[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 Fpl<2eBg4  
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！ qky{]qNW  

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)  v](7c2;  

Yhb=^)@))  
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。 w f,7  
AFF7fK  
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|EH`eu^i  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. qPK3"fzH  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of -o#HO_9  
%   order N and frequency M, evaluated at R.  N is a vector of AF g*  
%   positive integers (including 0), and M is a vector with the JV=d!Gi[C  
%   same number of elements as N.  Each element k of M must be a UQgOtqL3  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) , |CT|2D>  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is &~ QQZ]q6  
%   a vector of numbers between 0 and 1.  The output Z is a matrix 6UXa 5t  
%   with one column for every (N,M) pair, and one row for every In#V1[io  
%   element in R. D^W6Cq5\  
%
x]&V7Y  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- ;Oh4W<hH}  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is aT$q1!U`j2  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to 9JV(}v5[  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 x48Y#"'  
%   for all [n,m]. 6?b9~xRW  
% ;Y5"[C9|  
%   The radial Zernike polynomials are the radial portion of the L']EYK5  
%   Zernike functions, which are an orthogonal basis on the unit 7^P!@o$v!  
%   circle.  The series representation of the radial Zernike zA=gDuy3@  
%   polynomials is }A3(g$8KR  
% =|O`al  
%          (n-m)/2 9!zUv:;  
%            __ #80DM  
%    m      \       s                                          n-2s q$`{$RX  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r O'{UAb+-  
%    n      s=0 /PH+K24v~  
% i% 19|an  
%   The following table shows the first 12 polynomials. -H5n>j0!{  
% 2qLRcA=R  
%       n    m    Zernike polynomial    Normalization fEf",{I  
%       --------------------------------------------- h4N!zj[  
%       0    0    1                        sqrt(2)
uF_gfjR[m  
%       1    1    r                           2 rT9
<_<  
%       2    0    2*r^2 - 1                sqrt(6) )
F4H'  
%       2    2    r^2                      sqrt(6) xa#0y  
%       3    1    3*r^3 - 2*r              sqrt(8) ;A7
HEx  
%       3    3    r^3                      sqrt(8) Aq@_^mq1A  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Sr Z\]  
%       4    2    4*r^4 - 3*r^2            sqrt(10) 3CK4a,]Dm  
%       4    4    r^4                      sqrt(10) Oaf!\z}  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) >MZWm6M8  
%       5    3    5*r^5 - 4*r^3            sqrt(12) teH $hd-q  
%       5    5    r^5                      sqrt(12) s1.YH?A;  
%       --------------------------------------------- 0i/l2&x*k]  
% ]CsF} wr'z  
%   Example: E,&BP$B  
% 0(\ybppx  
%       % Display three example Zernike radial polynomials g N76  
%       r = 0:0.01:1; 9r=@S  
%       n = [3 2 5]; "W$,dWF  
%       m = [1 2 1]; 0j\?zt?  
%       z = zernpol(n,m,r); W}
p>jP}  
%       figure `p1szZD&  
%       plot(r,z) :bFCnV`Q  
%       grid on v1%rlP  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') )/kkvI()l  
% ]4wyuP,up  
%   See also ZERNFUN, ZERNFUN2. &^$dHr6v  
vJ9Uw  
% A note on the algorithm. ~&B{"d  
% ------------------------ N
)|mA)S)  
% The radial Zernike polynomials are computed using the series w=5D>]  
% representation shown in the Help section above. For many special u&Ts'j  
% functions, direct evaluation using the series representation can ,DsqKXSU  
% produce poor numerical results (floating point errors), because +>mbBu!7  
% the summation often involves computing small differences between m`CcU`s  
% large successive terms in the series. (In such cases, the functions U/'"w v1y  
% are often evaluated using alternative methods such as recurrence GADbXp3  
% relations: see the Legendre functions, for example). For the Zernike )\#w=P  
% polynomials, however, this problem does not arise, because the +M-x*;.  
