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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 F/3L^k]  
    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! +;M 5Sp  
     
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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 o4Cq  /K  
    function z = zernfun(n,m,r,theta,nflag) `%"x'B`mM  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. \okv}x^L=Z  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \NEk B&^n  
    %   and angular frequency M, evaluated at positions (R,THETA) on the 'J5F+, \Ka  
    %   unit circle.  N is a vector of positive integers (including 0), and -KH"2q  
    %   M is a vector with the same number of elements as N.  Each element m^3j|'mG  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) %e3E}m>  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, _\Z'Yl  
    %   and THETA is a vector of angles.  R and THETA must have the same dU2;   
    %   length.  The output Z is a matrix with one column for every (N,M) 9!Jt}n?!g  
    %   pair, and one row for every (R,THETA) pair. Oh>hy Y)}  
    % ~I%164B+/  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~(huUW  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), pV;0Hcy  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral E)f9`][  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \ym^~ Q|  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized n;$u%2t2  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ( ^@i(XQ  
    % =5V7212  
    %   The Zernike functions are an orthogonal basis on the unit circle. kWy@wPqms  
    %   They are used in disciplines such as astronomy, optics, and 9c }qVf-i  
    %   optometry to describe functions on a circular domain. %*wEzvt *  
    % ~J> ;l s1  
    %   The following table lists the first 15 Zernike functions. }#%Y eCA?  
    % :FtV~^Z  
    %       n    m    Zernike function           Normalization vw(ecs^C  
    %       -------------------------------------------------- jm@M"b'{  
    %       0    0    1                                 1 y'I m/{9U  
    %       1    1    r * cos(theta)                    2 s/s&d pT*  
    %       1   -1    r * sin(theta)                    2 -1d*zySL  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) c00rq ~<K  
    %       2    0    (2*r^2 - 1)                    sqrt(3) +PI}$c-|`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) gsM^Pu09ud  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) NA'45}fQ  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) {;& U5<NO  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) LL)t)  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) ",Vx.LV  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) SE@TY32T  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !Ko>   
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Mx`';z8~  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) B)1(  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) %N&W_.F6  
    %       -------------------------------------------------- i8-Y,&>V  
    % v1X[/\;U  
    %   Example 1: 6 R})KIG  
    % CI-za !T  
    %       % Display the Zernike function Z(n=5,m=1) jgG9?w)|u  
    %       x = -1:0.01:1; !K}W.yv,  
    %       [X,Y] = meshgrid(x,x); s@7hoU-+  
    %       [theta,r] = cart2pol(X,Y); Ut;4`>T  
    %       idx = r<=1; g52)/HM  
    %       z = nan(size(X)); G)t-W %D&  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); ty rP[y  
    %       figure xS5 -m6/  
    %       pcolor(x,x,z), shading interp j|K;Yi  
    %       axis square, colorbar $qdynKK  
    %       title('Zernike function Z_5^1(r,\theta)') 0H^*VUyW/  
    % `67i1w`  
    %   Example 2: Q~svtN  
    % .Wy'  
    %       % Display the first 10 Zernike functions 'ROz|iJ  
    %       x = -1:0.01:1; GN! R<9  
    %       [X,Y] = meshgrid(x,x); 5|K[WvG@Co  
    %       [theta,r] = cart2pol(X,Y); >(.|oT\Tb  
    %       idx = r<=1; <f8j^  
    %       z = nan(size(X)); \gPMYMd  
    %       n = [0  1  1  2  2  2  3  3  3  3]; Ry]9n.y  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; 0:u:#))1  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; nQ+5jGP1  
    %       y = zernfun(n,m,r(idx),theta(idx)); Uuu2wz3O0  
    %       figure('Units','normalized') BSg T 6K  
    %       for k = 1:10 jK*d  
    %           z(idx) = y(:,k); -aok]w m  
    %           subplot(4,7,Nplot(k)) zb!1o0, J  
    %           pcolor(x,x,z), shading interp ([>__c/Nd  
    %           set(gca,'XTick',[],'YTick',[]) };9s8VZE  
    %           axis square H{=G\N{  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `Ng Q>KV!  
