[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 T)\"Xj  
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！ HK+/:'Pu  

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)  /J)l
/oI  

rwIeqV{:  
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。 kX:tc  
v}^5Rp&m  
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) bAxTLIf  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. ydWtvFuS  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of I.}1JJF*  
%   order N and frequency M, evaluated at R.  N is a vector of T#:F]=  
%   positive integers (including 0), and M is a vector with the &;H{cv`  
%   same number of elements as N.  Each element k of M must be a e1 *__'  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) iZ[tHw||  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is Y ?]G}5  
%   a vector of numbers between 0 and 1.  The output Z is a matrix 9Z\z96O-  
%   with one column for every (N,M) pair, and one row for every guN4-gGDr<  
%   element in R.  Kn+=lCk  
% #`tD1T{;  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- `NwdbKX  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is _J?SIm  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to >Pe:I  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 E(+T*  
%   for all [n,m]. {g/wY%u=  
% o}8{Bh^  
%   The radial Zernike polynomials are the radial portion of the `0BdMK
jA  
%   Zernike functions, which are an orthogonal basis on the unit B{, Bno  
%   circle.  The series representation of the radial Zernike DOD6Liau{Q  
%   polynomials is &yH#s 8^8  
% g jG2  
%          (n-m)/2 RagiV6c  
%            __ 67<CbQZoN3  
%    m      \       s                                          n-2s ~] =?b)B  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r V*B0lI7`B  
%    n      s=0 &jts:^N>  
% Oo%!>!Lt,  
%   The following table shows the first 12 polynomials. 24@^{ }  
% Wb=Jj 9;  
%       n    m    Zernike polynomial    Normalization :q2YBa  
%       --------------------------------------------- _[E\=
 
%       0    0    1                        sqrt(2) f[/.I,9U^  
%       1    1    r                           2 H$!-f>Rxa  
%       2    0    2*r^2 - 1                sqrt(6) !Cj(A"uqY  
%       2    2    r^2                      sqrt(6) GXb47_b^  
%       3    1    3*r^3 - 2*r              sqrt(8) 5ouQQ)vA  
%       3    3    r^3                      sqrt(8) O4+F^+qN  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) +802`eax  
%       4    2    4*r^4 - 3*r^2            sqrt(10) QJ4AL3 ^6
 
%       4    4    r^4                      sqrt(10) gn5% F5W  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) (:?&G9k "  
%       5    3    5*r^5 - 4*r^3            sqrt(12) < tQc_  
%       5    5    r^5                      sqrt(12) *8!w&ME+.  
%       --------------------------------------------- 'XfgBJF=  
% kJ8vKcc  
%   Example: KVijs1q  
% >iy^$bqF  
%       % Display three example Zernike radial polynomials Vq}r_#!Q  
%       r = 0:0.01:1; Z*bC#s?  
%       n = [3 2 5]; yKDZ+3xK]  
%       m = [1 2 1]; \y*
j4 0  
%       z = zernpol(n,m,r); ;/Q6i  
%       figure oa0X5}D  
%       plot(r,z) _@prmSc  
%       grid on NiTJ}1 l  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') {'IFWD.5  
% u(9X  
%   See also ZERNFUN, ZERNFUN2. GoeIjuELR  
}'`xu9<  
% A note on the algorithm. 3_J>y  
% ------------------------ ="lI i$>O  
% The radial Zernike polynomials are computed using the series $9i9s4u^  
% representation shown in the Help section above. For many special T'R,vxP)\  
% functions, direct evaluation using the series representation can =x4a~=HX  
% produce poor numerical results (floating point errors), because 2Guvze_bU  
% the summation often involves computing small differences between a;&}zcc*  
% large successive terms in the series. (In such cases, the functions #{>uC&
jD  
% are often evaluated using alternative methods such as recurrence t{ H1u  
% relations: see the Legendre functions, for example). For the Zernike 6$z'wy/*  
% polynomials, however, this problem does not arise, because the @^wpAQfd4  
% polynomials are evaluated over the finite domain r = (0,1), and "A7<XN<  
% because the coefficients for a given polynomial are generally all `cO|RhD@  
% of similar magnitude. [aNhP;<  
% l:z};  
% ZERNPOL has been written using a vectorized implementation: multiple h2&y<Eg>  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] Doj(.wm~  
% values can be passed as inputs) for a vector of points R.  To achieve P!>g7X  
% this vectorization most efficiently, the algorithm in ZERNPOL T&4fBMBp,%  
% involves pre-determining all the powers p of R that are required to IozNjII$:.  
