[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 Lng. X8D  
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！ ^.6yzlY  

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)  S'!&,Dxq^  

> '=QBW  
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。 q\<l"b z  
p?L%'  
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) )yyH_Ax2  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. R{c~jjd  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 9~`#aQG T  
%   order N and frequency M, evaluated at R.  N is a vector of bK6^<,~  
%   positive integers (including 0), and M is a vector with the 8a*&,W  
%   same number of elements as N.  Each element k of M must be a 2n3&uvf'TL  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) 5 <k)tF%  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is zV}:~;w  
%   a vector of numbers between 0 and 1.  The output Z is a matrix eikZ~!@  
%   with one column for every (N,M) pair, and one row for every =)Hu
(;Yv  
%   element in R. *=oO3c0|b,  
% ;@S'8  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- WD\Yx~o  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is $B?8\>_?  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to %ud-3u52M8  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 MUbKlX  
%   for all [n,m]. 3!F^vZ.  
% T(u;<}e@[  
%   The radial Zernike polynomials are the radial portion of the 0
&)6mO  
%   Zernike functions, which are an orthogonal basis on the unit *@bz<{!  
%   circle.  The series representation of the radial Zernike d8;kM`U  
%   polynomials is Iq%<E:+GL  
% <5%*
"v  
%          (n-m)/2 n@tt.n!{l  
%            __ 1|8Bv0-b  
%    m      \       s                                          n-2s m7i_Iv  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ^[SW07o~  
%    n      s=0 \%r0'
1f  
% Y7+c/co  
%   The following table shows the first 12 polynomials. ftMlm_u  
% d1_kw A2y  
%       n    m    Zernike polynomial    Normalization 8nBYP+t,e  
%       --------------------------------------------- %
J-:%i
 
%       0    0    1                        sqrt(2) I(7GVYM  
%       1    1    r                           2 ,sSo\%  
%       2    0    2*r^2 - 1                sqrt(6) R"XycXn_$  
%       2    2    r^2                      sqrt(6) Fc[vs5
2  
%       3    1    3*r^3 - 2*r              sqrt(8) qN1fWU#$  
%       3    3    r^3                      sqrt(8) G9-ETj}  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ?ch?q~e)  
%       4    2    4*r^4 - 3*r^2            sqrt(10) dH5*%  
%       4    4    r^4                      sqrt(10) vTFG*\Cq  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) NsYEBT7f  
%       5    3    5*r^5 - 4*r^3            sqrt(12) s@$0!8sxm  
%       5    5    r^5                      sqrt(12) :vIJ>
6lIR  
%       --------------------------------------------- PeIi@0vA  
% ;bG?R0a  
%   Example: XK\n
OHLS  
% 3|w$gG;Y  
%       % Display three example Zernike radial polynomials wz3X;1l`c  
%       r = 0:0.01:1; Uu8ayN j  
%       n = [3 2 5]; o|d:rp!^  
%       m = [1 2 1]; /-!Fr:Ox>  
%       z = zernpol(n,m,r); xGr{ad.N  
%       figure yw:%)b{  
%       plot(r,z) u
9Adu`  
%       grid on VF11eZ"  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;]xc}4@=mg  
% ]:@{tX7c  
%   See also ZERNFUN, ZERNFUN2. HaL'/V~  
Wn6m$=  
% A note on the algorithm. u
SYI X  
% ------------------------ H(K!{k  
% The radial Zernike polynomials are computed using the series *YH!L{y  
% representation shown in the Help section above. For many special HOu$14g  
% functions, direct evaluation using the series representation can g&$5!ifgi  
% produce poor numerical results (floating point errors), because H0tu3Pqk  
% the summation often involves computing small differences between !21G$[H  
% large successive terms in the series. (In such cases, the functions 7
2RTEGy  
% are often evaluated using alternative methods such as recurrence a0]GQyIG  
% relations: see the Legendre functions, for example). For the Zernike L"vk ^>E6  
% polynomials, however, this problem does not arise, because the 'LG\]h>+)  
% polynomials are evaluated over the finite domain r = (0,1), and w5y.kc;  
% because the coefficients for a given polynomial are generally all GQ?FUFuIoW  
% of similar magnitude. bA0H  
% uPcx6X3]  
% ZERNPOL has been written using a vectorized implementation: multiple Mu:zWLM*M  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] $X:,Q,?  
