[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 =!`j7#:  
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！ }; f#^gz'  

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)  :OW;?{ ~j  

#?XQ7Im  
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。 1?| flK  
*FmTy|  
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) cv(PP-'\  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. k/03ZxC-  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of xP=/N!,#  
%   order N and frequency M, evaluated at R.  N is a vector of vfNAs>Xg"  
%   positive integers (including 0), and M is a vector with the fGv#s X  
%   same number of elements as N.  Each element k of M must be a |8bq>01~  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) Lw'9  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 2Sq_Tw3^  
%   a vector of numbers between 0 and 1.  The output Z is a matrix hb/]8mR  
%   with one column for every (N,M) pair, and one row for every xcJ`1*1N  
%   element in R. y}v+c%d  
% Bk}><H  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 2S8P}$mM  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is K
I]wm  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to "v}pdUW  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 +f0~D(d!_  
%   for all [n,m]. D{v8q)5r  
% 5/QRL\  
%   The radial Zernike polynomials are the radial portion of the f1PN|  
%   Zernike functions, which are an orthogonal basis on the unit "C?5f]T  
%   circle.  The series representation of the radial Zernike \7z^!m  
%   polynomials is .d?%;2*{q  
% _al|'obomy  
%          (n-m)/2 ICB~_O5  
%            __ 0GDvwy D1  
%    m      \       s                                          n-2s 0Y
]0!}  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r L&-hXGx=7  
%    n      s=0 y[@\j9Hq  
% ^+SkCO  
%   The following table shows the first 12 polynomials. #,(sAj  
% *[eL~oN.c  
%       n    m    Zernike polynomial    Normalization O9(r{Vu7u  
%       --------------------------------------------- as+GbstN  
%       0    0    1                        sqrt(2) zNSu  
%       1    1    r                           2 K1?Gmue#I  
%       2    0    2*r^2 - 1                sqrt(6) %g]vxm5?  
%       2    2    r^2                      sqrt(6) l$*=<tV  
%       3    1    3*r^3 - 2*r              sqrt(8) qEUT90  
%       3    3    r^3                      sqrt(8) ]v G{kAnH  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) "Dy'Kd%,%/  
%       4    2    4*r^4 - 3*r^2            sqrt(10) CJaKnz  
%       4    4    r^4                      sqrt(10) A\Txb_x  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) d
{2  
%       5    3    5*r^5 - 4*r^3            sqrt(12) *FR$vL
Gn  
%       5    5    r^5                      sqrt(12) MYe HS  
%       --------------------------------------------- jy2IZ o  
% ":Edu,6O  
%   Example: 'rb'7=z5  
% Wk4.%tpeO7  
%       % Display three example Zernike radial polynomials iP3Z  
%       r = 0:0.01:1; 9^F2$+T[:  
%       n = [3 2 5]; $!A:5jech  
%       m = [1 2 1]; uk`8X`'  
%       z = zernpol(n,m,r); s|bM%!$1  
%       figure Gi^Ha=?J%
 
%       plot(r,z) >i,iOx|E-  
%       grid on nL]^$J$  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 1U\$iy8}  
% Aw!gSf)  
%   See also ZERNFUN, ZERNFUN2. V_U'P>_I  
r!N]$lB  
% A note on the algorithm. *
B)yy[8j+  
% ------------------------ (y4#.vZh:  
% The radial Zernike polynomials are computed using the series RBGlz
k  
% representation shown in the Help section above. For many special {Z{o"56f  
% functions, direct evaluation using the series representation can %1M
cD{  
% produce poor numerical results (floating point errors), because TB aVW  
% the summation often involves computing small differences between |-2}j2'  
% large successive terms in the series. (In such cases, the functions Ek.&Sf$cd'  
% are often evaluated using alternative methods such as recurrence !{_yaVF  
% relations: see the Legendre functions, for example). For the Zernike |I[7,`C~  
% polynomials, however, this problem does not arise, because the \[wCp*;1}  
% polynomials are evaluated over the finite domain r = (0,1), and HO|-@yOF^  
% because the coefficients for a given polynomial are generally all Md;/nJO~{  
% of similar magnitude. K=u0nrG*  
% %NHYW\sKX  
% ZERNPOL has been written using a vectorized implementation: multiple u>T76,8|\  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] #$'"cfRxc  
% values can be passed as inputs) for a vector of points R.  To achieve &$fbP5uAZ  
% this vectorization most efficiently, the algorithm in ZERNPOL &;q<M_<  
% involves pre-determining all the powers p of R that are required to S9Y[4*//  
% compute the outputs, and then compiling the {R^p} into a single ,i`h x, Rg  
% matrix.  This avoids any redundant computation of the R^p, and `
QP ~  
% minimizes the sizes of certain intermediate variables. &bC}3D  
% Vj^dD9:  
%   Paul Fricker 11/13/2006 U_z2J(e~  
EH9Hpo  
+`| *s3M  
% Check and prepare the inputs: p_terD:  
% ----------------------------- 1-;?0en&0
 
