[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 E3gQ`+wNg?  
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！ -7pZRnv
 

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)  &%INfl>o7.  

Qs*g)Yr  
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。 J;G+6C$:  
!v94FkS>  
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) +r"{$'{^  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. pMKnA.|  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of >7p?^*&7;  
%   order N and frequency M, evaluated at R.  N is a vector of SBgBZm}%  
%   positive integers (including 0), and M is a vector with the 6`+dP"@  
%   same number of elements as N.  Each element k of M must be a V^As@P8,'(  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) F/IXqj  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is xJ:15eDC  
%   a vector of numbers between 0 and 1.  The output Z is a matrix ,dLh`t<\  
%   with one column for every (N,M) pair, and one row for every
K}L-$B*i  
%   element in R. 4%0eX]  
% u`O
xY  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 2I* 7?`  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is esIEi!d  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to /ZUKt  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 L#1YR}m  
%   for all [n,m]. ]<V[H  
% K?8{y
 
%   The radial Zernike polynomials are the radial portion of the ryg1o=1v/  
%   Zernike functions, which are an orthogonal basis on the unit yF8 av=<{  
%   circle.  The series representation of the radial Zernike aqSHo2]DX9  
%   polynomials is g[!t@K  
% }_vE lBh6$  
%          (n-m)/2 ,`ST Va-  
%            __ n*D-01vYP  
%    m      \       s                                          n-2s /'ccFm2  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r @`^+XPK\  
%    n      s=0 3Un q 9  
% J\^ZRu_K  
%   The following table shows the first 12 polynomials. e C?adCb  
% XCc/\  
%       n    m    Zernike polynomial    Normalization $vlq]6V8  
%       --------------------------------------------- R@N I  
%       0    0    1                        sqrt(2) Ri=>evx  
%       1    1    r                           2 rXPq'k'h#-  
%       2    0    2*r^2 - 1                sqrt(6) hy3j8?66  
%       2    2    r^2                      sqrt(6) B
&ItA76  
%       3    1    3*r^3 - 2*r              sqrt(8) aVN
BF`  
%       3    3    r^3                      sqrt(8) 8QDs4Bv|  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~za=yZo7(  
%       4    2    4*r^4 - 3*r^2            sqrt(10) `z=U-v'H)D  
%       4    4    r^4                      sqrt(10) `$vT
GkGpY  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) N}HQvlLkF9  
%       5    3    5*r^5 - 4*r^3            sqrt(12) Cp` [0v~0  
%       5    5    r^5                      sqrt(12) :8\!;!  
%       --------------------------------------------- \x P$m|Y3  
% ~w_4 nE  
%   Example: ,7&`V=C  
% ?f<JwF<  
%       % Display three example Zernike radial polynomials +LuGjDn0  
%       r = 0:0.01:1; +[Cdd{2  
%       n = [3 2 5]; ~47Bbom  
%       m = [1 2 1]; (C>FM8$J  
%       z = zernpol(n,m,r); Y /$`vgqs  
%       figure <ZGEmQ  
%       plot(r,z) `@1y|j:m  
%       grid on l$N b1&  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') ;T0F1  
% D;VQoO  
%   See also ZERNFUN, ZERNFUN2. d$Y3 a^O|  
o8Vtxnkg  
% A note on the algorithm. 3NAU|//J  
% ------------------------ ilJeI@  
% The radial Zernike polynomials are computed using the series Whp;wAz  
% representation shown in the Help section above. For many special .@KpN*`KH  
% functions, direct evaluation using the series representation can G8b/eWtP  
% produce poor numerical results (floating point errors), because [!1z; /  
% the summation often involves computing small differences between 5{/CqUIl  
% large successive terms in the series. (In such cases, the functions D#Fe\8!l  
% are often evaluated using alternative methods such as recurrence db#QA#^S  
% relations: see the Legendre functions, for example). For the Zernike =2!AK[KxX  
% polynomials, however, this problem does not arise, because the U ?'$E\  
% polynomials are evaluated over the finite domain r = (0,1), and XN65bq  
% because the coefficients for a given polynomial are generally all 65X31vU  
% of similar magnitude. &Uq++f6  
% O92a*)  
% ZERNPOL has been written using a vectorized implementation: multiple 64xq@_+  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] 5y-8_)y8o  
% values can be passed as inputs) for a vector of points R.  To achieve banie{ e  
% this vectorization most efficiently, the algorithm in ZERNPOL \)Jv4U\;  
% involves pre-determining all the powers p of R that are required to =Lx*TbsFYt  
% compute the outputs, and then compiling the {R^p} into a single E)
z[@Np  
% matrix.  This avoids any redundant computation of the R^p, and Pl^-]~  
% minimizes the sizes of certain intermediate variables. 7LMad%  
% ;ELQIHnD"  
%   Paul Fricker 11/13/2006 Y8!T4dkn  
uMOm<kn  
Cx$C+  
% Check and prepare the inputs: 6&V4W"k  
% ----------------------------- AdBF$nn[  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @gzm4  
    error('zernpol:NMvectors','N and M must be vectors.') eBiP\  
end \bAsn89O  
gNkx]bm  
if length(n)~=length(m) t95hI DtD  
    error('zernpol:NMlength','N and M must be the same length.') <K4`GT"n  
end R7aS{8nn  