% values can be passed as inputs) for a vector of points R.  To achieve X} V]3  
% this vectorization most efficiently, the algorithm in ZERNPOL FZU1WBNL%t  
% involves pre-determining all the powers p of R that are required to ~)$R'=  
% compute the outputs, and then compiling the {R^p} into a single Ff0V6j)ji  
% matrix.  This avoids any redundant computation of the R^p, and X ]&`"Z]  
% minimizes the sizes of certain intermediate variables. p`&{NR3+  
% ueU"v'h\  
%   Paul Fricker 11/13/2006 C$q-WoTM(  
ik?IC$*n3i  
0=7Ud<  
% Check and prepare the inputs: (&)uWjq `  
% ----------------------------- a'-xCV|^  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) lMW6D0^  
    error('zernpol:NMvectors','N and M must be vectors.') ":T"Y;  
end q2|z \  
xzz@Wc^_  
if length(n)~=length(m) o6 NmDv5  
    error('zernpol:NMlength','N and M must be the same length.') SZ4y\I  
end ;7E"@b,tPN  
WSeiW  
n = n(:); He4q-\ht  
m = m(:); Q ijO%)  
length_n = length(n); ~FI} [6Dd  
s$9ow<oi]  
if any(mod(n-m,2)) -KbO[b\V  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') S]T71W<i  
end }Dcpe M?  
/^{Q(R(X<  
if any(m<0) b;;y|H  
    error('zernpol:Mpositive','All M must be positive.') N0D5N(kH%  
end Z$Ps_Ik  
;
CL^2{  
if any(m>n) uVZm9Sp  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') <.lN'i;(  
end @:'E9J06  
/YwwG;1  
if any( r>1 | r<0 ) {Xpjm6a7  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') +&X>ul  
end
P:'y}a-  
eon(C|S7eK  
if ~any(size(r)==1) DVs$3RL  
    error('zernpol:Rvector','R must be a vector.') hI<$lEB  
end ;F,6]LH!  
$1Z3yb^  
r = r(:); )086u8w )y  
length_r = length(r); bDw\;bnG  
[sPLu)q2  
if nargin==4 \q-["W34  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); +C`vO5\0  
    if ~isnorm TxkvHiq2  
        error('zernpol:normalization','Unrecognized normalization flag.') _cfAJ)8=  
    end jP3~O  
else ^=cXL  
    isnorm = false; L(`q3>iC4.  
end 8p~[8}  
|])Ko08*tE  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G in  
% Compute the Zernike Polynomials OnW,R3eg  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +(q r{G?  
|>I4(''}  
% Determine the required powers of r: N~yGtnW  
% ----------------------------------- 99q$>nx,w  
rpowers = []; p_3VFKq>0  
for j = 1:length(n) K,HR=5  
    rpowers = [rpowers m(j):2:n(j)]; kA4kQ}q  
end ?0E-Lac=  
rpowers = unique(rpowers); .|kp`-F51  
U@:iN..  
% Pre-compute the values of r raised to the required powers, !.{{QwZ  
% and compile them in a matrix: pHDPj,lu  
% ----------------------------- |-AR)Smt  
if rpowers(1)==0 q*>|EJR^Rw  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `jFvG\aC  
    rpowern = cat(2,rpowern{:}); 3o__tU)B  
    rpowern = [ones(length_r,1) rpowern]; eY$Q}BcW  
else l]e7  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); dI8y}EbE~  
    rpowern = cat(2,rpowern{:}); !3at(+4  
end g!;
Hv  
Q#$dp  
% Compute the values of the polynomials: YC~kq?  
