[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 PX52a[wNDH  
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！ ;^lVIS%&{  

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,8a$Mdqh  

lV]hjt-L 2  
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。 ",(-AU!a)h  
0rxlN [Yp  
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) oJ ,t]e*q=  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. ;Z-xum{  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of n,hHh=.Fu  
%   order N and frequency M, evaluated at R.  N is a vector of oZHsCQ%  
%   positive integers (including 0), and M is a vector with the 0R\lm<&  
%   same number of elements as N.  Each element k of M must be a ]sE~gro  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) IFlDw}M!9  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is R9f*&lj  
%   a vector of numbers between 0 and 1.  The output Z is a matrix DPw"UY:  
%   with one column for every (N,M) pair, and one row for every }\|$8~  
%   element in R. 51;V#@CsQ  
% \`;FL\1+W  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 'b]GcAL  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is UpL?6)  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to U 3aY =8B  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ),v[.9!}:  
%   for all [n,m]. jY-{hW+r  
% HUuL3lYka  
%   The radial Zernike polynomials are the radial portion of the rbS67--]  
%   Zernike functions, which are an orthogonal basis on the unit P6&@fwJ<  
%   circle.  The series representation of the radial Zernike 4`)`%R$  
%   polynomials is 3+@
p
  
% U2\g Kg[-Q  
%          (n-m)/2 G&)A7WaC  
%            __ <]eWr:;  
%    m      \       s                                          n-2s SL zL/5s  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r R(@B4M2  
%    n      s=0 oFB~)}f<v  
% U+CZv1  
%   The following table shows the first 12 polynomials. ?wt%e;  
% } 1^/[?  
%       n    m    Zernike polynomial    Normalization jw?/@(AC6  
%       --------------------------------------------- cq>{  
%       0    0    1                        sqrt(2) |'9%vtbM  
%       1    1    r                           2 j2\bCGY  
%       2    0    2*r^2 - 1                sqrt(6) k"Y9Kc0XoU  
%       2    2    r^2                      sqrt(6) >p [|U`>{  
%       3    1    3*r^3 - 2*r              sqrt(8) -^]8wQU  
%       3    3    r^3                      sqrt(8) Gu}|CFL\  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) oXRmnt  
%       4    2    4*r^4 - 3*r^2            sqrt(10) +ObP[F  
%       4    4    r^4                      sqrt(10) E;.<'t>  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ? acm5dN  
%       5    3    5*r^5 - 4*r^3            sqrt(12) -'
BC*fVr  
%       5    5    r^5                      sqrt(12) /)sA{q 4  
%       --------------------------------------------- "aIiW VQ  
% A&*l
b7X  
%   Example: |b7v(Hx  
% FivgOa  
%       % Display three example Zernike radial polynomials 28[hp[<  
%       r = 0:0.01:1; CE]0OY  
%       n = [3 2 5]; @]P#]%^D2  
%       m = [1 2 1]; p \A^kX^5  
%       z = zernpol(n,m,r); 2a8ZU{wjn  
%       figure {R%v4#nk  
%       plot(r,z) A;7p  
%       grid on fgIzT!fyz  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') 1wP#?p)c  
% =cI -<0QSn  
%   See also ZERNFUN, ZERNFUN2. S&_Z,mT./  
2eo]D?}  
% A note on the algorithm. Vp{! Ft8>  
% ------------------------ xS?[v&"2  
% The radial Zernike polynomials are computed using the series j hf%ze  
% representation shown in the Help section above. For many special /?uA{/8  
% functions, direct evaluation using the series representation can iU"jV*P]  
% produce poor numerical results (floating point errors), because KI)jP((  
% the summation often involves computing small differences between (8qD'(@  
% large successive terms in the series. (In such cases, the functions WP[h@#7<  
% are often evaluated using alternative methods such as recurrence dZcRLLR  