% polynomials are evaluated over the finite domain r = (0,1), and |;3Ru vX?+  
% because the coefficients for a given polynomial are generally all Q-o}Xnj*!L  
% of similar magnitude. ; mnV)8:F  
% 'X&sH/>r  
% ZERNPOL has been written using a vectorized implementation: multiple lj0"2@z3"E  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] (PAkKY}  
% values can be passed as inputs) for a vector of points R.  To achieve dx}) 1%  
% this vectorization most efficiently, the algorithm in ZERNPOL !wy 
Qk  
% involves pre-determining all the powers p of R that are required to ~Z-M?8:  
% compute the outputs, and then compiling the {R^p} into a single 7pH`"$  
% matrix.  This avoids any redundant computation of the R^p, and )jkX&7x  
% minimizes the sizes of certain intermediate variables. 1Q1NircJ  
% dU%Q=r
8R  
%   Paul Fricker 11/13/2006 IGz92&y  
Im7<\ b@  
eaLSq  
% Check and prepare the inputs: JW[y  
% ----------------------------- 6)63Yp(  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >PdYQDyVS  
    error('zernpol:NMvectors','N and M must be vectors.') z%-Yz-G9  
end P__JN\{9  
QCB2&lN\&L  
if length(n)~=length(m) L1=+x^WQ  
    error('zernpol:NMlength','N and M must be the same length.') xL8r'gV@  
end 2z9\p%MX  
|hBX"  
n = n(:); h8@8Qw  
m = m(:); I^erMQn[ z  
length_n = length(n); q SR\=:$  
C "XvspJ  
if any(mod(n-m,2)) $D{KXkrd  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') H]&a}WQ_  
end G"w ?{W@  
+oa\'.~?  
if any(m<0)  1@Abs  
    error('zernpol:Mpositive','All M must be positive.') gzfs9e  
end xCU^4DO3p  
ZC}'! $r7  
if any(m>n) Y_m/? [:  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') wh4ik`S 1  
end 48;6C g  
}  IJ
 
if any( r>1 | r<0 ) {A2EGUmF2  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') $|+q9o\  
end #ra"(/)  
]WlE9z7:8  
if ~any(size(r)==1) HKu? J  
    error('zernpol:Rvector','R must be a vector.') Q9,H0r-%  
end k#mQLv  
)I7~<$w  
r = r(:); 0>@D{_}s  
length_r = length(r); /5cFa  
q@K8,=/.#  
if nargin==4
Ik[aiz  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ?,GCR1|4  
    if ~isnorm hP1}Do  
        error('zernpol:normalization','Unrecognized normalization flag.') ~*:{U   
    end 7{<:g!  
else 1Rrp#E}
 
    isnorm = false; * V;L|c  
end g\9I&z~?  
'a]4]d  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%eJolztKZ  
% Compute the Zernike Polynomials
#rZF4>c  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U*Q1(C  
F3BWi[Xh  
% Determine the required powers of r: IQn|0$':Z  
% ----------------------------------- h SGI  
rpowers = []; VVY#g%(K
 
for j = 1:length(n) ODS8bD0!i  
    rpowers = [rpowers m(j):2:n(j)]; vo48\w7[  
end &f12Q&jY7  
rpowers = unique(rpowers); K@uUe3  
,3 !D(&  
% Pre-compute the values of r raised to the required powers, \#1*r'V8  
% and compile them in a matrix: P .I<.e  
% ----------------------------- nR%ASUx:Y  
if rpowers(1)==0 e,j2#wjor  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fL3Px  
    rpowern = cat(2,rpowern{:}); CM$q{;y  
    rpowern = [ones(length_r,1) rpowern]; UO3Q
wZ4j;  
else SePPI.n  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j?!BHNs  
    rpowern = cat(2,rpowern{:}); 8sMDe'  
end _<;;CI3w  
-e#~CE
-  
% Compute the values of the polynomials: 9 Vn  
% -------------------------------------- )8BGN'jyi  
z = zeros(length_r,length_n); %V40I{1  
for j = 1:length_n l,z# :k  
    s = 0:(n(j)-m(j))/2; )-2sk@y  
    pows = n(j):-2:m(j); -)
cau-(X  
    for k = length(s):-1:1 FE}!I  
        p = (1-2*mod(s(k),2))* ... 7d9kr?3(U  
                   prod(2:(n(j)-s(k)))/          ... K})=&<M0  
                   prod(2:s(k))/                 ... N0Y$QWr_$  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... \BIa:}9O  
                   prod(2:((n(j)+m(j))/2-s(k))); a/})X[2  
        idx = (pows(k)==rpowers); jZRf{  
        z(:,j) = z(:,j) + p*rpowern(:,idx); ~|W0+&):  
    end @UbH;m  
     YH_m
WN\Wu  
