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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 正序阅读 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 :M22P`:  
    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! F-BJe]  
     
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    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)  EPc!p>  
    @4;&hP2Z:  
    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。 br b[})}  
    4[`[mE18.  
    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。
    www.rivolens.com
    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niuhelen
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    只看该作者 5楼 发表于: 2011-03-12
    function z = zernpol(n,m,r,nflag) EL7T'zJ$  
    %ZERNPOL Radial Zernike polynomials of order N and frequency M. r{S=Z~J  
    %   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of ?I[h~vr6.  
    %   order N and frequency M, evaluated at R.  N is a vector of Jn*Nao_)  
    %   positive integers (including 0), and M is a vector with the g5}lLKT  
    %   same number of elements as N.  Each element k of M must be a @^!\d#/M  
    %   positive integer, with possible values M(k) = 0,2,4,...,N(k) 9 ZGV%Tw  
    %   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Fj^AW v^/  
    %   a vector of numbers between 0 and 1.  The output Z is a matrix >$uUuiyL4  
    %   with one column for every (N,M) pair, and one row for every j;iL&eo>  
    %   element in R. akC>s8tqlA  
    % Y 9i][  
    %   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- E?Cj/o  
    %   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is i=L8=8B`  
    %   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to j&CZ=?K^c  
    %   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 hM>*a!)U  
    %   for all [n,m]. @8zT'/$  
    % ({e7U17[#  
    %   The radial Zernike polynomials are the radial portion of the ]G*$W+G]  
    %   Zernike functions, which are an orthogonal basis on the unit skR,-:"8  
    %   circle.  The series representation of the radial Zernike Szts<n5  
    %   polynomials is JT)k  
    % q 5p e~  
    %          (n-m)/2 ;& ~929  
    %            __ U Du~2%  
    %    m      \       s                                          n-2s o,FUfO}F  
    %   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 2r;GcjezH  
    %    n      s=0 <tuS,.  
    % _CE9B e\  
    %   The following table shows the first 12 polynomials. _2OuskL  
    % U:n*<l-k}  
    %       n    m    Zernike polynomial    Normalization h<Wg3o  
    %       --------------------------------------------- =/9<(Tt%m  
    %       0    0    1                        sqrt(2) h<% U["   
    %       1    1    r                           2 ^f|<R8`  
    %       2    0    2*r^2 - 1                sqrt(6) KjOi(YUnq7  
    %       2    2    r^2                      sqrt(6) G9Qe121m  
    %       3    1    3*r^3 - 2*r              sqrt(8) }9CrFTbx;  
    %       3    3    r^3                      sqrt(8) dB3N%pB^  
    %       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 8=F%+  
    %       4    2    4*r^4 - 3*r^2            sqrt(10) PN &|8_  
    %       4    4    r^4                      sqrt(10) v=p0 +J>  
    %       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 1lcnRHO  
    %       5    3    5*r^5 - 4*r^3            sqrt(12) %[ Zz0|A  
    %       5    5    r^5                      sqrt(12) oZ:{@ =  
    %       --------------------------------------------- 5B*qbM  
    % m7m \`;  
    %   Example: e$wt&^W  
    % !SF^a6jT  