% relations: see the Legendre functions, for example). For the Zernike DjY&)oce(  
% polynomials, however, this problem does not arise, because the -x)Oo`  
% polynomials are evaluated over the finite domain r = (0,1), and xO?w8*d  
% because the coefficients for a given polynomial are generally all |YCGWJaci  
% of similar magnitude. _^K)>  
% Xg
L-t~_  
% ZERNPOL has been written using a vectorized implementation: multiple C^9G \s'  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] |s[kY  
% values can be passed as inputs) for a vector of points R.  To achieve u`?MV2jU2  
% this vectorization most efficiently, the algorithm in ZERNPOL 29{Ep  
% involves pre-determining all the powers p of R that are required to >xrO W`p]  
% compute the outputs, and then compiling the {R^p} into a single q2 K@i*s  
% matrix.  This avoids any redundant computation of the R^p, and JrlDTNJj'  
% minimizes the sizes of certain intermediate variables. #tX\m;  
% S. my" j  
%   Paul Fricker 11/13/2006 _RI`I}&9Z  
c1 aCN  
=)_9GO  
% Check and prepare the inputs: _2uRY  
% ----------------------------- &j=FxF9o  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?pSb,kN}'  
    error('zernpol:NMvectors','N and M must be vectors.') kS_oj  
end ffyDi1Q  
U9^o"vT  
if length(n)~=length(m) ~*"]XE?M  
    error('zernpol:NMlength','N and M must be the same length.') pT3p!/pl3  
end ]^aOYtKX  
#9{N[t  
n = n(:); `;KU^dH  
m = m(:); F<FNZQ@<U  
length_n = length(n); Su"9`  
PF=BXY1<UL  
if any(mod(n-m,2)) Ja1[vO"YgP  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') K @x4>9 3n  
end eh4`a
<gC  
]`@= ;w  
if any(m<0) bu[PQsT  
    error('zernpol:Mpositive','All M must be positive.') _cPGS=Ew  
end `y"(\1  
t~_j+k0K#  
if any(m>n) EFpV  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') XS:W{tL!  
end 7b>FqW)%  
|#_IAN  
if any( r>1 | r<0 ) kpF")0qr  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') $glt%a  
end DJ&ni`  
mEK0ID\  
if ~any(size(r)==1) GxH]  
    error('zernpol:Rvector','R must be a vector.') >i&"{GZ  
end t<8vgdD  
RWyDX_z#<  
r = r(:); ZiR },F/  
length_r = length(r); RP!!6A6:  
4Js2/s  
if nargin==4 8&[Lr o9  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); dyH<D5  
    if ~isnorm 9, A(|g  
        error('zernpol:normalization','Unrecognized normalization flag.') 7Iz%Jty  
    end ;4(ULJ*  
else Kjw==5)}  
    isnorm = false; 6yn34'yw  
end hY*ylzr83  
`.oWmBey\  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Tt;h?  
% Compute the Zernike Polynomials MH wj
J  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x}^:Bs+j  
?=u/&3Cw  
% Determine the required powers of r: 7(8i~}  
% ----------------------------------- &#[w*t(A  
rpowers = []; c#)!-5E~H  
for j = 1:length(n) J\06j%d,  
    rpowers = [rpowers m(j):2:n(j)]; u92);1R  
end X667*L^  
rpowers = unique(rpowers); E&;[E  
[ADSGnw  
% Pre-compute the values of r raised to the required powers, Uz4!O  
% and compile them in a matrix: JD\yl[ac%  
% ----------------------------- p :v'"A}  
if rpowers(1)==0 qGlbO  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Fx@ovI- 5  
    rpowern = cat(2,rpowern{:}); !xE/  
    rpowern = [ones(length_r,1) rpowern]; ]n\Qa  
else Xu.Wdl/{Ra  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); LqYP0%7  
    rpowern = cat(2,rpowern{:}); c[IT?6J4  
end %yyvB5Y^  
FRBW(vKE  
% Compute the values of the polynomials: Ee~<PDzB  
% -------------------------------------- Jn|sS(Q}  
z = zeros(length_r,length_n); wo#,c(  
for j = 1:length_n 61aU~w11a  
    s = 0:(n(j)-m(j))/2;
Kl\g{>{Uz  
    pows = n(j):-2:m(j); "_
f~8f`y  
    for k = length(s):-1:1 nE
&`~  
        p = (1-2*mod(s(k),2))* ... Mzg zOM  
                   prod(2:(n(j)-s(k)))/          ... ~vbyX  
                   prod(2:s(k))/                 ...
N1',`L5  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... 5n0B`A  
                   prod(2:((n(j)+m(j))/2-s(k))); *+i1m`6Q  
        idx = (pows(k)==rpowers); MQ#nP_i  
        z(:,j) = z(:,j) + p*rpowern(:,idx); yv;KKQ
  