    if isnorm JCL+uEX4S  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1));
qG=?+em  
    end {VBn@^'s  
end N)F&c!anh  
1|]IWX|  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) nRP|Qt7>  
%ZERNFUN2 Single-index Zernike functions on the unit circle. =rw60B  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated % oPt],>  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive FU{$oCh/5  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, Yf=an`"  
%   and THETA is a vector of angles.  R and THETA must have the same VR8 kY&  
%   length.  The output Z is a matrix with one column for every P-value, vbo|q[z  
%   and one row for every (R,THETA) pair. 8R3x74fL  
% <7U\@si4  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike [uJfmrEH  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 8OS@gpz  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) J$aE:g6'  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 n\-nBrVSf  
%   for all p. fX ^hO+f
 
% {D6p?TL+  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 :\!D 6\o6  
%   Zernike functions (order N<=7).  In some disciplines it is q=EHB5!q  
%   traditional to label the first 36 functions using a single mode & bKl(,  
%   number P instead of separate numbers for the order N and azimuthal {7'Evfn)  
%   frequency M. @3^D[  
% QLs9W&PG  
%   Example: bv&#ay7  
% cEdf&*_-'I  
%       % Display the first 16 Zernike functions [~aRA'qJ{V  
%       x = -1:0.01:1; mp!S<m  
%       [X,Y] = meshgrid(x,x); %>z4hH,  
%       [theta,r] = cart2pol(X,Y); >/]` f8^  
%       idx = r<=1; C`J>Gm  
%       p = 0:15; 6#J>b[Q  
%       z = nan(size(X)); YaBZ#$r  
%       y = zernfun2(p,r(idx),theta(idx)); 2bs={p$}a  
%       figure('Units','normalized') qG6?k}\\  
%       for k = 1:length(p) NsPAWI|4  
%           z(idx) = y(:,k); '3p7ee&  
%           subplot(4,4,k) 6>yfm4o  
%           pcolor(x,x,z), shading interp 8[k:FGp>  
%           set(gca,'XTick',[],'YTick',[])  &x":  
%           axis square Lhts4D/V7  
%           title(['Z_{' num2str(p(k)) '}']) @QN(ouqQ  
%       end /wR,P  
% yd}1Mx  
%   See also ZERNPOL, ZERNFUN. ~6Xr^An/Z  
D2y[?RG  
%   Paul Fricker 11/13/2006 K9HXy*y49  
|3bCq(ZR\P  
nxjP4d>  
% Check and prepare the inputs: -MK9IO]i  
% ----------------------------- <hV%OrBz-  
if min(size(p))~=1 @^2?97i c  
    error('zernfun2:Pvector','Input P must be vector.') L0Ycf|[s,  
end JK/gq}c  
> u!# 4  
if any(p)>35 NimW=X;c  
    error('zernfun2:P36', ... d:L|BkQ7*  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... hr/H vB  
           '(P = 0 to 35).']) tP.jJC~  
end V0/PjD,jP  
+_T`tmQ  
% Get the order and frequency corresonding to the function number: SWLt5dV  
% ---------------------------------------------------------------- {@&%Bq*&  
p = p(:); +T/T\[  
n = ceil((-3+sqrt(9+8*p))/2); -cONC9=  
m = 2*p - n.*(n+2); Mb^E  
;ztt*py  
% Pass the inputs to the function ZERNFUN: }T~}W8H  
% ---------------------------------------- Xl %ax!/  
switch nargin qRcY(mb  
    case 3 !qe:M]C'l  
        z = zernfun(n,m,r,theta); BY5ODc$  
    case 4 ~-tKMc).X  
        z = zernfun(n,m,r,theta,nflag); fe4Ki  
    otherwise 6ec#3~ Y]  
        error('zernfun2:nargin','Incorrect number of inputs.') A-&XgOL  
end ccY! OSae  
1=Z, #r  
% 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 $1uT`>%  
function z = zernfun(n,m,r,theta,nflag) ".7\>8A#a  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +GvPJI  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  ae>B0#=  
%   and angular frequency M, evaluated at positions (R,THETA) on the &e\
UlM22  
%   unit circle.  N is a vector of positive integers (including 0), and 'w8p[h (,  
%   M is a vector with the same number of elements as N.  Each element '\% Kd+k  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 4q)+nh~s  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s4[PwD  
%   and THETA is a vector of angles.  R and THETA must have the same _KJ!C!  
