[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 **rA
/*Oc  
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！ SCl$+9E  

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)  \|nF55W [  

a'f"Zdh%w  
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。 Z`nHpmNM  
R%o:'-~  
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) E^w2IIw  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. I|69|^  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of lxb+
0fiN  
%   order N and frequency M, evaluated at R.  N is a vector of ,T@+QXh  
%   positive integers (including 0), and M is a vector with the &5puGnTZ  
%   same number of elements as N.  Each element k of M must be a %jz]s4u$5j  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) 0+MNu8t  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is k#Qav1_  
%   a vector of numbers between 0 and 1.  The output Z is a matrix >QO^h<.>  
%   with one column for every (N,M) pair, and one row for every Jb~$Vrdy  
%   element in R. 0JzH dz  
% O 4zD >O  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- |U{9Yy6p  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is li'h&!|]  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to G2 A#&86J{  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 0$)s? \  
%   for all [n,m]. FsQeyh>  
% .j?`U[V%a  
%   The radial Zernike polynomials are the radial portion of the 873$EiyXR  
%   Zernike functions, which are an orthogonal basis on the unit O ]o7  
%   circle.  The series representation of the radial Zernike p=%Vo@*]  
%   polynomials is XN9s!5A<L)  
% |,3s]b`  
%          (n-m)/2 M)S(:Il6Xx  
%            __ & $E[l'  
%    m      \       s                                          n-2s F.5'5%  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r e??tp]PLn  
%    n      s=0 Zjqa n  
% x`T  
%   The following table shows the first 12 polynomials. xCN6?  
% '%Og9Bgd+  
%       n    m    Zernike polynomial    Normalization e RY2.!  
%       --------------------------------------------- _8t5rF  
%       0    0    1                        sqrt(2) 9U[Gh97Sf  
%       1    1    r                           2 rR`'l=,t  
%       2    0    2*r^2 - 1                sqrt(6) *D`]7I~}  
%       2    2    r^2                      sqrt(6) a&:1W83  
%       3    1    3*r^3 - 2*r              sqrt(8) Gk_%WY*  
%       3    3    r^3                      sqrt(8) &"HxAK)f  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Mx9#YJ?t~  
%       4    2    4*r^4 - 3*r^2            sqrt(10) DUH\/<^g  
%       4    4    r^4                      sqrt(10) ?bFP'.  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) cUW>`F(S  
%       5    3    5*r^5 - 4*r^3            sqrt(12) ?LJ$:u  
%       5    5    r^5                      sqrt(12) *+(t2!yFmE  
%       --------------------------------------------- UNLmnj;-Q  
% CTawXHM  
%   Example: kc*zP=  
% ^n8ioL\*i  
%       % Display three example Zernike radial polynomials |OW/-&)  
%       r = 0:0.01:1; !ieMhJ5r  
%       n = [3 2 5]; N>h/!# ZC  
%       m = [1 2 1]; C]S~DK1  
%       z = zernpol(n,m,r); @ig'CF%(  
%       figure [/dGOl+  
%       plot(r,z) ?%RAX CK  
%       grid on fP 1V1ao  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') c:#<g/-{wM  
% 1{6BU!  
%   See also ZERNFUN, ZERNFUN2. ]vj.s/F~  
L{`S^'P<  
% A note on the algorithm. "FuOWI{in  
% ------------------------ U@t"o3E  
% The radial Zernike polynomials are computed using the series 0$=Uhi  
% representation shown in the Help section above. For many special EQQ/E!N8l  
% functions, direct evaluation using the series representation can wVegr  
% produce poor numerical results (floating point errors), because 5zk<s`h  
% the summation often involves computing small differences between SCwAAE9s]  
% large successive terms in the series. (In such cases, the functions ~ZrSoVP=  
% are often evaluated using alternative methods such as recurrence ggluQGA  
% relations: see the Legendre functions, for example). For the Zernike [3$L}m  
% polynomials, however, this problem does not arise, because the H~Z$pk%  
% polynomials are evaluated over the finite domain r = (0,1), and y{&k`H  
% because the coefficients for a given polynomial are generally all 4%!