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) zDBD.5R;  
    error('zernpol:NMvectors','N and M must be vectors.') ]= x 1`j  
end Aa(<L$e!`  
|DG@ht  
if length(n)~=length(m) 0~E 6QhV:  
    error('zernpol:NMlength','N and M must be the same length.') '?/
&n8J\  
end Q2'eQ0W{o  
: 1)}Epo,  
n = n(:); M?6;|-HH  
m = m(:); sB*o)8  
length_n = length(n); 9k2,3It  
whkJpK(  
if any(mod(n-m,2)) w{7ji}  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ] VG?+  
end {K.rl%_|N  
u35q,u=I  
if any(m<0) *=nO  
    error('zernpol:Mpositive','All M must be positive.') NtZ6$o<Y  
end Kr3];(w{  
c;V D}UD'  
if any(m>n) -
Ds|qzrN%  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') ;~tsF.=
 
end IKm&xzV-  
Yw"P)Zp  
if any( r>1 | r<0 ) ckwF|:e7*  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') ?n*fy  
end hLA;Bl  
!UNNjBBP7  
if ~any(size(r)==1) Wvr+
y!F  
    error('zernpol:Rvector','R must be a vector.') VO9f~>`(  
end R7aXR\ R  
x0x $  9  
r = r(:); 0$Ff#8  
length_r = length(r); K\sbt7~  
u6_jn
ZGB  
if nargin==4 k:0P+d  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ER<eX4oU  
    if ~isnorm 5#u.pu  
        error('zernpol:normalization','Unrecognized normalization flag.') 'O "kt T  
    end ec'tFL#u{  
else {})y^L  
    isnorm = false; X%J%A-k]  
end _7 `E[&v  
@&:VKpu\  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zz3 r<?#5  
% Compute the Zernike Polynomials hZF(/4Z2  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u9}!Gq  
+ U5U.f%  
% Determine the required powers of r: 3/tJDb5  
% ----------------------------------- twv lQ|  
rpowers = []; {,v: GMsm  
for j = 1:length(n) 22IYrk  
    rpowers = [rpowers m(j):2:n(j)]; $h]NXC6J  
end !rHx}n{rw  
rpowers = unique(rpowers); PN9^[X  
QZ0R:TY  
% Pre-compute the values of r raised to the required powers, $B ?? Ip?P  
% and compile them in a matrix: ?H0m<jO8~  
% ----------------------------- `(T!>QVW+g  
if rpowers(1)==0 [D9
:A  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); |$Xf;N37t  
    rpowern = cat(2,rpowern{:}); [Pqn3I[  
    rpowern = [ones(length_r,1) rpowern]; }z{wQ\  
else %#4 +!  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P8]ORQ6ZF  
    rpowern = cat(2,rpowern{:}); g 2#F_  
end yjv&4pIc1  
TMtI^mkB:  
% Compute the values of the polynomials: mrReast  
% -------------------------------------- aZxO/b^j  
z = zeros(length_r,length_n); f@*>P_t  
for j = 1:length_n rBD2Si=  
    s = 0:(n(j)-m(j))/2; KE#$+,
?  
    pows = n(j):-2:m(j); :5<#X8>d  
    for k = length(s):-1:1 @:IL/o*  
        p = (1-2*mod(s(k),2))* ... H\f/n`@,G  
                   prod(2:(n(j)-s(k)))/          ... 0w+5'lOg  
                   prod(2:s(k))/                 ... wJ(8}eI  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... l}+Cdy9>  
                   prod(2:((n(j)+m(j))/2-s(k))); 64b<0;~  
        idx = (pows(k)==rpowers); mOSCkp{<e  
        z(:,j) = z(:,j) + p*rpowern(:,idx); \086O9  
    end XP4jZCt9  
     jB/V{Y#y9@  
    if isnorm cyHhy_~R  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); E6JV}`hSk  
    end 0ZT 0
 
