[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 F+"_]  
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！ ]{oZn5F  

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)  !cwZ*eM  

J;C:nE|V  
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。 `ePC$Ovn  
p+CUYo(  
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)  is'V%q  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. al2t\Iq90  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of RB5SK#z  
%   order N and frequency M, evaluated at R.  N is a vector of sV\_DP/l  
%   positive integers (including 0), and M is a vector with the oBzl=N3<  
%   same number of elements as N.  Each element k of M must be a "y1Iu   
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) @~3--  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is W(,j2pU  
%   a vector of numbers between 0 and 1.  The output Z is a matrix .tngN<f  
%   with one column for every (N,M) pair, and one row for every h>N}M}8  
%   element in R. );5o13h2  
% z/@_?01T=  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 79\wjR!T  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ]v+<K63@T  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to h9vcN#2
2D  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 i5,iJe0cA  
%   for all [n,m]. NGx3f3 9  
% J2UQq7-y  
%   The radial Zernike polynomials are the radial portion of the zM'eqo>!c>  
%   Zernike functions, which are an orthogonal basis on the unit } M#e\neii  
%   circle.  The series representation of the radial Zernike /jbAf]"F;  
%   polynomials is 5KCB^`|b>t  
% Q;h.}N8W  
%          (n-m)/2 bO '\QtW9  
%            __ Sj9fq*  
%    m      \       s                                          n-2s )vp0X\3q`  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r K_7pr~D]@r  
%    n      s=0 ajYe?z  
% gP^2GnjHL8  
%   The following table shows the first 12 polynomials. &#r+a'  
% 8{ zX=  
%       n    m    Zernike polynomial    Normalization 6{Wo5O{!\  
%       --------------------------------------------- -YRIe<}E -  
%       0    0    1                        sqrt(2) )2}R1K>  
%       1    1    r                           2 rIyH/=;  
%       2    0    2*r^2 - 1                sqrt(6) pLMt2G  
%       2    2    r^2                      sqrt(6) qd`e:s*%  
%       3    1    3*r^3 - 2*r              sqrt(8) v^|U?  
%       3    3    r^3                      sqrt(8) i\R0+O{  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 5
]xuU.w'  
%       4    2    4*r^4 - 3*r^2            sqrt(10) 7|rH9Bc{U  
%       4    4    r^4                      sqrt(10) 3h@]cWp  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) .~z'm$s1o  
%       5    3    5*r^5 - 4*r^3            sqrt(12) @^{Hq6_`  
%       5    5    r^5                      sqrt(12) 
]hl*6  
%       --------------------------------------------- la!]Y-s)'4  
% h Yu6PWK  
%   Example: 1{}p_"s>  
% Jt~Ivn,  
%       % Display three example Zernike radial polynomials ZsmOn#`=^}  
%       r = 0:0.01:1; -<iP$,bq72  
%       n = [3 2 5]; -m@o\9Ic  
%       m = [1 2 1]; sNf& "C!;  
%       z = zernpol(n,m,r); >{#JIG.  
%       figure .RD<]BxJ  
%       plot(r,z) bIQ,=EA1  
%       grid on b#j:)PA0C  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Rr) 5[  
% ]#!uke Q  
%   See also ZERNFUN, ZERNFUN2. #Z&/w.D2  
'&>"`q  
% A note on the algorithm. O zAIz+`  
% ------------------------ kZ]H[\Fs  
% The radial Zernike polynomials are computed using the series %mI0*YRma  
% representation shown in the Help section above. For many special 1S{Biqi+  
% functions, direct evaluation using the series representation can j"W>fC/u  
% produce poor numerical results (floating point errors), because x*7@b8J  
% the summation often involves computing small differences between C]^Ep
  
% large successive terms in the series. (In such cases, the functions kY0HP a  
% are often evaluated using alternative methods such as recurrence [%W'd9`>  
% relations: see the Legendre functions, for example). For the Zernike Vl^(K_`(  
% polynomials, however, this problem does not arise, because the #3uv^m LGa  
% polynomials are evaluated over the finite domain r = (0,1), and NvK9L.K  
% because the coefficients for a given polynomial are generally all iL/
c^(
1  
% of similar magnitude. ycA<l"  
% 0<M-asI?  