    %       % Display three example Zernike radial polynomials 9m8ee&,  
    %       r = 0:0.01:1; ? )_7U  
    %       n = [3 2 5]; /{X_ .fv<v  
    %       m = [1 2 1]; Ae49n4J  
    %       z = zernpol(n,m,r); wmYvD<  
    %       figure d#\W hRE  
    %       plot(r,z) rk,p!}FqL  
    %       grid on  $Nu)E  
    %       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') +'XhC#:  
    % b Us|t  
    %   See also ZERNFUN, ZERNFUN2. e{C6by"j{S  
    "'A"U  
    % A note on the algorithm. |C D}<r(N  
    % ------------------------ @~o`#$*|  
    % The radial Zernike polynomials are computed using the series =i6:puf  
    % representation shown in the Help section above. For many special C).2gQ G  
    % functions, direct evaluation using the series representation can frqJN  
    % produce poor numerical results (floating point errors), because @^? XaU  
    % the summation often involves computing small differences between ,4Y*:JU4  
    % large successive terms in the series. (In such cases, the functions O&'/J8  
    % are often evaluated using alternative methods such as recurrence  I=[cZ;t  
    % relations: see the Legendre functions, for example). For the Zernike 5AOfp2O  
    % polynomials, however, this problem does not arise, because the xR908+>5  
    % polynomials are evaluated over the finite domain r = (0,1), and D$nK`r  
    % because the coefficients for a given polynomial are generally all DB@EVH  
    % of similar magnitude. r7I B{}>-  
    % <zfKC  
    % ZERNPOL has been written using a vectorized implementation: multiple If'2rE7J  
    % Zernike polynomials can be computed (i.e., multiple sets of [N,M] xK;e\^v  
    % values can be passed as inputs) for a vector of points R.  To achieve NHkL24ve  
    % this vectorization most efficiently, the algorithm in ZERNPOL Y {c5  
    % involves pre-determining all the powers p of R that are required to J&6:d  
    % compute the outputs, and then compiling the {R^p} into a single wFL3& *  
    % matrix.  This avoids any redundant computation of the R^p, and 8R xc&`_X  
    % minimizes the sizes of certain intermediate variables. ]W) jmw'mo  
    % Q)^g3J  
    %   Paul Fricker 11/13/2006 n )K6i7]xk  
    jvs[ /  
    tt4+m>/T  
    % Check and prepare the inputs: Fe$/t(  
    % ----------------------------- A/ 0qk  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) s#^pC*,'  
        error('zernpol:NMvectors','N and M must be vectors.') 7& 6Y  
    end f=I:DkR  
    2c:f<>r0y  
    if length(n)~=length(m) G;]:$J  
        error('zernpol:NMlength','N and M must be the same length.') Fu)Th|5GZ  
    end vv/J 5#^,\  
    ^l!L)iw  
    n = n(:); @vs@>CYdz  
    m = m(:); F~_;o+e;X  
    length_n = length(n); Z2p> n`D  
    L2|aHI1'l  
    if any(mod(n-m,2)) #@Y/{[s|@  
        error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Sz0+ <F#5  
    end wUp)JI  
    _;e\:7<m  
    if any(m<0) -<W2PY<  
        error('zernpol:Mpositive','All M must be positive.') O9]\Q@M.  
    end (@&I_>2Q  
    x / XkD]Hq  
    if any(m>n) 8$}OS-  
        error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 8/Rm!.8+~  
    end ^7u#30,}3~  
    $TG?4  
    if any( r>1 | r<0 ) \&q=@rJp(z  
        error('zernpol:Rlessthan1','All R must be between 0 and 1.') 4}yE+dRUK:  
    end =!<G!^  
    |}S1o0v{(a  
    if ~any(size(r)==1) ((MLM3zJ  
        error('zernpol:Rvector','R must be a vector.') F?4Sz#  
    end *fhX*e8y  
    T *P+Fh"  
    r = r(:); Y60"M4j  
    length_r = length(r); DO^y;y>  
    y1,5$0@G  
    if nargin==4 Tsz NlRxc  
        isnorm = ischar(nflag) & strcmpi(nflag,'norm'); h>3H7n.  
