[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 sG`||Kb;n  
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！ yPrp:%PS  

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)  m[ *)sm  

~_ss[\N  
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|OqM:J
;  
G)5w_^&%  
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) pzcV[E1  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. pw
;  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 7_t\wmvYp  
%   order N and frequency M, evaluated at R.  N is a vector of
lq0@)'D  
%   positive integers (including 0), and M is a vector with the S[!sJ-rG  
%   same number of elements as N.  Each element k of M must be a \-(.cj)?  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) ygt7;};!  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is AITV+=sN  
%   a vector of numbers between 0 and 1.  The output Z is a matrix Gvg)@VNr  
%   with one column for every (N,M) pair, and one row for every 'iy &%?  
%   element in R. ",wv*z)_>  
% paFiuQ  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- xWkCP2$?P  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is :4 9ttJl  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to #H9J/k_  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 E"LSM]^^<f  
%   for all [n,m]. jSMvZJX3n  
% r![RRa^  
%   The radial Zernike polynomials are the radial portion of the rv`kP"I  
%   Zernike functions, which are an orthogonal basis on the unit pfd||Z  
%   circle.  The series representation of the radial Zernike kMD:~V  
%   polynomials is jys1Ki  
% aXi5~,Ks_  
%          (n-m)/2 + 3+^J?N  
%            __ K/oC+Z;K  
%    m      \       s                                          n-2s CKJ9YKu{W  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r ?UD2

}D[M  
%    n      s=0 0*?/s\>PS;  
% n_G< /8  
%   The following table shows the first 12 polynomials. 02g!mJW>}y  
% 5Ym/'eT  
%       n    m    Zernike polynomial    Normalization *}BaO*
A  
%       --------------------------------------------- QwaCaYoh  
%       0    0    1                        sqrt(2) tqI]S X  
%       1    1    r                           2 X\X*-.]{  
%       2    0    2*r^2 - 1                sqrt(6) gFk~SJd  
%       2    2    r^2                      sqrt(6) 5sbMp;ZM  
%       3    1    3*r^3 - 2*r              sqrt(8) l2>ka~  
%       3    3    r^3                      sqrt(8) u=a5Z4N'  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) Af8&PhyrU  
%       4    2    4*r^4 - 3*r^2            sqrt(10) {(mT,}`4  
%       4    4    r^4                      sqrt(10) bs-O3w  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) f{ZOH<"Lo  
%       5    3    5*r^5 - 4*r^3            sqrt(12) HsA4NRF'7  
%       5    5    r^5                      sqrt(12) F8e]sa$K\  
%       --------------------------------------------- ^[]GsF  
% g\sW2qXEw  
%   Example: q}-q[p? 5  
% SM>V o+  
%       % Display three example Zernike radial polynomials jJ^p ?  
%       r = 0:0.01:1; nAc02lJh|  
%       n = [3 2 5]; h.\V;6ly  
%       m = [1 2 1]; DDdMWH^o7  
%       z = zernpol(n,m,r); dP8b\H  
%       figure QR'yZ45n4  
%       plot(r,z) z[
k
z[  
%       grid on :W'Yt9v)  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') Z i-)PK^  
% r:h\{DVf  
%   See also ZERNFUN, ZERNFUN2. 9qIdwDRY  
1mT3$Z  
% A note on the algorithm. +5n,/YjS`  
% ------------------------ f.?p"~!  
% The radial Zernike polynomials are computed using the series w2BIf[~t  
% representation shown in the Help section above. For many special V!!E)I  
% functions, direct evaluation using the series representation can j{)_&|^{  
% produce poor numerical results (floating point errors), because jq6BwUN  
% the summation often involves computing small differences between
sSVgDQ~q  
% large successive terms in the series. (In such cases, the functions 0cV=>|b>;  
% are often evaluated using alternative methods such as recurrence m.ib#Y)y  
% relations: see the Legendre functions, for example). For the Zernike 9b`J2_ ]k  
% polynomials, however, this problem does not arise, because the RS&l68[6  
% polynomials are evaluated over the finite domain r = (0,1), and T7f>u}
T  
% because the coefficients for a given polynomial are generally all (_^pX  
% of similar magnitude.
