[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 gV's=c
Q  
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！ Uiw2oi&_  

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)  "<1{9  

}&J q}j  
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。 ##>H&,Dp[  
1>h]{%I  
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) oUlVI*~ND  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. 3^yK!-Wp(  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Cp0=k  
%   order N and frequency M, evaluated at R.  N is a vector of N;`n@9BF  
%   positive integers (including 0), and M is a vector with the TM%%O :3  
%   same number of elements as N.  Each element k of M must be a w``U=sfmV  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) oEpFuWp%A  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is A.w.rVDD  
%   a vector of numbers between 0 and 1.  The output Z is a matrix SE*g;Cvg1  
%   with one column for every (N,M) pair, and one row for every yJIscwF  
%   element in R. 3u0RKLc\  
% cw <l{A  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- nX8v+:&}  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is LrpM\}t  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to TB31- ()  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 #Gi$DMW  
%   for all [n,m]. K{+2G&i  
% "3J}b?u_[  
%   The radial Zernike polynomials are the radial portion of the 0w7DsPdS  
%   Zernike functions, which are an orthogonal basis on the unit A,!-{/wc  
%   circle.  The series representation of the radial Zernike G' 1'/  
%   polynomials is "" EQE>d  
% -XG@'P
_  
%          (n-m)/2 TWX.D`W  
%            __ n+M<\  
%    m      \       s                                          n-2s 8LCb+^  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r f _:A0  
%    n      s=0 @
2i9n  
% <F'\lA9  
%   The following table shows the first 12 polynomials. uPvEwq* C  
% CTmT@A{  
%       n    m    Zernike polynomial    Normalization Dw"\/p:-3  
%       --------------------------------------------- r9XZ(0/p  
%       0    0    1                        sqrt(2) |DwZ{(R"W  
%       1    1    r                           2 +b6v!7_  
%       2    0    2*r^2 - 1                sqrt(6) Q,Eo mt  
%       2    2    r^2                      sqrt(6) [nh>vqum  
%       3    1    3*r^3 - 2*r              sqrt(8) /x *3}oI  
%       3    3    r^3                      sqrt(8) E{vbO/|kf  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 8{ I|$*nB  
%       4    2    4*r^4 - 3*r^2            sqrt(10) lU]nd[x  
%       4    4    r^4                      sqrt(10) 4<v&S2Yq  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) x?<FJ"8"k  
%       5    3    5*r^5 - 4*r^3            sqrt(12) 8zb/xP>  
%       5    5    r^5                      sqrt(12) |uJ
%5y#  
%       --------------------------------------------- ~V6D<  
% "J1 4C9u   
%   Example: '5tCz9}Y  
% yt2PU_),  
%       % Display three example Zernike radial polynomials U$UIN#  
%       r = 0:0.01:1; 1Z&(6cDY8M  
%       n = [3 2 5]; XK vi=0B  
%       m = [1 2 1]; wuo,kM  
%       z = zernpol(n,m,r); VxBo1\'  
%       figure 19] E 5'AI  
%       plot(r,z) 5lum$5  
%       grid on ugBCBr  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') M3au{6y  
% }QmqoCAE~m  
%   See also ZERNFUN, ZERNFUN2. 9tnD=A<PS  
'c~4+o4co  
% A note on the algorithm. %l%HHT  
% ------------------------ 1.>m@
Slr>  
% The radial Zernike polynomials are computed using the series ji="DYtL  
% representation shown in the Help section above. For many special 3(UVg!t  
% functions, direct evaluation using the series representation can 1 TXioDs=_  
% produce poor numerical results (floating point errors), because *NQ/UXE  
% the summation often involves computing small differences between to&m4+5?6  
% large successive terms in the series. (In such cases, the functions 8?C5L8)  
% are often evaluated using alternative methods such as recurrence mp3s-YfRc  
% relations: see the Legendre functions, for example). For the Zernike oL<St$1  
% polynomials, however, this problem does not arise, because the 
qJw_  
% polynomials are evaluated over the finite domain r = (0,1), and Yr|4Fl~U  
% because the coefficients for a given polynomial are generally all Qg/rRiV  
% of similar magnitude. E(|>Ddv B&  
% yCo.cd-  
% ZERNPOL has been written using a vectorized implementation: multiple ,"ql5Q4  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] eV~goj  
% values can be passed as inputs) for a vector of points R.  To achieve i@'dH3-kO  
% this vectorization most efficiently, the algorithm in ZERNPOL t$ *0{w E  
% involves pre-determining all the powers p of R that are required to T^q 0'#/  
% compute the outputs, and then compiling the {R^p} into a single FiU#T.`9'  
% matrix.  This avoids any redundant computation of the R^p, and Ir]
\|t  
% minimizes the sizes of certain intermediate variables. `$NP>%J-  
% fc@A0Hf  
%   Paul Fricker 11/13/2006 B7%U_F|m  
WEpoBP CL  
M^I(OuRMeI  
% Check and prepare the inputs: [00m/fT6  
% ----------------------------- -K$)DvV^(E  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :hA#m[  
    error('zernpol:NMvectors','N and M must be vectors.') yLcEX  
end DTs;{c  
eDB;cN  
if length(n)~=length(m) tnIX:6  
    error('zernpol:NMlength','N and M must be the same length.') "7`<~>9t.  
