[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 ,g^Bu{?  
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！ x)V.^-  

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 HOm^.gd  

u1<xt1K  
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。 -Fl3m  
6^ KDc  
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) (/N`Wu  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. _c #P  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9(
B)  
%   order N and frequency M, evaluated at R.  N is a vector of =
0v{+#}  
%   positive integers (including 0), and M is a vector with the [S9nF  
%   same number of elements as N.  Each element k of M must be a s&tr84u|  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?Ts Z_  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is =+"XV8Fi,  
%   a vector of numbers between 0 and 1.  The output Z is a matrix [hiOFmMJZ-  
%   with one column for every (N,M) pair, and one row for every ___+5r21\  
%   element in R. hpw;w}m  
% dkVVvK  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- xbmOch}j6  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is R'80{  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to y\%4Dir  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 5N[Y
2  
%   for all [n,m]. 1-b,X]i  
% FEP\5d>  
%   The radial Zernike polynomials are the radial portion of the a<HM|dcst  
%   Zernike functions, which are an orthogonal basis on the unit y24 0 +;a  
%   circle.  The series representation of the radial Zernike 3yZ@i<rfH  
%   polynomials is dA_s7),  
% /evh.S  
%          (n-m)/2 oF3#]6`;/  
%            __ %8$wod6  
%    m      \       s                                          n-2s |Rab'9U^  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r hz/5k%%UX  
%    n      s=0 =!{dKz-&  
% !}vz_6)  
%   The following table shows the first 12 polynomials. 
i\PN  
% lOE
bh  
%       n    m    Zernike polynomial    Normalization ,qr)}s-  
%       --------------------------------------------- Cf10 ud  
%       0    0    1                        sqrt(2) |epe;/  
%       1    1    r                           2 T8RQM1D_s  
%       2    0    2*r^2 - 1                sqrt(6) B)c.`cfr*\  
%       2    2    r^2                      sqrt(6) VX- f~  
%       3    1    3*r^3 - 2*r              sqrt(8) %b_zUFHPp  
%       3    3    r^3                      sqrt(8) lvFHr}W  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) g:*yjj  
%       4    2    4*r^4 - 3*r^2            sqrt(10) /Ia#udkNMp  
%       4    4    r^4                      sqrt(10) *F9uv)[kz  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) U}{r.MryFG  
%       5    3    5*r^5 - 4*r^3            sqrt(12) .rMGI"  
%       5    5    r^5                      sqrt(12) +V0uHpm  
%       --------------------------------------------- TRQva8d?  
% +-{HT+W  
%   Example: DLz~$TF^  
% 0_j!t  
%       % Display three example Zernike radial polynomials g;*~
xo  
%       r = 0:0.01:1; c5]1aFKz  
%       n = [3 2 5]; f"PApV9[  
%       m = [1 2 1];
=izB :  
%       z = zernpol(n,m,r); <2R=!n@b\  
%       figure z?K+LTf8  
%       plot(r,z) iKdC2m  
%       grid on M9iu#6P  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') PgxU;N7Y  
% Lu<'A4Q1  
%   See also ZERNFUN, ZERNFUN2. B,A/ -B\  
0/*z]2  
% A note on the algorithm. 0phGn+"R  
% ------------------------ . Bv;Zv  
% The radial Zernike polynomials are computed using the series &yP9vp="  
% representation shown in the Help section above. For many special |m?0h.O,  
% functions, direct evaluation using the series representation can bS0LjvY9g  
% produce poor numerical results (floating point errors), because rv\<Q-uQ8  
% the summation often involves computing small differences between UyvFR@  
% large successive terms in the series. (In such cases, the functions _@HMk"A  
% are often evaluated using alternative methods such as recurrence
Q#vur o  
% relations: see the Legendre functions, for example). For the Zernike he!e~5<@y  
% polynomials, however, this problem does not arise, because the .m4K ]^m  
% polynomials are evaluated over the finite domain r = (0,1), and 0BBWuNF.  
% because the coefficients for a given polynomial are generally all ZOU$do>O  
% of similar magnitude. {Ynr(J.  
% z43H]  
% ZERNPOL has been written using a vectorized implementation: multiple x2tx{Z  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] WJhI6lu  
% values can be passed as inputs) for a vector of points R.  To achieve 4sG^bZ,  
% this vectorization most efficiently, the algorithm in ZERNPOL qf'uXH  
% involves pre-determining all the powers p of R that are required to O!;!amvz  
% compute the outputs, and then compiling the {R^p} into a single ]ErAa"?  
