[求助]ansys分析后面型数据如何进行zernike多项式拟合？ [复制链接]

小弟不是学光学的，所以想请各位大侠指点啊！谢谢啦 >>,G3/Zd*  
就是我用ansys计算出了镜面的面型的数据，怎样可以得到zernike多项式的系数，然后用zemax各阶得到像差！谢谢啦！ cO*g4VL"[  

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)  E(r_mF7:  

,=u;1  
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。 c,~uurVi  
yxt"vm;  
07年就写过这方面的计算程序了。

我也一直想了解这个多项式的应用，还没用过呢

我也正在找啊

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！

function z = zernpol(n,m,r,nflag) pm6#azQ  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. o$No@~%v  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of U~x]2{}  
%   order N and frequency M, evaluated at R.  N is a vector of <\&9Odqc  
%   positive integers (including 0), and M is a vector with the \d@5*q  
%   same number of elements as N.  Each element k of M must be a hq=;ZI  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) :21d  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is +RooU?Aq  
%   a vector of numbers between 0 and 1.  The output Z is a matrix b>uD-CSA  
%   with one column for every (N,M) pair, and one row for every 5CY%h  
%   element in R. ok:uTeJI  
% 4IeCb?  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- E8PDIjp  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is mku@n;Hl_  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to xKUL}>8  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 5BVvT `<  
%   for all [n,m]. P 45Ir
ir  
% K<>kT4  
%   The radial Zernike polynomials are the radial portion of the F3|^b{'zO  
%   Zernike functions, which are an orthogonal basis on the unit ,PlH
|  
%   circle.  The series representation of the radial Zernike FNQ<k[#K'~  
%   polynomials is ,2M}qs"P7G  
% Z8SwW<{ $  
%          (n-m)/2 d[a(uWEl  
%            __ nR_Zrm  
%    m      \       s                                          n-2s CHgip&(.F  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r "hk#pQ  
%    n      s=0 o[ 5dR<  
% qf? "v;  
%   The following table shows the first 12 polynomials. pD<w@2K  
% bZ`v1d (r  
%       n    m    Zernike polynomial    Normalization vIV|y>;g  
%       --------------------------------------------- CWdsOS=  
%       0    0    1                        sqrt(2) @DysM~I  
%       1    1    r                           2 BSm"]!D8*  
%       2    0    2*r^2 - 1                sqrt(6) +z;*r8d<X  
%       2    2    r^2                      sqrt(6) H>TO8;5(  
%       3    1    3*r^3 - 2*r              sqrt(8) f[o~d`
z  
%       3    3    r^3                      sqrt(8) UoT`/.  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) As,`($=  
%       4    2    4*r^4 - 3*r^2            sqrt(10) Y1PR?c Q  
%       4    4    r^4                      sqrt(10) y'2|E+*V  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ^k6_j\5j  
%       5    3    5*r^5 - 4*r^3            sqrt(12) & zDuh[j}  
%       5    5    r^5                      sqrt(12) xM jn=\}  
%       --------------------------------------------- /!5Wd(:  
% )?rq8VO  
%   Example: h^3gYL7O6  
% 5u$.!l8Nl  
%       % Display three example Zernike radial polynomials R"t#dG]1t  
%       r = 0:0.01:1; '/>Mr!H#  
%       n = [3 2 5]; s#X/ F  
%       m = [1 2 1]; )iN;1>  
%       z = zernpol(n,m,r); 3aqH!?rVU  
%       figure \YvG+7a  
%       plot(r,z) hr vTFJ  
%       grid on '@{:FrG*U  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') e9LX0=  
% @)Vpj\jM-C  
%   See also ZERNFUN, ZERNFUN2. | eBwcC#^  
"Z@P&jl  
% A note on the algorithm. qDdO-fPev  
% ------------------------ Y
% iqSY  
% The radial Zernike polynomials are computed using the series m.<_WXH  
% representation shown in the Help section above. For many special 5H3o?x
 
