[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 Jjz:-Uqq2  
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！ ]OH
zE]Q  

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)  h2!We#  

;4Y@xS2M  
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。 g` 6Xrf  
;c_X ^"d  
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) ce\]o^4  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. b{=2#J-  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of (n05MwKu\  
%   order N and frequency M, evaluated at R.  N is a vector of yo`Jp$G  
%   positive integers (including 0), and M is a vector with the h!?7I=p~#  
%   same number of elements as N.  Each element k of M must be a $(H%|Oyn  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) Ra}%:  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is d_BECx<\  
%   a vector of numbers between 0 and 1.  The output Z is a matrix <LIL{g0eX  
%   with one column for every (N,M) pair, and one row for every Wjn1W;m&g  
%   element in R. 5m!FtHvm1  
% H-~V:OCB~  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- QM=M<~<Voh  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is `$\g8Mo  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to XMd-r8yYr  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 acz8 H0cS  
%   for all [n,m]. UB }n=  
% ~Jrtm7  
%   The radial Zernike polynomials are the radial portion of the xAw$bJj~s  
%   Zernike functions, which are an orthogonal basis on the unit im_WTZz2P  
%   circle.  The series representation of the radial Zernike U+F?b\  
%   polynomials is Sg(fZ' -  
%  iUJqAi1o  
%          (n-m)/2 eEePK~%c  
%            __  d!5C$C/x  
%    m      \       s                                          n-2s L}*:,&Y/  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r j-8v$0'  
%    n      s=0 dR<sBYo  
% 3zo]*6p0  
%   The following table shows the first 12 polynomials. 8Eyi`~cAiH  
% zs<W>
gBq  
%       n    m    Zernike polynomial    Normalization i\lur ET  
%       --------------------------------------------- D]S@U>]M!  
%       0    0    1                        sqrt(2) fhw.A5Ck  
%       1    1    r                           2 fNNkc[YTZI  
%       2    0    2*r^2 - 1                sqrt(6) O}I8P")m  
%       2    2    r^2                      sqrt(6) ~F[}*%iR  
%       3    1    3*r^3 - 2*r              sqrt(8) 
tfW/Mf  
%       3    3    r^3                      sqrt(8) 19Xc0ez  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) r!N)pt<g  
%       4    2    4*r^4 - 3*r^2            sqrt(10) G=W!$(:  
%       4    4    r^4                      sqrt(10)
@d]I3?`  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) j}7as&  
%       5    3    5*r^5 - 4*r^3            sqrt(12) mr dG-t(k  
%       5    5    r^5                      sqrt(12) e>vV8a\  
%       --------------------------------------------- v!n\A}^:  
% wegu1Ny  
%   Example: 2A7g}V  
% >Ig%|4Hw  
%       % Display three example Zernike radial polynomials _xT=AF9~o  
%       r = 0:0.01:1; }'mVD^<+  
%       n = [3 2 5]; 0g}+%5]yg  
%       m = [1 2 1]; .VG$`g"  
%       z = zernpol(n,m,r); vp@%wxl!:  
%       figure DFKFsu8s  
%       plot(r,z) M>W-lp^3  
%       grid on >Y=HP&A<  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') g^ZsV:D  
% ~WXT0-,  
%   See also ZERNFUN, ZERNFUN2. 6X2>zUHR  
}6ObQa43  
% A note on the algorithm. dR~4*59Bg  
% ------------------------ qD=o;:~Km  
% The radial Zernike polynomials are computed using the series ]H<5]({F  
% representation shown in the Help section above. For many special wQbN5*82  
% functions, direct evaluation using the series representation can  lc9aDt  
% produce poor numerical results (floating point errors), because |M]#D0v  
% the summation often involves computing small differences between Q;{[U!\:  
% large successive terms in the series. (In such cases, the functions ],CJSA!5F  
% are often evaluated using alternative methods such as recurrence ;AMbo`YK[  
% relations: see the Legendre functions, for example). For the Zernike eAKK uML  
% polynomials, however, this problem does not arise, because the d\Cx(Lb[  
% polynomials are evaluated over the finite domain r = (0,1), and 4Kjrk7GAx  