#=JCl  
% of similar magnitude. Zl,c+/  
% 7 s+j)  
% ZERNPOL has been written using a vectorized implementation: multiple X;2I' Kg  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] $~>3bik@  
% values can be passed as inputs) for a vector of points R.  To achieve '8%pEl^  
% this vectorization most efficiently, the algorithm in ZERNPOL #+VH]7]  
% involves pre-determining all the powers p of R that are required to 0!4;."S  
% compute the outputs, and then compiling the {R^p} into a single @|I:A  
% matrix.  This avoids any redundant computation of the R^p, and V[9#+l~#  
% minimizes the sizes of certain intermediate variables. }
E o\=>l7  
% Ufx^@%v  
%   Paul Fricker 11/13/2006 2bJqZ,@  
K)-Gv|*t  
[^N8v;O  
% Check and prepare the inputs: NxOiT#YH  
% ----------------------------- 8]SJ=c"}Xf  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [cJQ"G '  
    error('zernpol:NMvectors','N and M must be vectors.') Mn)>G36(  
end ,/m@<NyK  
!WTZ=|  
if length(n)~=length(m) .`I;qF  
    error('zernpol:NMlength','N and M must be the same length.') =J@M,mbHg  
end j@w+>h  
=1!,A  
n = n(:); Vgh;w-a  
m = m(:); OO7sj@  
length_n = length(n); 8`\^wG$W  
25bbuhss  
if any(mod(n-m,2)) 
"o| f  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') "hE/f~\  
end /6?A#%hc  
}kNbqwVP  
if any(m<0) v~l_6V}  
    error('zernpol:Mpositive','All M must be positive.') n jfh4}g:  
end /KL;%:7  
{c 82bFiv  
if any(m>n) os:/-A_m  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 6}V)\"u&  
end {"^LUw8fd  
,5Vc  
if any( r>1 | r<0 ) ywSV4ZtM  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') Sio> QL Y  
end '7'*+sgi$  
su?{Cj6*  
if ~any(size(r)==1) _oV;Y`_  
    error('zernpol:Rvector','R must be a vector.') qcNu9Ih  
end s!lLdR[g  
98c##NV(7|  
r = r(:); qVHXZdGL  
length_r = length(r); |igr3p5Fw  
PZT]H?  
if nargin==4 mYU7b8x_  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); [RAzKzC\M  
    if ~isnorm *qX!  
        error('zernpol:normalization','Unrecognized normalization flag.') +%O_xq
q  
    end
t:NYsL  
else G,{=sFX  
    isnorm = false; b`W2^/D  
end |"K<  
LnwI 7uvq  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2H,^i,  
% Compute the Zernike Polynomials 6%jv|\>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d9j+==S <  
uG5RE  
% Determine the required powers of r: O^Y}fo'  
% ----------------------------------- %=ZN2)7{  
rpowers = []; 8+7n"6GY2/  
for j = 1:length(n) .C6wsmQ  
    rpowers = [rpowers m(j):2:n(j)]; I.4o9Z[?  
end iY|zv|;]=  
rpowers = unique(rpowers); LTn@OhC  
 (0wQ [(  
% Pre-compute the values of r raised to the required powers, ^R g=*L  
% and compile them in a matrix: wqB 5KxO  
% ----------------------------- nnzfKn:J  
if rpowers(1)==0 6w?l I  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); jX9{Ki"  
    rpowern = cat(2,rpowern{:}); gv
6}GE
 
    rpowern = [ones(length_r,1) rpowern]; ak
SUk)}e  
else k;7R3O@  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); |9fvj6?Y  
    rpowern = cat(2,rpowern{:}); GlVb|O"  
end >}uDQwX8  
`4xnM`:L"  
% Compute the values of the polynomials: =DL |Q  
% -------------------------------------- @4O;dFOQ)  
z = zeros(length_r,length_n); I[x+7Y0k9  
for j = 1:length_n .wdWs tQ  
    s = 0:(n(j)-m(j))/2; E43Gk!/|(  
    pows = n(j):-2:m(j); Tz`O+fx&  
    for k = length(s):-1:1 TKwMgC}<[  
        p = (1-2*mod(s(k),2))* ... u|.c?fW'3  
                   prod(2:(n(j)-s(k)))/          ... o+wG69  
                   prod(2:s(k))/                 ... O<*l"fw3  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... <FkoWN  
                   prod(2:((n(j)+m(j))/2-s(k))); qe/|u3I<lF  
        idx = (pows(k)==rpowers); u|G&CV#r  
        z(:,j) = z(:,j) + p*rpowern(:,idx); nfldj33*  
    end >~%EB?8  
     rfz\DvVd  
    if isnorm wU"0@^k]<  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); |}FK;@'I6  
    end oP"X-I  
end hja;d1yH  
<[oPh(!V  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) x!i(M>P  
%ZERNFUN2 Single-index Zernike functions on the unit circle. {hNvCk  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated _MI8P/  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive i3SrsVSG  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, wnPg).  