% compute the outputs, and then compiling the {R^p} into a single CgoXZX  
% matrix.  This avoids any redundant computation of the R^p, and w-dI<s  
% minimizes the sizes of certain intermediate variables. ;Xh5oB\)W  
% BK*UR+,  
%   Paul Fricker 11/13/2006 ]-EN/V  
@H4wHlb  
<{ #<5 8  
% Check and prepare the inputs: vrO
%XvXW  
% ----------------------------- w06gY  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) bZXlJa`'S  
    error('zernpol:NMvectors','N and M must be vectors.') Wbd_aR (  
end <)J55++  
&z QWIv  
if length(n)~=length(m) 9/Wn!Ld  
    error('zernpol:NMlength','N and M must be the same length.') +WdL  
end * 2%oZXF  
rT<1S?jR  
n = n(:); wNR=?Z~  
m = m(:); |DGCdB|`G  
length_n = length(n);
d1lH[r!Z  
gQ,4xTX  
if any(mod(n-m,2)) ]wR6bEm7  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') T"B8;|  
end }Oh5Nm)  
E]?2!)mgce  
if any(m<0) VHj*aBHB  
    error('zernpol:Mpositive','All M must be positive.') _UaPwJ  
end LiF.w:
}  
!&6-(q9  
if any(m>n) G-sa L*  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') .:SfMr;G  
end MKe *f%  
"|\94  
if any( r>1 | r<0 )
qWkx:-g]  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') 
CCy.  
end lvG3<ls0K$  
DSTx#*  
if ~any(size(r)==1) qm~Kw!kV  
    error('zernpol:Rvector','R must be a vector.') R<t&F\>  
end HK?Foo?  
fA;x{0CAMX  
r = r(:); np= J:v4  
length_r = length(r); Zq9>VqGe  
KM EXT$p  
if nargin==4 @!Y.935/0  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); z{AM2Z  
    if ~isnorm
l; */M.B  
        error('zernpol:normalization','Unrecognized normalization flag.') EyzY2>"^  
    end x[Hhj'  
else xvHOY:  
    isnorm = false; ;,R[]B01u  
end zabw!@]  
>&g2 IvDS  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i^~sn `o  
% Compute the Zernike Polynomials La@\q[U{@  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A<]&JbIt  
t% <pbZO  
% Determine the required powers of r: /Lj%A  
% ----------------------------------- T$f:[ye]Z  
rpowers = []; PZ~`O  
for j = 1:length(n) F1zT )wW  
    rpowers = [rpowers m(j):2:n(j)]; rUGZjLIGqz  
end "Fo  
rpowers = unique(rpowers); rGGS]^  
elNB7%Y/  
% Pre-compute the values of r raised to the required powers, :A,O(   
% and compile them in a matrix:  `NTM%# w  
% ----------------------------- 3#7ENV`  
if rpowers(1)==0 wbn^R'  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZE{aS4c  
    rpowern = cat(2,rpowern{:}); ccIDMJ=2
 
    rpowern = [ones(length_r,1) rpowern]; `4se7{'UK`  
else eUi> Mp  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); NU BpIx&  
    rpowern = cat(2,rpowern{:}); z&\Il#'\m+  
end nYo&x'  
xn0s`I[  
% Compute the values of the polynomials: Ka[Sm|-q  
% -------------------------------------- c8H9_6  
z = zeros(length_r,length_n); )zo#1$C-  
for j = 1:length_n ^U R-#WaQ  
    s = 0:(n(j)-m(j))/2; oNh68
ON:c  
    pows = n(j):-2:m(j); 9;}L{yve  
    for k = length(s):-1:1 ]t
8{)r  
        p = (1-2*mod(s(k),2))* ... m4wPuW  
                   prod(2:(n(j)-s(k)))/          ... ly9x1`?$  
                   prod(2:s(k))/                 ... yd\5Z[iEp  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... f$~ _FX  
                   prod(2:((n(j)+m(j))/2-s(k))); cg>!<T*  
        idx = (pows(k)==rpowers); 3RBpbTNWp  
        z(:,j) = z(:,j) + p*rpowern(:,idx); 7O;BS}Lv=  
    end =ip~J<sw&  
     jAD+:@  
    if isnorm ]b5%?^Z#  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); #RCZA4>  
    end &ryl$!!3H  
end 6{Krw\0  
fU\k?'x_  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) y\r^\ S9%  
%ZERNFUN2 Single-index Zernike functions on the unit circle. &etL&s v  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated =rf)yp-D  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive j3sz*:  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, s0X/1Cq  
%   and THETA is a vector of angles.  R and THETA must have the same '7RR2f>V  
%   length.  The output Z is a matrix with one column for every P-value, nm{'HH-4  
%   and one row for every (R,THETA) pair. 0{^l2?mgSb  
% <=5,(a5g  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike \UkNE5  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) e{qp!N1!  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) y{&{=1#  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 k8S
u/U  
%   for all p. t wa(M?  