end [{/$9k-aF?  
79a9L{gso  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) "TUPYFK9  
%ZERNFUN2 Single-index Zernike functions on the unit circle. uGM>C"  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated `{%-*f^  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive &6
Ns7w6*z  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, jB`7T^bU  
%   and THETA is a vector of angles.  R and THETA must have the same ` -yhl3si  
%   length.  The output Z is a matrix with one column for every P-value, ^b:Xo"q#H  
%   and one row for every (R,THETA) pair. aDXpkG0E  
% >b3@>W  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Q^vGj</u  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ` v>
/  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) .$UTH@;7  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 C1n??Y[  
%   for all p. e{:86C!d)  
% S'|lU@PCl  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 J&'>IA  
%   Zernike functions (order N<=7).  In some disciplines it is $m{{,&}k  
%   traditional to label the first 36 functions using a single mode oO8]lHS?@  
%   number P instead of separate numbers for the order N and azimuthal xP
42xv9U  
%   frequency M. x Ridc^  
% }Z^FEd"y  
%   Example: l'W3=,G[?  
% @h!U  
%       % Display the first 16 Zernike functions |e~u!V\m  
%       x = -1:0.01:1; uF+);ig  
%       [X,Y] = meshgrid(x,x); >'ie!VW@  
%       [theta,r] = cart2pol(X,Y); <xXiJU+  
%       idx = r<=1; >y&[BB7S6  
%       p = 0:15; 4(m/D>6:  
%       z = nan(size(X)); w4NZt|>5j;  
%       y = zernfun2(p,r(idx),theta(idx)); mf+K{y,L  
%       figure('Units','normalized') +}&pVe\t  
%       for k = 1:length(p) #)Ep(2  
%           z(idx) = y(:,k); hT\p)w  
%           subplot(4,4,k) _F! :(@}  
%           pcolor(x,x,z), shading interp mi*:S%;h  
%           set(gca,'XTick',[],'YTick',[]) Y"r3i]  
%           axis square ?Ozk^#H[  
%           title(['Z_{' num2str(p(k)) '}']) *oKgP8CF  
%       end =7*oC  
% "tqS|ok.  
%   See also ZERNPOL, ZERNFUN. t)YFT
O"Jj  
22l|!B%o  
%   Paul Fricker 11/13/2006 E=$7ieW  
IiG4ib>)W  
niXHK$@5  
% Check and prepare the inputs: ^H f+du  
% ----------------------------- 1!K!oY  
if min(size(p))~=1 FEge
+`{,  
    error('zernfun2:Pvector','Input P must be vector.') wa9'2a1?  
end Y+|L3'H  
mvUVy1-c  
if any(p)>35 ?,.HA@T%  
    error('zernfun2:P36', ... 40`9t Xn  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... x&mz-  
           '(P = 0 to 35).']) mABwM$_  
end B7NmET4  
iuvtj]/  
% Get the order and frequency corresonding to the function number: R^n* o  
% ---------------------------------------------------------------- H[>klzh6 !  
p = p(:); K * xM[vO  
n = ceil((-3+sqrt(9+8*p))/2); J"m%q\'  
m = 2*p - n.*(n+2); DW'0j$;  
uJ2C+$=Ul  
% Pass the inputs to the function ZERNFUN: >FK)p   
% ---------------------------------------- wFKu
Sd  
switch nargin ]w1BJZa36  
    case 3 r4]hS`X~%  
        z = zernfun(n,m,r,theta); Om&{4a\  
    case 4 wa-_
O<  
        z = zernfun(n,m,r,theta,nflag); HYa$EE2  
    otherwise Pf^Ly97  
        error('zernfun2:nargin','Incorrect number of inputs.') 75QXkJu  
end X^?|Sz<^E  
')Dp%"\?  
% 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 [~Z'xY y  
function z = zernfun(n,m,r,theta,nflag) vUodp#s  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. zx_O"0{5  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N #NVF\  
%   and angular frequency M, evaluated at positions (R,THETA) on the qCxD{-9x{  
%   unit circle.  N is a vector of positive integers (including 0), and N4Fy8qU;  
%   M is a vector with the same number of elements as N.  Each element T9U2j-lA?  
%   k of M must be a positive integer, with possible values M(k) = -N(k) ,_5YaX:<4  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, TCEXa?,L  
%   and THETA is a vector of angles.  R and THETA must have the same {8*d;[X50  
%   length.  The output Z is a matrix with one column for every (N,M) 5pKvNLy.t  
%   pair, and one row for every (R,THETA) pair. {{4p{  
% .5#tB*H  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `lV  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f2SU5e2  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral U||w6:W5  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c],frhmyd  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized AD!<%h:  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J6^Ct  