% matrix.  This avoids any redundant computation of the R^p, and A}W&=m8!  
% minimizes the sizes of certain intermediate variables. ;Cv x48  
% ?}O\'Fa8
 
%   Paul Fricker 11/13/2006 o^lKM?t  
i)eub`uMy  
S=o Ab&  
% Check and prepare the inputs: F_@PSA+  
% ----------------------------- sl`\g1<{`  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zg"
<N  
    error('zernpol:NMvectors','N and M must be vectors.') Vw+U?  
end 5B"j\TwQ  
( vgoG5  
if length(n)~=length(m) #IgY'L  
    error('zernpol:NMlength','N and M must be the same length.') 9.>v ;:vL  
end XN??^1{J}]  
M$|^?U>cm  
n = n(:); S_1R]n1/  
m = m(:); 
^e)KEkh  
length_n = length(n); |wWBV{^  
yn`H}@`k  
if any(mod(n-m,2)) bluhiiATd  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ~6E `6;`  
end #dU-*wmJ  
3>c
<E1   
if any(m<0) 
Gi?"  
    error('zernpol:Mpositive','All M must be positive.') `WX @1]m  
end tP7l ;EX4  
0~)cAKus  
if any(m>n) L%I@HB9-Q0  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') "1WwSh}Z  
end c]#F^(-A`  
\M<C6m5  
if any( r>1 | r<0 ) e=Kf<ZQt  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') 4E<iIA\
x  
end D&:,,Dp  
{rf.sN~M  
if ~any(size(r)==1) \"|E8A6/  
    error('zernpol:Rvector','R must be a vector.') -n+=[M  
end 4h|sbB"t  
0LeR#l:I  
r = r(:); Xw_AZ-|1D  
length_r = length(r); &O7]e3Ej  
Xu2:yf4No*  
if nargin==4 hZ[,.  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); aF]4%E  
    if ~isnorm .\".}4qQ  
        error('zernpol:normalization','Unrecognized normalization flag.') *FmY4w  
    end ?45bvkCT  
else nj]l'~Y0  
    isnorm = false; .T#h5[S2x  
end }f?$QSF  
sZx
f.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h3[^uYe  
% Compute the Zernike Polynomials :Z3Tyj}4  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X
y5#wDRC  
g\ilK:r}  
% Determine the required powers of r: PuYAoKG  
% ----------------------------------- dtTQY  
rpowers = []; F-D9nI
4{X  
for j = 1:length(n) : M=0o<  
    rpowers = [rpowers m(j):2:n(j)]; p48mk  
end 0go{gUI  
rpowers = unique(rpowers); vz[oy|{F  
`bY>f_5+  
% Pre-compute the values of r raised to the required powers, leR-oeSO  
% and compile them in a matrix: DP_ ]\V<sT  
% ----------------------------- Z8IY!d  
if rpowers(1)==0 # 3UrGom  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Dc-v`jZ@)  
    rpowern = cat(2,rpowern{:}); KW`^uoY$  
    rpowern = [ones(length_r,1) rpowern]; @{n"/6t  
else e98f+,E/  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b\^X1eo  
    rpowern = cat(2,rpowern{:}); ( y0  
end Kg?(Ax4  
5e1;m6  
% Compute the values of the polynomials: v,, .2UR4  
% -------------------------------------- icS%])3LF  
z = zeros(length_r,length_n); !p #m?|Km  
for j = 1:length_n \USl9*E  
    s = 0:(n(j)-m(j))/2; 2 8>  
    pows = n(j):-2:m(j); `X)y5*##wq  
    for k = length(s):-1:1 Z{XF!pS%H  
        p = (1-2*mod(s(k),2))* ... BRSIg]  
                   prod(2:(n(j)-s(k)))/          ... \D67J239E  
                   prod(2:s(k))/                 ... 5y^I~"_i  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ...