    %       end p!^K.P1 '  
    % 5  >0\=  
    %   See also ZERNPOL, ZERNFUN2. z+6PVQ  
     .nrbd#i-  
    %   Paul Fricker 11/13/2006 NiW9/(;xB  
    iO?^y(phC  
    ,&S0/j  
    % Check and prepare the inputs: S qb>a j  
    % ----------------------------- n9={D  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) KhB775  
        error('zernfun:NMvectors','N and M must be vectors.') Q. O4R_H  
    end X5 or5v  
    UhS:tT]7  
    if length(n)~=length(m) K&NH?  
        error('zernfun:NMlength','N and M must be the same length.') 0LL0\ly]  
    end 63Gq5dF  
    u_9c>  
    n = n(:); x} c  
    m = m(:); } f&=}  
    if any(mod(n-m,2)) $ [fqTh  
        error('zernfun:NMmultiplesof2', ... d!R+-Fp  
              'All N and M must differ by multiples of 2 (including 0).') sV{\IgH/x  
    end +<F3}]]  
    i^.eX VV/  
    if any(m>n) a4~B  
        error('zernfun:MlessthanN', ... a<r,LE  
              'Each M must be less than or equal to its corresponding N.') X5J)1rL  
    end (E00T`@t0i  
    t7x<=rW7u  
    if any( r>1 | r<0 ) ly*v|(S&  
        error('zernfun:Rlessthan1','All R must be between 0 and 1.') )/)u.$pi  
    end ]9/A=p?J@  
    L{F]uz_[x  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j0{`7n  
        error('zernfun:RTHvector','R and THETA must be vectors.') %?gG-R  
    end Tt~[hC h  
    SIrNZ^I  
    r = r(:); fTy:Re  
    theta = theta(:); Icg-rwa<Z  
    length_r = length(r); X0P +[.i  
    if length_r~=length(theta) c8uw_6#r(D  
        error('zernfun:RTHlength', ... E#rQJ  
              'The number of R- and THETA-values must be equal.') #n|5ng|CJ  
    end }O@>:?U  
    *a CVkFp  
    % Check normalization: qX-5/;n  
    % -------------------- hui #<2{  
    if nargin==5 && ischar(nflag) Sj(>G;  
        isnorm = strcmpi(nflag,'norm'); MW rhVn{R  
        if ~isnorm ,(x` zpp _  
            error('zernfun:normalization','Unrecognized normalization flag.') <H60rON  
        end ^il$t]X5-  
    else mp$IhJ6#  
        isnorm = false; HLPRTta.  
    end 6z U  
    A9BoH[is7  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \^dse  
    % Compute the Zernike Polynomials ~%>i lWaHB  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v; #y^O  
    >KrI}>!9r  
    % Determine the required powers of r: ms}o[Z@n  
    % ----------------------------------- ma*#*4  
    m_abs = abs(m); h]&  
    rpowers = []; (!{*@?S  
    for j = 1:length(n) |Sjy   
        rpowers = [rpowers m_abs(j):2:n(j)]; aanS^t0  
    end QlMLWi  
    rpowers = unique(rpowers); fG>3gS6&  
    8TB|Y  
    % Pre-compute the values of r raised to the required powers, d9TTAaf  
    % and compile them in a matrix: (jU_lsG  
    % ----------------------------- A? B +  
    if rpowers(1)==0 '1b8>L  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aIa<,  
        rpowern = cat(2,rpowern{:}); nD eVYK  
        rpowern = [ones(length_r,1) rpowern]; EL3X8H  
    else 5Q8 H8!^  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,iao56`E  
        rpowern = cat(2,rpowern{:}); +jB;  
    end !zOj`lx  
    [#@lsI  
    % Compute the values of the polynomials: X5.9~  
    % -------------------------------------- w#A\(z%;x  
    y = zeros(length_r,length(n)); 7M~/ q.  
    for j = 1:length(n) MFa/%O_*  
        s = 0:(n(j)-m_abs(j))/2; NCi~. I  
        pows = n(j):-2:m_abs(j); 2=K|kp5  
        for k = length(s):-1:1 !^F_7u@Q  
            p = (1-2*mod(s(k),2))* ... BSHS)_xs  
                       prod(2:(n(j)-s(k)))/              ... c$ib-  
                       prod(2:s(k))/                     ... &)Qq%\EP4  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =Y|( }92  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); dYD;Z<l  
            idx = (pows(k)==rpowers); T$u'+* Xx  
            y(:,j) = y(:,j) + p*rpowern(:,idx); 7$%G3Q|)L  
        end $-UVN0=  
         +=9iq3<yfS  
        if isnorm fNAW4I I}  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cFq<x=S  
        end qZ[HILh!  