end QSj]ZA  
2"~8Z(0  
n = n(:); mA}"a<0  
m = m(:); A)KZa
"EX  
length_n = length(n); |7Kbpj  
B-ESFATc  
if any(mod(n-m,2)) oXS}IL og'  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') (iGTACoF  
end |nF8gh~}  
/7LR;>Bj  
if any(m<0) |'2d_vR  
    error('zernpol:Mpositive','All M must be positive.') hzC>~Ub5  
end <7$1kGlA  
 C.QO#b  
if any(m>n) -.3w^D"l  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') L8n|m!MOD  
end "h ^Z  
k_R"CKd  
if any( r>1 | r<0 ) ze;KhUPRm  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') RT5T1K08I  
end 1nOCQ\$l  
(I}v[W  
if ~any(size(r)==1) Np)lI
GE  
    error('zernpol:Rvector','R must be a vector.') ]{LjRSV
 
end RGX=)  
cS+>J@L  
r = r(:); |D
.ND%K&  
length_r = length(r); Xm2'6f,  
u2[w#  
if nargin==4 U%<Inb}ad  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); iyog`s c  
    if ~isnorm Xx(T">
]vJ  
        error('zernpol:normalization','Unrecognized normalization flag.') .[ mRM  
    end wdZ/Xp9]  
else PxE3K-S)G  
    isnorm = false; L_s:l9!r  
end 8.~kK<)!  
PYzv
Cf`?  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~v"L!=~G;a  
% Compute the Zernike Polynomials C8\^#5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bJ;'`sw1  
-`t^7pr  
% Determine the required powers of r: [fIg{Q  
% ----------------------------------- 'P}0FktP`  
rpowers = []; m#F`] {  
for j = 1:length(n)
8JD,u  
    rpowers = [rpowers m(j):2:n(j)]; ]0\MmAJRn  
end CWS4lx  
rpowers = unique(rpowers); 4H<lm*!^  
v9->nVc-  
% Pre-compute the values of r raised to the required powers, FsryEHz  
% and compile them in a matrix: ?R#)1{(8d~  
% ----------------------------- j8`BdKg  
if rpowers(1)==0 C6y&#uX\  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); E+JqWR5  
    rpowern = cat(2,rpowern{:}); Oc; G(l(  
    rpowern = [ones(length_r,1) rpowern]; @ry_nKr9  
else SZ$Kz n  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GM<-&s!Uj  
    rpowern = cat(2,rpowern{:}); fd2T=fz-  
end tnG# IU *  
)>- =R5ZV  
% Compute the values of the polynomials: K96<M);:g  
% -------------------------------------- r>U@3%0&  
z = zeros(length_r,length_n); m9Hit8f@Q  
for j = 1:length_n VAu&@a`  
    s = 0:(n(j)-m(j))/2; 3%ZOKb"D*  
    pows = n(j):-2:m(j); ZQ0F$J)2~  
    for k = length(s):-1:1 DDH:)=;z  
        p = (1-2*mod(s(k),2))* ... '08=yqy4N  
                   prod(2:(n(j)-s(k)))/          ... #Vha7  
                   prod(2:s(k))/                 ... '6Q=#:mc\  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... :4%k9BGAj"  
                   prod(2:((n(j)+m(j))/2-s(k))); |H+Wed|  
        idx = (pows(k)==rpowers); 8*T=Xei8  
        z(:,j) = z(:,j) + p*rpowern(:,idx); ^ovR7+V  
    end aAAU{EWW  
     (ICd}  
    if isnorm ,WB{i^TD  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); iW /}#  
    end 5o8EC" 0  
end /~f'}]W  
/gkX38  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) ?m?::R
H  
%ZERNFUN2 Single-index Zernike functions on the unit circle. DZPPJ2}  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 8eHyL  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive u^qT2Ss0  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, ~1vDV>dpE  
%   and THETA is a vector of angles.  R and THETA must have the same xjj6WED  
%   length.  The output Z is a matrix with one column for every P-value, _t #k,;  
%   and one row for every (R,THETA) pair. c|@bwat4  
% d,n 'n  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike Y#P%6Fy  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) g~A`N=r;h  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) K}MK<2vU  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ByNn  
%   for all p. DG:Z=LuJr  
% &AbNWtCV+G  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !NvI:C_4|  
%   Zernike functions (order N<=7).  In some disciplines it is oEKvl3Hz_  
%   traditional to label the first 36 functions using a single mode U0N 60  
%   number P instead of separate numbers for the order N and azimuthal }oGA-Qc}B  
%   frequency M. 2)HuZda  
% k5.Ln
a  
%   Example: EE'io5\et  
% T!WT;A  
%       % Display the first 16 Zernike functions xBi' X  
%       x = -1:0.01:1; 9H`XeQ.  