% functions, direct evaluation using the series representation can 65LtCQ}  
% produce poor numerical results (floating point errors), because o#qdgZ  
% the summation often involves computing small differences between j)J |'b|  
% large successive terms in the series. (In such cases, the functions dseI~}  
% are often evaluated using alternative methods such as recurrence j yHa}OT  
% relations: see the Legendre functions, for example). For the Zernike f{9+,z   
% polynomials, however, this problem does not arise, because the ^to*ET{0  
% polynomials are evaluated over the finite domain r = (0,1), and o
5Oig  
% because the coefficients for a given polynomial are generally all owNwj  
% of similar magnitude. #49l\>1z  
% $?GggP d  
% ZERNPOL has been written using a vectorized implementation: multiple tc~gn!"  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] vXKL<  
% values can be passed as inputs) for a vector of points R.  To achieve lNe4e6  
% this vectorization most efficiently, the algorithm in ZERNPOL I!/32* s1t  
% involves pre-determining all the powers p of R that are required to ,3:f4e\<  
% compute the outputs, and then compiling the {R^p} into a single !u7KgB<=/F  
% matrix.  This avoids any redundant computation of the R^p, and {it.F4.  
% minimizes the sizes of certain intermediate variables. NpVL;6?7T  
% S`-I-VS=L  
%   Paul Fricker 11/13/2006 lelmX  
i c{I  
F,_L}  
% Check and prepare the inputs: k0Vri$x  
% ----------------------------- xw*e`9vAe  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @<W

` w  
    error('zernpol:NMvectors','N and M must be vectors.') 8
\{!*?9!  
end uda++^y:  
pm O9mWq   
if length(n)~=length(m) k^7!iOK2  
    error('zernpol:NMlength','N and M must be the same length.') }IygU 6{G  
end 'o&d!  
w:zo \  
n = n(:); *f+s  
m = m(:); ;+75"=[YT  
length_n = length(n); n@pwOHQn<|  
_9BL7W $;  
if any(mod(n-m,2)) y[McdlH m  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') SK}jhm"y  
end Luao?;|U  
aImzK/  
if any(m<0) Z}O]pm>=G  
    error('zernpol:Mpositive','All M must be positive.') z83v J*.  
end
Jt$YSp=!!  
~~yng-3)1  
if any(m>n) +?\JQ|  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ld(60?z>FH  
end }+jB5z'w  
JE O$v|X  
if any( r>1 | r<0 ) [#KY.n  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') xOkdu
k]  
end Y1cL dQn  
]t<=a6<P  
if ~any(size(r)==1) IJf%OA>v  
    error('zernpol:Rvector','R must be a vector.') >33=0<  
end ;Tbo \Wp9  
4QQt 0u0  
r = r(:); K+Him] b  
length_r = length(r); 'bbw0aB4  
45biy(qa  
if nargin==4 aQoB1qd8  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); @Z/jaAjUC  
    if ~isnorm 8cO?VH,nk  
        error('zernpol:normalization','Unrecognized normalization flag.') '6zZ`Ll9  
    end NLZ5 5yo$  
else _sy{rnaqvb  
    isnorm = false; w,P@@Q E  
end 10Eun }  
B2r[oT R  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TX5??o  
% Compute the Zernike Polynomials 4$^mLD$>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kO)Y|zQ  
O n0!
>-b,  
% Determine the required powers of r:  Ht.P670  
% ----------------------------------- B-g uz  
rpowers = []; o"Xv)#g&  
for j = 1:length(n) Op0*tj2i),  
    rpowers = [rpowers m(j):2:n(j)]; o;c"-^>  
end <Ve0PhK  
rpowers = unique(rpowers); DWtITO>  
38sLyoG=i  
% Pre-compute the values of r raised to the required powers, @Yt394gA%\  
% and compile them in a matrix: uWx<J3~q.  
% ----------------------------- qBF|' .$^  
if rpowers(1)==0 6!i`\>I]  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ((Av3{05H&  
    rpowern = cat(2,rpowern{:}); IS"[<  
    rpowern = [ones(length_r,1) rpowern]; {zZ)JWM<w  
else `Fcr`[  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z1b@JCWE  
    rpowern = cat(2,rpowern{:}); nql1I<I  
end :PV3J0pB~  
xjBY6Ylz  
% Compute the values of the polynomials: ok6t| 7sq  
% -------------------------------------- RQ0^ 1 R  
z = zeros(length_r,length_n); 7zzFM  
for j = 1:length_n 6`-<N!  
    s = 0:(n(j)-m(j))/2; !{,2uQXe  
    pows = n(j):-2:m(j); 2r2:  
    for k = length(s):-1:1 xw{K,;WeO  
        p = (1-2*mod(s(k),2))* ... =zH)R0!eG  
                   prod(2:(n(j)-s(k)))/          ... ,#N}Ni:  
                   prod(2:s(k))/                 ... mfj%-)l9  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... *_YH}U  
                   prod(2:((n(j)+m(j))/2-s(k))); @D[+@N  
        idx = (pows(k)==rpowers); ?h1g$SBxk  
        z(:,j) = z(:,j) + p*rpowern(:,idx); a{ke%
W$*P  
    end U;kNo3=  
     kJ%a;p`O  
    if isnorm l`#rhuy`  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); dB~A4pZa  
    end K:e[#b8:R  
end L_T+KaQCH  
$$Tf1hIg  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) Vo[.^0  
%ZERNFUN2 Single-index Zernike functions on the unit circle. hVyeHbx  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated OI0@lSAo<  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive N`d%4)|{  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, uzb|yV'B  
%   and THETA is a vector of angles.  R and THETA must have the same >B``+Z^2  
%   length.  The output Z is a matrix with one column for every P-value, ,Y| ;V  
%   and one row for every (R,THETA) pair. OW6dK#CFt  
% <}.!G>X  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike CXuMNa  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (I6Q"&h]  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 9*~";{O.Oa  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 >`[+24e  
%   for all p. jT]R"U/Q  
% d%L/[.&  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 n@H;*nI|  
%   Zernike functions (order N<=7).  In some disciplines it is EqBTN07dZS  
%   traditional to label the first 36 functions using a single mode }!r pH{y  
%   number P instead of separate numbers for the order N and azimuthal sSk qU  
%   frequency M. ~-G_
c=E?  
% cb|hIn\>7  
%   Example: vV=rBO0a?  
% cM<08-:v  
%       % Display the first 16 Zernike functions OrL4G `O  
%       x = -1:0.01:1; KRR)pT  
%       [X,Y] = meshgrid(x,x); GbQg(%2F  
%       [theta,r] = cart2pol(X,Y); *Jt+-ZM  
%       idx = r<=1; f6\4,()  
%       p = 0:15; pI.8Ip_r  
%       z = nan(size(X)); fGA#0/_`  
%       y = zernfun2(p,r(idx),theta(idx)); !M)] 1Y
 