% &ttv4BC^r  
%   The Zernike functions are an orthogonal basis on the unit circle. SCt=OdP=  
%   They are used in disciplines such as astronomy, optics, and iz%A0Z+`bg  
%   optometry to describe functions on a circular domain. 35N/v G0  
% %M0mwty]  
%   The following table lists the first 15 Zernike functions. fEv<W  
% HN~v&,  
%       n    m    Zernike function           Normalization aJa^~*N/Aa  
%       -------------------------------------------------- ou,=MpXx*  
%       0    0    1                                 1 4HJZ^bq9|  
%       1    1    r * cos(theta)                    2 #.<F5  
%       1   -1    r * sin(theta)                    2 !=h|&Vta  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 9,EaN{GM  
%       2    0    (2*r^2 - 1)                    sqrt(3) vACsppa>#  
%       2    2    r^2 * sin(2*theta)             sqrt(6) P9tQS"Rs  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) u8k{N  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) k,*#I<($  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 5[j!\d}U  
%       3    3    r^3 * sin(3*theta)             sqrt(8)
0Z);.l^  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) |q.:hWYFpM  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %Dr4~7=7a  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ;~gd<KK  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Oih2UrF  
%       4    4    r^4 * sin(4*theta)             sqrt(10) yzM+28}L<I  
%       -------------------------------------------------- ]Re~V{uh  
% C +?@iMh  
%   Example 1: mP$G9R  
% N5rG.6K  
%       % Display the Zernike function Z(n=5,m=1) =`\,2Nb  
%       x = -1:0.01:1; D`~{[cv)\  
%       [X,Y] = meshgrid(x,x); >&TnTv?I  
%       [theta,r] = cart2pol(X,Y); kj3o1Y  
%       idx = r<=1; }MavI'  
%       z = nan(size(X)); ^tKOxW# a  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); /4B4IT  
%       figure MkNURy>
n&  
%       pcolor(x,x,z), shading interp D"] [&m  
%       axis square, colorbar C"Y]W-Mgg  
%       title('Zernike function Z_5^1(r,\theta)') cVHE}0Xd(  
% M}oFn}-T9a  
%   Example 2: 2bn@:71`  
% UK<DcM~n  
%       % Display the first 10 Zernike functions `TlUJ]d)  
%       x = -1:0.01:1; R,5$ 0_]|+  
%       [X,Y] = meshgrid(x,x); o?O,nD 6  
%       [theta,r] = cart2pol(X,Y); mv%:[+!  
%       idx = r<=1; >5@vY?QXO  
%       z = nan(size(X)); > v!c\  
%       n = [0  1  1  2  2  2  3  3  3  3]; j.'"CU  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; &<P^Tvqq&  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Qdr-GODx  
%       y = zernfun(n,m,r(idx),theta(idx)); wAOVH].  
%       figure('Units','normalized') 5f*'
wA  
%       for k = 1:10 L|1zHDxQ  
%           z(idx) = y(:,k); Nb!6YY=Ez-  
%           subplot(4,7,Nplot(k)) F3 l^^Mc  
%           pcolor(x,x,z), shading interp O"^a.`27  
%           set(gca,'XTick',[],'YTick',[]) PUZXmnB  
%           axis square \;:@=9`  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) pn%|;  
%       end aq,)6P`  
% u r.T YKF  
%   See also ZERNPOL, ZERNFUN2. n`T[eb~  
=O'%)Y&  
%   Paul Fricker 11/13/2006 AUjTcu>i  
'kg]|"M  
#Xw[
i  
% Check and prepare the inputs: L%O8vn^3  
% ----------------------------- (:HbtrI  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Cz);mOb%M%  
    error('zernfun:NMvectors','N and M must be vectors.') y3[)zv  
end 4x{ti5Y0  
|Sv#f2`  
if length(n)~=length(m) {ZM2WFpE  
    error('zernfun:NMlength','N and M must be the same length.') No&[ \;  
end iN4'jD^oP  
~5!TV,>ls  
n = n(:); g#%FY1xp  
m = m(:); L8tLW09  
if any(mod(n-m,2)) <d&)|W  
    error('zernfun:NMmultiplesof2', ... 8Pdnw/W  
          'All N and M must differ by multiples of 2 (including 0).') g7z9i[  
end ^t ldm7{_  
ftH%, /,  
if any(m>n) "sx&8H"  
    error('zernfun:MlessthanN', ... ,Y8X"~{A  
          'Each M must be less than or equal to its corresponding N.') :aqskeT  
end LLY;IUK!R  
*#^1rKGWK  
if any( r>1 | r<0 ) @d^h/w  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') )9jQ
_  
end Jb.u^
3R@  
|< FCt-U  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^
QQNJ  
    error('zernfun:RTHvector','R and THETA must be vectors.') ?[B[ F  
end ~
tuFjj^  