%   and THETA is a vector of angles.  R and THETA must have the same 0DZ}8"2  
%   length.  The output Z is a matrix with one column for every P-value, 17..  
%   and one row for every (R,THETA) pair. p'fD:M:  
% M'gL_Xsei  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike +HpPVuV  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) zK_+UT  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 6^Q/D7U;s  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ^p}S5,  
%   for all p. nYvx[ zq?^  
% {\P`-'C  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 fecV[  
%   Zernike functions (order N<=7).  In some disciplines it is "R!)"B==  
%   traditional to label the first 36 functions using a single mode 7<Yf  
%   number P instead of separate numbers for the order N and azimuthal G9|w o)N  
%   frequency M. C3hQT8~  
% !Z}d^$  
%   Example: zZhA]J  
% f)b+>!  
%       % Display the first 16 Zernike functions O^L#(8bC  
%       x = -1:0.01:1; ;/79tlwq  
%       [X,Y] = meshgrid(x,x); yPmo@aw]1  
%       [theta,r] = cart2pol(X,Y); 5.TeH@(  
%       idx = r<=1; BPwn!ii|  
%       p = 0:15; }}Kjb  
%       z = nan(size(X)); ~Q
3y3,x  
%       y = zernfun2(p,r(idx),theta(idx)); g2|qGfl{C  
%       figure('Units','normalized') nF#1B4b>  
%       for k = 1:length(p) A#@9|3  
%           z(idx) = y(:,k); je[1>\3W  
%           subplot(4,4,k) ;WqWD
-C  
%           pcolor(x,x,z), shading interp d OYEl<!J  
%           set(gca,'XTick',[],'YTick',[]) })#SjFq<V  
%           axis square 3$yOv"`  
%           title(['Z_{' num2str(p(k)) '}']) vTk\6o q  
%       end d8p<f+  
% ?B5934X  
%   See also ZERNPOL, ZERNFUN. P2t{il
 
>%?kp[  
%   Paul Fricker 11/13/2006 h
@H8oZ[  
j]X$7  
p7{%0
  
% Check and prepare the inputs: S!r,p
};  
% ----------------------------- 4]P5k6nV  
if min(size(p))~=1 VHbQLJ0  
    error('zernfun2:Pvector','Input P must be vector.') 'Y;M
%  
end #=81`u  
ulAOQGZ  
if any(p)>35 `Jv~.EF%  
    error('zernfun2:P36', ... R?E< }\!  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... kHhxR;ymA7  
           '(P = 0 to 35).']) G
Lpl  
end 5nA *'($j  
v&]k8Hc-  
% Get the order and frequency corresonding to the function number: Gp.XTz#=  
% ---------------------------------------------------------------- 0g{`Qd  
p = p(:); m5'nqy F  
n = ceil((-3+sqrt(9+8*p))/2); }9FAM@x1K&  
m = 2*p - n.*(n+2); *]#(?W.$w  
d>wpG^"w  
% Pass the inputs to the function ZERNFUN:  qH9bo-6  
% ---------------------------------------- 5?=haGn  
switch nargin $E,,::oJ  
    case 3 :g~X"C1s  
        z = zernfun(n,m,r,theta); {n'+P3\T:  
    case 4 9[@K4&  
        z = zernfun(n,m,r,theta,nflag); 5%#V>|@e#  
    otherwise KM:k<pvi  
        error('zernfun2:nargin','Incorrect number of inputs.') +f"q^RIU  
end rFLm!J]  
z^z,_?q;  
% 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 ;QS(`SK l  
function z = zernfun(n,m,r,theta,nflag) !V O^oD7  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~bnyk%S o  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^/M-*U8ab  
%   and angular frequency M, evaluated at positions (R,THETA) on the WFm\ bZ.  