        if ~isnorm y1C/v:;  
            error('zernpol:normalization','Unrecognized normalization flag.') ;<-7*}Dj  
        end SrZ50Se  
    else xzk}[3P{  
        isnorm = false; 25em[Q:  
    end 5"=:#zN  
    bK=c@GXS  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gq4 . d  
    % Compute the Zernike Polynomials - ]Y wl  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6P+DnS[]  
    B8~= RmWLl  
    % Determine the required powers of r: *K)0UKBr  
    % ----------------------------------- u~'OcO  
    rpowers = []; \A(5;ZnuD  
    for j = 1:length(n) pP\h6b+B  
        rpowers = [rpowers m(j):2:n(j)]; `K?1L{p'4  
    end (I=6Nnt'  
    rpowers = unique(rpowers); g42T#p8^  
    \ X$)vK  
    % Pre-compute the values of r raised to the required powers, yNva1I  
    % and compile them in a matrix: {tt$w>X  
    % ----------------------------- \"d?=uFe  
    if rpowers(1)==0 J Jy{@[m  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D#v?gPo4  
        rpowern = cat(2,rpowern{:}); 295w.X(J  
        rpowern = [ones(length_r,1) rpowern]; cU+>|'f &  
    else YzV(nEW  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); O5-;I,)H  
        rpowern = cat(2,rpowern{:}); &=zU611,  
    end :_H>SR:  
    %dmfBf Ev  
    % Compute the values of the polynomials: Yu3_=: <C  
    % -------------------------------------- ]{Ek[Av  
    z = zeros(length_r,length_n); jG8;]XP  
    for j = 1:length_n v@_in(dk  
        s = 0:(n(j)-m(j))/2; dS$ji#+d$  
        pows = n(j):-2:m(j); %F5 =n"  
        for k = length(s):-1:1 oT->^4WY  
            p = (1-2*mod(s(k),2))* ... jo0Pd_W8&  
                       prod(2:(n(j)-s(k)))/          ... sqG`"O4W  
                       prod(2:s(k))/                 ... D5` (}  
                       prod(2:((n(j)-m(j))/2-s(k)))/ ... Zh,]J `  
                       prod(2:((n(j)+m(j))/2-s(k))); \n0gTwiO%  
            idx = (pows(k)==rpowers); !$r9C/k  
            z(:,j) = z(:,j) + p*rpowern(:,idx); F+*Q <a4  
        end "Ot{^ _e  
         :bv|Ah  
        if isnorm Qa?aL  
            z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); mV zu~xym  
        end $ !5f"<FCB  
    end <E}N=J'uJ  
    C BlXC7_Mi  
    % EOF zernpol
    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) G>0)I  
    %ZERNFUN2 Single-index Zernike functions on the unit circle. 9K$]h2  
    %   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Y)OBTX  
    %   at positions (R,THETA) on the unit circle.  P is a vector of positive 4W-"|Z_x  
    %   integers between 0 and 35, R is a vector of numbers between 0 and 1, 7@~QkTH~y  
    %   and THETA is a vector of angles.  R and THETA must have the same .C?GW1[c~@  
    %   length.  The output Z is a matrix with one column for every P-value, ,D8&q?a  
    %   and one row for every (R,THETA) pair. %,;gP.dh7  
    % GBr,LN  
    %   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike >{F!ntEj  
    %   functions, defined such that the integral of (r * [Zp(r,theta)]^2) QQ~23TlA  
    %   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;NG1{]|Z  
    %   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 @<5Tba>SC  
    %   for all p. InN{^uN  
    % X~zRZ0  
    %   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 mQ=sNZ-d]  
    %   Zernike functions (order N<=7).  In some disciplines it is U2UyN9:6F  
    %   traditional to label the first 36 functions using a single mode 0Jg+sUs{  
    %   number P instead of separate numbers for the order N and azimuthal sW'6} ^Q  
    %   frequency M. raF] k0{  
    % 6Ty 3e|do  
    %   Example: :ln/`_  
    % Z>h{` X\2  
    %       % Display the first 16 Zernike functions L ]*`4 L  
    %       x = -1:0.01:1; WP!il(Gr  
    %       [X,Y] = meshgrid(x,x); UEN56@eCNf  
    %       [theta,r] = cart2pol(X,Y); g@(4ujOT  
    %       idx = r<=1; Y2D >tpqNw  
    %       p = 0:15; 8? F 2jv  
    %       z = nan(size(X)); nj"m^PmWo3  
    %       y = zernfun2(p,r(idx),theta(idx)); "?[7#d])  
    %       figure('Units','normalized') j"0rkN3$J  
    %       for k = 1:length(p) 1~L\s}|2d  
    %           z(idx) = y(:,k); l{QC}{Ejc2  
    %           subplot(4,4,k) :fW.-^"VP  
    %           pcolor(x,x,z), shading interp L,W:,i/C  
    %           set(gca,'XTick',[],'YTick',[]) )WclV~  
    %           axis square <dS5|||  
    %           title(['Z_{' num2str(p(k)) '}']) v!KJ|c@m  
    %       end GqMB^Ad  
    % 18rp; l{  
    %   See also ZERNPOL, ZERNFUN. \eT/%$  
    SlT7L||Ww  
    %   Paul Fricker 11/13/2006 _E (x2BS?  
