[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 7|vB\[s  
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！ :>G3N+A)  

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)  c/c%-=  

gUrb&#\X  
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。 7%(|)
3"V  
v7l4g&  
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) blZiz2F  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. iSxxy1R  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of +a&-'`7g  
%   order N and frequency M, evaluated at R.  N is a vector of =+{.I,g}g@  
%   positive integers (including 0), and M is a vector with the %r5&CUE5?  
%   same number of elements as N.  Each element k of M must be a sBIqee'T  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) ?6   
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is F9e$2J)C  
%   a vector of numbers between 0 and 1.  The output Z is a matrix 'f`~"@  
%   with one column for every (N,M) pair, and one row for every Z'GOp?  
%   element in R. 0k5Zl?  
% h<*l=`#  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- *DX6m  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ,_T,B'a:  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to O0"i>}g4  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 )_T
[thf]  
%   for all [n,m]. RR:m<9l  
% uNnwz%w  
%   The radial Zernike polynomials are the radial portion of the qH6DZ|  
%   Zernike functions, which are an orthogonal basis on the unit -8tWc]c |4  
%   circle.  The series representation of the radial Zernike rsfA.o  
%   polynomials is 5;V#Z@S  
% IxCEE5+`%  
%          (n-m)/2 Cc]s94  
%            __ d@"eWvnlZ  
%    m      \       s                                          n-2s *&e+z-E  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Bh*~I_Ta>  
%    n      s=0 d{0w4_x  
% w;' F;j~  
%   The following table shows the first 12 polynomials. #hd<5+$U}l  
% 0|{U"\  
%       n    m    Zernike polynomial    Normalization mf@YmKbp  
%       ---------------------------------------------  | qHWM  
%       0    0    1                        sqrt(2) V#!ypX]AB[  
%       1    1    r                           2 44?5]C7  
%       2    0    2*r^2 - 1                sqrt(6) h^>kjMM  
%       2    2    r^2                      sqrt(6) 5vYh~|  
%       3    1    3*r^3 - 2*r              sqrt(8) KLqu[{y.'  
%       3    3    r^3                      sqrt(8) a-Cp"pKlVY  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) pP?J(0Q~  
%       4    2    4*r^4 - 3*r^2            sqrt(10) > Q@*o  
%       4    4    r^4                      sqrt(10) da!N0\.1T  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Rv q_Zsm  
%       5    3    5*r^5 - 4*r^3            sqrt(12) c ~YD|l  
%       5    5    r^5                      sqrt(12) S M987Y!B  
%       --------------------------------------------- @D"#B@j  
% 1elcP`N1  
%   Example: 6FSw_[)  
% wXZ.D}d  
%       % Display three example Zernike radial polynomials M)EKS  
%       r = 0:0.01:1; :M|c,SQK  
%       n = [3 2 5]; gKb4n Nt  
%       m = [1 2 1]; (L|SE4  
%       z = zernpol(n,m,r); 2I?HBz1v  
%       figure AXPUJ?V  
%       plot(r,z) 9qXHdpb#g"  
%       grid on r'&9'rir2  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') /l+x&xYD  
% 0 nWV1)Q0=  
%   See also ZERNFUN, ZERNFUN2. K^{`8E&A  
S;ulJ*qv  
% A note on the algorithm. OM!ES%c,  
% ------------------------ %/etoK  
% The radial Zernike polynomials are computed using the series ~8pf.^,fi  
% representation shown in the Help section above. For many special -ZQ3^'f:0J  
% functions, direct evaluation using the series representation can ZFW}Vnl  
% produce poor numerical results (floating point errors), because #4
na>G|  
% the summation often involves computing small differences between V]k!]  