%   unit circle.  N is a vector of positive integers (including 0), and `"s*'P398  
%   M is a vector with the same number of elements as N.  Each element jV 982Y  
%   k of M must be a positive integer, with possible values M(k) = -N(k) :v Do{My^1  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ~zO>Q4-k  
%   and THETA is a vector of angles.  R and THETA must have the same ?K!^[aO}=  
%   length.  The output Z is a matrix with one column for every (N,M) Bbj%RF2,  
%   pair, and one row for every (R,THETA) pair. aUYq~E tj  
% MY w3+B+Jj  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +LhV4@zC  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), KSgYf;  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral
VOkSR6  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, $_Kcm"oj  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized x"83[0ib  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )[np{eF.k  
% N?j#=b+D  
%   The Zernike functions are an orthogonal basis on the unit circle. Y2d(HD@  
%   They are used in disciplines such as astronomy, optics, and 08MY=PC~R  
%   optometry to describe functions on a circular domain. `P *wz<
 
% AO~f=GW  
%   The following table lists the first 15 Zernike functions. kesuM3  
% 76eF6N+%}t  
%       n    m    Zernike function           Normalization ^hRx{A  
%       -------------------------------------------------- FnWN]9  
%       0    0    1                                 1 @aC9O9|~  
%       1    1    r * cos(theta)                    2 m'PU0x  
%       1   -1    r * sin(theta)                    2 i1JVvNMQ,  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) KJYcP72P  
%       2    0    (2*r^2 - 1)                    sqrt(3) Rc2JgV  
%       2    2    r^2 * sin(2*theta)             sqrt(6) _uq[D`=  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 4d63+iM+}  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) W!o|0u!D  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) mhW*rH*m  
%       3    3    r^3 * sin(3*theta)             sqrt(8) IcJQC  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) t b>At*tO  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S.R|Bwj}(Y  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) /I48jO^2  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mkuK$Mj  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 
yN{TcX  
%       -------------------------------------------------- 7fXta|eP0  
% pB:/oHV  
%   Example 1: F S!D  
% Y|nC_7&Bv  
%       % Display the Zernike function Z(n=5,m=1) tRVz4
fk[G  
%       x = -1:0.01:1; `DS7J\c$  
%       [X,Y] = meshgrid(x,x); ESmWK;7b  
%       [theta,r] = cart2pol(X,Y); t_kRYdW9  
%       idx = r<=1; VM}7 ~  
%       z = nan(size(X)); RMs+pN<5  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ^E&WgXlb  
%       figure *X\J[$!  
%       pcolor(x,x,z), shading interp cq"#[y$r  
%       axis square, colorbar U28frRa  
%       title('Zernike function Z_5^1(r,\theta)') ]XjL""EbC  
% 8 -YC#&  
%   Example 2: 9?tG?b0  
% w&x$RP  
%       % Display the first 10 Zernike functions v(P5)R,  
%       x = -1:0.01:1; 821;;]H  
%       [X,Y] = meshgrid(x,x); YB]{gm2  
%       [theta,r] = cart2pol(X,Y); R q`j|tY  
%       idx = r<=1; 8}K4M(  
%       z = nan(size(X)); |ngv{g  
%       n = [0  1  1  2  2  2  3  3  3  3]; D}~uxw;[^  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; O"~CZh,:r}  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; *h
M5pw  
%       y = zernfun(n,m,r(idx),theta(idx)); q,T4- E  
%       figure('Units','normalized') |+Cd2[hN  
%       for k = 1:10 9xOTR#B:_V  
%           z(idx) = y(:,k); @tlWyUju  
%           subplot(4,7,Nplot(k)) zALtG<_t  
%           pcolor(x,x,z), shading interp f~:wI
9  
%           set(gca,'XTick',[],'YTick',[]) UsgrI>|l  
%           axis square y' RQ_Gi  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -"6Z@8=  
%       end }"M5"?  
% }=p+X:k=  
%   See also ZERNPOL, ZERNFUN2. 'fPDODE  

IL{tm0$r  
%   Paul Fricker 11/13/2006 m,)o&ix1  
g\1|<jb3  
uj@d {AQ  
% Check and prepare the inputs: CU@}{}Yl  
% ----------------------------- Gq-~zmg  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^$s&bH'8  
    error('zernfun:NMvectors','N and M must be vectors.') &l0,q=T  
end H'}6Mw%ra  
Y)2#\ F  
if length(n)~=length(m) hA1p#  
    error('zernfun:NMlength','N and M must be the same length.') -I[KIeF  
end "R]wPF5u  
nh+Hwj#(x  
n = n(:); dP?QPk
y{9  
m = m(:); _R}yZ=di  
if any(mod(n-m,2)) (0["|h32,  
    error('zernfun:NMmultiplesof2', ... ` <u2 N  
          'All N and M must differ by multiples of 2 (including 0).') $r)NL  
end %+oqAYm+s  
x(A8FtG  
if any(m>n) 0YAH[YF  
    error('zernfun:MlessthanN', ... m(`O>zS  
          'Each M must be less than or equal to its corresponding N.') [lGxys)J  
end aU*}.{<!  