Z#
uxa  
                   prod(2:((n(j)+m(j))/2-s(k))); e\)r"!?H`  
        idx = (pows(k)==rpowers); IX+!+XC"U  
        z(:,j) = z(:,j) + p*rpowern(:,idx); c`,'[Q5(O  
    end K4U_sCh#f  
     pz4lC=H%o  
    if isnorm +6~ut^YiM.  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ~ p~  
    end @3*S:;x  
end {gT4Oq__  
-8zdkm8k  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) >Py:9~g,  
%ZERNFUN2 Single-index Zernike functions on the unit circle. u(W>HVEG  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated xYp-Y"a.  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive awB+B8^s  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, Se}&2 R  
%   and THETA is a vector of angles.  R and THETA must have the same x1`4hB  
%   length.  The output Z is a matrix with one column for every P-value, e+~@"^|  
%   and one row for every (R,THETA) pair. 4|/}~9/  
% vJj
}$AlI  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike )ko[_OJj  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Xk] uXx:TN  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) H
-iCaX
T  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ()^tw5e'^  
%   for all p. )tm%0z7R  
% )">uI\bi  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Cu!S|Xj.  
%   Zernike functions (order N<=7).  In some disciplines it is ]P*H,&I`#  
%   traditional to label the first 36 functions using a single mode j4wsDtmAU  
%   number P instead of separate numbers for the order N and azimuthal @6kkt~>:  
%   frequency M. mr
QT:B\8  
% M{t/B-'4  
%   Example: IOd
du2.(  
% K%.t%)A_3  
%       % Display the first 16 Zernike functions gq~K(Q<O<  
%       x = -1:0.01:1; HDi_|{2^  
%       [X,Y] = meshgrid(x,x); 'YB{W8bR  
%       [theta,r] = cart2pol(X,Y); 8;d./!|'&g  
%       idx = r<=1; /$d#9Uv  
%       p = 0:15; 9K>~9Za  
%       z = nan(size(X)); Nd He::  
%       y = zernfun2(p,r(idx),theta(idx)); cTja<*W^xv  
%       figure('Units','normalized') LFAefl\  
%       for k = 1:length(p) ~^/BAc  
%           z(idx) = y(:,k); o'_
eLp  
%           subplot(4,4,k) Z|B`n SzH  
%           pcolor(x,x,z), shading interp ;w;+<Rd  
%           set(gca,'XTick',[],'YTick',[]) =4u
O"o  
%           axis square  p ~pl|  
%           title(['Z_{' num2str(p(k)) '}']) 0 s@>e  
%       end bE!z[j]  
% JLGC'mbJ  
%   See also ZERNPOL, ZERNFUN. -amNz.`[PR  
JMfv|>=  
%   Paul Fricker 11/13/2006 z s\N)LyM  
pmiC|F83!8  
!jR 1!i  
% Check and prepare the inputs: 
z $iI  
% ----------------------------- _xM}*_<VP  
if min(size(p))~=1 ]P2Wa   
    error('zernfun2:Pvector','Input P must be vector.') xB_78X1  
end VVe^s|~Z  
jA3xDbM  
if any(p)>35 G[+{[W  
    error('zernfun2:P36', ... 5Nt9'
"  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 4|hfzCjMI  
           '(P = 0 to 35).']) b?-KC\}v  
end o,qUf  
U9XOs)^  
% Get the order and frequency corresonding to the function number: Ny$N5/b!!  
% ---------------------------------------------------------------- qgxGq(6K  
p = p(:); cS>xT cj  
n = ceil((-3+sqrt(9+8*p))/2); ybc
Cq]cgt  
m = 2*p - n.*(n+2); @=?#nB&  
RijFN.s  
% Pass the inputs to the function ZERNFUN: ^V"
08  
% ---------------------------------------- ;vUw_M{P=)  
switch nargin Dc3bG@K*G  
    case 3 {3BWT  
        z = zernfun(n,m,r,theta); (r-PkfXvIf  
    case 4 |~+bbN|b  
        z = zernfun(n,m,r,theta,nflag); gb26Y!7%  
    otherwise ;Ouu+#s  
        error('zernfun2:nargin','Incorrect number of inputs.') vv
 D515i  
end A<-3u  
^x_+&  
% 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 \; zix(N[5  
function z = zernfun(n,m,r,theta,nflag) (o8?j^ -v  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cKt8e^P  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 51puR8AG>  
%   and angular frequency M, evaluated at positions (R,THETA) on the G}`Hu_ [\)  
%   unit circle.  N is a vector of positive integers (including 0), and
R-5e9vyS  
%   M is a vector with the same number of elements as N.  Each element JjG>$z  
%   k of M must be a positive integer, with possible values M(k) = -N(k) &z"yls  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (oB9$Zz!t  