% -------------------------------------- j~9,Ct  
z = zeros(length_r,length_n); ;V~~lcD&Y`  
for j = 1:length_n iuq%Q\0@w  
    s = 0:(n(j)-m(j))/2; _UeIzdV9  
    pows = n(j):-2:m(j); Op<|Oz$Q|l  
    for k = length(s):-1:1 F a'2i<  
        p = (1-2*mod(s(k),2))* ... w.0]>/C  
                   prod(2:(n(j)-s(k)))/          ... B7qiCX}pD  
                   prod(2:s(k))/                 ... #T)gKp  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... $3n@2 N`  
                   prod(2:((n(j)+m(j))/2-s(k))); EabZ7zFoN  
        idx = (pows(k)==rpowers); ,7Lu7Q  
        z(:,j) = z(:,j) + p*rpowern(:,idx); oG;;='*  
    end ODqWXw#  
     BcTV5Wc
r  
    if isnorm ViT$]Nv  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); s*pgR=dZZ  
    end Z,Tv8;  
end D3%`vqu&  
U5wO;MA  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag)  Dy@f21+  
%ZERNFUN2 Single-index Zernike functions on the unit circle. L$+ap~ld  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated xem:#>&r  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive .<`Rq'  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, :xT=uE.I  
%   and THETA is a vector of angles.  R and THETA must have the same 9f4#b8  
%   length.  The output Z is a matrix with one column for every P-value, =r:-CRq(  
%   and one row for every (R,THETA) pair. 7L:$Amb_F  
% pJ#R :#P  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ,2%>e"%  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) q|%(47}z  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 29#;;n}p  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1
v(t?d  
%   for all p. A%s"WSx,  
% r`L$[C5I  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 lLT;V2=osX  
%   Zernike functions (order N<=7).  In some disciplines it is *lIK?"mo  
%   traditional to label the first 36 functions using a single mode JtU/%s  
%   number P instead of separate numbers for the order N and azimuthal oY{r83h{  
%   frequency M. ZIx,?E+eJ  
% ^8nK x<&5  
%   Example:  3y?ig2  
% h^5'i}@u  
%       % Display the first 16 Zernike functions HBL)_c{/O  
%       x = -1:0.01:1; E0}jEl/
{  
%       [X,Y] = meshgrid(x,x); <c6C+OWT,  
%       [theta,r] = cart2pol(X,Y); }_L@CpG  
%       idx = r<=1; U:E:"  
%       p = 0:15; :R<n{%~  
%       z = nan(size(X)); Rd5r~iT
 
%       y = zernfun2(p,r(idx),theta(idx)); #$Z|)i]w  
%       figure('Units','normalized') @"H+QVJ@  
%       for k = 1:length(p) -)aBS3  
%           z(idx) = y(:,k); mYN|)QVKy  
%           subplot(4,4,k) fV_(P_C  
%           pcolor(x,x,z), shading interp G~e`O,
+  
%           set(gca,'XTick',[],'YTick',[]) Ng,#d`Br  
%           axis square *"Ipu"G5?  
%           title(['Z_{' num2str(p(k)) '}']) S3Tww
]q  
%       end W&6P%0G/  
% g4I&3 M  
%   See also ZERNPOL, ZERNFUN. xU^Flw,4  
s"G6aM  
%   Paul Fricker 11/13/2006 EpCUL@+  
;#!`cgAh  
#uT-_L}sw  
% Check and prepare the inputs: l\*}  
% ----------------------------- 3M(:}c  
if min(size(p))~=1 r$6z{Na\[  
    error('zernfun2:Pvector','Input P must be vector.') H>`?S{J  
end :D?%!Q 0  
0%+TU4Xx  
if any(p)>35 N@^
?J@#V  
    error('zernfun2:P36', ... ;EE*#"IJ  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 5Y)!q?#H  
           '(P = 0 to 35).']) #T n~hnW  
end e4ajT  
?PSm) ~Oa  
% Get the order and frequency corresonding to the function number: 8Hs>+Udl  
% ---------------------------------------------------------------- s&M6DFlA  
p = p(:); A[!Fg0X0  
n = ceil((-3+sqrt(9+8*p))/2); ^[8e|,U  
m = 2*p - n.*(n+2); } CJQC
 
Zi2NgVF  
% Pass the inputs to the function ZERNFUN: JB'q_dS}  
% ---------------------------------------- ?4_^}B9  
switch nargin zie])_8|h  
    case 3 %n9}P , ?  