%       figure('Units','normalized') Z:<w
B#G  
%       for k = 1:length(p) -glGOTk  
%           z(idx) = y(:,k); S|KUh|=Q  
%           subplot(4,4,k) 2[1t )EW  
%           pcolor(x,x,z), shading interp uK#2vgT  
%           set(gca,'XTick',[],'YTick',[]) B7Zi|-F
  
%           axis square 4$mtc*tzT  
%           title(['Z_{' num2str(p(k)) '}']) !?J-Y  
%       end -2u)orWP  
% zmu+un"\j  
%   See also ZERNPOL, ZERNFUN. 8N |K
  
$tc1te  
%   Paul Fricker 11/13/2006 uvR l`"Y  
CbxWK#aMmB  
UxF9Ko( ]d  
% Check and prepare the inputs:  V_
e  
% ----------------------------- b>#=7;  
if min(size(p))~=1 nWK7*  
    error('zernfun2:Pvector','Input P must be vector.') ;bHS^  
end {6
1Y;  
2 p}I  
if any(p)>35 zN)).a  
    error('zernfun2:P36', ... / $s(OFbi#  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... P0sAq7"  
           '(P = 0 to 35).']) f`}/^*D  
end +T4}wm  
ZCBF&.!  
% Get the order and frequency corresonding to the function number: ?'H+u[1.  
% ---------------------------------------------------------------- `}L{gssv  
p = p(:); '.gi@Sr5  
n = ceil((-3+sqrt(9+8*p))/2); ^ rUq{  
m = 2*p - n.*(n+2); z yp3+|  
%] :ZAmN  
% Pass the inputs to the function ZERNFUN: mfffOG  
% ---------------------------------------- k!bJ&} Q(b  
switch nargin 19[!9ci  
    case 3 _I3v"d  
        z = zernfun(n,m,r,theta); 8X278^ #  
    case 4 2}*8( 32  
        z = zernfun(n,m,r,theta,nflag); D dCcsYm,  
    otherwise -0]%#(E%`h  
        error('zernfun2:nargin','Incorrect number of inputs.') &m\Uc  
end 5:5d=7WX  
f<|*^+  
% EOF zernfun2