% ue^?/{OuT  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 F1{?]>G  
%   Zernike functions (order N<=7).  In some disciplines it is &UJTy'  
%   traditional to label the first 36 functions using a single mode 14@q$}sf  
%   number P instead of separate numbers for the order N and azimuthal /:*R -VdF  
%   frequency M. wHo#%Y,Nmi  
% _^ CQ*+F  
%   Example: ]XpU'/h>q;  
% dcrJ,>i}  
%       % Display the first 16 Zernike functions i"r.>X'Z  
%       x = -1:0.01:1; 8`rAE_n`%  
%       [X,Y] = meshgrid(x,x); y<'2BTf  
%       [theta,r] = cart2pol(X,Y); Z7KB?1{G  
%       idx = r<=1; V;[__w  
%       p = 0:15; gs`27Gih  
%       z = nan(size(X)); 3LmBV\["  
%       y = zernfun2(p,r(idx),theta(idx)); (A
y4B*|!  
%       figure('Units','normalized') >g
S5[`xRE  
%       for k = 1:length(p) ]VHdE_7
)  
%           z(idx) = y(:,k); +i q+  
%           subplot(4,4,k) |+$j(
YuH  
%           pcolor(x,x,z), shading interp ~3*ZG  
%           set(gca,'XTick',[],'YTick',[]) am$-sh72  
%           axis square 7Da^Jv k  
%           title(['Z_{' num2str(p(k)) '}']) gl(6m`a>  
%       end 8YJqM,t5)  
% kHo;9j-U  
%   See also ZERNPOL, ZERNFUN. [w#x5Xs
n  
zYgK$u^H  
%   Paul Fricker 11/13/2006 *fuGVA  
46.q anh  
8
en#PH }  
% Check and prepare the inputs: !z4Hj{A_  
% ----------------------------- 0F;(_2V-  
if min(size(p))~=1 40l#'< y;  
    error('zernfun2:Pvector','Input P must be vector.') tKqCy\-q  
end gYH:EuY,  
XM5;AcD  
if any(p)>35 +_|cZlQ&  
    error('zernfun2:P36', ... (>Q9
jN
W  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... i5~ /+~  
           '(P = 0 to 35).']) @u'27c_<d3  
end GO:1 Z?^  
83 ^,'Z
 
% Get the order and frequency corresonding to the function number: KSpC%_LC  
% ---------------------------------------------------------------- zf!\wY"`  
p = p(:); iYfLo">  
n = ceil((-3+sqrt(9+8*p))/2); Lh9>8@ jf  
m = 2*p - n.*(n+2); o4I!VK(C#s  
;HLMU36q  
% Pass the inputs to the function ZERNFUN: k~s>8N:&G  
% ---------------------------------------- 9|kE
q>d  
switch nargin smLDm  
    case 3 |yl0}.()  
        z = zernfun(n,m,r,theta); +EB,7<5<  
    case 4 G]3ML)l  
        z = zernfun(n,m,r,theta,nflag); Z@aL"@2]a  
    otherwise GzZ|T7fm  
        error('zernfun2:nargin','Incorrect number of inputs.') 5)zh@aJ@  
end |~"A:gf  
>J75T1PH=  
% 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 KL\=:iWA  
function z = zernfun(n,m,r,theta,nflag) L"vG:Mq@D  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 
d~f0]O  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N QO`SnN}  
%   and angular frequency M, evaluated at positions (R,THETA) on the '*{Rn7B5  
%   unit circle.  N is a vector of positive integers (including 0), and 0~L8yMM  
%   M is a vector with the same number of elements as N.  Each element ppo$&W &z  
%   k of M must be a positive integer, with possible values M(k) = -N(k) A5H8+gATK  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, Wes"t}[25  
%   and THETA is a vector of angles.  R and THETA must have the same #Uk6Fmu]  
%   length.  The output Z is a matrix with one column for every (N,M) ]=XL9MI  
%   pair, and one row for every (R,THETA) pair. ]~x/8%e76  
% 28qWC~/9  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike exMPw;8  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >U Ich  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral j tkPi)QR  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, FZ.Yn  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized n_NG~/x  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?;7>`F6ld  
% cw-JGqLx  
%   The Zernike functions are an orthogonal basis on the unit circle. \c^jaK5  
%   They are used in disciplines such as astronomy, optics, and 73Zs/  
%   optometry to describe functions on a circular domain. 6!PX! UkF  
% ^>}[[:(6/  
%   The following table lists the first 15 Zernike functions. FHPZQC8  
% *E q7r>[  
%       n    m    Zernike function           Normalization ;? QAPTz  
%       -------------------------------------------------- <y
aw9k+P  
%       0    0    1                                 1 b0CaoSWo  
%       1    1    r * cos(theta)                    2 8n p>#V  
%       1   -1    r * sin(theta)                    2 ?U[nYp}"v  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ~=]@],{  
%       2    0    (2*r^2 - 1)                    sqrt(3) Gkvd{G?F  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 6#63D>OWp  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) >bP7}T  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) e$|)wOwU  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) PsT v\!  