    .=CH!{j  
    s_S$7N`ocS  
    % Check and prepare the inputs: DEcGFRgN~  
    % ----------------------------- 5E0wn'  
    if min(size(p))~=1 xlk5Gob*  
        error('zernfun2:Pvector','Input P must be vector.') K9^"NS3  
    end oo\IS\  
    f:&)"  
    if any(p)>35 sX Z4U0 #  
        error('zernfun2:P36', ... C,(j$Id  
              ['ZERNFUN2 only computes the first 36 Zernike functions ' ... C&e8a9*,(a  
               '(P = 0 to 35).']) ePi Z  
    end Z|}H^0~7S  
    IE|, ~M2  
    % Get the order and frequency corresonding to the function number: lqauk)(A0  
    % ---------------------------------------------------------------- =^#0.  
    p = p(:); [Zh2DNp  
    n = ceil((-3+sqrt(9+8*p))/2); ps "9;4P  
    m = 2*p - n.*(n+2); k 5t{  
    JLml#Pu4  
    % Pass the inputs to the function ZERNFUN: :)j7U3u  
    % ---------------------------------------- :ET x*c  
    switch nargin T:Bzz)2/  
        case 3 50COL66:7  
            z = zernfun(n,m,r,theta); " xlJs93c  
        case 4 7WXiG0  
            z = zernfun(n,m,r,theta,nflag); @ }ZGY^  
        otherwise Kjv2J;Xuh  
            error('zernfun2:nargin','Incorrect number of inputs.') @PKAz&0  
    end a!&bc8J7  
    80dSQ"y  
    % EOF zernfun2
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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 Tq.MubaO  
    function z = zernfun(n,m,r,theta,nflag) kBQ5]Q"  
    %ZERNFUN Zernike functions of order N and frequency M on the unit circle. cn@03&dAl  
    %   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N LAj}kW~  
    %   and angular frequency M, evaluated at positions (R,THETA) on the QziN]  
    %   unit circle.  N is a vector of positive integers (including 0), and ^tQPJ  
    %   M is a vector with the same number of elements as N.  Each element kx:c*3q.k  
    %   k of M must be a positive integer, with possible values M(k) = -N(k) \dC.%#  
    %   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, O 0Vn";Q 4  
    %   and THETA is a vector of angles.  R and THETA must have the same 8.:B=A  
    %   length.  The output Z is a matrix with one column for every (N,M) ]6jHIk|  
    %   pair, and one row for every (R,THETA) pair. nZ"{y  
    % xV"6d{+  
    %   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zuK/(qZ  
    %   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), F">Nrj-bs  
    %   with delta(m,0) the Kronecker delta, is chosen so that the integral qn5y D!1  
    %   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, s@/B*r9  
    %   and theta=0 to theta=2*pi) is unity.  For the non-normalized ,w,ENU0~f  
    %   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :y4)qF  
    % K-N]h  
    %   The Zernike functions are an orthogonal basis on the unit circle. PWADbu{+  
    %   They are used in disciplines such as astronomy, optics, and XYz,NpK  
    %   optometry to describe functions on a circular domain. iK$Vd+Lgc  
    % d;;>4}XJ]  
    %   The following table lists the first 15 Zernike functions. =>gyc;{2K<  
    % rhlW  
    %       n    m    Zernike function           Normalization Z'7 c^c7_  
    %       -------------------------------------------------- QC$=Fs5+  
    %       0    0    1                                 1 SS l8  
    %       1    1    r * cos(theta)                    2 h;cB_6vt  
    %       1   -1    r * sin(theta)                    2 hY\Eh.  