        z = zernfun(n,m,r,theta); $d!Sl a  
    case 4 >NW /0'/  
        z = zernfun(n,m,r,theta,nflag); wI}5[m  
    otherwise ."PR Z,  
        error('zernfun2:nargin','Incorrect number of inputs.') :j vx-jQ  
end *n2Q_o  
Jnm{i|6N  
% 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 _$
wWKJy9  
function z = zernfun(n,m,r,theta,nflag) n@5pS3qZ  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /^#k/z  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0d:t$2~C  
%   and angular frequency M, evaluated at positions (R,THETA) on the z>&Py(  
%   unit circle.  N is a vector of positive integers (including 0), and o]}b#U8S  
%   M is a vector with the same number of elements as N.  Each element R',|Jf=`  
%   k of M must be a positive integer, with possible values M(k) = -N(k) W2yNEiH  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Q]w;o
&eo  
%   and THETA is a vector of angles.  R and THETA must have the same ,^,Vq]$3  
%   length.  The output Z is a matrix with one column for every (N,M) -t?S:9[w  
%   pair, and one row for every (R,THETA) pair. &EmxSYL>  
% . %tc7`k8  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /!JpmI  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RXt`y62yK  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral u$&7fmZ  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, phbdV8$L  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 3oxQ[.o  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. t\{q,4  
% Otn,(j;u  
%   The Zernike functions are an orthogonal basis on the unit circle. eOD;@4lR  
%   They are used in disciplines such as astronomy, optics, and '7wI 2D  
%   optometry to describe functions on a circular domain. @p|[7'  
% E`XUK,b  
%   The following table lists the first 15 Zernike functions. 2j4VW0:  
% p."pI Bd  
%       n    m    Zernike function           Normalization _7]5Q  
%       -------------------------------------------------- 88pz<$  
%       0    0    1                                 1 0d`s(b54;O  
%       1    1    r * cos(theta)                    2 {NKDmeg:D  
%       1   -1    r * sin(theta)                    2 /.$n>:XR  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 05DK-Wh?  
%       2    0    (2*r^2 - 1)                    sqrt(3) MrLDe{^C2  
%       2    2    r^2 * sin(2*theta)             sqrt(6) nrwb6wj  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ,, ]y 8P  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) N~\1yQT  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Nh]eZ3O  
%       3    3    r^3 * sin(3*theta)             sqrt(8) H=Yl @  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Si6%6rAhj  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WsJ3zZc  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) YqR MVWcnk  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \ zhT1#O  
%       4    4    r^4 * sin(4*theta)             sqrt(10) -:a 9'dT  
%       -------------------------------------------------- 4zpprh+`K  
% f Nm Sx  
%   Example 1: /Kw
o^Q{  
% bX|Z||img  
%       % Display the Zernike function Z(n=5,m=1) HP.
j.  
%       x = -1:0.01:1; 1U@
qRU  
%       [X,Y] = meshgrid(x,x); K}`.?6O  
%       [theta,r] = cart2pol(X,Y); &Zd{ElM  
%       idx = r<=1; Q++lgVh)E  
%       z = nan(size(X));  7I^(vQ  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); !ygh`]6V  
%       figure -7'>Rw  
%       pcolor(x,x,z), shading interp ztgSd8GGE  
%       axis square, colorbar  Cj_cu  
%       title('Zernike function Z_5^1(r,\theta)') 9d#-;qV  
% '2uQ  
%   Example 2: Sw%=/g  
% f/*Xw{s#  
%       % Display the first 10 Zernike functions >Ah [uM  
%       x = -1:0.01:1; C[& \Xq  
%       [X,Y] = meshgrid(x,x); `cy_@Z5A  
%       [theta,r] = cart2pol(X,Y); -?2ThvT  
%       idx = r<=1; {]Nvq9?  
%       z = nan(size(X)); |"5NI'X?  
%       n = [0  1  1  2  2  2  3  3  3  3]; h[ba$S,T  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &=<x&4H+  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; p8%x@%k  
%       y = zernfun(n,m,r(idx),theta(idx)); E2LpQNvN%g  
%       figure('Units','normalized') dL |D  
%       for k = 1:10 `L]cJ0tAs  
%           z(idx) = y(:,k); Pqo"~&Y|~  
%           subplot(4,7,Nplot(k)) -+Kx^V#'R  
%           pcolor(x,x,z), shading interp \[B5j0vV,  
%           set(gca,'XTick',[],'YTick',[]) =>)l6**UE  