20[_eu)  
% l7G&[\~  
% ZERNPOL has been written using a vectorized implementation: multiple 7CfHL;+m<4  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] %T:~N<8)  
% values can be passed as inputs) for a vector of points R.  To achieve $~M#msK9  
% this vectorization most efficiently, the algorithm in ZERNPOL _yje"  
% involves pre-determining all the powers p of R that are required to }S{#DgZ@X  
% compute the outputs, and then compiling the {R^p} into a single <0,c{e  
% matrix.  This avoids any redundant computation of the R^p, and r+8%oWj  
% minimizes the sizes of certain intermediate variables. _jmkAmeu  
% |2mm@):  
%   Paul Fricker 11/13/2006 Xy{\>}i]N  
3Qt-%=b&  
V+7x_>!&)  
% Check and prepare the inputs: N}0-L$@SL  
% ----------------------------- _8$arjx=  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LfD70r\  
    error('zernpol:NMvectors','N and M must be vectors.') yLfb'Ba  
end {Lj]++`fB]  
M7R.?nk  
if length(n)~=length(m) DgOO\
 
    error('zernpol:NMlength','N and M must be the same length.') a4gJ-FE  
end %X(iAoxbj  
v]CH L# |  
n = n(:); 1`1U'ibhe  
m = m(:); cPcp@Dp  
length_n = length(n); T_}
9b   
"F/%{0d  
if any(mod(n-m,2))
6C6<,c
 
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') w i,}sEoM  
end mfHZGk[[  
<Wgp$qt;  
if any(m<0) Yj+p^@{S2P  
    error('zernpol:Mpositive','All M must be positive.') RJ/4T#b"+  
end uveby:dh  
Ba$&4?8  
if any(m>n) V4]t=3>  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 9&R. <I  
end feOX
]g#  
VfS&V*un  
if any( r>1 | r<0 ) xij`Mr  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') ;eYm+e^?.  
end ~>:uMXyV2t  
1-`Il]@?8  
if ~any(size(r)==1) 2l5>>yY  
    error('zernpol:Rvector','R must be a vector.') E/MD]ox  
end ?kfLOJQ:I  
sem:"  
r = r(:); LadE4:oy  
length_r = length(r); V=%j]`Os  
6?an._ C  
if nargin==4 {DzOXTI[Y  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); p^u;]~JO  
    if ~isnorm 5>{S^i~!  
        error('zernpol:normalization','Unrecognized normalization flag.') WEgJ_dB  
    end xVOoYr>O  
else !]1'?
8  
    isnorm = false; i7hWBd4wK  
end r+6=b"  
oWg"f*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |KF_h^  
% Compute the Zernike Polynomials Fk01j;k.H  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @LQe[`  
[ +w=  
% Determine the required powers of r: WCc7 MK  
% ----------------------------------- .xnJT2uu'  
rpowers = []; <Co\?h/<  
for j = 1:length(n) +6dq+8msF  
    rpowers = [rpowers m(j):2:n(j)]; 0s>ozAJ  
end D?yiK=:08`  
rpowers = unique(rpowers); VKI`@rY4  
J]|Zh  
% Pre-compute the values of r raised to the required powers, sFhmp  
% and compile them in a matrix: 1ztL._Td  
% ----------------------------- QahM)Gb  
if rpowers(1)==0 \]bAXa{ p  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2$G,pT1J  
    rpowern = cat(2,rpowern{:}); ;[pY>VJ(  
    rpowern = [ones(length_r,1) rpowern]; vwA d6Tm  
else !% 'dyj  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); W!@*3U]2R  
    rpowern = cat(2,rpowern{:}); C!a
#M{:  
end E?,O>bCJ5  
l_Eeus  
% Compute the values of the polynomials: 0O,Q]P 82f  
% -------------------------------------- &-l(nr]h]  
z = zeros(length_r,length_n); 1DzI@c~X  
for j = 1:length_n QLr.5Wcg>  
    s = 0:(n(j)-m(j))/2; ,OGXH2!h  
    pows = n(j):-2:m(j); '3h"Ol{b  
    for k = length(s):-1:1 IEbk_-h[  
        p = (1-2*mod(s(k),2))* ... P
ra,r9h,  
                   prod(2:(n(j)-s(k)))/          ... J.%%]-f=&  
                   prod(2:s(k))/                 ... )SJ18 no|l  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... =a!w)z_rw  
                   prod(2:((n(j)+m(j))/2-s(k))); ?H7YmN  
        idx = (pows(k)==rpowers); IYZ$a/{P  
        z(:,j) = z(:,j) + p*rpowern(:,idx); ZT;8Wvo  
    end 9d5|rk8VS  
     WoYXXYP/E  
    if isnorm st91rV$y?  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Ze
?(N~  
    end m]XG7:}V0  
end v'fX'/  
Pl2ZA)[g  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) A9:dHOmT^U  
%ZERNFUN2 Single-index Zernike functions on the unit circle. ^nPk;%`0  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated f_{OU E  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive *_Sx^`"X`l  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, @'D ,