% ZERNPOL has been written using a vectorized implementation: multiple 05UN <l]  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] i~B?p[  
% values can be passed as inputs) for a vector of points R.  To achieve -I< >Ab  
% this vectorization most efficiently, the algorithm in ZERNPOL -D^I;[j_  
% involves pre-determining all the powers p of R that are required to 5Xy(za  
% compute the outputs, and then compiling the {R^p} into a single ,67Q!/O  
% matrix.  This avoids any redundant computation of the R^p, and _nGx[1G( 5  
% minimizes the sizes of certain intermediate variables. 7h' C"rH  
% ChBf:`e  
%   Paul Fricker 11/13/2006 F.s$Y+c!6  
C{)1#<`  
?hoOSur+  
% Check and prepare the inputs: [8V;Q  
% ----------------------------- Cq5.gkS<  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ULx:2jz  
    error('zernpol:NMvectors','N and M must be vectors.') 'nmGHorp  
end 0uy'Py@2<  
e=amh  
if length(n)~=length(m) kc'$4 J4Tw  
    error('zernpol:NMlength','N and M must be the same length.') X9>fE{)!  
end I}$`gUXX8x  
r&=ulg  
n = n(:); s{^98*  
m = m(:); cXweg;  
length_n = length(n); q~{) {t;  
w\C1Bh!  
if any(mod(n-m,2)) !z?   
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') RB>=#03  
end )W\)37=.  

]4~Yi1]  
if any(m<0)  3@Ndn  
    error('zernpol:Mpositive','All M must be positive.') 2- iY:r  
end e02Hf{eOfw  
HcRw9,I'  
if any(m>n) 7w )?s@CD  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') S!K<kn`E3  
end ,GOIg|51  
tFU4%c7V  
if any( r>1 | r<0 ) fe .=Z&  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') +%5L2/n7  
end =<\22d5L  
,%!m%+K9a  
if ~any(size(r)==1) XG#?fr}L  
    error('zernpol:Rvector','R must be a vector.') w4 yrAj 2  
end T!/o^0w  
A%w9Da?B  
r = r(:); ,fjY|ip  
length_r = length(r); B>{%$@4  
qI'pjTMDY  
if nargin==4 7cc^
n\c?Y  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ;<~f-D,  
    if ~isnorm E}wT5t;u  
        error('zernpol:normalization','Unrecognized normalization flag.') lHiWzt u  
    end nDi^s{  
else zC50 @S3|  
    isnorm = false; , ['}9:f9  
end hcVu`Bn  
iXWzIb}CJ-  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8W3zrnc  
% Compute the Zernike Polynomials B*/!s7c.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0E\#!L  
}6Pbjm*  
% Determine the required powers of r: .!1[I{KU  
% ----------------------------------- (
KI9j7  
rpowers = []; m.++nF  
for j = 1:length(n) _'D(>e?  
    rpowers = [rpowers m(j):2:n(j)]; Z+B*V)a=  
end MlTC?Rp#  
rpowers = unique(rpowers); x'EEmjJ  
Kp7DI0~  
% Pre-compute the values of r raised to the required powers, ,ye}p1M  
% and compile them in a matrix: cb-IRGF  
% ----------------------------- MkW=sD_  
if rpowers(1)==0 tE%g)hL-  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fz=8"cDR  
    rpowern = cat(2,rpowern{:}); MKbcJZe  
    rpowern = [ones(length_r,1) rpowern]; QC'Ru'8S  
else ;R=n<=Axa  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (iKJ~bJ  
    rpowern = cat(2,rpowern{:}); xLed];2G  
end
S(@kdL  
|GMo"[  
% Compute the values of the polynomials: bo<P%$(D  
% -------------------------------------- *VsGa<V  
z = zeros(length_r,length_n); _DxHJl  
for j = 1:length_n AL":j6!OQ  
    s = 0:(n(j)-m(j))/2; L#SW!  