% large successive terms in the series. (In such cases, the functions tO[+O=d  
% are often evaluated using alternative methods such as recurrence ?]9uHrdsN}  
% relations: see the Legendre functions, for example). For the Zernike 2z0HB+Y}x  
% polynomials, however, this problem does not arise, because the h%=b"x  
% polynomials are evaluated over the finite domain r = (0,1), and N%r}0  
% because the coefficients for a given polynomial are generally all lBG*P>;  
% of similar magnitude. }lpcbm  
% ~O1*]  
% ZERNPOL has been written using a vectorized implementation: multiple #(aROTV5a  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] 3<mv9U(  
% values can be passed as inputs) for a vector of points R.  To achieve ~d5"<`<^o  
% this vectorization most efficiently, the algorithm in ZERNPOL M5ZWcD.1  
% involves pre-determining all the powers p of R that are required to "bf8[D  
% compute the outputs, and then compiling the {R^p} into a single K7f-g]Ibdn  
% matrix.  This avoids any redundant computation of the R^p, and )7j"OE  
% minimizes the sizes of certain intermediate variables. BLaXp0  
% ejr"(m(Xe  
%   Paul Fricker 11/13/2006 GE5@XT  
lh#GD"^(w&  
^G2vA8%  
% Check and prepare the inputs: -S,dG|  
% ----------------------------- /$eEj  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Qgx~'
9
 
    error('zernpol:NMvectors','N and M must be vectors.') e/Q[%y.X  
end Q.yKbO<[  
r`B+ KQ4  
if length(n)~=length(m) U1q$B32  
    error('zernpol:NMlength','N and M must be the same length.') p\-.DRwT`  
end f "&q~V4?  
~!&[;EM<bm  
n = n(:); M9&tys[KX  
m = m(:); oTa! F;I  
length_n = length(n); q!ZmF1sU  
zfo.S[R@  
if any(mod(n-m,2)) 1|. 0]~0  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') |@ldXuYb  
end aS
F&^/j  
=~0XdS/1  
if any(m<0) I^ >zr.zA  
    error('zernpol:Mpositive','All M must be positive.') |Q I3H]T7  
end hPk+vvXtK  
=OHDp7GXO>  
if any(m>n) ix#  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') KdR&OBm  
end kW:!$MX!  
}jk^M|Z"Oz  
if any( r>1 | r<0 ) 4xYo2X,B  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') zp9 ?Ia  
end Wey\GQ`"8  
A!Yqj~  
if ~any(size(r)==1) 3
+$O#>  
    error('zernpol:Rvector','R must be a vector.') 8n:D#`K  
end (gmB$pwS  
mPD'"  
r = r(:); r9t{/})A  
length_r = length(r); W ,U'hk%  
K5ph x  
if nargin==4 .^NV e40O  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); jF5JpyOc  
    if ~isnorm U^YPL,m1  
        error('zernpol:normalization','Unrecognized normalization flag.') gU%GM  
    end Y~:7l5C  
else ""a8eB6  
    isnorm = false; <'B^z0I,  
end 1k~jVC
2VA  
$-0u`=!  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~k%\ LZ3s  
% Compute the Zernike Polynomials 0x &^{P~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PaZYs~EO  
Iymz2  
% Determine the required powers of r: #Nd+X@j  
% ----------------------------------- d{m
0uX56  
rpowers = []; 3TtW2h
>M  
for j = 1:length(n) HkN +:  
    rpowers = [rpowers m(j):2:n(j)]; *o#`lH  
end >6dgf
`U  
rpowers = unique(rpowers); 3OZ}&[3  
[KKo
EZ  
% Pre-compute the values of r raised to the required powers, t(yv   
% and compile them in a matrix: [~o3S$C&7  
% ----------------------------- 7.t$#fzi  
if rpowers(1)==0 ^v'Lu!\f  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,rdM{ r  
    rpowern = cat(2,rpowern{:}); OG+$F  
    rpowern = [ones(length_r,1) rpowern]; H:_`]X"  
else 5 9vGLN!L  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); UGMdWq  
    rpowern = cat(2,rpowern{:}); *Tlv'E.M  
end vKt_z@{{L  
%Fv)$ :b  
% Compute the values of the polynomials: E*l"uV  
% -------------------------------------- 6p@ts`#  
z = zeros(length_r,length_n); 88K*d8m  
for j = 1:length_n g;h&Xkp  
    s = 0:(n(j)-m(j))/2; J\*d4I<(Rt  
    pows = n(j):-2:m(j); upr
Qy<I@  
    for k = length(s):-1:1 $3ILVT  
        p = (1-2*mod(s(k),2))* ... ;gyE5n-{  
                   prod(2:(n(j)-s(k)))/          ... fjFy$NX&>  
                   prod(2:s(k))/                 ... 5-*]PAC  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... "mf;k^sqS  
                   prod(2:((n(j)+m(j))/2-s(k))); ;'p'8lts  
        idx = (pows(k)==rpowers); 7`}z7nk  
        z(:,j) = z(:,j) + p*rpowern(:,idx); +\%zy=  
    end Xwi&uyvU&  
     #L)rz u  
    if isnorm Z7^}G=*  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 1#(1Bs6X  
    end f-<6T
 
end UU;:x"4  
EHZSM5hu  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) !UTJ) &  
%ZERNFUN2 Single-index Zernike functions on the unit circle. lNb\^b  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 3o>t~Sfi  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive ^ne8~
;Q  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, S2)S/ nf  
%   and THETA is a vector of angles.  R and THETA must have the same }U9jsm  
%   length.  The output Z is a matrix with one column for every P-value, Qx;A; n!lw  
%   and one row for every (R,THETA) pair. jvQ"cs$.  