%   length.  The output Z is a matrix with one column for every (N,M) 6FkBb!ASk  
%   pair, and one row for every (R,THETA) pair. \P;2s<6i\  
%
&7r73~TXm  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ECk* H  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), n.7-$1  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral -oT3`d3  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o/hj~;(]  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized LUzn7FZk  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %j/}e>$"Nk  
% WXQ+`OH7  
%   The Zernike functions are an orthogonal basis on the unit circle. 6E{(_i  
%   They are used in disciplines such as astronomy, optics, and P?h
B`5X  
%   optometry to describe functions on a circular domain. PJL [En*  
% ]@uuB\u  
%   The following table lists the first 15 Zernike functions. 4x-
K0  
% oc"
7|YG  
%       n    m    Zernike function           Normalization 97k}{tG  
%       -------------------------------------------------- zG)vmysJf  
%       0    0    1                                 1 _t>[gB,  
%       1    1    r * cos(theta)                    2 Vt(s4  
%       1   -1    r * sin(theta)                    2 uvl>Z= "  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) .Vrl:  
%       2    0    (2*r^2 - 1)                    sqrt(3) snYyxi

  
%       2    2    r^2 * sin(2*theta)             sqrt(6) Ot{~mMDp  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) C@WdPjxj  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) xEg@Y"NQ  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 8GeJ%^0o}  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 0"{-<Wot}  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Mq='|0,  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )|6OPR@(#/  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _+OCI%=:  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) P9)L1l<3I  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ~;}uYJ  
%       -------------------------------------------------- -TS
5g1  
% W[:CCCDL  
%   Example 1: q1Ad"r
m  
% |W*5<2Q9  
%       % Display the Zernike function Z(n=5,m=1) S1#5oy2  
%       x = -1:0.01:1; yN/g;bQ  
%       [X,Y] = meshgrid(x,x); pM9M8d  
%       [theta,r] = cart2pol(X,Y); ?}U?Q7vx@@  
%       idx = r<=1; ZgfhNI\
 
%       z = nan(size(X)); B,]
AfH  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); +g;{c+Kw:  
%       figure 3Ww 37V>h  
%       pcolor(x,x,z), shading interp &F8*>F^7  
%       axis square, colorbar LqLhZBU9  
%       title('Zernike function Z_5^1(r,\theta)') A8g_BLj!e  
% <}G/x*N  
%   Example 2: yL
%88,/  
% g> m)XY  
%       % Display the first 10 Zernike functions /VD[:sU7  
%       x = -1:0.01:1; M)~sL1)  
%       [X,Y] = meshgrid(x,x); 1amEQ  
%       [theta,r] = cart2pol(X,Y); r>gf&/Pl  
%       idx = r<=1; U6.hH%\}@  
%       z = nan(size(X)); s>)?MB*vb  
%       n = [0  1  1  2  2  2  3  3  3  3]; N'CWSf.e  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; o]WcODJdl  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Z3U%Afl2{  
%       y = zernfun(n,m,r(idx),theta(idx)); Vh
a,r
Ii  
%       figure('Units','normalized') 4X!4S6JfB  
%       for k = 1:10 Wt.['`c<  
%           z(idx) = y(:,k); u$FL(m4  
%           subplot(4,7,Nplot(k))
p W
@Yr  
%           pcolor(x,x,z), shading interp L)qUBp@MW  
%           set(gca,'XTick',[],'YTick',[]) qHvU4v  
%           axis square cG&@PO]+.  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !<out4Mz"  
%       end ?*.:*A  
% z4(`>z2a  
%   See also ZERNPOL, ZERNFUN2. raZkH8  
=!)x`1j!S  
%   Paul Fricker 11/13/2006 SLNq%7apx  
KWM.e1(  
UCj:]!P  
% Check and prepare the inputs: m6'9Id-:L  
% ----------------------------- mDk6@Gd@U  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) MS:,I?  
    error('zernfun:NMvectors','N and M must be vectors.') @urZ  
end ky&wv+7  
%`~+^{Wp  
if length(n)~=length(m) t|s(V-Wq  
    error('zernfun:NMlength','N and M must be the same length.') V5p^]To!  