"EhO )lR  
r = r(:); T<?BIQz(}  
theta = theta(:); 7<o;3gR7Kj  
length_r = length(r); vGHYB1=~  
if length_r~=length(theta) @CI6$  
    error('zernfun:RTHlength', ... A":b_!sW  
          'The number of R- and THETA-values must be equal.') W8h\ s {  
end 5g>kr<K  
p}7&x[fTLk  
% Check normalization: %ys}Q!gR  
% -------------------- pDq_nx9  
if nargin==5 && ischar(nflag) y+afUJT  
    isnorm = strcmpi(nflag,'norm'); }z-  
    if ~isnorm PSR`8z n
 
        error('zernfun:normalization','Unrecognized normalization flag.') +M&S  
    end oz-I/g3go  
else T5_Cu9>ax  
    isnorm = false; swL|Ff`$  
end (+ anTA=  
a`iAA1HJ  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I'b]s~u  
% Compute the Zernike Polynomials .{Oq)^!ot  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >!.9g  
#de^~  
% Determine the required powers of r: t3g!5  
% ----------------------------------- p=gUcO8  
m_abs = abs(m); 4yv31QG$  
rpowers = []; oa !P]r  
for j = 1:length(n) g"?D>}@=  
    rpowers = [rpowers m_abs(j):2:n(j)]; d( g_y m*  
end beZ| i 1:  
rpowers = unique(rpowers); iRHQRdij  
@2*6+w_Ae  
% Pre-compute the values of r raised to the required powers, MXV4bgltT  
% and compile them in a matrix: fEv36xb2S  
% ----------------------------- ]X|G+[Ujv  
if rpowers(1)==0 *7ro [  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); kk+8NwM1  
    rpowern = cat(2,rpowern{:}); ITlkw~'G  
    rpowern = [ones(length_r,1) rpowern]; S\!E;p  
else
c (8J  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j)qh>y)  
    rpowern = cat(2,rpowern{:}); <t37DnCgI  
end uwA3!5  
dwMwd@*j  
% Compute the values of the polynomials: \hN2w]e  
% -------------------------------------- t&]Mt7  
y = zeros(length_r,length(n)); :q1r2&ne  
for j = 1:length(n) N&`ay{&`:  
    s = 0:(n(j)-m_abs(j))/2; 6E]rxps}"  
    pows = n(j):-2:m_abs(j); qDb}b d5  
    for k = length(s):-1:1 uK5x[m
 
        p = (1-2*mod(s(k),2))* ... Mwc3@  
                   prod(2:(n(j)-s(k)))/              ... ?='9YM  
                   prod(2:s(k))/                     ... ZE`{J=,  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >K%x44|  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); &[5az/Hj*  
        idx = (pows(k)==rpowers); c.v)M\:  
        y(:,j) = y(:,j) + p*rpowern(:,idx); K_n%`
5  
    end EPy/6-5b  
     Zh
^w)}(W  
    if isnorm bp,CvQ'}a  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); _s8_i6 Y  
    end
?~IZ{!  
end PM7/fv*,  
% END: Compute the Zernike Polynomials UXHFti/A<  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 55p=veq \  
`0:@`)&g1  
% Compute the Zernike functions: (Lnh> '2  
% ------------------------------ n]Y _C^  
idx_pos = m>0; f lB2gr^  
idx_neg = m<0; I&Y(]S,cU  
3(5Y-.aK}^  
z = y; z?,5v`,t2  
if any(idx_pos) e_TDO   
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); y{JkY\g  
end :^a$ve3(Jq  
if any(idx_neg) YyIt-fPZ  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 7i.aZ2a%  
end ( Iew%U  
)3sb2 #  
% 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