%   and THETA is a vector of angles.  R and THETA must have the same jxZd =%7Q  
%   length.  The output Z is a matrix with one column for every (N,M)
*%QTv3{  
%   pair, and one row for every (R,THETA) pair. Es+BV+x[.c  
% G=>LW1E|  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike vUgo)C#<  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Cc` )P>L  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral tSq`_[@  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @W!cC#u  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized mTZgvPJ!  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. z.*=3   
% yQ+C}8r5  
%   The Zernike functions are an orthogonal basis on the unit circle. "3VMjF\  
%   They are used in disciplines such as astronomy, optics, and E"b"VB  
%   optometry to describe functions on a circular domain. / Hexv#3  
% 67dp)X  
%   The following table lists the first 15 Zernike functions. 3
o^oq  
% A@OSh6/{h  
%       n    m    Zernike function           Normalization oW8 hC  
%       -------------------------------------------------- }@jT-t]P  
%       0    0    1                                 1 eX9H/&g  
%       1    1    r * cos(theta)                    2 qjd8Q  
%       1   -1    r * sin(theta)                    2 u9)<i]2  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) QP4`r#,  
%       2    0    (2*r^2 - 1)                    sqrt(3) .p(~/MnO  
%       2    2    r^2 * sin(2*theta)             sqrt(6) uM9RlI5  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) \S"YLRn"  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) Fd@
:*ER  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 06vxsT@
 
%       3    3    r^3 * sin(3*theta)             sqrt(8) h
h"=|c  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) "K-2y^Dl  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yX;v  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) B$%7U><'  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0Xw3h^%  
%       4    4    r^4 * sin(4*theta)             sqrt(10) r2 o-/$  
%       -------------------------------------------------- K~aIY0=<  
% -(+/u .  
%   Example 1: WjvD C"  
% IBr|A  
%       % Display the Zernike function Z(n=5,m=1) =o+))R4  
%       x = -1:0.01:1; \%N | X  
%       [X,Y] = meshgrid(x,x); 3re|=_ Hy  
%       [theta,r] = cart2pol(X,Y); 5\$8"/H  
%       idx = r<=1; o%\p
I%  
%       z = nan(size(X)); j{u!/FD  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); kKRZ79"7s  
%       figure -g]g  
%       pcolor(x,x,z), shading interp M/mUY  
%       axis square, colorbar 0`dMT>&I  
%       title('Zernike function Z_5^1(r,\theta)') B?)=d,E  
% GwaU7[6  
%   Example 2: F,-S&d  
% ghd*EXrF H  
%       % Display the first 10 Zernike functions &r Lg/UEV-  
%       x = -1:0.01:1; *eo<5YUHt  
%       [X,Y] = meshgrid(x,x); jPf*qe>U  
%       [theta,r] = cart2pol(X,Y); -w:F8k ~  
%       idx = r<=1; s8]9OG3g  
%       z = nan(size(X)); < l%3P6|  
%       n = [0  1  1  2  2  2  3  3  3  3]; aD:vNX  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; -] .Y";  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; \#; -C<[b  
%       y = zernfun(n,m,r(idx),theta(idx)); "' hc)58y  
%       figure('Units','normalized') $}G03G@  
%       for k = 1:10 =?/RaK/ w  
%           z(idx) = y(:,k); x\Det$3Kx  
%           subplot(4,7,Nplot(k)) UR&Uwa&.  
%           pcolor(x,x,z), shading interp 6{r^3Hz  
%           set(gca,'XTick',[],'YTick',[]) $G5;y>  
%           axis square &S"o
jbb  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) er?'o1M  
%       end k~st;FO  
% V_g9
oR_  
%   See also ZERNPOL, ZERNFUN2. `2@t) :  
eSgCS*}0$z  
%   Paul Fricker 11/13/2006 AZCbUkq  
^"h
`U'YC  
FV&
&  
% Check and prepare the inputs: t$z FsFTQ  
% ----------------------------- jtk2>Ol  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {1y-*@yU(  
    error('zernfun:NMvectors','N and M must be vectors.') ^rc!X]C9  
end nKJJ7 RL  
G2%%$7Jj  
if length(n)~=length(m) ~
YKBxt  
    error('zernfun:NMlength','N and M must be the same length.') n(gw%w+\7  
end .1.Bf26}d  
_tg&_P+kV  
n = n(:); ?[\(i)]  
m = m(:); &r6VF/  
if any(mod(n-m,2)) 0.