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 rce._w }  
function z = zernfun(n,m,r,theta,nflag) Dbq/t^  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. OQKc_z'"  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \I<R.49oW  
%   and angular frequency M, evaluated at positions (R,THETA) on the vfXNN F  
%   unit circle.  N is a vector of positive integers (including 0), and 28c6~*Te#  
%   M is a vector with the same number of elements as N.  Each element &#gh :5  
%   k of M must be a positive integer, with possible values M(k) = -N(k) $"MVr5q6  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, wf\7sz  
%   and THETA is a vector of angles.  R and THETA must have the same 8K8jz9.s  
%   length.  The output Z is a matrix with one column for every (N,M) WB<MU:.Vc  
%   pair, and one row for every (R,THETA) pair. .=d40m  
% )~ &gBX  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {X_
I>)Wg  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), fBz|-I:k +  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral :qj;f];|  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?_p!teb  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized H5 :,hrZY  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. pjoyMHWK  
% esQ`6i  
%   The Zernike functions are an orthogonal basis on the unit circle. K)+]as  
%   They are used in disciplines such as astronomy, optics, and C+%eT&
OO  
%   optometry to describe functions on a circular domain. @,c`#,F/  
% n6M#Xc'JA  
%   The following table lists the first 15 Zernike functions. X?&{< vz  
% %W=BdGr[8z  
%       n    m    Zernike function           Normalization C@zG(?X  
%       -------------------------------------------------- ._<, Eodv  
%       0    0    1                                 1 sX3qrRY  
%       1    1    r * cos(theta)                    2 Qnt9x,
1m_  
%       1   -1    r * sin(theta)                    2 Uq{$j5p8  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) `_i|\}tl  
%       2    0    (2*r^2 - 1)                    sqrt(3) piuM#+Y\'S  
%       2    2    r^2 * sin(2*theta)             sqrt(6) Dsc0;7~6  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) rwio>4=  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1w7XM0SHcn  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) .}Ys+d1b9c  
%       3    3    r^3 * sin(3*theta)             sqrt(8) q4G$I?4  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) d<HO~+9  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W\5 -Yg(@  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) P{:Zxli0  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) . &`Y
lK  
%       4    4    r^4 * sin(4*theta)             sqrt(10) N`3^:EJL8  
%       -------------------------------------------------- \&ZEIAe  
% 7'Hh^0<
 