T^I  
%   and THETA is a vector of angles.  R and THETA must have the same p|q}z/  
%   length.  The output Z is a matrix with one column for every P-value, ysHmi{V~  
%   and one row for every (R,THETA) pair. xRTr
@  
% =r?#,'a  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike .+.BNS   
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) KMU2PoqD  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) T?!D?YV  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 0\/cTNN  
%   for all p. y,YK Mc  
% bOvMXj/HV=  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36
3$~6+i  
%   Zernike functions (order N<=7).  In some disciplines it is *{#l0My  
%   traditional to label the first 36 functions using a single mode :\Pk>a  
%   number P instead of separate numbers for the order N and azimuthal &I=27!S  
%   frequency M. v"Z`#Bi  
% vc r5  
%   Example: K0YUN^St  
% . #7B10  
%       % Display the first 16 Zernike functions <E&[sQ|3  
%       x = -1:0.01:1; Iz^lED  
%       [X,Y] = meshgrid(x,x); koTb{UL  
%       [theta,r] = cart2pol(X,Y); da,;IE{1u  
%       idx = r<=1; 4I
q-4IG(  
%       p = 0:15; :)PAj  
%       z = nan(size(X)); 'K8emt$d+  
%       y = zernfun2(p,r(idx),theta(idx)); 7y/Pch  
%       figure('Units','normalized') -_4ZT^.Lna  
%       for k = 1:length(p) 2u=Nb0  
%           z(idx) = y(:,k); O]/BNacS  
%           subplot(4,4,k) p3f>;|uh_  
%           pcolor(x,x,z), shading interp L)mb.U$`c|  
%           set(gca,'XTick',[],'YTick',[]) :t'*fHi~  
%           axis square }BR@vY'd  
%           title(['Z_{' num2str(p(k)) '}']) {&qB!axj  
%       end <dd(i  
% b+6%Mu}o  
%   See also ZERNPOL, ZERNFUN. kr=&x)Wy!  
.3Ag6YI0N  
%   Paul Fricker 11/13/2006 #&oL iz=hZ  
p1mY@  
c}l?x \/  
% Check and prepare the inputs: >axf_k  
% ----------------------------- / }tMb  
if min(size(p))~=1 _$f XK  
    error('zernfun2:Pvector','Input P must be vector.') hj<h]dhp  
end kv)IG$S0  
j,%<16f^A  
if any(p)>35 x9lG$0k:V  
    error('zernfun2:P36', ... X / {;  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... }ag -J."5M  
           '(P = 0 to 35).']) Z~WUILx,  
end a-9Y &#U  
FFvF4]|L  
% Get the order and frequency corresonding to the function number: hG8!aJo  
% ---------------------------------------------------------------- <"SOH;w  
p = p(:); b5Sgf'B^  
n = ceil((-3+sqrt(9+8*p))/2); 13lJq:bM  
m = 2*p - n.*(n+2); "y5LojdCs  
$ M8ZF(W  
% Pass the inputs to the function ZERNFUN: p^yuz (  
% ---------------------------------------- YR\pt8(z?  
switch nargin ~|>q)4is6a  
    case 3 O:hCUr  
        z = zernfun(n,m,r,theta); $vQ#ah/k  
    case 4 LKx<hl$O  
        z = zernfun(n,m,r,theta,nflag); $
7~T+fmF  
    otherwise 555*IT3b
 
        error('zernfun2:nargin','Incorrect number of inputs.') e2@{Ab  
end }r)T75_1  
4,yS7l  
% 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 tzKIi_2  
function z = zernfun(n,m,r,theta,nflag) qVpV ZH!  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5Lo\[K>j  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +wwb+aG6{  
%   and angular frequency M, evaluated at positions (R,THETA) on the nB#m?hK  
%   unit circle.  N is a vector of positive integers (including 0), and R[l9f8  
%   M is a vector with the same number of elements as N.  Each element x?*)  
%   k of M must be a positive integer, with possible values M(k) = -N(k) n&j@7R  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,
x'c%w:  
%   and THETA is a vector of angles.  R and THETA must have the same <x^Ab#K"  
%   length.  The output Z is a matrix with one column for every (N,M) I1Jo8s  
%   pair, and one row for every (R,THETA) pair. 04u^Q  
% *G6Py,- !f  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zlw+=NX  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f7mN,_Lt  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral `ecIy_O3P&  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, _3_kvs  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized N"+o=nS  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :nYnTo`  
% ,'KS:`m!  