% :!$z1u8R  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike PS6`o  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) J~q+G  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) 8:xo ~Vc  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 l'QR2r7&.  
%   for all p. F6p1 VFs  
% .TC `\mV  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 i1\2lh$  
%   Zernike functions (order N<=7).  In some disciplines it is p( *3U[1  
%   traditional to label the first 36 functions using a single mode {O)&5  
%   number P instead of separate numbers for the order N and azimuthal 1!3kAcBP  
%   frequency M. $4 Uy3C+6  
% mx~sxYa  
%   Example: k5D'RD  
% KU,w9<~i(  
%       % Display the first 16 Zernike functions $x;h[,y   
%       x = -1:0.01:1; Tm\[q  
%       [X,Y] = meshgrid(x,x); BA,6f?ktXS  
%       [theta,r] = cart2pol(X,Y); 4-_lf(#i  
%       idx = r<=1; x[UO1% _o-  
%       p = 0:15; VU9P\|c@<  
%       z = nan(size(X)); #~;8#!X  
%       y = zernfun2(p,r(idx),theta(idx)); x-&v|w'  
%       figure('Units','normalized') v
Lv@Mo  
%       for k = 1:length(p) t[hocl/6  
%           z(idx) = y(:,k); "Q{~Bj~  
%           subplot(4,4,k) `Xdxg\|  
%           pcolor(x,x,z), shading interp A@(h!Cq  
%           set(gca,'XTick',[],'YTick',[]) e"#D){k#  
%           axis square 1m;*fs  
%           title(['Z_{' num2str(p(k)) '}']) Z4ioXl  
%       end !"%sp6Wc  
% l-}5@D[  
%   See also ZERNPOL, ZERNFUN. z\>X[yNpA  
$?AA"Nz  
%   Paul Fricker 11/13/2006 @T1+b"TC  
]31XX=  
9ox|.68q  
% Check and prepare the inputs: h;qy5KS  
% ----------------------------- 8G&+  
if min(size(p))~=1 GA.bRN2CI2  
    error('zernfun2:Pvector','Input P must be vector.') n~u3  
end I0+wczW,^  
o MkY#<Q}  
if any(p)>35 ggc?J<Dv  
    error('zernfun2:P36', ... 3DC%I79  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... VHr7GAmU  
           '(P = 0 to 35).']) 9zYiG3 d  
end QhQ"OVFr#  
9+ l3$  
% Get the order and frequency corresonding to the function number: LG{inhbp  
% ---------------------------------------------------------------- nDB 2>J  
p = p(:); o Xi}@  
n = ceil((-3+sqrt(9+8*p))/2); U!?gdX  
m = 2*p - n.*(n+2); OP`Jc$|6  
nVn|$ "r  
% Pass the inputs to the function ZERNFUN: exO#>th1  
% ---------------------------------------- 7[v@*/W@  
switch nargin dP7Vsa+  
    case 3 92]ZiL?k  
        z = zernfun(n,m,r,theta); m+2`"1IE[  
    case 4 ct4 [b|  
        z = zernfun(n,m,r,theta,nflag); |
W*@}D  
    otherwise Fra>|;do  
        error('zernfun2:nargin','Incorrect number of inputs.') <o!&Kk9  
end UlNfI}#X  
M'zS7=F!:  
% 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 VP%i1|XZJ  
function z = zernfun(n,m,r,theta,nflag) Fu4EEi  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z@,PZ  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >*= =wlOB  
%   and angular frequency M, evaluated at positions (R,THETA) on the 7AO3-; l]  
%   unit circle.  N is a vector of positive integers (including 0), and {[,Wn:  
%   M is a vector with the same number of elements as N.  Each element %x}&=zx0*1  
%   k of M must be a positive integer, with possible values M(k) = -N(k) !/6\m!e|1R  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, UiR,^/8ED  
%   and THETA is a vector of angles.  R and THETA must have the same ,<$YVXe/  
%   length.  The output Z is a matrix with one column for every (N,M) wD6!#t k  
%   pair, and one row for every (R,THETA) pair. FL`1yD^2  
% w3<"g&n|  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ln=>@  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), | 7 m5P
@X  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral sB( `[5I  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J0Hm)*  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized qcTmsMpj  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z .bit_(  