    %       2   -2    r^2 * cos(2*theta)             sqrt(6) Jnh;;<  
    %       2    0    (2*r^2 - 1)                    sqrt(3) <~IH`  
    %       2    2    r^2 * sin(2*theta)             sqrt(6) fvAh?<Ul  
    %       3   -3    r^3 * cos(3*theta)             sqrt(8) /\IAr,w[  
    %       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) d A_S"Zc  
    %       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) B:R7[G;1  
    %       3    3    r^3 * sin(3*theta)             sqrt(8) wgSR*d>y*9  
    %       4   -4    r^4 * cos(4*theta)             sqrt(10) |G`4"``]k  
    %       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u|_I Twk  
    %       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) / K2.V@T  
    %       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) | TQedC  
    %       4    4    r^4 * sin(4*theta)             sqrt(10) ,kGw;8X  
    %       -------------------------------------------------- rf1wS*uU+  
    % ZuF4N=;  
    %   Example 1: /xX,   
    % v*C+U$_3\1  
    %       % Display the Zernike function Z(n=5,m=1) SK2J`*  
    %       x = -1:0.01:1; ?{ 8sT-Z-L  
    %       [X,Y] = meshgrid(x,x); 'O\d<F.c$2  
    %       [theta,r] = cart2pol(X,Y); #z-iL!?  
    %       idx = r<=1; r{Qs9  
    %       z = nan(size(X)); =F^->e0N  
    %       z(idx) = zernfun(5,1,r(idx),theta(idx)); X:xC>4]gG'  
    %       figure iOX4Kl  
    %       pcolor(x,x,z), shading interp jm#F*F vL  
    %       axis square, colorbar d[P>jl%7  
    %       title('Zernike function Z_5^1(r,\theta)') T@d_ t  
    % XNkQk0i;g&  
    %   Example 2: = C$ @DNEc  
    % 5'{qEZs^QU  
    %       % Display the first 10 Zernike functions 1?e>x91  
    %       x = -1:0.01:1; c'TiWZP~  
    %       [X,Y] = meshgrid(x,x); k#IS ,NKE  
    %       [theta,r] = cart2pol(X,Y);  "df13U"  
    %       idx = r<=1; }Uqa8&  
    %       z = nan(size(X)); MQbNWUi  
    %       n = [0  1  1  2  2  2  3  3  3  3]; }v'PY/d.  
    %       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &4[#_(pk  
    %       Nplot = [4 10 12 16 18 20 22 24 26 28]; lD;,I^Lt6  
    %       y = zernfun(n,m,r(idx),theta(idx)); x[.z"$T@  
    %       figure('Units','normalized') j]ln :?\  
    %       for k = 1:10 Im' :sJ31  
    %           z(idx) = y(:,k); Yt O@n@1  
    %           subplot(4,7,Nplot(k)) =yhn8t7@]  
    %           pcolor(x,x,z), shading interp _($-dJ {  
    %           set(gca,'XTick',[],'YTick',[]) ZB_16&2Ow  
    %           axis square =?`y(k4a  
    %           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) c9ov;Bw6S  
    %       end %x'bo>h@  
    % \/x)BE,  
    %   See also ZERNPOL, ZERNFUN2. :y O,  
    cV=_G E  
    %   Paul Fricker 11/13/2006 #BT6bH08X  
    {Y@-*pL]  
    B;D:9K  
    % Check and prepare the inputs: _?(hWC"0  
    % ----------------------------- Q 2SSJ  
    if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S'%cf7Z  
        error('zernfun:NMvectors','N and M must be vectors.') zoJkDr=jn  
    end x9o^9QJh  
    @cTZ`bg  
    if length(n)~=length(m) l~Wk07r3  
        error('zernfun:NMlength','N and M must be the same length.') Y9co?!J 5M  
    end ]\9B?W(#  
    ncw?;  
    n = n(:); ixJ20A7  
    m = m(:); 5(MZ%-~l  
    if any(mod(n-m,2)) [nPzh Xs  
        error('zernfun:NMmultiplesof2', ... ,d [b"]Zy  
              'All N and M must differ by multiples of 2 (including 0).') ]stLC; nI  
    end ,C'w(af@}  
    ,C@hTOT  
    if any(m>n) d;jJe0pH  
        error('zernfun:MlessthanN', ... A2O_pbQti  
              'Each M must be less than or equal to its corresponding N.') V jB`~  
    end {+UNjKQC  
    kO`3ENN  
    if any( r>1 | r<0 ) (.XDf3   
        error('zernfun:Rlessthan1','All R must be between 0 and 1.')  eS@!\H x  
    end zs]/Y2  