%   The Zernike functions are an orthogonal basis on the unit circle. 5Wyo!pRi  
%   They are used in disciplines such as astronomy, optics, and >Fzs%]
M  
%   optometry to describe functions on a circular domain. ks}J ke>  
% }#0i1]n$D  
%   The following table lists the first 15 Zernike functions. D (>,#F  
% |6ZH+6[  
%       n    m    Zernike function           Normalization WaaF;|,(  
%       -------------------------------------------------- R[%ZyQ_  
%       0    0    1                                 1 ^E)*i#."4  
%       1    1    r * cos(theta)                    2 \9Z1'W  
%       1   -1    r * sin(theta)                    2 V5ySOgzw,  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 19r4J(pV  
%       2    0    (2*r^2 - 1)                    sqrt(3) mw[  
%       2    2    r^2 * sin(2*theta)             sqrt(6) ~g6`Cp`  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) H;eGB
Vi  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) O>h,u[0  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) X*Qtbm,  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 0pC}+ +  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 4IT`8n~  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ixf~3Y8  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) cg]\R1Gm  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^uDNArDmj5  
%       4    4    r^4 * sin(4*theta)             sqrt(10) %YH+=b:uW  
%       -------------------------------------------------- MPtn$@  
% ['*{f(AI  
%   Example 1: ,"@Tm01os  
% 8BHtN  
%       % Display the Zernike function Z(n=5,m=1) Q7~9~  
%       x = -1:0.01:1; -$; h+9BO  
%       [X,Y] = meshgrid(x,x); +i@r-OL  
%       [theta,r] = cart2pol(X,Y); Hju7gP=y}  
%       idx = r<=1; !bPsJbIo>  
%       z = nan(size(X)); {#Lj,o  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); _`H2CXGg  
%       figure !' D1aea5
 
%       pcolor(x,x,z), shading interp 4F-r}Fj3  
%       axis square, colorbar 0c$0<2D%  
%       title('Zernike function Z_5^1(r,\theta)') #JOWiO0>  
% sp2"c"_+  
%   Example 2: :nt 7jm,  
% _>6xUt  
%       % Display the first 10 Zernike functions \L-K}U>
J  
%       x = -1:0.01:1; B5nzkJV<X  
%       [X,Y] = meshgrid(x,x); %y{
f]m  
%       [theta,r] = cart2pol(X,Y); BotGPk><c  
%       idx = r<=1; cIm_~HH  
%       z = nan(size(X)); TSl:a &  
%       n = [0  1  1  2  2  2  3  3  3  3]; -$@$  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; zE~{}\J  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; &EELq"5K  
%       y = zernfun(n,m,r(idx),theta(idx)); t7t?xk!2  
%       figure('Units','normalized') WRq:xDRn0  
%       for k = 1:10 uA'S8b%C  
%           z(idx) = y(:,k); )YKnFSm  
%           subplot(4,7,Nplot(k)) :75$e%'A  
%           pcolor(x,x,z), shading interp TpHvZ]c  
%           set(gca,'XTick',[],'YTick',[]) H
P$GI  
%           axis square ')bas#=uP  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ="k9 y  
%       end 015 ;'V#we  
% gJ; *?Uq(  
%   See also ZERNPOL, ZERNFUN2. xbN)
z  
sULCYiT|Hn  
%   Paul Fricker 11/13/2006 4;rt|X77  
xla64Qld  
CJDnHuozc  
% Check and prepare the inputs: \z~wm&  
% ----------------------------- q{fgsc8v\  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) e%Sw(=a  
    error('zernfun:NMvectors','N and M must be vectors.') z]^u@]@NC  
end U)f;*{U  
t#fbagTON  
if length(n)~=length(m) y@T0 jI  
    error('zernfun:NMlength','N and M must be the same length.') ^:Mal[IR  
end YqJ `eLu  
/M0A9ZT[  
n = n(:); oPqWL9]  
m = m(:); h^H~q<R[T  
if any(mod(n-m,2)) 3:S>MFRn.3  
    error('zernfun:NMmultiplesof2', ... 2"'<Yk9  