    pows = n(j):-2:m(j); 1$RJzHS  
    for k = length(s):-1:1 &~2m@X(o  
        p = (1-2*mod(s(k),2))* ... fXWy9 #M  
                   prod(2:(n(j)-s(k)))/          ... <T>s;b  
                   prod(2:s(k))/                 ... "{8j!+]4i  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... {.Qv1oOa  
                   prod(2:((n(j)+m(j))/2-s(k))); D%+yp  
        idx = (pows(k)==rpowers); !aSj1 2J  
        z(:,j) = z(:,j) + p*rpowern(:,idx); /KvJjt'8  
    end k86TlQRh  
     HGAi2+&  
    if isnorm
DpggZ|
J  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); np2&W'C/i  
    end +yI$4MY  
end ZK;/~9KU  
WVD48}HF-  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) oeg Bk  
%ZERNFUN2 Single-index Zernike functions on the unit circle. wU|@fm"  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated zG$5g^J  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive !p$p 7   
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, %|e)s_%XE  
%   and THETA is a vector of angles.  R and THETA must have the same ^?RH<z  
%   length.  The output Z is a matrix with one column for every P-value, 1UK= t  
%   and one row for every (R,THETA) pair. ^mn!;nu  
% W`PJflr|  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike i.'"`pn_  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) T; tY7;<  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) p _[,P7  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 .v(GVkE}  
%   for all p. JXL?.{'A  
% M6
&=-  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 T^Ia^B-%}g  
%   Zernike functions (order N<=7).  In some disciplines it is tTBDb  
%   traditional to label the first 36 functions using a single mode F%<*a,m6g  
%   number P instead of separate numbers for the order N and azimuthal N ;=zo-8  
%   frequency M. 2*Qi4%s#  
% y5F+~z}{  
%   Example: "LTw;& y  
% ef^GJTv&k  
%       % Display the first 16 Zernike functions ]7}!3m  
%       x = -1:0.01:1; 6HZtdRQF  
%       [X,Y] = meshgrid(x,x); kJmw
R  
%       [theta,r] = cart2pol(X,Y); 1q(Qr h  
%       idx = r<=1; (1|wM+)"  
%       p = 0:15; Yw#fQFm  
%       z = nan(size(X)); rX)&U4#[m  
%       y = zernfun2(p,r(idx),theta(idx)); 0?$|F0U"J  
%       figure('Units','normalized') K?J_cnJ`  
%       for k = 1:length(p) C*ep8{B  
%           z(idx) = y(:,k); }Q

4Vy  
%           subplot(4,4,k) r QiRhp  
%           pcolor(x,x,z), shading interp VOD-< "|  
%           set(gca,'XTick',[],'YTick',[]) Hmr f\(x  
%           axis square )Mdddz4  
%           title(['Z_{' num2str(p(k)) '}']) /%g9g_rt#  
%       end a%.W9=h=M(  
% w^Y/J4 I0  
%   See also ZERNPOL, ZERNFUN. N#Rb8&G)b  
!b_(|~7Lc  
%   Paul Fricker 11/13/2006 joskKik^  
#M|lBYdW}  
c45s #6  
% Check and prepare the inputs: B>c$AS\5y  
% ----------------------------- xgMh@@e  
if min(size(p))~=1 rmzzbLTu  
    error('zernfun2:Pvector','Input P must be vector.') `$Rgn3  
end lXTE#,XVf  
C0[U}Y/r2  
if any(p)>35 'UhHcMh:  
    error('zernfun2:P36', ... .F8[;+  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ^Zz^h@+  
           '(P = 0 to 35).']) B?i#m^S  
end z(A[xN@/W<  
[-*&ZYp  
% Get the order and frequency corresonding to the function number: %\ i&g$  
% ----------------------------------------------------------------
]UUa/ep
-  
p = p(:); )>{.t=#  
n = ceil((-3+sqrt(9+8*p))/2); V5(_7b#z``  
m = 2*p - n.*(n+2); \4wMv[;7  
_M/N_Fm  
% Pass the inputs to the function ZERNFUN: OJpfiZ@Q_  
% ---------------------------------------- F:q4cfL6  
switch nargin sR1_L/.  
    case 3 ]uox ^HC  
        z = zernfun(n,m,r,theta); `{:Nt#7  
    case 4 KxK,en4)+  
        z = zernfun(n,m,r,theta,nflag); pi"M*$  
    otherwise )9"^ D  
        error('zernfun2:nargin','Incorrect number of inputs.') ]TT >3"Dw7  
end Q// @5m_  
KV$&qM.  
% 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 syBpF:`-W  
function z = zernfun(n,m,r,theta,nflag) =!q]0#  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /al56n  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ck )W=  
%   and angular frequency M, evaluated at positions (R,THETA) on the aC[G_ACwc  
%   unit circle.  N is a vector of positive integers (including 0), and _y[C52,  
%   M is a vector with the same number of elements as N.  Each element 9SsVJ<9,R  
%   k of M must be a positive integer, with possible values M(k) = -N(k) |p[Mp:^^  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 6:G&x<{  
%   and THETA is a vector of angles.  R and THETA must have the same `.J)Z=o  
%   length.  The output Z is a matrix with one column for every (N,M) g7]S  
%   pair, and one row for every (R,THETA) pair. #aL.E(%  
% y\^zxG*]'  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "b`#RohCi  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), e2c'Wab  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral yZ6WbI8n  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, }{N#JTmjB#  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized V.:,Q  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [T r7SU#x  
% b
\ED<'  
%   The Zernike functions are an orthogonal basis on the unit circle. f4 S:L&  
%   They are used in disciplines such as astronomy, optics, and DQY1oM)D!  