%       [X,Y] = meshgrid(x,x); XG{zlOD+  
%       [theta,r] = cart2pol(X,Y); 54R#W:t  
%       idx = r<=1; Xg!{K3OS  
%       p = 0:15; T&u5ki4NE  
%       z = nan(size(X)); xH"/1g  
%       y = zernfun2(p,r(idx),theta(idx)); "Nbq#w\  
%       figure('Units','normalized') CSq4x5!_7>  
%       for k = 1:length(p) )g#T9tx2D  
%           z(idx) = y(:,k); *@=/qkaJaI  
%           subplot(4,4,k) }.m<  
%           pcolor(x,x,z), shading interp pm0{R[:T7  
%           set(gca,'XTick',[],'YTick',[]) JL}_72gs  
%           axis square V_}"+&W9  
%           title(['Z_{' num2str(p(k)) '}']) ywm8N%]v  
%       end %^GfS@t  
% lbl?k5  
%   See also ZERNPOL, ZERNFUN. =BAW[%1b  
0e ~JMUb  
%   Paul Fricker 11/13/2006 ;m{1_1  
jc[Y}gd,  
-Xm'dwm  
% Check and prepare the inputs: vJc-6EO
 
% ----------------------------- ']z{{UNUN  
if min(size(p))~=1 ZWU)\}}_R  
    error('zernfun2:Pvector','Input P must be vector.') -g Sa_8R  
end D_^ nI:  
gANuBWh8T  
if any(p)>35 Z<y I\1  
    error('zernfun2:P36', ...  zC@o  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... $f=J2&D,Cz  
           '(P = 0 to 35).'])
d#rf5<i  
end aPfO$b:  
6J6BF%  
% Get the order and frequency corresonding to the function number: 1 A !bE  
% ---------------------------------------------------------------- Jg\zdi:t  
p = p(:); 1&evG-#<:  
n = ceil((-3+sqrt(9+8*p))/2); 6x[}g  
m = 2*p - n.*(n+2); j94=hJVKi  
C>j@,G4  
% Pass the inputs to the function ZERNFUN: a /l)qB#  
% ---------------------------------------- Ln<`E|[29  
switch nargin lC("y' ::  
    case 3 E}Z/*lX  
        z = zernfun(n,m,r,theta); L Mbn  
    case 4 ex9g?*Q  
        z = zernfun(n,m,r,theta,nflag); Ou!2[oe@M  
    otherwise |w1Bq  
        error('zernfun2:nargin','Incorrect number of inputs.') 2%@4]  
end #TX/aKr:  
Cc' 37~6~P  
% 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 v/0QOp  
function z = zernfun(n,m,r,theta,nflag) B=y
qW  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. E$:*NSXj  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]kG"ubHV?h  
%   and angular frequency M, evaluated at positions (R,THETA) on the Vb4#,  
%   unit circle.  N is a vector of positive integers (including 0), and ^aMg/.