          'All N and M must differ by multiples of 2 (including 0).') d*Wg>8|  
end &D/@H1fBe  
FLb Q#c\  
if any(m>n) L"_l(<g  
    error('zernfun:MlessthanN', ... _#jR6g TY  
          'Each M must be less than or equal to its corresponding N.') DCv=*=6w  
end c2tf7fkH  
9{A[n}  
if any( r>1 | r<0 ) U= Gw(  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') \8KAK3i'  
end l{2Y[&%  
+K@wh  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /"f4aF[  
    error('zernfun:RTHvector','R and THETA must be vectors.') 8Hdm(>  
end vFz#A/1  
&e-MOM2&  
r = r(:); dr54D  
theta = theta(:); y{?wxg9  
length_r = length(r); 6]Vf`i  
if length_r~=length(theta) q JdC5z\[  
    error('zernfun:RTHlength', ... =k{ n! e  
          'The number of R- and THETA-values must be equal.') daX$=n  
end (]Pr[xB  
t&oNC6  
% Check normalization: Z{MR#.I  
% -------------------- Z [aKic  
if nargin==5 && ischar(nflag) IwTAM9n  
    isnorm = strcmpi(nflag,'norm'); Wv4x^nJ  
    if ~isnorm 4U;Zs3  
        error('zernfun:normalization','Unrecognized normalization flag.') 'Avp16zg  
    end fH>I/%  
else .$rt>u,8<  
    isnorm = false; ;P
A^.RB  
end q#6K'
=AC  
Y*KP1=Md  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ac2G;}B|  
% Compute the Zernike Polynomials 3SeM:OYq]s  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $YPU(y  
kwM1f=!-  
% Determine the required powers of r: ZQVr]/W^r  
% ----------------------------------- -(Z%?]+  
m_abs = abs(m); t=6[FK  
rpowers = []; RyN}Gz/YN  
for j = 1:length(n) d~>d\K%v  
    rpowers = [rpowers m_abs(j):2:n(j)]; ZJod=^T  
end G<MX94?  
rpowers = unique(rpowers); m|c5X)}-  
ZDhl$m[m  
% Pre-compute the values of r raised to the required powers, : U Yn  
% and compile them in a matrix: p bT sn  
% ----------------------------- HTa]T'  
if rpowers(1)==0  hb,G'IU  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X`+8r
O[  
    rpowern = cat(2,rpowern{:}); NCKhrDd&  
    rpowern = [ones(length_r,1) rpowern]; n{@^ne4m  
else ,t!K? Y  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); "h84D&V  
    rpowern = cat(2,rpowern{:}); 
Ln4zy*v{  
end "A>/m"c]*  
fPj*qi  
% Compute the values of the polynomials: ?S~@Ea8/M  
% -------------------------------------- kzb%=
EI  
y = zeros(length_r,length(n)); < 9 vS  
for j = 1:length(n) }23#z  
    s = 0:(n(j)-m_abs(j))/2; #% 1|$V*:  
    pows = n(j):-2:m_abs(j); Pi!3wy  
    for k = length(s):-1:1 zg[.Pws:E  
        p = (1-2*mod(s(k),2))* ... / Ml d.  
                   prod(2:(n(j)-s(k)))/              ... ^gu;  
                   prod(2:s(k))/                     ... FZi'#(y  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... W3h{5\d!  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); O\5q_>]  
        idx = (pows(k)==rpowers); IuW5LS  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ).8i*Ys,:  
    end {<k}U;uiO  
     %ylpn7I\6  
    if isnorm g:&V9~FR  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /yd<+on^  
    end l(fStpP  
end l`' lqnhv  
% END: Compute the Zernike Polynomials W4ygJL7 6  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;'fn{j6C  
gT=RJB  
% Compute the Zernike functions: L>PPAI  
% ------------------------------ ~=#jr0IZ  
idx_pos = m>0; 3v
:c".O2O  
idx_neg = m<0; n#fc=L1U  
mz<wYV*  
z = y; }w
%W A&"W  
if any(idx_pos) E{#Y=  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >.}ewz&9o  
end B*,Qw_3dG  
if any(idx_neg) #ozQF~
 
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); OpT0V]k^"9  
end 5"cYZvGkJ  
-y1t;yU.L  
% 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