%           axis square }/SbmW8(1  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ~>k<I:BtrT  
%       end &h
`s:Y  
% c,!Ijn\;(  
%   See also ZERNPOL, ZERNFUN2.  zy  
s34{\/'D+  
%   Paul Fricker 11/13/2006 CS:j->  
Wf-i)oc4I  
//3iai  
% Check and prepare the inputs: 7xO =:*  
% -----------------------------  pzg|?U  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (6^v`SZ  
    error('zernfun:NMvectors','N and M must be vectors.') Owo2DsT t  
end Q\<C9%a  
q$(aMO&J  
if length(n)~=length(m) DJS0;!# |O  
    error('zernfun:NMlength','N and M must be the same length.') i`z1if6O  
end d2Z5HFtY  
2
h IM!wQ  
n = n(:); +Hc[5WL  
m = m(:); X#Y0g`muW  
if any(mod(n-m,2)) A Ns.`S  
    error('zernfun:NMmultiplesof2', ... }/4 AT  
          'All N and M must differ by multiples of 2 (including 0).') 4;<?ec(dc  
end Z[)t34EY"  
`J'xVq#O  
if any(m>n) Xjw>Qws  
    error('zernfun:MlessthanN', ... $.a<b^.Xi  
          'Each M must be less than or equal to its corresponding N.') M56^p,  
end r?nvJHP  
|cEJRs@B  
if any( r>1 | r<0 ) p^3]Q  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5k)QjZo  
end )M<"YI)g  
 sX.L  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~pO6C*"  
    error('zernfun:RTHvector','R and THETA must be vectors.') Ur6UE2  
end Qj.]I0d  
(+`pEDD{X  
r = r(:); ai)S:2  
theta = theta(:); :1^ R$0d  
length_r = length(r); ,|D_? D)U  
if length_r~=length(theta) umaF}}-Q{  
    error('zernfun:RTHlength', ... Nj$3Ig"l  
          'The number of R- and THETA-values must be equal.') k@L},Td  
end N7Kq$G2O  
JR8 b[Oj.S  
% Check normalization: "1FPe63\*O  
% -------------------- {_&'tXL  
if nargin==5 && ischar(nflag) EiQX*v  
    isnorm = strcmpi(nflag,'norm'); ;IZ*o<_  
    if ~isnorm = NHuj.  
        error('zernfun:normalization','Unrecognized normalization flag.') j]U sb
_7  
    end ELfcZfJ  
else ROlef;/A  
    isnorm = false; Zyt,D|eWj  
end 3=5K7F  
ajC'C!"^Ty  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UCG8=+t5T  
% Compute the Zernike Polynomials o=}}hE\H  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^,*ED Yz  
f4UnLig  
% Determine the required powers of r: JL[$B1  
% ----------------------------------- 0zQ"5e?qy  
m_abs = abs(m); O=~8+sa  
rpowers = []; Ir&rTGFN  
for j = 1:length(n) W; yNg  
    rpowers = [rpowers m_abs(j):2:n(j)]; d` %8qLIW  
end t5t,(^;f  
rpowers = unique(rpowers); Oxo?\ :T  
~QgyhJM_h=  
% Pre-compute the values of r raised to the required powers, %IrR+f+H  
% and compile them in a matrix: _&V%idz!0  
% ----------------------------- K.)ionb  
if rpowers(1)==0 5.m&93P  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); x93h{Kf  
    rpowern = cat(2,rpowern{:}); [Jv0^"]  
    rpowern = [ones(length_r,1) rpowern]; w0qrh\3du  
else wJyrF  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); B7PkCS&X  
    rpowern = cat(2,rpowern{:}); I><B6pIR  
end Hdvtgss!  
t/55tL  
% Compute the values of the polynomials: e_RLKFv7  
% -------------------------------------- 8(-V pU  
y = zeros(length_r,length(n)); <A|X4;  
for j = 1:length(n) ?QA![  
    s = 0:(n(j)-m_abs(j))/2; paKur%2u  
    pows = n(j):-2:m_abs(j); V"Cx5#\7C  
    for k = length(s):-1:1 bfo..f-0/Y  
        p = (1-2*mod(s(k),2))* ... 7egE."  
                   prod(2:(n(j)-s(k)))/              ... LGnb"ZN  
                   prod(2:s(k))/                     ... SeuC7!q{  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... m=Mb'<  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); L& =a(  
        idx = (pows(k)==rpowers); 9mE6Cp.Wv  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ba5,?FVI~  
    end (=A61]yB  
     .8o?`  
    if isnorm "PgVvm#w'  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); l5h+:^#M5c  
    end L`'#}#O l  
end 9S'u1%  
% END: Compute the Zernike Polynomials /q|r!+  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nm"]q`(K  
`iHyGfm  
% Compute the Zernike functions: zM+4<k_dH]  
% ------------------------------ JJ:pA_uX  
idx_pos = m>0; EdL2t``  
idx_neg = m<0; 5D]3I=kj  
1G}f83yR  
z = y; 1`hmD1d  
if any(idx_pos) 7eNLs  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); IRhi1{K$"  
end @},|i*H/  
if any(idx_neg) 5!QT }Um  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); #|PPkg%v
<  
end WCNycH+1  
rn$G.SMgz  
% 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