end @R<z=n"  
<oi'yr  
n = n(:); X"9N<)C  
m = m(:); 6"NtVfui  
if any(mod(n-m,2)) *>2e4j]  
    error('zernfun:NMmultiplesof2', ... 7rYB
FSp  
          'All N and M must differ by multiples of 2 (including 0).') 5$Kd<ky  
end `+0dz,  
@t0T+T3  
if any(m>n) 0$0 215  
    error('zernfun:MlessthanN', ... IVy<>xpt  
          'Each M must be less than or equal to its corresponding N.') HCCq9us  
end #;5Qd'  
$|@pY| f  
if any( r>1 | r<0 ) )a5ON8?  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') bxzx@sF2l  
end @eutp`xoT\  
Jd?qvE>Pp  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9_4(}|"N|  
    error('zernfun:RTHvector','R and THETA must be vectors.') 6QJ.=.>b  
end =qbN?a/?2  
L8H:,} 2  
r = r(:); FS=LpvOG)  
theta = theta(:); n).*=YLN  
length_r = length(r); IuA4eDr^Y%  
if length_r~=length(theta) ti$60Up  
    error('zernfun:RTHlength', ... m`):= ^nC  
          'The number of R- and THETA-values must be equal.') oRJ!TAbD  
end 'Z:wEt!  
o4OB xHKy  
% Check normalization: 2(x| %  
% -------------------- w^=(:`  
if nargin==5 && ischar(nflag) f$9|qfW'$  
    isnorm = strcmpi(nflag,'norm'); *B\ @L  
    if ~isnorm 3,`M\#z%K  
        error('zernfun:normalization','Unrecognized normalization flag.') =v'Aub  
    end +'+Nr<  
else CBNt _y  
    isnorm = false; 2b,edJVt?  
end mq:WBSsV  
%Ofw"W  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  zY7M]Az  
% Compute the Zernike Polynomials SAj#+_db  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,qo^G0XO  
[8q`~S%-]  
% Determine the required powers of r: H "5,To  
% ----------------------------------- 'n1$Y%t  
m_abs = abs(m); cui%r!D  
rpowers = []; k}I65 ^l#  
for j = 1:length(n) (C1~>7L  
    rpowers = [rpowers m_abs(j):2:n(j)]; xWqV~NnE  
end }Y|M+0   
rpowers = unique(rpowers); Slj U=,  
tIV{uVM[|D  
% Pre-compute the values of r raised to the required powers, T8)X?>CIW  
% and compile them in a matrix: mdQe)>  
% ----------------------------- ~c] q:pU2  
if rpowers(1)==0 !`4ie  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2VUN  
    rpowern = cat(2,rpowern{:}); k.2GIc:5  
    rpowern = [ones(length_r,1) rpowern]; Q[aF"5h%  
else eK5~gnv,  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ssS"X@VZ \  
    rpowern = cat(2,rpowern{:}); mPqKk  
end h-sO7M0E]  
c0HPS9N\  
% Compute the values of the polynomials: jUl_ToX  
% -------------------------------------- Nn-k hl|11  
y = zeros(length_r,length(n)); Y2'HP)tfIw  
for j = 1:length(n) ]Hq,Pr_+  
    s = 0:(n(j)-m_abs(j))/2; `B&=ya|bl  
    pows = n(j):-2:m_abs(j); M(:bM1AD`u  
    for k = length(s):-1:1 _?y3&4N)  
        p = (1-2*mod(s(k),2))* ... ZMr[:,Jp  
                   prod(2:(n(j)-s(k)))/              ... oM^vJ3  
                   prod(2:s(k))/                     ... mF!4*k  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... =R 4]Kf  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); {O).!  
        idx = (pows(k)==rpowers); EYZ&%.Sy5  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 64'QTF{D  
    end f2pA+j5[  
     *JZ9'|v_H  
    if isnorm tS5J{j>T  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); L[<Y6u>m!1  
    end S 1^t;{"  
end 4p+Veo6B  
% END: Compute the Zernike Polynomials "#gS?aS  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZR0 OqSp]  
?(Tin80=r  
% Compute the Zernike functions: )F\
kGe  
% ------------------------------ u|.|dv'mbp  
idx_pos = m>0; @$L
|  
idx_neg = m<0; 6R8>w,  
/*BK6hc  
z = y; ?azLaAG  
if any(idx_pos) ~Ym*QSD  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {Y=k`t,  
end &iq'V*+-\  
if any(idx_neg) !FyO5`v  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); . XbDb  
end n[qnrk*3 %  
lKU{jWA  
% 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