[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 /^eemx  
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！ ^?]H$e  

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)  ),,vu  

q/?_djv  
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。 5K{h)* *5  
o7zfD94I  
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) V;$lgTs|'  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. rie1F,  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of SLW1]ZaG  
%   order N and frequency M, evaluated at R.  N is a vector of yDPek*#^"q  
%   positive integers (including 0), and M is a vector with the ZEW`?6  
%   same number of elements as N.  Each element k of M must be a <R2bz1!h.  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) t^q/'9Ai&J  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is IySlu^a  
%   a vector of numbers between 0 and 1.  The output Z is a matrix  M`
bK   
%   with one column for every (N,M) pair, and one row for every t<4+CC2H  
%   element in R. bp }~{]:b  
% f.!cR3XgV  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- s;>jy/o0 s  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is ;lGjj9we>  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to *h`zV<j  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 8$1<N  
%   for all [n,m]. kc}e},k  
% AtSEKpKc  
%   The radial Zernike polynomials are the radial portion of the 8kk$:8  
%   Zernike functions, which are an orthogonal basis on the unit <MoWS9s!yb  
%   circle.  The series representation of the radial Zernike 1Eh(U  
%   polynomials is gP`8hNwR  
% h
W(Mf  
%          (n-m)/2 K?) &8S
 
%            __ NI3_wV  
%    m      \       s                                          n-2s <}G7#xg  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Wfp[)MM;  
%    n      s=0 {@k5e) Q  
% d&F8nBIM5  
%   The following table shows the first 12 polynomials. IG 6yt  
% ]"^U  
%       n    m    Zernike polynomial    Normalization ;:f.a(~c  
%       --------------------------------------------- +Ibcc8Qud  
%       0    0    1                        sqrt(2) IF<pT)  
%       1    1    r                           2 S-*4HV_l  
%       2    0    2*r^2 - 1                sqrt(6) Pr9$(6MX  
%       2    2    r^2                      sqrt(6) PE0A`  
%       3    1    3*r^3 - 2*r              sqrt(8) 4Z,MqG>  
%       3    3    r^3                      sqrt(8) l@%MS\{  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ckS.j)@.c  
%       4    2    4*r^4 - 3*r^2            sqrt(10) voEg[Gg4%I  
%       4    4    r^4                      sqrt(10) O)n"a\LD  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)  [td)v,  
%       5    3    5*r^5 - 4*r^3            sqrt(12) pO[ @2tF  
%       5    5    r^5                      sqrt(12) P;C3{>G9  
%       --------------------------------------------- CNwIM6
t  
% kZfa8wL]P  
%   Example: Z_Qs^e$  
% {1Z8cV  
%       % Display three example Zernike radial polynomials ],V_"\ATD  
%       r = 0:0.01:1; 2r4owB?  
%       n = [3 2 5]; B&3oo  
%       m = [1 2 1]; yY+)IU.  
%       z = zernpol(n,m,r); yPs4S?<s  
%       figure MEf`&<t  
%       plot(r,z) %5Q5xw]w3  
%       grid on >]s\%GO  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') QQ;<L"VW  
% {xJq F4  
%   See also ZERNFUN, ZERNFUN2. /P { Zo  
>zx]% W  
% A note on the algorithm. YW9r'{(D(I  
% ------------------------ 7/C,<$Ep  
% The radial Zernike polynomials are computed using the series P&I%!'<   
% representation shown in the Help section above. For many special J'{69<`Dl  
% functions, direct evaluation using the series representation can q=Xda0c  
% produce poor numerical results (floating point errors), because ;J[ed>v;3  
% the summation often involves computing small differences between KT'Ebb]  
% large successive terms in the series. (In such cases, the functions WRIOjQ:  
% are often evaluated using alternative methods such as recurrence RgTm^?Ex  
% relations: see the Legendre functions, for example). For the Zernike }#'I,?_k  
% polynomials, however, this problem does not arise, because the q+<<Ku(20  
% polynomials are evaluated over the finite domain r = (0,1), and wwmHr!b:6  
% because the coefficients for a given polynomial are generally all n@1;5)&k~  
% of similar magnitude.