    end /Q7q2Ne^*  
    % END: Compute the Zernike Polynomials diu"Nt  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4s:M}=]N  
    Z HZxr  
    % Compute the Zernike functions: Hm>cKPZ)  
    % ------------------------------ )N- '~<N  
    idx_pos = m>0; @R`6j S_gK  
    idx_neg = m<0; z0+JMZ/  
    >i  
    z = y; ? Pi|`W   
    if any(idx_pos) '/UT0{2;rS  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); b1#C,UWK  
    end K!9K^h  
    if any(idx_neg) (Ox&B+\v+v  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Pi5MFw'v  
    end ly34aD/p~,  
    .^=I&X/P  
    % 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) U1/I( w  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. U}X'RCM  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated %Bm{ctf#)  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive "zNS6I?rzE  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, ku*k+4rz  
    %   and THETA is a vector of angles.  R and THETA must have the same T7mT:z>:  
    %   length.  The output Z is a matrix with one column for every P-value, 6<\dQ+~  
    %   and one row for every (R,THETA) pair. 9A4n8,&sm  
    % |=:@<0.'  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike :kI[Pf!z  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Z ;~%!  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 8wd["hga<%  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 B0yGr\KJ  
    %   for all p. 1yF9zKs&_  
    % ]!S#[Wt {k  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ={cM6F}a@  
    %   Zernike functions (order N<=7).  In some disciplines it is y~]D402Cx  
    %   traditional to label the first 36 functions using a single mode D +0il=5  
    %   number P instead of separate numbers for the order N and azimuthal "dv\ 9O  
    %   frequency M. j8e=],sQ  
    % W?E01"p  
    %   Example: L T`T~|pz  
    % mp sX4  
    %       % Display the first 16 Zernike functions 9(HGe+R4o  
    %       x = -1:0.01:1; 6@# =z  
    %       [X,Y] = meshgrid(x,x); 2Q(ZW@0  
    %       [theta,r] = cart2pol(X,Y); |j'@no_rv  
    %       idx = r<=1; H&*&n}vh5y  
    %       p = 0:15; }T}c%p  
    %       z = nan(size(X)); {-7ovH?  
    %       y = zernfun2(p,r(idx),theta(idx)); T7ShE-X  
    %       figure('Units','normalized') '.pGkXyQ  
    %       for k = 1:length(p) .Gr"| uII  
    %           z(idx) = y(:,k); 7W},5c  
    %           subplot(4,4,k) 6w3[PNd  
    %           pcolor(x,x,z), shading interp ={o4lFe3v(  
    %           set(gca,'XTick',[],'YTick',[]) =HMCNl  
    %           axis square lws.;abm%n  
    %           title(['Z_{' num2str(p(k)) '}']) .XK3o .ZhW  
    %       end ~yXDN4s  
    % (K6vXq.;\\  
    %   See also ZERNPOL, ZERNFUN. d/oD]aAEr  
    ~IO'"h'w  
    %   Paul Fricker 11/13/2006 E 8W*^^z(  
    {7IZN< e  
    w H=7pS"s  
    % Check and prepare the inputs: _z]v;Q  
    % -----------------------------  K\ pZ  
    if min(size(p))~=1 `_neYT  
        error('zernfun2:Pvector','Input P must be vector.') m|?1HCRXRI  
    end ; 1?L  
    E<B/5g!  
    if any(p)>35 ,{iMF (Nj  
        error('zernfun2:P36', ... $@{ d\@U  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... B[w.8e5  
               '(P = 0 to 35).']) UZiL NKc  
    end 1M_6X7PH  
    %|/\Qu  
    % Get the order and frequency corresonding to the function number: vqUYr  
    % ---------------------------------------------------------------- OS L~a_  
    p = p(:); VWXyN  
    n = ceil((-3+sqrt(9+8*p))/2); j}@LiH'Q  
    m = 2*p - n.*(n+2); 33}p02#  
    ^N ;TCn  
    % Pass the inputs to the function ZERNFUN: 'R$/Qt;uA  
    % ---------------------------------------- A;/,</  
    switch nargin =VWH8w.3  
        case 3 CIwI1VR^  
            z = zernfun(n,m,r,theta); 4yMi9Ri4H  
        case 4 I L&PN`#  
            z = zernfun(n,m,r,theta,nflag); {}Afah  
        otherwise W1M Bk[:Q  
            error('zernfun2:nargin','Incorrect number of inputs.') _iqaKYT$  
    end ==KDr 0|G  
    NQ '|M  
    % 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) 70R_O&f-k  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. (G>g0(;D-  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of |hvclEu,  
    %   order N and frequency M, evaluated at R.  N is a vector of >n&+<06  
    %   positive integers (including 0), and M is a vector with the Q`=d5Uvw  
    %   same number of elements as N.  Each element k of M must be a >IKIe  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) &]KA%Db2  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &6\E'bBt  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix sw(|EZ7F  
    %   with one column for every (N,M) pair, and one row for every \%W"KLP  
    %   element in R. *'@T+$3s  
    % /dR:\ffz2  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- (x[z=_I%`  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ``h* A  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 2tp95E`(O  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 eN  TKX  
    %   for all [n,m]. >/-Bg:  
    % c5eimA%`  
    %   The radial Zernike polynomials are the radial portion of the 2) Q/cH\g  
    %   Zernike functions, which are an orthogonal basis on the unit x)kp*^/  
    %   circle.  The series representation of the radial Zernike ~MK%^5y?  