%       3    3    r^3 * sin(3*theta)             sqrt(8) +GtGyp  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) gG>^h1_o~  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N28?JQha  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) _@?Jx/`;bk  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 90k|u'ikOp  
%       4    4    r^4 * sin(4*theta)             sqrt(10) siZ_JJW  
%       -------------------------------------------------- #EK8Qe_  
% 4T\/wyq0  
%   Example 1: /3%xQK>%  
% | (9FV^_  
%       % Display the Zernike function Z(n=5,m=1) eC:Q)%$%l  
%       x = -1:0.01:1; &8L\FAY0%9  
%       [X,Y] = meshgrid(x,x); m|gd9m$,?  
%       [theta,r] = cart2pol(X,Y); nezbmpL4  
%       idx = r<=1; _jKVA6_E  
%       z = nan(size(X)); n,LKkOG  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); JNCtsfd
 
%       figure epyYo&x}  
%       pcolor(x,x,z), shading interp eV}Tx;1|}  
%       axis square, colorbar -%$ dFq  
%       title('Zernike function Z_5^1(r,\theta)') \>azY g  
% tO;W?g  
%   Example 2: `;GGuJb \  
% '0rwNEg  
%       % Display the first 10 Zernike functions r}Av"  
%       x = -1:0.01:1; T<
GD!j(  
%       [X,Y] = meshgrid(x,x); mQuaO# I,  
%       [theta,r] = cart2pol(X,Y); 4'|:SyOm  
%       idx = r<=1; "$YLU}S9  
%       z = nan(size(X)); 1D DOUV  
%       n = [0  1  1  2  2  2  3  3  3  3]; HKw4}FC*  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; Bq`kVfx  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; Jtk(yp{Zz  
%       y = zernfun(n,m,r(idx),theta(idx)); Lxrn#Z eM  
%       figure('Units','normalized') =%G[vm/-)  
%       for k = 1:10 'mR+W{r  
%           z(idx) = y(:,k); $oH,:x?}  
%           subplot(4,7,Nplot(k)) C{^@.8:  
%           pcolor(x,x,z), shading interp S*:w\nXP~  
%           set(gca,'XTick',[],'YTick',[]) (j"MsCwE  
%           axis square %W@IB8]Vr  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _@76eZd  
%       end 3f8Z?[Bb@  
% Jx?>1q=M  
%   See also ZERNPOL, ZERNFUN2. ,Yz+?SmSZ&  
``Rb-.Fq,  
%   Paul Fricker 11/13/2006 >Sah\u`  
!7?wd^C'f  
NQ=YTRU  
% Check and prepare the inputs: G"wQ(6J@  
% ----------------------------- `^{P,N>X  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ZeV)/g,w  
    error('zernfun:NMvectors','N and M must be vectors.') 6>J#M  
end 4f,x@:Jw  
L,L7WObA  
if length(n)~=length(m) F tjm@:X  
    error('zernfun:NMlength','N and M must be the same length.') GrC")Z|3u  
end net9KX4\  
rfpxE>_|G  
n = n(:); uD3_'a  
m = m(:); ![%,pip2/&  
if any(mod(n-m,2)) m"]ys#  
    error('zernfun:NMmultiplesof2', ... A4h/oMis  
          'All N and M must differ by multiples of 2 (including 0).') ry"zec B  
end 1YL5 ![T  
F{tSfKy2  
if any(m>n) n Lb 9$&  
    error('zernfun:MlessthanN', ... 5Bo)j_Qo  
          'Each M must be less than or equal to its corresponding N.') v^'~
-^s  
end c-d}E!C:  
Xi.?9J
`@  
if any( r>1 | r<0 ) YvX I  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') |e>-v  
end ;"z>p25=T  
X3yr6J[ ^  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (=9&"UH  
    error('zernfun:RTHvector','R and THETA must be vectors.') V;jz0B  
end g!