k#"Pv"  
n = n(:); Q|eRek  
m = m(:); q)JG_Y.p  
length_n = length(n); 2}`Q9?  
N_S>%Z+  
if any(mod(n-m,2))  FkJa+ZA  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') [XFZ2'OO  
end 86d*  
BOqq=WY  
if any(m<0) aIaydu+\  
    error('zernpol:Mpositive','All M must be positive.')
0_JbE  
end g*$
0G  
^$3 ~;/|  
if any(m>n) PRmZ3  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') N!6{c
~^  
end d5<@WI:wz  
.UNh\R?r  
if any( r>1 | r<0 ) ~N+lI\K  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') 8>q:Q<BB2  
end ^9T6Ix{=  
6_/oVvd  
if ~any(size(r)==1) $$YLAgO4  
    error('zernpol:Rvector','R must be a vector.') %8iA0t+  
end -,jJ{Y~  

y@g{:/cmO  
r = r(:); rXo2MX@u  
length_r = length(r); ?#]K54?  
1xK'T_[  
if nargin==4 [;B_ENV  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); rC]jz$sle  
    if ~isnorm Cnf;5/  
        error('zernpol:normalization','Unrecognized normalization flag.') -F,o@5W>Y  
    end 2cfzLW(  
else }i_[wq{E&  
    isnorm = false; D[{p~x^  
end |E@Gsw  
Avw"[~
Xd  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u&l2s&i  
% Compute the Zernike Polynomials vj0`[X   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /:dVW"A|  
*|AnL}GJ  
% Determine the required powers of r: GV)#>PL  
% ----------------------------------- [:$j<}UmB  
rpowers = []; [ d<|Cde  
for j = 1:length(n) A`u04Lm7  
    rpowers = [rpowers m(j):2:n(j)]; ;}IF'ANA  
end 77/y{#Sk  
rpowers = unique(rpowers); W#'c6Hq2c  
Y5LESZWo  
% Pre-compute the values of r raised to the required powers, {neE(0c  
% and compile them in a matrix: FsZM_0>/s  
% ----------------------------- f ;|[  
if rpowers(1)==0 !$h%$se  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _%l+v
  