    end JI3x^[(Z  
     ?lPn{oB9"  
    if isnorm 7Mj:bm&9  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); P Nf_{4  
    end a23XrX  
end '5)PYjMnH  
)K}-z+$)k  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) '>^!a!<G  
%ZERNFUN2 Single-index Zernike functions on the unit circle. 2CF5qn}T  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated Wt M1nnJp  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive "?M)2,:A  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, Y6E0-bL@Fe  
%   and THETA is a vector of angles.  R and THETA must have the same 1xD?cA\vu  
%   length.  The output Z is a matrix with one column for every P-value, 8yC/:_ML  
%   and one row for every (R,THETA) pair. W9G1w
U  
% h J H  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ujf]@L?  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) 1wg#4h43l  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi)
,Dy9-o  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 98rO]rg  
%   for all p. v8y !zo'  
% 0F%/R^mw  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Y'+mC  
%   Zernike functions (order N<=7).  In some disciplines it is =&"a:l  
%   traditional to label the first 36 functions using a single mode 0B]c`$"aD  
%   number P instead of separate numbers for the order N and azimuthal b:Tv Ta  
%   frequency M. iOB*K)U1  
% ^ AJ_  
%   Example:  WjsmLb:5  
% *AG01# ZF  
%       % Display the first 16 Zernike functions x
qpq|U  
%       x = -1:0.01:1; %%T?LRv  
%       [X,Y] = meshgrid(x,x); .3CQFbHF  
%       [theta,r] = cart2pol(X,Y); &U_T1-UR2  
%       idx = r<=1; H-qbgd6&>R  
%       p = 0:15; pM-mZ/?  
%       z = nan(size(X)); oi7Y
?hTj  
%       y = zernfun2(p,r(idx),theta(idx)); "uP~hFA7M  
%       figure('Units','normalized') =bOMtQ]  
%       for k = 1:length(p) Hbl&)!I  
%           z(idx) = y(:,k); Ov;q]Vn>  
%           subplot(4,4,k) =>-W!Of  
%           pcolor(x,x,z), shading interp e
8 c.&j3m  
%           set(gca,'XTick',[],'YTick',[]) 2Mu3]2>  
%           axis square Rxq4Diq5k  
%           title(['Z_{' num2str(p(k)) '}']) re fAgS!=q  
%       end @GWlo\rM6^  
% #!C|~=  
%   See also ZERNPOL, ZERNFUN. s_P[lbHt
.  
u/apnAW@M  
%   Paul Fricker 11/13/2006 ul{D)zm\D  
Gce[RB:  
}vd72PB  
% Check and prepare the inputs: (@NW
2  
% ----------------------------- a5/r|BiBK  
if min(size(p))~=1 v.53fx  
    error('zernfun2:Pvector','Input P must be vector.') ?L"x>$  
end 2S//5@~_m  
gNC'kCx0c  
if any(p)>35 J4Ca0Ag  
    error('zernfun2:P36', ... }_D{|!!!T  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... 5R6QZVc  
           '(P = 0 to 35).']) 
ep
I~w  
end [W99}bi$  
Pf~0JNnc  
% Get the order and frequency corresonding to the function number: Rl'xEtaN  
% ---------------------------------------------------------------- k- exqM2x=  
p = p(:); ab[V->>%  
n = ceil((-3+sqrt(9+8*p))/2); tILnD1q  
m = 2*p - n.*(n+2); ~FVbL-2  
P]7s1kgaS  
% Pass the inputs to the function ZERNFUN: OI:T#uk5  
% ---------------------------------------- 0zk054F'  
switch nargin 1[-RIN;U8  
    case 3 |!J_3*6$>*  
        z = zernfun(n,m,r,theta); ;x&3tN/I  
    case 4 ?4t~z 1.f  
        z = zernfun(n,m,r,theta,nflag); GL^ j |1  
    otherwise @ev^e!B  
        error('zernfun2:nargin','Incorrect number of inputs.') }OSfC~5P  
end yMOYTN@]  
KP!7hJhw  
% 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 lEpPi@2PK  
function z = zernfun(n,m,r,theta,nflag) 7N0m7SC  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle.
zu1gP/  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Pd(n|t3[8  
%   and angular frequency M, evaluated at positions (R,THETA) on the Si|8xq$E;  
%   unit circle.  N is a vector of positive integers (including 0), and Q
zYaxNGv  
%   M is a vector with the same number of elements as N.  Each element K4^B~0~  
%   k of M must be a positive integer, with possible values M(k) = -N(k) '=IuwCB|;  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ^fM
=|.?  
%   and THETA is a vector of angles.  R and THETA must have the same N]|U-fN\  
%   length.  The output Z is a matrix with one column for every (N,M) qt%/0  
%   pair, and one row for every (R,THETA) pair. &jDRRT3  
% 1'5!")r  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +7K]5p;!~  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), cr{dl\Na  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral 2aQ}| `  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Vb2")+*:  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized cH7D@p}  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J1Y3>40  
% GF
Rd:e  
%   The Zernike functions are an orthogonal basis on the unit circle. ,qlFk|A|  
%   They are used in disciplines such as astronomy, optics, and EtB56FU\  
%   optometry to describe functions on a circular domain. iainl@3Qj  
% L^nS%lm  
%   The following table lists the first 15 Zernike functions. zdDJcdbGd1  
% J~G"D-l<9/  
%       n    m    Zernike function           Normalization .]Z,O>N
 