n`v;S>aT  
% F/8="dM  
% ZERNPOL has been written using a vectorized implementation: multiple }enS'Fpf`  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] cl\Gh  
% values can be passed as inputs) for a vector of points R.  To achieve -O&u;kh4g  
% this vectorization most efficiently, the algorithm in ZERNPOL \Dn47V{7-  
% involves pre-determining all the powers p of R that are required to ^lw0} i  
% compute the outputs, and then compiling the {R^p} into a single &>%R)?SZh  
% matrix.  This avoids any redundant computation of the R^p, and /i!3Fr"  
% minimizes the sizes of certain intermediate variables. yLFZo"r  
% ioJ~k[T  
%   Paul Fricker 11/13/2006 \}:RG^*m  
3P}^Wu  
/Ko{S_3<I  
% Check and prepare the inputs: H0dHW;U<1  
% ----------------------------- xbTvv>'U  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =P}BAJ  
    error('zernpol:NMvectors','N and M must be vectors.') 6cQ)*,Q  
end 4hQ.RO   
n7A %y2  
if length(n)~=length(m) n "J+?~9  
    error('zernpol:NMlength','N and M must be the same length.') GS*Mv{JJ  
end #&$a7L}  
XT"-  
n = n(:); 9y$"[d27;+  
m = m(:); W]TO%x{  
length_n = length(n); Sd9%tO9mf  
Lh_Q@>k  
if any(mod(n-m,2)) s7 K](T4  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') I\y=uC  
end n){F FM  
rzk-_AFR  
if any(m<0) WVMkLMg8d  
    error('zernpol:Mpositive','All M must be positive.') :~PzTUz  
end )a;ou>u  
xRiWg/Z~  
if any(m>n) "TQ3{=j{  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') bMjE@S&  
end _?m%i]~o
 
Fx]}<IudA^  
if any( r>1 | r<0 ) B098/`r  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') ]ysEj3  
end Ojwhcb^  
M[}aQWT$v  
if ~any(size(r)==1) "I n[= 2w  
    error('zernpol:Rvector','R must be a vector.') <<iwJ U%:  
end &}."sGK  
pf%B  
r = r(:); c 0/vB  
length_r = length(r); W6y-~  
,T<q"d7-#  
if nargin==4 P-25]-  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); iQ7S*s+l5O  
    if ~isnorm x./l27}6  
        error('zernpol:normalization','Unrecognized normalization flag.') 5
p]Cwj<u  
    end NJqjW  
else Zk .V   
    isnorm = false; 3;t{V$  
end ynQ+yW74Z  
s(u,mtG  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }s>.Fh  
% Compute the Zernike Polynomials 4 >2g&);B  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  |tVWmm^m  
*41 2)zEy  
% Determine the required powers of r: ~;ZT<eCIA  
% ----------------------------------- ~bsL W:.'  
rpowers = []; K'tck
J#%  
for j = 1:length(n) !L|PDGD  
    rpowers = [rpowers m(j):2:n(j)]; }]K^b1Fs5  
end VQe@H8>3  
rpowers = unique(rpowers);  >M-ZjT>  
8-clL\bm  
% Pre-compute the values of r raised to the required powers, ;mtv  
% and compile them in a matrix: hY5tBL  
% ----------------------------- 2z+-vT%  
if rpowers(1)==0 [/'=M h  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *v6 j7<H  
    rpowern = cat(2,rpowern{:}); 4Y!_tZ>  
    rpowern = [ones(length_r,1) rpowern]; k}tTl 2  
else 9u%S<F"  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3`3`iN!8\@  
    rpowern = cat(2,rpowern{:}); DwBKqhu  
end -!JnyD  
`j1(GQt  
% Compute the values of the polynomials: 8*[Q{:'.  