    rpowern = cat(2,rpowern{:}); ;nj'C1  
    rpowern = [ones(length_r,1) rpowern]; t&SJ!>7_c  
else M "p6xp/  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x@.iDP@(  
    rpowern = cat(2,rpowern{:}); G/(oQA  
end )?'sw5C  
M4M 4*o  
% Compute the values of the polynomials: `{I,!to  
% -------------------------------------- H_;Dq*  
z = zeros(length_r,length_n); F']Vg31c  
for j = 1:length_n 8s8q`_.)(  
    s = 0:(n(j)-m(j))/2; 3f's>+,#%  
    pows = n(j):-2:m(j); 3leg,qd  
    for k = length(s):-1:1 #f.@XIt'  
        p = (1-2*mod(s(k),2))* ... )ACa0V>*p  
                   prod(2:(n(j)-s(k)))/          ... v)N6ZOj*C  
                   prod(2:s(k))/                 ... pvy;L[c  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... m|~,#d@  
                   prod(2:((n(j)+m(j))/2-s(k))); R2Tvo?xI7  
        idx = (pows(k)==rpowers); 3D\.Sj%  
        z(:,j) = z(:,j) + p*rpowern(:,idx); DWJ%r"aN  
    end ~'fa,XZ<  
     k;zbq  
    if isnorm w,8 M  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); i40r}?-  
    end hv*n";V   
end /G[y 24 Q  
y(.WK8  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) O-T/H-J`  
%ZERNFUN2 Single-index Zernike functions on the unit circle. QH\*l~;B\  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated (!X:[Ah*$  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive .v1rrH?  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, 5tq$SF42X  
%   and THETA is a vector of angles.  R and THETA must have the same yv
Dzxu  
%   length.  The output Z is a matrix with one column for every P-value, SVq7qc9K?  
%   and one row for every (R,THETA) pair. 3%EwA\V(  
% S"3g 1yU^_  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike DvOg|XUU0  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ?U-p jjM  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ;;nmF#  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 m(OBk;S~   
%   for all p. )1x333.[c  
% LiV]!*9$KG  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 fI$,?>  
%   Zernike functions (order N<=7).  In some disciplines it is >MQW{^  
%   traditional to label the first 36 functions using a single mode *"zE,Bp"  
%   number P instead of separate numbers for the order N and azimuthal APc@1="#J  
%   frequency M. !uO|T'u0a  
% STOE=TC>  
%   Example: ae!_u \$  
% IL
%&*B  
%       % Display the first 16 Zernike functions ~o # NOfYi  
%       x = -1:0.01:1; R:ar85F  
%       [X,Y] = meshgrid(x,x); #='#`5_5  
%       [theta,r] = cart2pol(X,Y); b/5~
VY*T  
%       idx = r<=1; UVI=&y]c,p  
%       p = 0:15; Yhsb$wu  
%       z = nan(size(X)); .FMF0r>l  
%       y = zernfun2(p,r(idx),theta(idx)); HPCA,*YR`  
%       figure('Units','normalized') hcf>J6ZLT  
%       for k = 1:length(p) T2(+
HI2  
%           z(idx) = y(:,k); hR`dRbBi%  
%           subplot(4,4,k) El+]}D"  
%           pcolor(x,x,z), shading interp *M`[YG19!e  
%           set(gca,'XTick',[],'YTick',[]) %xk]y&jv  
%           axis square 5N|77AAxK  
%           title(['Z_{' num2str(p(k)) '}']) "R30oA#m  
%       end }Ql;%7  
% SfR!q4b=  
%   See also ZERNPOL, ZERNFUN.  E;|\?>
 
G:
&Q)_  
%   Paul Fricker 11/13/2006 ?b7vc^E&  
%vThbP#mR|  
K 6Gn  
% Check and prepare the inputs:
(f.A5~e  
% ----------------------------- eBN>|mE4N  
if min(size(p))~=1 >C&!# 3  
    error('zernfun2:Pvector','Input P must be vector.') ,}|V'y  
end tllg$CQ5  
2 rBF<z7  
if any(p)>35 d8r
+UP@#  
    error('zernfun2:P36', ... ypifXO;m7  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... ^IId =V=2  
           '(P = 0 to 35).']) Rd!.8K[  
end gSv<.fD"  
l3MH+o  
% Get the order and frequency corresonding to the function number: i)p__Is  
% ---------------------------------------------------------------- SwL\=nq+~  
p = p(:); WUV Q_<i+  
n = ceil((-3+sqrt(9+8*p))/2); hg&AQk  
m = 2*p - n.*(n+2); _HL3XT  
nbRg<@  
% Pass the inputs to the function ZERNFUN: \G"/Myi  
% ---------------------------------------- q
>X:z0H  
switch nargin d+w<y~\ q  
    case 3 h}d7M55#|  
        z = zernfun(n,m,r,theta); Umd!j,  
    case 4 $R5-JvJJH  
        z = zernfun(n,m,r,theta,nflag); -|T^  
    otherwise VcL  
        error('zernfun2:nargin','Incorrect number of inputs.') -Tt}M#W  
end 0jS/U|0  
x?F{=\z
/o  
% 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 uj@<_|7  
function z = zernfun(n,m,r,theta,nflag) }X?*o`sW  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. LNb![Rq  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N P:TpB6.=q  
%   and angular frequency M, evaluated at positions (R,THETA) on the ]3,0 8JW=  
%   unit circle.  N is a vector of positive integers (including 0), and +g[B &A!d+  
%   M is a vector with the same number of elements as N.  Each element w;(gi  
%   k of M must be a positive integer, with possible values M(k) = -N(k) :&%;s*-9  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, 9|`@czw  
%   and THETA is a vector of angles.  R and THETA must have the same yM2&cMHH~  
%   length.  The output Z is a matrix with one column for every (N,M) ChGM7uu2  
%   pair, and one row for every (R,THETA) pair. m [g}vwS  
% ""d>f4,S  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike v\eBL&WK  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SDwSlwf  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral "=(;l3-o  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E-D5iiF  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized
_XZ=4s  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. B`aAvD`7  
% NjxW A&[ng  
%   The Zernike functions are an orthogonal basis on the unit circle. SS~Q;
9o  
%   They are used in disciplines such as astronomy, optics, and sdWl5 "  
%   optometry to describe functions on a circular domain. xNkY'4%  
%
"BRE0Ir:  
%   The following table lists the first 15 Zernike functions. 
Z]f2&  
% >B  
%       n    m    Zernike function           Normalization OpLSjr  
%       -------------------------------------------------- nS4S[|w"  
%       0    0    1                                 1 8tMt
e!E  
%       1    1    r * cos(theta)                    2 02[II_< 1  
%       1   -1    r * sin(theta)                    2 )mdNvb[*n  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) s>\g03=  
%       2    0    (2*r^2 - 1)                    sqrt(3) pG6-.F;  
%       2    2    r^2 * sin(2*theta)             sqrt(6) BT3O_X`u  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) ntV>m*^  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) =fG8YZ(  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) LDeVNVM  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 63S1ed[  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) :$aW@?zAY  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L@r.R_*H?s  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 17)M.(qmuP  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HW726K*
 