L_q3m-x0h  
if any( r>1 | r<0 ) hQeG#KQ  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') z"-oD*ICw  
end g3f;JB  
<m~{60{  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]f>0P3O5&  
    error('zernfun:RTHvector','R and THETA must be vectors.') M(vX.kF  
end gYBMi)`RT  
~ ReX$
9  
r = r(:); Ol;DJV
 
theta = theta(:); uU=!e&3  
length_r = length(r); tIS.,CEQF  
if length_r~=length(theta) ={;7WB$  
    error('zernfun:RTHlength', ... CSY-{  
          'The number of R- and THETA-values must be equal.') e.fxB  
end [5]n,toAh  
x[xRqC vL  
% Check normalization: 3H|drj:KV  
% -------------------- 8nwps(3  
if nargin==5 && ischar(nflag) Zv(6VVj  
    isnorm = strcmpi(nflag,'norm'); c Qe3  
    if ~isnorm 0lV;bVa%  
        error('zernfun:normalization','Unrecognized normalization flag.') Q%524%f$  
    end GK;IY=8W  
else m\70&%v  
    isnorm = false; +ViL"  
end x_CY`Y  
;*0nPhBw0>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eAStpG"*  
% Compute the Zernike Polynomials Tv6y+l  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1OV] W f  
6s'n r7'0  
% Determine the required powers of r: q[9N4nj$<  
% ----------------------------------- a_-@rceU  
m_abs = abs(m); nw_s:  
rpowers = []; &TL"Hd  
for j = 1:length(n) y06xl:iQwF  
    rpowers = [rpowers m_abs(j):2:n(j)]; ?#Y:2LqPC  
end 5nTcd
@lX  
rpowers = unique(rpowers); '$rCV,3q  
?J-
\}X  
% Pre-compute the values of r raised to the required powers, TZGk[u^*  
% and compile them in a matrix: "$D'gSoYe  
% ----------------------------- sWB@'P:x  
if rpowers(1)==0 0+u>"7T  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5=I"bnIU  
    rpowern = cat(2,rpowern{:}); bZr,jLEf  
    rpowern = [ones(length_r,1) rpowern]; RAnF=1[v  
else , &
n"
#  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e-OKv#]  
    rpowern = cat(2,rpowern{:}); _#MKpH  
end yPY{ZADkQ  
UWhJkJsX  
% Compute the values of the polynomials: i=1crJ:  
% -------------------------------------- *K|ah:(r1\  
y = zeros(length_r,length(n));
&=kb>*  
for j = 1:length(n) \ \Tz'>[\  
    s = 0:(n(j)-m_abs(j))/2; W\j)Vg__e  
    pows = n(j):-2:m_abs(j); y0ObcP.MA  
    for k = length(s):-1:1 ,7k-LAA  
        p = (1-2*mod(s(k),2))* ... hg#O_4D  
                   prod(2:(n(j)-s(k)))/              ... >#'?}@FWQN  
                   prod(2:s(k))/                     ... cx ("F/Jm  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3o0ZS^#eB  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Fn iht<  
        idx = (pows(k)==rpowers); p&5>j\uJ1&  
        y(:,j) = y(:,j) + p*rpowern(:,idx); jVZ<i}h0B  
    end dsj}GgG?Z  
     W/b)OlG"2  
    if isnorm Jgg<u#  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t%J1(H  
    end Lis>Qr  
end ekU%^R<  
% END: Compute the Zernike Polynomials Jz3,vVfQ:  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% H5&._  
Ok|Dh;1_  
% Compute the Zernike functions: L &hw-.Q  
% ------------------------------ KV$4}{  
idx_pos = m>0; 3Zl:rYD?  
idx_neg = m<0; hvQXYo>TZx  
XogCq?_m  
z = y; jwBJG7\  
if any(idx_pos) yTh%[k  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X,#~[%h$-=  
end eG8l^[  
if any(idx_neg) )7[#Ti  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 1_A_)l11  
end UqyW8TCf?  
p\F%Nj,  
% 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