%   optometry to describe functions on a circular domain. HjA~3l7  
% Hj>9#>b  
%   The following table lists the first 15 Zernike functions. >KuN
HuHu  
% P1[.[q/-e  
%       n    m    Zernike function           Normalization k*;U?C!  
%       -------------------------------------------------- c;]\$#2  
%       0    0    1                                 1 )8oyo~4?  
%       1    1    r * cos(theta)                    2 5V/&4$.U!  
%       1   -1    r * sin(theta)                    2 u;$qJj
S N  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 6<W^T9}v@/  
%       2    0    (2*r^2 - 1)                    sqrt(3) !O"2)RU1  
%       2    2    r^2 * sin(2*theta)             sqrt(6) HE+'fQ!R  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) >I@&"&d  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e*T^:2oRl  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) dY
ISjk@  
%       3    3    r^3 * sin(3*theta)             sqrt(8) (9]1p;  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) }Q:
CZ  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aQ(P#n>a2  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 1oO(;--u_  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @
xdtl{5G  
%       4    4    r^4 * sin(4*theta)             sqrt(10)  dHx4yFS  
%       -------------------------------------------------- 1sg:8AA  
% WVyDE1K<  
%   Example 1: {D8opepO)  
% ~s&r.6DW  
%       % Display the Zernike function Z(n=5,m=1) <7`k[~)VB  
%       x = -1:0.01:1; %R4 \[e  
%       [X,Y] = meshgrid(x,x); (enr{1  
%       [theta,r] = cart2pol(X,Y); VE]TT><  
%       idx = r<=1; AAfU]4
u0S  
%       z = nan(size(X)); A v>v\ :.>  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); f`ibP6%  
%       figure ,$t1LV;o=  
%       pcolor(x,x,z), shading interp #Db^*  
%       axis square, colorbar rEp\ld  
%       title('Zernike function Z_5^1(r,\theta)') [H\0 '  
% 9 D.
wW  
%   Example 2: bJPKe]spJ=  
% h(kPf]0  
%       % Display the first 10 Zernike functions ;rL>{UhG  
%       x = -1:0.01:1; }~LGq.H  
%       [X,Y] = meshgrid(x,x); 4j0;okQWV'  
%       [theta,r] = cart2pol(X,Y); pWE(?d_M{G  
%       idx = r<=1; {w3<dfJ  
%       z = nan(size(X)); O6$,J12l  
%       n = [0  1  1  2  2  2  3  3  3  3]; nnhI]#,a{  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; uDG>m7(}/h  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; GTbV5{Ss  
%       y = zernfun(n,m,r(idx),theta(idx)); U= GJuixy  
%       figure('Units','normalized') U4dfO=  
%       for k = 1:10 /NB|N*}O)  
%           z(idx) = y(:,k); ^vh!1"T  
%           subplot(4,7,Nplot(k)) sE]z.Po=  
%           pcolor(x,x,z), shading interp O=}  
%           set(gca,'XTick',[],'YTick',[]) Zt41fPQ  
%           axis square ,^ ,R .T  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) < (9 BO&  
%       end %Qj$@.*:  
% I3.JAoB>!  
%   See also ZERNPOL, ZERNFUN2. 
7?g(
{]  
(ZF~   
%   Paul Fricker 11/13/2006 DJdhOLx  
dL'oIBp  
w$s6NBF7  
% Check and prepare the inputs: IV1O/lGp
 
% ----------------------------- ShtV2}s|  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FDF DB  
    error('zernfun:NMvectors','N and M must be vectors.') \COoU("  
end cfBlHeYE  
 7a_u=\,  
if length(n)~=length(m) DI-&P3iGx  
    error('zernfun:NMlength','N and M must be the same length.') n|.eL8lX.<  
end ^Hf?["m^@  
ErA*a3  
n = n(:); qMVuB
v  
m = m(:); 3&[d.,/  
if any(mod(n-m,2)) 0ZD)(ps|  
    error('zernfun:NMmultiplesof2', ... 3^H-,b0^  
          'All N and M must differ by multiples of 2 (including 0).') wmbG$T%k  
end JC$_Pg
!  