% -------------------------------------- blHJhB&8  
z = zeros(length_r,length_n); Uc>$w?oA  
for j = 1:length_n T 7EkRcb  
    s = 0:(n(j)-m(j))/2; JrhDqyk*  
    pows = n(j):-2:m(j); LkA_M'G  
    for k = length(s):-1:1 ?Gr2@,jlD  
        p = (1-2*mod(s(k),2))* ... kntM  
                   prod(2:(n(j)-s(k)))/          ... uU;]/  
                   prod(2:s(k))/                 ... e/*T,ZJ  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... `zmjiC  
                   prod(2:((n(j)+m(j))/2-s(k))); SZ)AO8&  
        idx = (pows(k)==rpowers); p#D
Jow  
        z(:,j) = z(:,j) + p*rpowern(:,idx); >56I`[)  
    end 0'Y'K6hG`  
     8n`O{8:fi  
    if isnorm p.50BcDg  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ?r"QJa>  
    end /Q*o6Gys0  
end [Kc"L+H\  
/rQ[Ik$|  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) R<-u`uXnP  
%ZERNFUN2 Single-index Zernike functions on the unit circle. md.#n  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated =iZj&B X  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive h'D-e5i  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, RqnT*  
%   and THETA is a vector of angles.  R and THETA must have the same q[rBu9  
%   length.  The output Z is a matrix with one column for every P-value, ^P| K2at  
%   and one row for every (R,THETA) pair. )08mG_&atL  
% D;RZE  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike djG
zJLH  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) wy)I6`v  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) |}YeQl  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 .&Rj2d  
%   for all p. COC6H'F  
% 6gSo>F4=  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 "h>B`S  
%   Zernike functions (order N<=7).  In some disciplines it is p~h)@  
%   traditional to label the first 36 functions using a single mode rElbzL"&<  
%   number P instead of separate numbers for the order N and azimuthal 5*
he  
%   frequency M. Cc7YjsRW
 
% ,*Vt53@E  
%   Example: @}Y,A~  
% $2?10}
mrx  
%       % Display the first 16 Zernike functions kOycS  
%       x = -1:0.01:1; fNkN  
%       [X,Y] = meshgrid(x,x); @&7|Laa  
%       [theta,r] = cart2pol(X,Y); SW^/\cJ^  
%       idx = r<=1; X9lh@`3  
%       p = 0:15; ^{vf|zZ _  
%       z = nan(size(X)); in/ITy-  
%       y = zernfun2(p,r(idx),theta(idx)); c,\!<4  
%       figure('Units','normalized') iHp@R-g  
%       for k = 1:length(p) ~sd+ch*  
%           z(idx) = y(:,k); x
K3 xiR  
%           subplot(4,4,k) [^U#ic>cT  
%           pcolor(x,x,z), shading interp m^O9G?  
%           set(gca,'XTick',[],'YTick',[]) `(RQh@H  
%           axis square
;T6x$e  
%           title(['Z_{' num2str(p(k)) '}']) yLo{^4a.  
%       end $h({x~Oj
9  
% w# t[sI"IT  
%   See also ZERNPOL, ZERNFUN. -]uN16\ F  
{APsi7HYBr  
%   Paul Fricker 11/13/2006 T2nbU6H  

KvFGwq"X  
~}c`r4  
% Check and prepare the inputs: qrFC4\q}  
% ----------------------------- Pmj]"7Vd[  
if min(size(p))~=1 .Q,IOCHk  
    error('zernfun2:Pvector','Input P must be vector.') &*}NN5Sv  
end <By6%<JTn  
5 W<\J  
if any(p)>35 !Cm<K*c"&E  
    error('zernfun2:P36', ... 6~ *w~U  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... T:$zNX<f  
           '(P = 0 to 35).']) >bf29tr  
end Q]';1#J\  
He)<S?X-6  
% Get the order and frequency corresonding to the function number: C] <K s  
% ---------------------------------------------------------------- %SRUHx[D  
p = p(:); _]"5]c&*3  
n = ceil((-3+sqrt(9+8*p))/2); |g5B==KI  
m = 2*p - n.*(n+2); KzQ\A!qG  
oKIry 8'^N  
% Pass the inputs to the function ZERNFUN: 't$(Ruw  
% ---------------------------------------- |~mi6 lJ6  
switch nargin #clPao?r  
    case 3 g)*[W>M  
        z = zernfun(n,m,r,theta); D6iHkD
Tg  
    case 4 $73j*@EQA  
        z = zernfun(n,m,r,theta,nflag); DiK@>$v  
    otherwise oYNP,8r^  
        error('zernfun2:nargin','Incorrect number of inputs.') 3K2`1+kBVG  
end ^|@t2Rp@  
)
bZS0f-  
% 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 T]d9tX-  
function z = zernfun(n,m,r,theta,nflag) [z$th  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. EnXNTat})  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1BK-uv:  
%   and angular frequency M, evaluated at positions (R,THETA) on the R]e?<,"X  
%   unit circle.  N is a vector of positive integers (including 0), and fOEw]B#@  
%   M is a vector with the same number of elements as N.  Each element &*\wr}a!  