% HkdN=q  
%   The Zernike functions are an orthogonal basis on the unit circle. ,](v?v.[4  
%   They are used in disciplines such as astronomy, optics, and "*w)puD  
%   optometry to describe functions on a circular domain. _,_8X7   
% <AMb!?Obh  
%   The following table lists the first 15 Zernike functions. 4|6&59?pnc  
% L19MP  
%       n    m    Zernike function           Normalization 'h^-t^:<>b  
%       -------------------------------------------------- -@ZzG uS(  
%       0    0    1                                 1 Ht|",1yr+  
%       1    1    r * cos(theta)                    2 #
vj#! 1  
%       1   -1    r * sin(theta)                    2 +urS5c* j  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 3}B5hht"D  
%       2    0    (2*r^2 - 1)                    sqrt(3) ~V2ajM1Z&O  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 5S%C~iB  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ua`6M  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) -BA"3 S  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) -DP8NTl"  
%       3    3    r^3 * sin(3*theta)             sqrt(8) IXZ(]&we
 
%       4   -4    r^4 * cos(4*theta)             sqrt(10) # 0GGc.  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L$1K7<i.  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 2{t)DUs  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) [d4,gEx`Q\  
%       4    4    r^4 * sin(4*theta)             sqrt(10) PwW^y#96  
%       -------------------------------------------------- ,`<^F:xl  
% ':l"mkd+`  
%   Example 1: A\".t=+7  
% (R_CUH  
%       % Display the Zernike function Z(n=5,m=1) e0f":Vct  
%       x = -1:0.01:1; 61/)l0<;  
%       [X,Y] = meshgrid(x,x); ,b<9?PM  
%       [theta,r] = cart2pol(X,Y); h/I@_?k+  
%       idx = r<=1; Abj97S  
%       z = nan(size(X)); 2GSgG.%SSM  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ?"{QK:`  
%       figure },DyU  
%       pcolor(x,x,z), shading interp \F 3C=M@:  
%       axis square, colorbar lPY@{1W  
%       title('Zernike function Z_5^1(r,\theta)') Zc-#;/b3T  
% }{ n\tzR  
%   Example 2: Bk@)b`WR  
% u_N\iCYp  
%       % Display the first 10 Zernike functions aZ`<PdA  
%       x = -1:0.01:1; p?Ed- S  
%       [X,Y] = meshgrid(x,x); `#ul,%  
%       [theta,r] = cart2pol(X,Y); ispkj'  
%       idx = r<=1; pT4qPta,2  
%       z = nan(size(X)); sNm,Fmuz:  
%       n = [0  1  1  2  2  2  3  3  3  3]; CN7k?JO<  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !w q4EV  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; @l'G[jN5  
%       y = zernfun(n,m,r(idx),theta(idx)); "H>.':c"+3  
%       figure('Units','normalized') {3hqp*xl  
%       for k = 1:10 qAqoZMpI|;  
%           z(idx) = y(:,k); eGLO!DdxZ  
%           subplot(4,7,Nplot(k)) vQUZVq5M  
%           pcolor(x,x,z), shading interp <eY%sFq,  
%           set(gca,'XTick',[],'YTick',[]) re;Lg C  
%           axis square CoU3S,;*  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A2y6UzLYD  
%       end 25d\!3#E  
% Pg/T^n&  
%   See also ZERNPOL, ZERNFUN2. 4E
2yH6l  
YMT8p\#rp  
%   Paul Fricker 11/13/2006 8O6_iGTBh  
{O)YwT$`  
%y>+1hakkX  
% Check and prepare the inputs: wa!zv^;N*  
% ----------------------------- wX ,h<\7  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) gmY/STN   
    error('zernfun:NMvectors','N and M must be vectors.') 9`B0fv Q&  
end 5G#$c'A{4  
Jen%}\  
if length(n)~=length(m) id^|\hDR  
    error('zernfun:NMlength','N and M must be the same length.') F?z
<xL@  
end |8H_-n  
ob E:kNE9  
n = n(:); 0#QKVZq2>  
m = m(:); 7{pIPmJ  
if any(mod(n-m,2)) #$FrFU;ZR  
    error('zernfun:NMmultiplesof2', ... !6HuFf  
          'All N and M must differ by multiples of 2 (including 0).') 1 tPVP  
end R:~(Z?  
ocF>LR%P  
if any(m>n)
IU|kNBo  
    error('zernfun:MlessthanN', ... 