j  
%   M is a vector with the same number of elements as N.  Each element
9T}pT{~V  
%   k of M must be a positive integer, with possible values M(k) = -N(k) KL:j
?.0  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, *1 ]uH e  
%   and THETA is a vector of angles.  R and THETA must have the same 7he,?T)vD  
%   length.  The output Z is a matrix with one column for every (N,M) z(exA  
%   pair, and one row for every (R,THETA) pair. /-ch`u md  
% |`Ntv}  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c74.< @w  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =J]]EoX/  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral z8~NZ;A  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +EAsW(F1  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized FLCexlv^  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .b&t;4q  
% wd^':  
%   The Zernike functions are an orthogonal basis on the unit circle.  ZrxD`1L  
%   They are used in disciplines such as astronomy, optics, and _AYK435>N  
%   optometry to describe functions on a circular domain.  &)Tdc  
% Ic:(Gi- %  
%   The following table lists the first 15 Zernike functions. Ovt.!8  
% M~#gRAUJ  
%       n    m    Zernike function           Normalization # E^1|:  
%       -------------------------------------------------- y$F'(b|)  
%       0    0    1                                 1 ^qvbqfh  
%       1    1    r * cos(theta)                    2 } FlT%>Gw  
%       1   -1    r * sin(theta)                    2 [0[i5'K:  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) =:,g  
%       2    0    (2*r^2 - 1)                    sqrt(3) b8V
To lJ  
%       2    2    r^2 * sin(2*theta)             sqrt(6) He/8=$c%  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) hh)`645=x  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) cAqLE\h  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) uR4z&y  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ksqQM
 
%       4   -4    r^4 * cos(4*theta)             sqrt(10) UA0Bzoky;  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Lpz>>}  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) c|B
('3h  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o>i4CCU+  
%       4    4    r^4 * sin(4*theta)             sqrt(10) q&-`,8#  
%       -------------------------------------------------- k&q;JyUi  
% K5VWt)Z#  
%   Example 1: 7P5)Z-K[  
% Z1f8/?`W  
%       % Display the Zernike function Z(n=5,m=1) K.nHii   
%       x = -1:0.01:1; FZ<gpIv!NS  
%       [X,Y] = meshgrid(x,x); [{,T.;'<j  
%       [theta,r] = cart2pol(X,Y); 4Zddw0|2  
%       idx = r<=1; 82qoGSD.  
%       z = nan(size(X)); fS:&Ak ];  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); y`5 9A  
%       figure #PW9:_BE  
%       pcolor(x,x,z), shading interp c(m<h+2VL  
%       axis square, colorbar !bx;Ta.  
%       title('Zernike function Z_5^1(r,\theta)') Y;Dp3v!  
% G1tY)_-8[  
%   Example 2: 6qpJUkd  
% l -mfFN  
%       % Display the first 10 Zernike functions (k)v!O-  
%       x = -1:0.01:1; Z'W=\rl  
%       [X,Y] = meshgrid(x,x); :T$|bc  
%       [theta,r] = cart2pol(X,Y); S-b/S5  
%       idx = r<=1; zO
IDU
  
%       z = nan(size(X)); $am$EU?s  
%       n = [0  1  1  2  2  2  3  3  3  3]; beGa#JH,  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; EhvX)s  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; e@07  
%       y = zernfun(n,m,r(idx),theta(idx)); b<ZIWfs  
%       figure('Units','normalized') u8g~
  
%       for k = 1:10 JPUW6e07o  
%           z(idx) = y(:,k); ^j7Vt2-  
%           subplot(4,7,Nplot(k)) ({)+3]x  
%           pcolor(x,x,z), shading interp fk>aqm7D!  
%           set(gca,'XTick',[],'YTick',[]) .},'~NM]  
%           axis square su(1<S}  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >J?fl8  
%       end `r':by0M  
% [Ek7b*  
%   See also ZERNPOL, ZERNFUN2. >dD@j:Qc  
FUb\e-Q=  
%   Paul Fricker 11/13/2006 nEy&>z  
X-Kh(Z  
MYvY]Jx3  
% Check and prepare the inputs: <w9JRpFY  
% ----------------------------- 9
YyLf;  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (gU!=F?#m  
    error('zernfun:NMvectors','N and M must be vectors.') S Lj!v&'  
end $6 9&O  
y9GoPC`z  
if length(n)~=length(m) 50wulGJud  
    error('zernfun:NMlength','N and M must be the same length.')
}?i0 I  
end !hy-L_wL]  
MrFQ5:=  
n = n(:); }C?'