%   k of M must be a positive integer, with possible values M(k) = -N(k) A+
*M<W  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, X4LU/f<f  
%   and THETA is a vector of angles.  R and THETA must have the same W'x/Kg,w-  
%   length.  The output Z is a matrix with one column for every (N,M) 1
w}%>e
-S  
%   pair, and one row for every (R,THETA) pair. e[f}Lxln  
% '+LbFGrO3  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike uc]]zI6  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 45e-A{G~  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral m[6?v;w  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .}Va~[0j  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized !t/I j~o  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o.IJ4'}aN  
% nen(  
%   The Zernike functions are an orthogonal basis on the unit circle. ygoA/*s  
%   They are used in disciplines such as astronomy, optics, and T$[50~  
%   optometry to describe functions on a circular domain. <7-:flQz~  
% qyzmjV6J2  
%   The following table lists the first 15 Zernike functions. |P!7T.  
% ]E/^(T-O  
%       n    m    Zernike function           Normalization *Ii_dpJ  
%       -------------------------------------------------- J:g4ES-/  
%       0    0    1                                 1 h=tzG KI  
%       1    1    r * cos(theta)                    2 0;9X`z J  
%       1   -1    r * sin(theta)                    2 %0 cFs'  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) i+rh&,  
%       2    0    (2*r^2 - 1)                    sqrt(3) {/|RKV83  
%       2    2    r^2 * sin(2*theta)             sqrt(6) }7)iLfi  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) a`/\0~  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8)
r{oRN  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) H!N`hEEj>  
%       3    3    r^3 * sin(3*theta)             sqrt(8) v+\&8)W=  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) yhTC?sf<  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) JK.<(=y\  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) r xlKoa  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z'hHXSXM  
%       4    4    r^4 * sin(4*theta)             sqrt(10) R3 Zg,YM  
%       -------------------------------------------------- 2iX57-6Ub  
% \3K%>   
%   Example 1: \tCxz(vKz  
% 2g0_[$[m  
%       % Display the Zernike function Z(n=5,m=1) ~;)H |R5kV  
%       x = -1:0.01:1; Pi/V3D)B  
%       [X,Y] = meshgrid(x,x); $0[t<4K`yn  
%       [theta,r] = cart2pol(X,Y); /&>vhpZ}  
%       idx = r<=1; |[+/ ]Y  
%       z = nan(size(X)); :<QmG3F  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); 1r9.JS  
%       figure DH IC:6EY  
%       pcolor(x,x,z), shading interp =PM6:3aKh  
%       axis square, colorbar ,#V}qSKUS  
%       title('Zernike function Z_5^1(r,\theta)') j3t,Cx  
% c9/&A  
%   Example 2: 9YQYg@+R  
% `zoC++hx  
%       % Display the first 10 Zernike functions UlD]!5NO  
%       x = -1:0.01:1; pP|LSrY!  
%       [X,Y] = meshgrid(x,x); ,^n5UA`PK  
%       [theta,r] = cart2pol(X,Y); 96#aGh>  
%       idx = r<=1; wSPwa,)7s  
%       z = nan(size(X)); ^| r6>b  
%       n = [0  1  1  2  2  2  3  3  3  3]; -Cc2|~n  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; /6@$^paB  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; o,yZ1"  
%       y = zernfun(n,m,r(idx),theta(idx)); GOU>j"5}2  
%       figure('Units','normalized') Be
nUyv1d  
%       for k = 1:10 $ISx0l~  
%           z(idx) = y(:,k); d`sIgll&n  
%           subplot(4,7,Nplot(k)) .|c=]_{  
%           pcolor(x,x,z), shading interp aB^`3J  
%           set(gca,'XTick',[],'YTick',[]) njGZ#{"eC  
%           axis square Y+Cqc.JBQ  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) I4Rd2G_  
%       end Mh"vH0\Lj  
% j<PpCL_8%  
%   See also ZERNPOL, ZERNFUN2. zL=PxFw0  
Wu@v%!0  
%   Paul Fricker 11/13/2006 20`QA u)'  
5c 69M5  
Z"N}f ,  
% Check and prepare the inputs: 9iM[3uyO  
% ----------------------------- 3FsX3K,_X  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) xY@<<  
    error('zernfun:NMvectors','N and M must be vectors.') [WUd9fUL  
end 2'-o'z<  
,r,$x4*  
if length(n)~=length(m) n0vhc;d  
    error('zernfun:NMlength','N and M must be the same length.') 19*
D*dkBR  
end V]6CHE:BS  
H|s,;1#  
n = n(:); 4%>2>5  
m = m(:); W;QU6z>  
if any(mod(n-m,2)) 1+9}Xnxb  
    error('zernfun:NMmultiplesof2', ... I9hZ&ed16  
          'All N and M must differ by multiples of 2 (including 0).')