% because the coefficients for a given polynomial are generally all EV{kd.=f  
% of similar magnitude. 1cPm $=B  
% ;/+VHZP;  
% ZERNPOL has been written using a vectorized implementation: multiple mi9BC9W(  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] 7KIOI,qb6  
% values can be passed as inputs) for a vector of points R.  To achieve WM$)T6M  
% this vectorization most efficiently, the algorithm in ZERNPOL ^% y<7>%  
% involves pre-determining all the powers p of R that are required to #v!(uuq,  
% compute the outputs, and then compiling the {R^p} into a single <YX)am'\y  
% matrix.  This avoids any redundant computation of the R^p, and (qd$wv^h  
% minimizes the sizes of certain intermediate variables. ?w'a^+H  
% 4/YE
kD  
%   Paul Fricker 11/13/2006 W:D'k^u  
@V{s'V   
AZ' "M{wiI  
% Check and prepare the inputs: mUW|4zl i}  
% ----------------------------- 3Mw
\}q  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) VK\ Bjru9
 
    error('zernpol:NMvectors','N and M must be vectors.') 5sJi- ^  
end ngF5ywIG  
z d 9Gi5&  
if length(n)~=length(m) Q`mw2$zv  
    error('zernpol:NMlength','N and M must be the same length.') !u~h.DrvZ  
end BDoL)}bRE  
|O0=Q,<m  
n = n(:); xbJ@z{  
m = m(:); SN2X{Q|*  
length_n = length(n); :M" NB+T  
Y"qKe,
  
if any(mod(n-m,2)) P|}~=2J  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') N)'oX3?x  
end R^zTgyr  
C =fs[  
if any(m<0) "0x"Xw#I  
    error('zernpol:Mpositive','All M must be positive.') v&g(6~b_>  
end *p!K9$4  
3/a$oO
 
if any(m>n) g7*cwu  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') &t)dE7u
5  
end YrAaL"20  
lK;/97Ze  
if any( r>1 | r<0 ) {SH+lX0]{  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') 8joJe>9VJ  
end "=)i'x"0"  
gM=oH   
if ~any(size(r)==1) 'G1~\CT  
    error('zernpol:Rvector','R must be a vector.') .#n1p:}[  
end U9:?d>7  
2`4
'Y.Qf  
r = r(:); &sbA:xZBA  
length_r = length(r); fsc^8  
:`BZ,j_  
if nargin==4 G<.p".o4  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 4u5^I;4pL  
    if ~isnorm l:NEK`>i  
        error('zernpol:normalization','Unrecognized normalization flag.') 9/Q_Jv-Q  
    end S0.  
else M$O}roOa  
    isnorm = false; _%WJ7~>  
end 4>]^1J7Wz  
;)ffGg>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;u;YfOr  
% Compute the Zernike Polynomials |a@$K
F$  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s=`1wkh0  
gE8=#%1<  
% Determine the required powers of r: w <zO  
% ----------------------------------- =\s(v-8  
rpowers = []; =x "N0p  
for j = 1:length(n) [uO
W\)`  
    rpowers = [rpowers m(j):2:n(j)]; l'YpSO~l7  
end 6hKavzSi
 
rpowers = unique(rpowers); $I(2}u?1+d  
9:0JWW^so  
% Pre-compute the values of r raised to the required powers, <qH>[
\  
% and compile them in a matrix: d!) &@k  
% ----------------------------- uiq)?XUKv  
if rpowers(1)==0 k,0RpE  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xM85^B'  
    rpowern = cat(2,rpowern{:}); 7NG^X"N{Ul  
    rpowern = [ones(length_r,1) rpowern]; ^T\JFzV  
else w91gM*A  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (n7v $A  
    rpowern = cat(2,rpowern{:}); GCul6,w  
end T1m097  
eN]0]9JO  
% Compute the values of the polynomials: 1fRYXqx  
% -------------------------------------- `+]9+:tS  
z = zeros(length_r,length_n); V
.J[Uwf  
for j = 1:length_n * bmdY=#7  
    s = 0:(n(j)-m(j))/2; `WF?87l1  
    pows = n(j):-2:m(j); w2y{3O"p=  
    for k = length(s):-1:1 jM1|+o*Wr  
        p = (1-2*mod(s(k),2))* ... 7V?]Qif~  
                   prod(2:(n(j)-s(k)))/          ... Ni IX^&N1  
                   prod(2:s(k))/                 ... 7SYU^GD  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... 3$+|nP:U  
                   prod(2:((n(j)+m(j))/2-s(k))); ^!H8"CdC3  
        idx = (pows(k)==rpowers); %w7J
0p  
        z(:,j) = z(:,j) + p*rpowern(:,idx); !]qwRB$5  
    end ]t_AXKd  
     vvw6 GB,M  
    if isnorm ewB&PR  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); *{:FPmDU  
    end p"ht|x  
end Uj}iMw,  
Z[KXDQn8  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) 'wz*GMGWC  
%ZERNFUN2 Single-index Zernike functions on the unit circle. F>&Q5Kl R  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 5~4I.+~8  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive 4W}8?&T  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, 20;M