%       4    4    r^4 * sin(4*theta)             sqrt(10) M[u3]dN  
%       -------------------------------------------------- ,~xU>L^  
% ]ECZU  
%   Example 1: ;;!{m(;LS}  
% Rk%M~D*-  
%       % Display the Zernike function Z(n=5,m=1) o$VH,2 QF  
%       x = -1:0.01:1; 3gy;$}Lq T  
%       [X,Y] = meshgrid(x,x); *^6xt7  
%       [theta,r] = cart2pol(X,Y); +c`C9RXk  
%       idx = r<=1; "NH+qQhs  
%       z = nan(size(X)); [!?,TGM}^  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); [9om"'  
%       figure ZHlin#"  
%       pcolor(x,x,z), shading interp Z(mn U;9{v  
%       axis square, colorbar .oj"ru  
%       title('Zernike function Z_5^1(r,\theta)') KHz838C]  
% g/Jj]X#r  
%   Example 2: D{c>i`\G  
% Z'dI!8(Nf  
%       % Display the first 10 Zernike functions 8M+F!1-#  
%       x = -1:0.01:1; :TYzzl43  
%       [X,Y] = meshgrid(x,x); zl 0^EltiU  
%       [theta,r] = cart2pol(X,Y); up3<=u{>  
%       idx = r<=1; MVP)rugU  
%       z = nan(size(X)); \Ntdl:fSw  
%       n = [0  1  1  2  2  2  3  3  3  3]; ({kGK0
 
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ?>jA
rzI  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 50bP&dj&  
%       y = zernfun(n,m,r(idx),theta(idx)); efkie}  
%       figure('Units','normalized') [pgkY!R?)  
%       for k = 1:10 sk6
|_  
%           z(idx) = y(:,k); yn":!4U1  
%           subplot(4,7,Nplot(k)) "rDzrz  
%           pcolor(x,x,z), shading interp [I<'E LX  
%           set(gca,'XTick',[],'YTick',[]) q\y#  
%           axis square T>Rf?%o  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ajW$d!  
%       end FJ,\?ooGf  
% S%s|P=u  
%   See also ZERNPOL, ZERNFUN2. 'A(-MTd%  
m\Fb ,  
%   Paul Fricker 11/13/2006 Ldj^O9p(  
&R FM d=  
us,,W(q  
% Check and prepare the inputs: ~K#_'Ldrd  
% ----------------------------- \3(|c#c  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?CW^*So  
    error('zernfun:NMvectors','N and M must be vectors.') qMdtJ(gq  
end t2%@py*bU  
_KhEwd  
if length(n)~=length(m) 'j<:FUDJ  
    error('zernfun:NMlength','N and M must be the same length.') 0/00W6r0  
end [xs)u3b  
m>-^K  
n = n(:); ^AjYe<RU}  
m = m(:); (=tF2YBV  
if any(mod(n-m,2)) aU]O$Pg{  
    error('zernfun:NMmultiplesof2', ... g yH7((#i  
          'All N and M must differ by multiples of 2 (including 0).') a0/n13c?G  
end t"bPKFRy9E  
iMeRQYW  
if any(m>n) -f;j1bQ  
    error('zernfun:MlessthanN', ... CbH T #  
          'Each M must be less than or equal to its corresponding N.') {{[jC"4AY  
end k1Mxsd  
GKsL~;8"  
if any( r>1 | r<0 ) B/9<b{6  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') JXRf4QmG  
end 0@e}hv;  
>[%.h(h/%  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6[3Ioh  
    error('zernfun:RTHvector','R and THETA must be vectors.') Rr;LV<q+  
end qfP"UAc{/  
D.&eM4MZ  
r = r(:); 5IE+
M  
theta = theta(:); mLk6!&zN  
length_r = length(r); z1SMQLk  
if length_r~=length(theta) )<x;ra^  
    error('zernfun:RTHlength', ... kSDa\l!W]  
          'The number of R- and THETA-values must be equal.') NtA|#"^  
end eYD9#y  
ZaUcP6[h  
% Check normalization: Yr"!&\[oz  
% -------------------- J.e8UQ@=5  
if nargin==5 && ischar(nflag) j#nO6\&o  
    isnorm = strcmpi(nflag,'norm'); x+*L5$;h  
    if ~isnorm "U5Ln2X{J
 