    3NxaOO`  
    if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 'B<qG<>  
        error('zernfun:RTHvector','R and THETA must be vectors.') M_.,c Vk  
    end j+B5m:ExfI  
    O]%m{afM  
    r = r(:); luz%FY:  
    theta = theta(:); `Py= ?[cD  
    length_r = length(r); ^q N1~v=hS  
    if length_r~=length(theta) 8$jT#\_  
        error('zernfun:RTHlength', ... +_XbHjhN/  
              'The number of R- and THETA-values must be equal.') Z"%O&O  
    end &_^*rD~  
    )6R#k8'ERr  
    % Check normalization: ]gZ8b- 2O  
    % -------------------- Ccd7|L1  
    if nargin==5 && ischar(nflag) "KI,3g _V  
        isnorm = strcmpi(nflag,'norm'); T# _n-b>  
        if ~isnorm $I6eHjYT  
            error('zernfun:normalization','Unrecognized normalization flag.') 'hi\98y  
        end JQh s=Xg  
    else m<'xlF  
        isnorm = false; zf;[nz  
    end )w 8lusa  
    *I1W+W`G  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pA,EUh| H  
    % Compute the Zernike Polynomials [.Wt,zrE  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0S5C7df  
    >=W#z  
    % Determine the required powers of r: ~md|k  
    % ----------------------------------- ]3y5b9DuW  
    m_abs = abs(m); MbFe1U]B  
    rpowers = []; *_K-T#  
    for j = 1:length(n) 0V,MDX}#_  
        rpowers = [rpowers m_abs(j):2:n(j)]; ;pyJ O_R[  
    end |mE +f]7$  
    rpowers = unique(rpowers); wukos5  
    9-iB?a7{.  
    % Pre-compute the values of r raised to the required powers, m^x\@!N:(  
    % and compile them in a matrix: 4Hpu EV8Q  
    % ----------------------------- 95 .'t}  
    if rpowers(1)==0 @N_H]6z4  
        rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Qp.!U~  
        rpowern = cat(2,rpowern{:}); /Vd#q)b%T  
        rpowern = [ones(length_r,1) rpowern]; /\$|D&e  
    else IS4K$Ac.  
        rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0#{]!>R  
        rpowern = cat(2,rpowern{:}); o3=S<|V  
    end A-Pwi.$  
    9}0Jc(B/x  
    % Compute the values of the polynomials: Lz{T8yvZ  
    % -------------------------------------- q[+: t   
    y = zeros(length_r,length(n)); fQ_8{=<-&X  
    for j = 1:length(n) $,zM99  
        s = 0:(n(j)-m_abs(j))/2; aq\TO?  
        pows = n(j):-2:m_abs(j); .-~% w  
        for k = length(s):-1:1 uvv-lAbjw  
            p = (1-2*mod(s(k),2))* ... UVmyOC[Y{  
                       prod(2:(n(j)-s(k)))/              ... _v=WjN  
                       prod(2:s(k))/                     ... [K@!JY  
                       prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... vfTG*jG  
                       prod(2:((n(j)+m_abs(j))/2-s(k))); 9.#R?YP$  
            idx = (pows(k)==rpowers); sH,)e'0  
            y(:,j) = y(:,j) + p*rpowern(:,idx); `[:1!I.}-  
        end mf_'| WDs  
         Wy>\KrA1  
        if isnorm /D]r "-  
            y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "<$vU_  
        end [Q_| 6Di  
    end s^R2jueR  
    % END: Compute the Zernike Polynomials 7:q-NzE\6  
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *"sDaN0@R  
    poFjhq /#(  
    % Compute the Zernike functions: tUF]f6  
    % ------------------------------ 6U9Fa=%>}  
    idx_pos = m>0; Ns8NaD  
    idx_neg = m<0; ! e,(Zz5  
    ~EkGG .  
    z = y; uOqDJM'RM  
    if any(idx_pos) j =%-b]  
        z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); %_O>Hy|p  
    end 8B+^vF   
    if any(idx_neg) 3^yWpSC  
        z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 'wV26Dm  
    end G*p.JsZP  
    GJfNO-  
    % EOF zernfun
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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
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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