXh9%x<  
    end e<gx~N9l'  
    AH{^spD{7,  
    if length(n)~=length(m) _|isa]u\ z  
        error('zernpol:NMlength','N and M must be the same length.') u@FsLHn  
    end j nwQV  
    n<V1|X  
    n = n(:); qX>Q+_^  
    m = m(:); L&Qi@D0P  
    length_n = length(n); %Ny) ?B  
    lj&>cScC  
    if any(mod(n-m,2)) {,O`rW_eS  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') CBD_a#K{  
    end nb dGt  
    fAj2LAK  
    if any(m<0) s ?l%L!  
        error('zernpol:Mpositive','All M must be positive.') qJ[@:&:  
    end :Eh'(   
    :\V,k~asl  
    if any(m>n) DpL8'Dib  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') lUh*?l  
    end Na!za'qk[o  
    J+<p+(^*v  
    if any( r>1 | r<0 ) @Hr+/52B  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') |LYKc.xo  
    end wFlV=!>,  
    P0\eB S  
    if ~any(size(r)==1) M/jb}*xDR  
        error('zernpol:Rvector','R must be a vector.') L{ ^4DznI  
    end ekzjF\!y  
    VfSGCe  
    r = r(:); %]Cjhs"v  
    length_r = length(r); K%,$ V,#  
    /B HepD}  
    if nargin==4 IKf`[_,t]  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); StQ@g  
        if ~isnorm u2qV6/  
            error('zernpol:normalization','Unrecognized normalization flag.') @oH[SWx  
        end kN'Thq/ZE  
    else z<a2cQ?XQ  
        isnorm = false; Ob&W_D^=N  
    end >,g5Hkmqr  
    A_r<QYq0|  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0U&d q#  
    % Compute the Zernike Polynomials I5pp "*u  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]O@"\_}  
    _p4}<pG  
    % Determine the required powers of r: mCb 9*|  
    % ----------------------------------- [n:PNB  
    rpowers = []; F RH&B5w  
    for j = 1:length(n) SgSk !lj  
        rpowers = [rpowers m(j):2:n(j)]; $Qq_qTJu?G  
    end >rRf9wO1l  
    rpowers = unique(rpowers); r>3^kL5UI  
    F_PTMl=Q|J  
    % Pre-compute the values of r raised to the required powers, q,,j',8kq/  
    % and compile them in a matrix: T]2U fi.  
    % ----------------------------- +sn2Lw!^  
    if rpowers(1)==0 T7GQ^WnA  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); eti9nPjG  
        rpowern = cat(2,rpowern{:}); ]%XK)[:5_=  
        rpowern = [ones(length_r,1) rpowern]; HU[oR4E  
    else ?Leyz  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %1jdiHTaL  
        rpowern = cat(2,rpowern{:}); rUFFF'm\*a  
    end (n=Aa;  
    /oDpgOn  
    % Compute the values of the polynomials: g5TkD~w"  
    % -------------------------------------- }vsO^4Sjc  
    z = zeros(length_r,length_n); ]piM/v\  
    for j = 1:length_n 9[f%;WaS  
        s = 0:(n(j)-m(j))/2; :1BM=_WwI  
        pows = n(j):-2:m(j); ,|x\MHd?t_  
        for k = length(s):-1:1 9%TT> 2#  
            p = (1-2*mod(s(k),2))* ... 5byeWH0n3  
                       prod(2:(n(j)-s(k)))/          ... y$h"ty{g  
                       prod(2:s(k))/                 ... {jG.=}/Dk  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... As}eUm)B5c  
                       prod(2:((n(j)+m(j))/2-s(k))); WJcVQM s  
            idx = (pows(k)==rpowers); '8Qw:fh  
            z(:,j) = z(:,j) + p*rpowern(:,idx); z"av|(?d  
        end :tlE`BIp  
         k1wr/G'H[  
        if isnorm r:#Q9EA  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); . z].:$J&  
        end X4 Y  
    end V@Kn24''  
    NE[y|/  
    % 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)  y>@v>S  
    B{;11 u  
    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。 P)Z/JHB  
    tDEXm^B2Sv  
    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)