        error('zernfun:normalization','Unrecognized normalization flag.') 0q>NE
<L  
    end K@j^gF/0B  
else mb~=Xyk&  
    isnorm = false; zmL~]!~&  
end DvRA2(M  
S `m-5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y5AXL5  
% Compute the Zernike Polynomials =
6Kv`  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4<3?al&  
Z*vpQBbu  
% Determine the required powers of r: d[>N6?JA/  
% ----------------------------------- 
O2'bNR  
m_abs = abs(m); ll<9f)  
rpowers = []; `3sy>GU?  
for j = 1:length(n) B=Zukg1G  
    rpowers = [rpowers m_abs(j):2:n(j)]; 9OQ0Yc!3  
end UP~WP@0F  
rpowers = unique(rpowers); 7k`*u) Q  
-M>K4*%K  
% Pre-compute the values of r raised to the required powers, S4{Mu(^xT  
% and compile them in a matrix: K5)yM @cq  
% ----------------------------- g@k#J"Q'[  
if rpowers(1)==0 4*D fI  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Dc&9emKI  
    rpowern = cat(2,rpowern{:}); M]4=(Vv+5  
    rpowern = [ones(length_r,1) rpowern]; 7{Ki;1B[w  
else V$-~%7@>;9  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ].k+Nzf_  
    rpowern = cat(2,rpowern{:}); J2oWssw"  
end *b6I%MZn  
^ =/?<C4  
% Compute the values of the polynomials: >TlW]st  
% -------------------------------------- O7'<I|aD  
y = zeros(length_r,length(n)); ;Oi[:Ck  
for j = 1:length(n) |Uy e>%*}4  
    s = 0:(n(j)-m_abs(j))/2; Ha=_u+@  
    pows = n(j):-2:m_abs(j); _\4`  
    for k = length(s):-1:1 n,&/D  
        p = (1-2*mod(s(k),2))* ... Uxk[O  
                   prod(2:(n(j)-s(k)))/              ... hr_9;,EPh  
                   prod(2:s(k))/                     ... :~ZqB\>i  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]< s\V-y  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); j]!7BHC  
        idx = (pows(k)==rpowers); 9k=U0]!ch  
        y(:,j) = y(:,j) + p*rpowern(:,idx); n2xLgK=  
    end (<bm4MPf  
     xb+RRTgj  
    if isnorm `x{.z=xC  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); (b/A|hl  
    end wQD0vsD  
end MG7 ?N #  
% END: Compute the Zernike Polynomials Q)LXL.0h  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d}{LM!s  
Ci7P%]9  
% Compute the Zernike functions: O6m.t%*  
% ------------------------------ {) :%WnM9  
idx_pos = m>0; %]a @A8o0  
idx_neg = m<0; ;~Q  
JQKC;p  
z = y; /~3N@J  
if any(idx_pos) b 0LGH. z4  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); &v5G92  
end v`#j  
if any(idx_neg) "3{#d9Gs  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @uI?  
end w(76H^e  
Vs#"SpH{'  
% 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