-Wx  
%   and THETA is a vector of angles.  R and THETA must have the same hu[=9#''$  
%   length.  The output Z is a matrix with one column for every P-value, "v4;m\g&:  
%   and one row for every (R,THETA) pair. ,6PV"E)_  
% /Q5pAn-u  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ORlz1&hW  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) D:'|poH  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) #B:J7&@fn  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 2qKo|'gL`  
%   for all p. <I'kJ{"
 
% !cT#G  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 w(n&(5FzB<  
%   Zernike functions (order N<=7).  In some disciplines it is 5gZ0a4  
%   traditional to label the first 36 functions using a single mode 1t=Y+|vA9  
%   number P instead of separate numbers for the order N and azimuthal 5+:b#B  
%   frequency M. $bh2zKB)  
% wcP0PfY  
%   Example: +pc_KR  
% l*("[?>I  
%       % Display the first 16 Zernike functions U#1T HO`  
%       x = -1:0.01:1; @^} % o-:  
%       [X,Y] = meshgrid(x,x); %k{~Fa  
%       [theta,r] = cart2pol(X,Y); g"
( vl-Uw  
%       idx = r<=1; cH'*J/  
%       p = 0:15; :h0as!2@dp  
%       z = nan(size(X));  IPa08/  
%       y = zernfun2(p,r(idx),theta(idx)); neJNMdv@T  
%       figure('Units','normalized') ;r>?V2,tm  
%       for k = 1:length(p) =|S8.|r+  
%           z(idx) = y(:,k); :2Qm*Y&_$V  
%           subplot(4,4,k) -% PUY(  
%           pcolor(x,x,z), shading interp kmNY ;b6Y$  
%           set(gca,'XTick',[],'YTick',[]) Y}'C'PR  
%           axis square Tz+2g&+  
%           title(['Z_{' num2str(p(k)) '}']) ?2OT:/I,  
%       end 4z Af|Je  
% "2+>!G RQ  
%   See also ZERNPOL, ZERNFUN. V\%;S  
`da6}Vqj:  
%   Paul Fricker 11/13/2006 ?(>7v[=iT  
!tv3.:eT  
UF=5k~7<b  
% Check and prepare the inputs:
YTQ|Hg6jO  
% ----------------------------- W
!.UMmw`  
if min(size(p))~=1 $$<9tqA  
    error('zernfun2:Pvector','Input P must be vector.') R}Uvi9?  
end PV(bJ7&R  
4O9HoX#-?  
if any(p)>35 vLDMa>  
    error('zernfun2:P36', ... 5|x&Z/hL  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... =t<!W  
           '(P = 0 to 35).']) |^C?~g  
end r-'\<d(J$  
?RiW:TQ*  
% Get the order and frequency corresonding to the function number: je9[S_Z:Y  
% ---------------------------------------------------------------- 2,/("lV@0  
p = p(:); B@e,3:  
n = ceil((-3+sqrt(9+8*p))/2); c%>t(ce`Tl  
m = 2*p - n.*(n+2); 9 }|Bs=q  
=r3Yt9  
% Pass the inputs to the function ZERNFUN: Thn-8DT  
% ---------------------------------------- xO9,,w47  
switch nargin p]*$m=
t0r  
    case 3 JxiLjvIq  
        z = zernfun(n,m,r,theta);  UB&ofO  
    case 4 bPC {4l  
        z = zernfun(n,m,r,theta,nflag); muT+H(Zp}  
    otherwise %Gl,V5z&  
        error('zernfun2:nargin','Incorrect number of inputs.') 4?><x[l2{  
end i|Lir{vW  
6=Kl[U0Y  
% 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 3F5r3T6j}  
function z = zernfun(n,m,r,theta,nflag) S"CsY2;  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 37?%xQ!  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }EE  
%   and angular frequency M, evaluated at positions (R,THETA) on the 9Vxsv*OR,  
%   unit circle.  N is a vector of positive integers (including 0), and xVuGeanCv  
%   M is a vector with the same number of elements as N.  Each element jeN_ sm81b  
%   k of M must be a positive integer, with possible values M(k) = -N(k) %((F}9_6  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, !\JG]2 \  
%   and THETA is a vector of angles.  R and THETA must have the same S-gL]r3G8  
%   length.  The output Z is a matrix with one column for every (N,M) l@q.4hT  
%   pair, and one row for every (R,THETA) pair. ]]d9\fw  
% R)Wv
U4+U  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @bmu4!"d  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9x?" %b  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral $ n`<,;^l  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \|@]XNSN  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^>.?kh9z  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. szF[LRb  
% 1Nx%uz  
%   The Zernike functions are an orthogonal basis on the unit circle. ~/!Zh  