%   Example 1: mh`uvqY  
% q8;MPXSG3  
%       % Display the Zernike function Z(n=5,m=1) x*=m'IM[  
%       x = -1:0.01:1; JP5en  
%       [X,Y] = meshgrid(x,x); $/5\Hg1  
%       [theta,r] = cart2pol(X,Y); v0=v1G*rvJ  
%       idx = r<=1; yHlQKI  
%       z = nan(size(X)); i_l{#*t  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); :F#^Q%-IS  
%       figure *tk=DsRW  
%       pcolor(x,x,z), shading interp hx8pg,X  
%       axis square, colorbar h(J$-SUs  
%       title('Zernike function Z_5^1(r,\theta)') %hw4IcWJ|  
% }|N88PN  
%   Example 2: DHuvHK0#  
% ["Tro;K#  
%       % Display the first 10 Zernike functions :RJo#ape  
%       x = -1:0.01:1; .a(G=fk  
%       [X,Y] = meshgrid(x,x); dTu*%S1Z  
%       [theta,r] = cart2pol(X,Y); T<b*=i  
%       idx = r<=1; :A:7^jrhi  
%       z = nan(size(X)); *qAG0EM|  
%       n = [0  1  1  2  2  2  3  3  3  3]; =h +SZXe<r  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; m|x_++3  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 0R`>F">  
%       y = zernfun(n,m,r(idx),theta(idx)); ^,vFxN--q  
%       figure('Units','normalized') MU2kA&LH  
%       for k = 1:10 m .(\u?J  
%           z(idx) = y(:,k); f7!48
,(fB  
%           subplot(4,7,Nplot(k)) T /IX(b'<  
%           pcolor(x,x,z), shading interp 9) $[W  
%           set(gca,'XTick',[],'YTick',[]) {hN<Ot  
%           axis square &y|PseH"  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) d)D!np=  
%       end P?c V d2Y  
% U 0~BcFpD  
%   See also ZERNPOL, ZERNFUN2. F9r/ M"5  
%6^nb'l'C  
%   Paul Fricker 11/13/2006 lcy+2)+  
*P]]7DR  
D+! S\~u  
% Check and prepare the inputs: 3 Fy CD4#  
% ----------------------------- <RbfW'<G  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tlg}"lY  
    error('zernfun:NMvectors','N and M must be vectors.') nhC8Tq[m  
end %H&WihQ  
i O?f&u  
if length(n)~=length(m) PNo:vRtsq  
    error('zernfun:NMlength','N and M must be the same length.') [q_62[-X  
end b\o>4T  
r
|\{!;7  
n = n(:); ahCwA}  
m = m(:); \v<S:cTf  
if any(mod(n-m,2)) ht>/7.p]  
    error('zernfun:NMmultiplesof2', ... iycceZ  
          'All N and M must differ by multiples of 2 (including 0).') yD.(j*bMK;  
end >hq{:m  
q@XJ,e1A  
if any(m>n) *icaKy3  
    error('zernfun:MlessthanN', ... _5(p=Zc  
          'Each M must be less than or equal to its corresponding N.') _y>drvg  
end F$1{w"&  
!TY0;is  
if any( r>1 | r<0 ) S%Ky+0  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8
9{;R  
end u;1[_~  
! 9*l!(  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) be]/ROP>H  
    error('zernfun:RTHvector','R and THETA must be vectors.') i[FY
R;C  
end GE=S.P;  
vkR~nIp  
r = r(:); On!+7is'  
theta = theta(:); !v9`oL26  
length_r = length(r); m?Cb^WgcF  
if length_r~=length(theta) p}/D{|xO  
    error('zernfun:RTHlength', ... @W @,8e]c  
          'The number of R- and THETA-values must be equal.') -a~n_Z>_  
end n&|N=zh  
4!xRA''  
% Check normalization: 1
{d;Ngx  
% -------------------- vUO[V$rx  
if nargin==5 && ischar(nflag) F:jtzy"  
    isnorm = strcmpi(nflag,'norm'); i!3*)-a\~`  
    if ~isnorm -wl&~}%M  
        error('zernfun:normalization','Unrecognized normalization flag.') V:P]Ved  
    end ./0wt+  
else "zTy_0[;  
    isnorm = false; hy%5LV<(  
end &sBD0R(a  
v.TgB)  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *mWl=J;u  
% Compute the Zernike Polynomials ~=[5X,Ta  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7,Z<PE  
88[u^aC  
% Determine the required powers of r: yIngenr$  
% ----------------------------------- 3W#E$^G_v  
m_abs = abs(m); 4t/?b  
rpowers = []; $9X?LGUz  
for j = 1:length(n) g=qaq  
    rpowers = [rpowers m_abs(j):2:n(j)]; EowzEGq!a5  
end :5T=y @  
rpowers = unique(rpowers); bXXX-Xc  
'X6Y!VDd  
% Pre-compute the values of r raised to the required powers, }opMf6`w  
% and compile them in a matrix: /Lm~GmPt  
% ----------------------------- di9OQ*6a7  
if rpowers(1)==0 eK*oV}U-k  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l.Ev]G/5  
    rpowern = cat(2,rpowern{:}); {+d)M  
    rpowern = [ones(length_r,1) rpowern]; VSV]6$~H  
else yuJ>xsM  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7w8UnPuM  
    rpowern = cat(2,rpowern{:}); _G.!^+)kEm  
end 5M5vxJ)Lh  
=Bm|9A1  
% Compute the values of the polynomials: \*b  .f  
% -------------------------------------- P7bb2"_9  
y = zeros(length_r,length(n)); ;8eGf'  
for j = 1:length(n) zOFHdd ,"g  
    s = 0:(n(j)-m_abs(j))/2; .q4$)8[Pg  
    pows = n(j):-2:m_abs(j); 5ZH3}B^L$  
    for k = length(s):-1:1 GJ2ZK=/  
        p = (1-2*mod(s(k),2))* ... P{_%p<:V  
                   prod(2:(n(j)-s(k)))/              ... ~%M*@fm  
                   prod(2:s(k))/                     ... g"Ueo'd*  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... EfpMzD7/(  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); o:3(J}  
        idx = (pows(k)==rpowers); Hy,""Py  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 1-PlRQs.1  
    end }YM\IPsPu  
     xaoR\H  
    if isnorm B>=D$*_  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _~C1M&b(X3  
    end En\q. 3 5  
end g"m9[R=]6  
% END: Compute the Zernike Polynomials LYd:S  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FeO1%#2<y  
.8%b;b  
% Compute the Zernike functions: M l@F  
% ------------------------------ `#8kJt  
idx_pos = m>0; IhZn  
idx_neg = m<0; 7ZyP  
wQd8/&mmk  
z = y; )S`[ gK  
if any(idx_pos) )rAJ>;  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >G%oWRk  
end S^p^) fAmF  
if any(idx_neg) #-+Q]}fB4  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5$Kj#9g-#  
end V rx,'/IS8  
q+*\'H>  
% EOF zernfun

泽尼克多项式的前9项对应象差的

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