H_8PK$c;  
if any(m>n)
(G{:O  
    error('zernfun:MlessthanN', ... .pxUO3g  
          'Each M must be less than or equal to its corresponding N.') x^`P[>  
end ooa"Th<  
NU.4_cixb  
if any( r>1 | r<0 ) u1'l4VgT  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') NP\/9 8|1  
end B&!>& Rbx  
{6)H.vpP  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @Z""|H"0  
    error('zernfun:RTHvector','R and THETA must be vectors.') !*qQ7  
end n[a%*i6x  
Xa'b@*o&  
r = r(:); W`#E[g?]  
theta = theta(:); -idbR[1{?  
length_r = length(r); +
:C
.G[+  
if length_r~=length(theta) W+V &  
    error('zernfun:RTHlength', ... _w <6o<@  
          'The number of R- and THETA-values must be equal.') G!F_Q7|-  
end ?6\A$
?  
? R[GSS1  
% Check normalization: 5@bmm]  
% -------------------- 0LHge7482  
if nargin==5 && ischar(nflag) $ JCOL  
    isnorm = strcmpi(nflag,'norm'); \@NnL\t u  
    if ~isnorm cE,,9M@^  
        error('zernfun:normalization','Unrecognized normalization flag.') ZD?LsD3  
    end >Zm|R|{BE  
else 8"wavh|g4  
    isnorm = false; IQ~EL';<w  
end f0{tBD!%  
YpSK|(  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nl-tJ.MU"  
% Compute the Zernike Polynomials pug;1UZ  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .'1]2/ad  
bHs},i6  
% Determine the required powers of r: 'A/{7*,  
% ----------------------------------- S U P  
m_abs = abs(m); lz#@_F|.*  
rpowers = []; V8%( h[  
for j = 1:length(n) ]MMXpj,9h  
    rpowers = [rpowers m_abs(j):2:n(j)]; <;Td8T;  
end :7qJ[k{g  
rpowers = unique(rpowers); ]4_)WUS.c  
^S(["6OJ(  
% Pre-compute the values of r raised to the required powers, 2+\@0j[q  
% and compile them in a matrix: ARB
^]  
% ----------------------------- gEq";B%?  
if rpowers(1)==0 _#E@&z".L  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !V0)
eC50  
    rpowern = cat(2,rpowern{:}); N?s5h?  
    rpowern = [ones(length_r,1) rpowern]; +227SPLd  
else 9aKCO4  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); s(Fxi|v;  
    rpowern = cat(2,rpowern{:}); /
T<,vR  
end 2s`~<EF N  
iS8yJRy  
% Compute the values of the polynomials: w}(Ht_6q{  
% -------------------------------------- HO8x:2m  
y = zeros(length_r,length(n)); Oufdi3
h  
for j = 1:length(n) 7/c9azmC  
    s = 0:(n(j)-m_abs(j))/2; ;MKfssG  
    pows = n(j):-2:m_abs(j); +&)&Ny$W  
    for k = length(s):-1:1 3)~z~p7  
        p = (1-2*mod(s(k),2))* ... NK(; -~{P  
                   prod(2:(n(j)-s(k)))/              ... u*!/J R  
                   prod(2:s(k))/                     ... Gc:oSvm  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... S6|L !pO  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); cD4H@!=a  
        idx = (pows(k)==rpowers); wArtg'=X  
        y(:,j) = y(:,j) + p*rpowern(:,idx); }mQh^  
    end TSYe~)I  
     q;qY#wD@  
    if isnorm sJcwN.s  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kF"G {5  
    end P*8DM3':  
end %-, -:e  
% END: Compute the Zernike Polynomials T#G (&0J5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R`emI7|  
C'//(gjQ-G  
% Compute the Zernike functions: 2sqNTuO6,|  
% ------------------------------ vWpkU<&3|  
idx_pos = m>0; <-a6'g2y  
idx_neg = m<0; ePwoza  
j[YO1q*  
z = y; 7S]akcT/  
if any(idx_pos) !T8h+3I  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); km#Rh^  
end yBwCFn.uP-  
if any(idx_neg) 73d7'Fw  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); XnI)s^  
end >Sh"/3%q  
3eS *U`_  
% 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