fa2hQJ02  
end 8?G534*r@2  
d#u*NwY}  
if any(m>n) <4RP:2#  
    error('zernfun:MlessthanN', ... 2SJ|$VsLaE  
          'Each M must be less than or equal to its corresponding N.') a=AP*adx8  
end n7iIY4gZ  
PGJkQsp0  
if any( r>1 | r<0 ) P*3PDa@  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0<u
(!iL  
end )5Ofr-Y  
Qkx}A7sK  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qq
r]S^WW  
    error('zernfun:RTHvector','R and THETA must be vectors.') W7?f_E\>W  
end !xz{X?  
b =R9@!  
r = r(:); RZTC+ylj  
theta = theta(:); 3R`eddenF  
length_r = length(r); /<)kI(gf  
if length_r~=length(theta) 'WcP+4c  
    error('zernfun:RTHlength', ... Tu7sA.73k  
          'The number of R- and THETA-values must be equal.') cnR18NK  
end xF7q9'/F  
xv~E
wT)  
% Check normalization: ?f4jqF~Fh  
% -------------------- 2sYOO>  
if nargin==5 && ischar(nflag) f]DO2r  
    isnorm = strcmpi(nflag,'norm'); _aK4[*jnqh  
    if ~isnorm W'f)W4D$6  
        error('zernfun:normalization','Unrecognized normalization flag.') +=g9T`YbE  
    end #6F/:j;  
else OuV f<@a  
    isnorm = false; ~.&2NUr  
end mH5[(?  
V8+8?5'l  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YZQF*fj  
% Compute the Zernike Polynomials %G/j+Pf  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UwVc!Lys  
CK#SD|~:  
% Determine the required powers of r: ;
/)u/[KAv  
% ----------------------------------- P"ATqQG%D  
m_abs = abs(m); uH=^ILN.  
rpowers = []; OVhtU+r  
for j = 1:length(n) b,o@m  
    rpowers = [rpowers m_abs(j):2:n(j)]; RZ GD5`n  
end
kbKGGn4u  
rpowers = unique(rpowers); J>%uak<  
_0 $W;8X  
% Pre-compute the values of r raised to the required powers, jgd^{!  
% and compile them in a matrix: K f}h{X  
% ----------------------------- +Qo]'xKr  
if rpowers(1)==0 +-OnO7f  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); IPEJ7n49  
    rpowern = cat(2,rpowern{:}); 3Q_L6Wj~  
    rpowern = [ones(length_r,1) rpowern]; -:
NFF'  
else J~(M%] &k^  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C|H/x\?zRv  
    rpowern = cat(2,rpowern{:}); ;+Uc}=  
end 8zWKKcf7t  
-naoM  
% Compute the values of the polynomials: Kta7xtu  
% -------------------------------------- OF/DI)j3  
y = zeros(length_r,length(n)); 2j(]Bt:  
for j = 1:length(n) (J,^)!g7  
    s = 0:(n(j)-m_abs(j))/2; /%9CR'%*c  
    pows = n(j):-2:m_abs(j); g_2EH
  
    for k = length(s):-1:1 -lNT"9  
        p = (1-2*mod(s(k),2))* ... G$_=rHt_%  
                   prod(2:(n(j)-s(k)))/              ... TOvpv@?-  
                   prod(2:s(k))/                     ... R<FW?z*  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Z9vJF.clO  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); f{j(H?5  
        idx = (pows(k)==rpowers); -&3mOn& (1  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 3D*vNVI  
    end |uRZT3bGyj  
     p:@JCsH=  
    if isnorm -|aNHZr  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); !3 j@gi2  
    end @}B,l.Tj  
end i!+Wv-  
% END: Compute the Zernike Polynomials |E=8  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'hn=X7  
\yNe5  
% Compute the Zernike functions: g (:%E  
% ------------------------------ Wo[*P\8  
idx_pos = m>0; _b(y"+k  
idx_neg = m<0; *4oj'}  
F^bzE5#  
z = y; ^N`bA8
 
if any(idx_pos) eTrIN,4  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); d9>k5!  
end f+o%N  
if any(idx_neg) @+(TM5Ub  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]BiLLDz(  
end WUnmUW[/  
JE$aYs<(TF  
% 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