    %   polynomials is KW36nY\7  
    % -0*z"a9<p8  
    %          (n-m)/2 U]E~7C  
    %            __ jpiBHi]5+  
    %    m      \       s                                          n-2s ?j8_j  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r s<LYSrd  
    %    n      s=0 t8#u}u  
    % ?[X^'zz}  
    %   The following table shows the first 12 polynomials. QIcc@PGT9a  
    % >\1j`/ :ZI  
    %       n    m    Zernike polynomial    Normalization QyEoWKu;  
    %       --------------------------------------------- 2J6(TrQ  
    %       0    0    1                        sqrt(2) ?ieC>cr  
    %       1    1    r                           2 cD<5~`l  
    %       2    0    2*r^2 - 1                sqrt(6) (;nh?"5  
    %       2    2    r^2                      sqrt(6) ~Y7:08  
    %       3    1    3*r^3 - 2*r              sqrt(8) K3J,f2Cn$  
    %       3    3    r^3                      sqrt(8) @$|bMH*1:  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 5&Le?-/\  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) Bc?KAK  
    %       4    4    r^4                      sqrt(10) sH{(=N  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 8SO(pw9  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) tNDv[IF  
    %       5    5    r^5                      sqrt(12) ShU1RQk  
    %       --------------------------------------------- `+T"^{ Z  
    % ";w"dfC^  
    %   Example: GaOM|F'>  
    % rn-CQ2{?  
    %       % Display three example Zernike radial polynomials ^r 9  
    %       r = 0:0.01:1; 7MwS[N%#  
    %       n = [3 2 5]; _VLA2#V>   
    %       m = [1 2 1]; AKUmh  
    %       z = zernpol(n,m,r); `R_;n#3F0  
    %       figure 9.l*#A^  
    %       plot(r,z) &jE@i#  
    %       grid on N?mQ50o~C  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') yH',vC.  
    % p) m0\  
    %   See also ZERNFUN, ZERNFUN2. Uf\U~wM<  
    y9Q.TL>=[  
    % A note on the algorithm. t$ 3/ZTx  
    % ------------------------ m}sh (W5\  
    % The radial Zernike polynomials are computed using the series ,-5|qko=  
    % representation shown in the Help section above. For many special Gx h1wqLR  
    % functions, direct evaluation using the series representation can ;0:[X+"(  
    % produce poor numerical results (floating point errors), because :z%q09.)  
    % the summation often involves computing small differences between U~Rs?JmTdD  
    % large successive terms in the series. (In such cases, the functions C.?~D*Q  
    % are often evaluated using alternative methods such as recurrence '0jn|9l58  
    % relations: see the Legendre functions, for example). For the Zernike _ v3VUm#  
    % polynomials, however, this problem does not arise, because the ECvTmU'=  
    % polynomials are evaluated over the finite domain r = (0,1), and AP/#?   