%   They are used in disciplines such as astronomy, optics, and _p?lRU8  
%   optometry to describe functions on a circular domain. igOjlg_Q  
% p W:[Q\rSj  
%   The following table lists the first 15 Zernike functions. qdCa]n!d  
%  8yOzD  
%       n    m    Zernike function           Normalization :)g=AhBF  
%       -------------------------------------------------- <uU AAHi  
%       0    0    1                                 1 1FjA  
%       1    1    r * cos(theta)                    2 -869$  
%       1   -1    r * sin(theta)                    2 09_3`K.*  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) i:&Y{iPQp  
%       2    0    (2*r^2 - 1)                    sqrt(3) 8n?P'iM  
%       2    2    r^2 * sin(2*theta)             sqrt(6) n/pM[gI  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) 9:!n'mn  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) t.j q]L  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ~uqJ@#o{  
%       3    3    r^3 * sin(3*theta)             sqrt(8) W6K]jIQ  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) l4O}>#  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)  M)Yu^  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) wS%I.  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x(hUQu 6  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 2xni! *T+  
%       -------------------------------------------------- qp"gD-,-o  
% U/&?rY^|  
%   Example 1: ykRKZYfsw(  
% b?CmKiM%  
%       % Display the Zernike function Z(n=5,m=1) *=MC+4E  
%       x = -1:0.01:1; AXH4jQw  
%       [X,Y] = meshgrid(x,x); @>qzRo  
%       [theta,r] = cart2pol(X,Y); #fG!dD42  
%       idx = r<=1; W`eYd|+C  
%       z = nan(size(X)); )qn =  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); <gZC78}E  
%       figure |}QDC/  
%       pcolor(x,x,z), shading interp er+m:XuV  
%       axis square, colorbar 6^mO<nB   
%       title('Zernike function Z_5^1(r,\theta)') A0oC*/  
% }dAb}0XK.  
%   Example 2: CYEqH2"3  
% 'iXjt MX  
%       % Display the first 10 Zernike functions [LL"86D  
%       x = -1:0.01:1; H6/@loO!Xy  
%       [X,Y] = meshgrid(x,x); MGX,JW>L  
%       [theta,r] = cart2pol(X,Y); :?@d\c'  
%       idx = r<=1; $*b>c:  
%       z = nan(size(X)); &%fy  
%       n = [0  1  1  2  2  2  3  3  3  3]; kzLj1Ix2  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; _Y|k \|'  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; e|):%6#  
%       y = zernfun(n,m,r(idx),theta(idx)); @gE +T37x2  
%       figure('Units','normalized') h[C!cX  
%       for k = 1:10 ^-4mZXAy1|  
%           z(idx) = y(:,k); 17$JBQ,[  
%           subplot(4,7,Nplot(k)) "0`r]5 5d  
%           pcolor(x,x,z), shading interp ?j?{}Z  
%           set(gca,'XTick',[],'YTick',[]) UG Fx  
%           axis square ~)tMR9=wX  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) fR5 NiH  
%       end G/Kz_Y,  
% fT[6Cw5w`  
%   See also ZERNPOL, ZERNFUN2. x\3 ` W  
%ghQ#dZ]&  
%   Paul Fricker 11/13/2006 , *e^,|#  
xPQO}wKa  
_UjAct]6  
% Check and prepare the inputs: wjq f u /  
% ----------------------------- OZ![9l  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) V/"0'H\"1  
    error('zernfun:NMvectors','N and M must be vectors.') .oaW#f}0P  
end -R~;E[ {%  
YDi_Gl$  
if length(n)~=length(m) a}
M7"
v9  
    error('zernfun:NMlength','N and M must be the same length.') dI>)4()  
end wC
MsaW  
z;#}uC  
n = n(:); Hi,_qlc+  
m = m(:); : 60PO  
if any(mod(n-m,2)) []3xb`<&  
    error('zernfun:NMmultiplesof2', ... ]8+%57:E  
          'All N and M must differ by multiples of 2 (including 0).') baR{  
end qAR~js`5  
?Sn$AS I  
if any(m>n) duXv [1  
    error('zernfun:MlessthanN', ... 6\4oHRJC  
          'Each M must be less than or equal to its corresponding N.') kzJNdYtdH  
end w]tv<U={  
0m7J'gm{  
if any( r>1 | r<0 ) eJ'2CM6  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') XXuU@G6Z7$  
end "}7K>|a  
TjD`<k  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 75!IzJG  
    error('zernfun:RTHvector','R and THETA must be vectors.') b[GZ sXD-  
end v}xz`]MW<,  
>bz}IcZP  
r = r(:); wA.YEI|CSj  
theta = theta(:); T-fW[][&$  
length_r = length(r);
(}4tj4d  
if length_r~=length(theta) 6by5VESx  
    error('zernfun:RTHlength', ... Yx(?KN7V?  