O~27/  
          'Each M must be less than or equal to its corresponding N.') kn}zgSO  
end oV9z(!X/  
>SoO4i8  
if any( r>1 | r<0 ) ~^&R#4J  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') C/G]v*MBQ  
end :&qhJtGo  
o)&"Rf  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yHNuU)Ft  
    error('zernfun:RTHvector','R and THETA must be vectors.') O$qtq(Q%  
end ydQ!4  
R,Fgl2  
r = r(:); ([R")~`(l2  
theta = theta(:); _A{+H^,  
length_r = length(r); f#$|t>  
if length_r~=length(theta) dv~pddOs  
    error('zernfun:RTHlength', ... ;F5"}x  
          'The number of R- and THETA-values must be equal.') s\gp5MT  
end R4{-Qv#8 q  
jvHFFSK  
% Check normalization: lpy:3`ti  
% -------------------- 19Ww3PvQ;  
if nargin==5 && ischar(nflag) i%;"[M  
    isnorm = strcmpi(nflag,'norm'); JJ?I>S N!  
    if ~isnorm +j{Y,t{4  
        error('zernfun:normalization','Unrecognized normalization flag.') 0(y:$  
    end GqLq gns  
else Zw{MgoJ0Z  
    isnorm = false; =gjDCx$|  
end CqFeF?xd8h  
8#X_#  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _?`&JF?*  
% Compute the Zernike Polynomials khx.y
Rx  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O9s?h3  
?Go!j?#a  
% Determine the required powers of r: gEwd &J  
% ----------------------------------- VUtXxvH  
m_abs = abs(m); 0[xpEiDx  
rpowers = []; ])w[

  
for j = 1:length(n) /_t|Dry015  
    rpowers = [rpowers m_abs(j):2:n(j)]; \X|sU:g  
end tfYB_N  
rpowers = unique(rpowers); Kqg!,Sn|  
[AS}
RV  
% Pre-compute the values of r raised to the required powers, NHm]`R,  
% and compile them in a matrix: (R*j|HAw`X  
% ----------------------------- l_{8+\`!  
if rpowers(1)==0 YoKs:e2/:  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }Fa%%}  
    rpowern = cat(2,rpowern{:}); ,Na^%A@TJ  
    rpowern = [ones(length_r,1) rpowern]; &nj&:?w  
else DyO$P#~?  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CnISe^h  
    rpowern = cat(2,rpowern{:}); i47j lyH  
end <a(}kk}  
S($Su7g%_  
% Compute the values of the polynomials: ]:ZdV9`  
% -------------------------------------- n
,$z>  
y = zeros(length_r,length(n)); Bv6K$4  
for j = 1:length(n) LWnR?Qve<  
    s = 0:(n(j)-m_abs(j))/2; -WJ?:?'  
    pows = n(j):-2:m_abs(j); ^D{lPu 3  
    for k = length(s):-1:1 ABh&X+YD  
        p = (1-2*mod(s(k),2))* ... #%lo;W~IY  
                   prod(2:(n(j)-s(k)))/              ... (R!hjw~  
                   prod(2:s(k))/                     ... IkPN?N  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... aEt/NwgiQ  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); @? c2)0
 
        idx = (pows(k)==rpowers); RY9V~8|M  
        y(:,j) = y(:,j) + p*rpowern(:,idx); `aC){&
AP(  
    end 5PT5#[  
     9X$ma/P[  
    if isnorm YW/QC'_iC  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); PcT?<HU  
    end tDg}Ys=4K>  
end 9$qm
>,o  
% END: Compute the Zernike Polynomials Az;t"  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r!PpUwod  
#OO>rm$  
% Compute the Zernike functions: PwB1]p=  
% ------------------------------ t. ='/`!N  
idx_pos = m>0; *)M49a*UD  
idx_neg = m<0; v59dh (:`Z  
;r[@v347  
z = y; BZ!v%4^9  
if any(idx_pos) aJ") <_+  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); gKYfQ+  
end %a+mk E  
if any(idx_neg) VHJM*&5  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f y:,_#  
end .Z:zZ_Ev  
,'xYlH3s  
% 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