+"K}  
    error('zernfun:NMmultiplesof2', ... s wdW70  
          'All N and M must differ by multiples of 2 (including 0).') MEQ:[;1  
end m>gok0{pm  
Syn>;FX  
if any(m>n) 05\A7.iy  
    error('zernfun:MlessthanN', ... p AzPi  
          'Each M must be less than or equal to its corresponding N.') r`|/qP:T[  
end 9IKFrCO9,  
)jK"\'cK  
if any( r>1 | r<0 ) {ZH9W
 
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') )POuH*j  
end bGvALz'  
0)V<)"i  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) J(0.eD91v  
    error('zernfun:RTHvector','R and THETA must be vectors.') T1pA <6  
end oXgKuR  
lK%pxqx  
r = r(:); ;$G.?r  
theta = theta(:); |Ebwl]X2  
length_r = length(r); j(!M  
if length_r~=length(theta) J'O</o@e  
    error('zernfun:RTHlength', ... m9UI3fBX  
          'The number of R- and THETA-values must be equal.') zxtx~XO  
end  =uZ[  
m<wng2`NTv  
% Check normalization: 31LXzQvFG  
% -------------------- qWf7k+7G  
if nargin==5 && ischar(nflag) E
?(  
    isnorm = strcmpi(nflag,'norm'); NamBJ\2E1[  
    if ~isnorm 5tg  
        error('zernfun:normalization','Unrecognized normalization flag.') 9cAb\5c|  
    end %_wX9ZT  
else }+0{opY4R  
    isnorm = false; r>S?,qr  
end |A0LYKni  
^zHBDRsb2F  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k+2~=#  
% Compute the Zernike Polynomials |b{XnD_g  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TdI5{?sW  
C`3}7qi|C  
% Determine the required powers of r: 1@C0c%  
% ----------------------------------- u=feR0|8  
m_abs = abs(m); >[=q9k  
rpowers = []; G}CzeL
w  
for j = 1:length(n) ow*) 1eo  
    rpowers = [rpowers m_abs(j):2:n(j)]; 4;_{*U-  
end 716JnG>  
rpowers = unique(rpowers); *z`_U]tP  
"jzU`  
% Pre-compute the values of r raised to the required powers, gk\IivPb  
% and compile them in a matrix: 5YaTE<G  
% ----------------------------- DPJ#Y -0  
if rpowers(1)==0 a|`Pg1j#  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kJqgY|  
    rpowern = cat(2,rpowern{:}); u-3A6Q  
    rpowern = [ones(length_r,1) rpowern]; Fd-PjW/E8  
else - *!R  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '<~l%q  
    rpowern = cat(2,rpowern{:}); _6&x$*O  
end [
k.|iCD  
9hEIf,\  
% Compute the values of the polynomials: @Hj5ZJ 3  
% -------------------------------------- v!T%xUb0  
y = zeros(length_r,length(n)); 2e zQX2q  
for j = 1:length(n) D\]gIXg  
    s = 0:(n(j)-m_abs(j))/2; `3^*K/K\  
    pows = n(j):-2:m_abs(j); D)XF@z;  
    for k = length(s):-1:1 *{8Kb>D  
        p = (1-2*mod(s(k),2))* ... QWv+Ja  
                   prod(2:(n(j)-s(k)))/              ... Zf}]sW$H  
                   prod(2:s(k))/                     ... ,qV8(`y_  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... T>s~bIzL*e  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Vo<V!G
{  
        idx = (pows(k)==rpowers); WY#A9i5Ge  
        y(:,j) = y(:,j) + p*rpowern(:,idx); W/9dT^1y4'  
    end * F%Wf  
     N"/jn_>+j  
    if isnorm l=U@j T  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 5G cdz  
    end u HqPb8  
end = ;#?CAa:  
% END: Compute the Zernike Polynomials $5ZBNGr  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XRCiv   
x?Doe`/6?  
% Compute the Zernike functions: f/RzE  
% ------------------------------ 72R|zR  
idx_pos = m>0; yB\}e'J^  
idx_neg = m<0; Tz3 L#0:j  
,\q9>cZ!  
z = y; >&3M #s(w  
if any(idx_pos) &{NN!X  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \\XvVi:B  
end Yo3my>N&g  
if any(idx_neg) 2{Nv&ZX?  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); z^+f3-Z  
end &p=Uus  
a]-F,MJ  
% 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