          'The number of R- and THETA-values must be equal.') OjJlGElw  
end */nb%QV  
q$:T<mFK$  
% Check normalization: $o/?R]h  
% -------------------- nt "VH5  
if nargin==5 && ischar(nflag) 6/nhz6=  
    isnorm = strcmpi(nflag,'norm'); ZS>}NN  
    if ~isnorm
t.lm`
=  
        error('zernfun:normalization','Unrecognized normalization flag.') W qci51y>#  
    end G A2S  
else "E/F{6NH  
    isnorm = false; ecA0z c~  
end +c}fDrr)  
4xtbP\=
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -M%n<,XN0  
% Compute the Zernike Polynomials U+~0m!|4  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .Ks&r  
:'1ePq  
% Determine the required powers of r: W zy8  
% ----------------------------------- bi#o1jR  
m_abs = abs(m); l:j9lBS  
rpowers = []; hd/5*C{s  
for j = 1:length(n)  yZmQBh$  
    rpowers = [rpowers m_abs(j):2:n(j)]; !x;T2l  
end PovPO  
rpowers = unique(rpowers); Z%(aBz7Et  
z)43+8;  
% Pre-compute the values of r raised to the required powers, A-ir   
% and compile them in a matrix: FT`y3~
 
% ----------------------------- +r4US or  
if rpowers(1)==0
[rqq*_eB  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (zk'i13#6  
    rpowern = cat(2,rpowern{:}); 9e=F  
    rpowern = [ones(length_r,1) rpowern]; jY87NHg  
else *bmk(%g  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3_^w/-7`B  
    rpowern = cat(2,rpowern{:}); d cPh@3  
end "^22Y}VB  
53BXz= k  
% Compute the values of the polynomials: UupQ*,dJ  
% -------------------------------------- u"X8(\pOn  
y = zeros(length_r,length(n)); [A*vl9=  
for j = 1:length(n) | 2p\M?@  
    s = 0:(n(j)-m_abs(j))/2; MZv&$KG4m@  
    pows = n(j):-2:m_abs(j); t!D=oBCro  
    for k = length(s):-1:1 4?jhZLBU  
        p = (1-2*mod(s(k),2))* ... YDs/BF Z  
                   prod(2:(n(j)-s(k)))/              ... .Zf#L'Rf  
                   prod(2:s(k))/                     ... *=^
_K`y  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... AGK+~EjL@  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 6tzZ j:yq  
        idx = (pows(k)==rpowers); () b0Sh=  
        y(:,j) = y(:,j) + p*rpowern(:,idx); aOWbIS[8  
    end AWD &K!  
     >=C)\Yfu)  
    if isnorm 1hi^  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); nHyWb6  
    end z7HC6{g%X  
end /\ ~{  
% END: Compute the Zernike Polynomials -_>c P  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q_cqjly<  
]y-r
 I  
% Compute the Zernike functions: N|1J@"H  
% ------------------------------ L?Wl#wP\;*  
idx_pos = m>0; )bPNL$O  
idx_neg = m<0; 5jx{O${u  
gJvc<]W8!  
z = y; sE>'~+1_O  
if any(idx_pos) ,2&'8:B  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^C<dr}8  
end -lb}}z+/  
if any(idx_neg) z-krL:A  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); xv4nYm9  
end .(1=iL_3e  
`Bkba:  
% 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