ww;_  
1O4"MeF  
r = r(:); wP*Z/}Uum+  
theta = theta(:); Pa<X^&  
length_r = length(r); ;\N*iN#K  
if length_r~=length(theta) vKf=t&gqr  
    error('zernfun:RTHlength', ... /+msrrpD  
          'The number of R- and THETA-values must be equal.') $}fA;BP  
end 5@ug1F&  
ig{5]wZ(  
% Check normalization: C+5nft6:  
% -------------------- bE~
lc}%  
if nargin==5 && ischar(nflag) _L":Wux  
    isnorm = strcmpi(nflag,'norm'); FQ%mNowuj  
    if ~isnorm \Z':hw  
        error('zernfun:normalization','Unrecognized normalization flag.')
X[<9+Q
-&  
    end x#D=?/~/Kv  
else "RLb wm~  
    isnorm = false; L%FL{G  
end s?Kn,6Y  
P>|2~YxjU  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9&cZIP  
% Compute the Zernike Polynomials \BL9}5y  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <=Qk^Y2k  
jxvVp*-=<j  
% Determine the required powers of r: 5oS\uX|  
% ----------------------------------- eAMT72_  
m_abs = abs(m); ,"o\_{<z  
rpowers = []; il~,y8WTU{  
for j = 1:length(n) lS^0*(Y  
    rpowers = [rpowers m_abs(j):2:n(j)]; o9i\[Ul  
end OjZ@_V:  
rpowers = unique(rpowers); JFZ p^{  
iweP3u##  
% Pre-compute the values of r raised to the required powers, W=
!f  
% and compile them in a matrix: #82B`y<<y/  
% ----------------------------- $Tg$FfD6&  
if rpowers(1)==0 ;Peyo1  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); KVuv%?  
    rpowern = cat(2,rpowern{:}); Z>l>@wNm  
    rpowern = [ones(length_r,1) rpowern]; |{ kB`  
else :Lx]`dSk  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kbY@Y,:w  
    rpowern = cat(2,rpowern{:}); iX=*qiVX  
end jkq+j^  
|($pXVLH`  
% Compute the values of the polynomials: Q*he%@w  
% -------------------------------------- k;sUDmrO  
y = zeros(length_r,length(n)); k{~5pxd-t  
for j = 1:length(n) O%r<I*T^r  
    s = 0:(n(j)-m_abs(j))/2; PsLCO(26  
    pows = n(j):-2:m_abs(j); "q$M\jK#V  
    for k = length(s):-1:1 }49?Z3  
        p = (1-2*mod(s(k),2))* ... pfT7  
                   prod(2:(n(j)-s(k)))/              ... Ev%\YI!MaY  
                   prod(2:s(k))/                     ... /
y}  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d1-QkW^0y  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); W>~V?%F&'  
        idx = (pows(k)==rpowers); .qZ<ROZ  
        y(:,j) = y(:,j) + p*rpowern(:,idx); <Wpz\U  
    end $)U RY~;i  
     <6@Db$-  
    if isnorm
G.Q+"+*^  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Sz =z TPnO  
    end ,0~=9dR  
end 2,+H;Ypi!  
% END: Compute the Zernike Polynomials \m<*3eS  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fc91D]c  
wNlp4Z'[  
% Compute the Zernike functions: }sFHb[I &  
% ------------------------------ [(C lvGx  
idx_pos = m>0; ?^dyQhb  
idx_neg = m<0; 4 QWHGh"  
[lf[J&}X  
z = y; 5q\]]LV>  
if any(idx_pos) DD1S]m  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); H_{Yr+p  
end Q-\: u~  
if any(idx_neg) 1peN@Yk2W  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); )lZb=t  
end \|Mz'*  
fIu/*PFPVY  
% 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