BRX  
m = m(:); Tv=mgH=b  
if any(mod(n-m,2)) P>D)7V9Hh  
    error('zernfun:NMmultiplesof2', ... #A/  
          'All N and M must differ by multiples of 2 (including 0).') >T-u~i$s  
end "m8^zg hL  
6l x>>J!H  
if any(m>n) :\c ^*K(9  
    error('zernfun:MlessthanN', ... ]:-mbgW  
          'Each M must be less than or equal to its corresponding N.') o#Dk& cH  
end 6;d*r$0Fc  
FVbb2Y?R  
if any( r>1 | r<0 ) pE0Sw}A:9  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') _6hQ %hv8  
end #p&qUw  
|aS.a&vwR  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) H$t_Xw==  
    error('zernfun:RTHvector','R and THETA must be vectors.') xm~`7~nFR  
end ksUcx4;a@F  
k]|~>9eY]  
r = r(:); s!zx} 5  
theta = theta(:); '<)n8{3Q5w  
length_r = length(r); .`H5cuF`  
if length_r~=length(theta) my1@41 H  
    error('zernfun:RTHlength', ... ET*SB  
          'The number of R- and THETA-values must be equal.') )2o?#8J  
end J]'zIOQ  
f'RX6$}\1X  
% Check normalization:  |>^JRx  
% -------------------- |YWD8 +  
if nargin==5 && ischar(nflag) ^z*t%<@[Q  
    isnorm = strcmpi(nflag,'norm'); Dx?,=~W9  
    if ~isnorm n( yn<  
        error('zernfun:normalization','Unrecognized normalization flag.') a58H9w"u)  
    end 2l'6
.  
else vh%B[brUJ  
    isnorm = false; ,ZNq,$j  
end oZgjQM$YP  

H%tdhu\e  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F/{!tx  
% Compute the Zernike Polynomials ="H`V V_  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C{rcs'  
0#hlsfc]\  
% Determine the required powers of r: !f[_+CD  
% ----------------------------------- q?yVR3]M  
m_abs = abs(m); 8TKnL\aar  
rpowers = []; >+1duAC  
for j = 1:length(n) U7F!Z( 9  
    rpowers = [rpowers m_abs(j):2:n(j)]; tcI*a>  
end h[Y1?ln&h  
rpowers = unique(rpowers); vvMT}-!  
UI0VtR]   
% Pre-compute the values of r raised to the required powers, (w3YvG.  
% and compile them in a matrix: wwZ,;\  
% ----------------------------- b8UO,fY q  
if rpowers(1)==0 *i%d,w0+  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4+8@`f>s  
    rpowern = cat(2,rpowern{:}); 1GcE)e!>  
    rpowern = [ones(length_r,1) rpowern]; g!|kp?  
else 0{D'n@veP  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %tGO?JMkd  
    rpowern = cat(2,rpowern{:}); n_A3#d<9  
end oG\Vxg*  
6H$FhJF  
% Compute the values of the polynomials: S,UDezxg  
% -------------------------------------- "!^
"[mX4  
y = zeros(length_r,length(n)); I\ob7X'Xu!  
for j = 1:length(n) A;M'LM-M  
    s = 0:(n(j)-m_abs(j))/2; _Fl9>C"u  
    pows = n(j):-2:m_abs(j); >kVz49j  
    for k = length(s):-1:1 Y$_B1_  
        p = (1-2*mod(s(k),2))* ... 3=j"=-=  
                   prod(2:(n(j)-s(k)))/              ... rV#ch(  
                   prod(2:s(k))/                     ... onzxx4bax  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 4!?eRY  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Fx.=#bVX7  
        idx = (pows(k)==rpowers); m{HS0l'  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 4tBYR9|  
    end B]tQ(s~  
     e\L8oOk#r  
    if isnorm iYy1!\  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 26h21Z16q  
    end F)eelPZ+,  
end 4kx N<]  
% END: Compute the Zernike Polynomials 'H;*W|:-]  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xA*<0O\V  
Km$\:Xo  
% Compute the Zernike functions: x.$FNt(9  
% ------------------------------ gPPkT"  
idx_pos = m>0; k<?b(&`J  
idx_neg = m<0; i/Zd8+.n$  
[7y]n;Fy  
z = y; ckCE1e>s  
if any(idx_pos) ~t~|"u"P  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); WpvhTX  
end &};zvo~P.  
if any(idx_neg) ;$g?T~v7  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); p`qgrI`  
end kAUymds;O  
ECmW`#Otb)  
% 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