    % because the coefficients for a given polynomial are generally all OCbwV7q:  
    % of similar magnitude. g3vR\?c`  
    % 8QFg6#"O  
    % ZERNPOL has been written using a vectorized implementation: multiple o9c?)KQ  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] - ~`)V`@  
    % values can be passed as inputs) for a vector of points R.  To achieve %"BJW  
    % this vectorization most efficiently, the algorithm in ZERNPOL %HcCe[d5l  
    % involves pre-determining all the powers p of R that are required to AkVgFQg" n  
    % compute the outputs, and then compiling the {R^p} into a single .%+y_.l  
    % matrix.  This avoids any redundant computation of the R^p, and P?j;&@$^e  
    % minimizes the sizes of certain intermediate variables. C?t!Uvs  
    % FZ=xy[q]~  
    %   Paul Fricker 11/13/2006 {Z^q?~zC[  
    \MB$Cwc  
    r5'bt"K\>  
    % Check and prepare the inputs: 3?bTs =  
    % ----------------------------- ?=V;5H.  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) kJ.0|l0  
        error('zernpol:NMvectors','N and M must be vectors.') ]q3.^F  
    end V ^hR%*i'  
    )H[Pz.'ah0  
    if length(n)~=length(m) o&#!W(   
        error('zernpol:NMlength','N and M must be the same length.') b-HELS`nX  
    end U,7}VdO  
    5b;~&N4~  
    n = n(:); :HkX sZ  
    m = m(:); O*ER3  
    length_n = length(n); ;_p!20.(  
    EfGy^`,'G  
    if any(mod(n-m,2)) r&Qq,koE  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') y=SVS3D  
    end $Ahe Vps@@  
    }mOo=)C!  
    if any(m<0) my%MXTm2  
        error('zernpol:Mpositive','All M must be positive.') >IJX=24Rc  
    end sI6coe5n  
    C!W0L`r  
    if any(m>n)  N}KL'  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') U}DLzn|w  
    end Y |9  
    (ty&$  
    if any( r>1 | r<0 ) `j!XWh*$  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') LyRW\\z2  
    end KRb'kW  
    2&XNT-Qm  
    if ~any(size(r)==1) L"}tJM.d  
        error('zernpol:Rvector','R must be a vector.') Q2<v: *L  
    end Ix+eP|8F  
    vF1Fcp.@  
    r = r(:); x.Tulo0/  
    length_r = length(r); }mpFo 2  
    moL3GV%]Gq  
    if nargin==4 &1ZUMc  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); sq?js#C5  
        if ~isnorm 2=uwGIF  
            error('zernpol:normalization','Unrecognized normalization flag.') 'zOB!QqA`v  
        end _RE;}1rb,  
    else zJov*^T-C  
        isnorm = false; ( @V_47o  
    end 3!vnSX(iv  
    lN*beOj  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (#8B  
    % Compute the Zernike Polynomials lcK4 Uq\q  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `RXlqj#u  
    wlgR = l  
    % Determine the required powers of r: @D>qo=KPM  
    % ----------------------------------- 8J:=@X^}  
    rpowers = []; r+;k(HMY}[  
    for j = 1:length(n) OAf}\  
        rpowers = [rpowers m(j):2:n(j)]; aNP\Q23D  
    end ]A%~bQ7  
    rpowers = unique(rpowers); 2"_5Yyb  
    o pTH6a  
    % Pre-compute the values of r raised to the required powers, Wg,7k9I  
    % and compile them in a matrix: 2S-f5&o  
    % ----------------------------- [:+f Y[4==  
    if rpowers(1)==0 a(X V~o  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); n'[>h0  
        rpowern = cat(2,rpowern{:}); 5= T$h;O  
        rpowern = [ones(length_r,1) rpowern]; 6j95>}@  
    else 7^tYtMm|U  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NS-0-o|4#  
        rpowern = cat(2,rpowern{:}); S0M i  
    end ~RLWr.pK  
    #x`K4f)  
    % Compute the values of the polynomials: ~F%sO'4!  
    % -------------------------------------- mZO-^ct4  
    z = zeros(length_r,length_n); "z*.Bk  
    for j = 1:length_n ZG-#YF.1  
        s = 0:(n(j)-m(j))/2; f3S 8~!  
        pows = n(j):-2:m(j); t~ <HFY*w  
        for k = length(s):-1:1 &j,# 5f(  
            p = (1-2*mod(s(k),2))* ... >~_z#2PA  
                       prod(2:(n(j)-s(k)))/          ... bM.$D-?dF*  
                       prod(2:s(k))/                 ... <KfR)7I$0a  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... yzZzaYv "/  
                       prod(2:((n(j)+m(j))/2-s(k))); +s^nT{B@\  
            idx = (pows(k)==rpowers); ;e.8EL  
            z(:,j) = z(:,j) + p*rpowern(:,idx); &XCP@@T  
        end [5ncBY*A7  
         O p1TsRm5L  
        if isnorm %a FZbLK  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); #+i:s92],  
        end 6l T< lzT  
    end |?nYs>K  
    cQ'x]u_  
    % 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)  |oSt%l Q1  
    e 2N F.  
    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。 YGM7?o  
    bA Yp }  
    07年就写过这方面的计算程序了。
    让光学不再神秘,让光学变得容易,快速实现客户关于光学的设想与愿望。