%       -------------------------------------------------- fGLOXbsA  
%       0    0    1                                 1 [g*]u3s  
%       1    1    r * cos(theta)                    2 bRAf!<3  
%       1   -1    r * sin(theta)                    2 )^'wcBod,  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 9$'Edi=6  
%       2    0    (2*r^2 - 1)                    sqrt(3) iAWoKW  
%       2    2    r^2 * sin(2*theta)             sqrt(6) BcoE&I?[m|  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) +<I1@C  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) /h%MWCZWm^  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) @)8C
 
%       3    3    r^3 * sin(3*theta)             sqrt(8) Jh:-<xy)  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) 1')/BM2  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _'oy C(:}  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) dUJNr_  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h Tn^:%(  
%       4    4    r^4 * sin(4*theta)             sqrt(10) @.i
OFY  
%       -------------------------------------------------- rQ$A|GJL  
% f1>^kl3@P  
%   Example 1:
w02HSQ  
% ^ihXM]1{G  
%       % Display the Zernike function Z(n=5,m=1) `ionMTZY  
%       x = -1:0.01:1; osX23T~-  
%       [X,Y] = meshgrid(x,x); I_ .;nU1xA  
%       [theta,r] = cart2pol(X,Y); 7"JU)@ U]  
%       idx = r<=1; C5RDP~au  
%       z = nan(size(X)); =-pss 47  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); :7>Si%  
%       figure MgMLfgt"V  
%       pcolor(x,x,z), shading interp UmgLH Cz  
%       axis square, colorbar <p0$Q!^dK=  
%       title('Zernike function Z_5^1(r,\theta)')
-{b1&  
% p go\(K0  
%   Example 2: L kq>>?T=  
% c8"I]Qc7  
%       % Display the first 10 Zernike functions Sc~kO4  
%       x = -1:0.01:1; |f?C*t',  
%       [X,Y] = meshgrid(x,x); *E)Y?9u"  
%       [theta,r] = cart2pol(X,Y); e<^4F%jSK  
%       idx = r<=1; T*T.\b  
%       z = nan(size(X)); M<~F>(wxA  
%       n = [0  1  1  2  2  2  3  3  3  3]; G[>-@9_b  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; hy)RV=X  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; #=.h:_9  
%       y = zernfun(n,m,r(idx),theta(idx)); 'qd")  
%       figure('Units','normalized') l*m|b""].u  
%       for k = 1:10 t+(CAP|,  
%           z(idx) = y(:,k); tl^[MLQa  
%           subplot(4,7,Nplot(k)) 0\~Zg  
%           pcolor(x,x,z), shading interp +tN-X'u##  
%           set(gca,'XTick',[],'YTick',[]) `A^} X  
%           axis square YYvs~?bAy  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) z"O-d<U5  
%       end M{4_BQ4$  
% ]Ojt3)fB  
%   See also ZERNPOL, ZERNFUN2. x+TNF>%'D  
hW+Dko(s  
%   Paul Fricker 11/13/2006 j5)qF1W,  
Elq8WtS  
)nk>*oE  
% Check and prepare the inputs: >PJ-Z~O'   
% ----------------------------- ,/ : )FV  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &L?Dogo  
    error('zernfun:NMvectors','N and M must be vectors.') t]o gn(  
end n{yjH*\Z  
M:SxAo-D2  
if length(n)~=length(m) ]\ezES  
    error('zernfun:NMlength','N and M must be the same length.') U+i[r&{gb  
end UiEB?X]-l'  
XHg%X  
n = n(:); #"M Pe4  
m = m(:); t;1NzI$^  
if any(mod(n-m,2)) e.GzGX  
    error('zernfun:NMmultiplesof2', ... Ja&%J:  
          'All N and M must differ by multiples of 2 (including 0).') {LeEnh-  
end ]O\W<'+V  
"%]dC{  
if any(m>n) X m3t xp#  
    error('zernfun:MlessthanN', ... ^Bb_NcU  
          'Each M must be less than or equal to its corresponding N.') GT.^u#r  
end e`rY]X  
FTfA\/tl(;  
if any( r>1 | r<0 ) 7GUJ&U)J  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') !tdfTf$  
end xVyUUzXs  
%E\%nTV  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) yBj)#m5!  
    error('zernfun:RTHvector','R and THETA must be vectors.') B# fzMaC  
end D=>^m=?0  
bH{aI:9Fb
 
r = r(:); ;^*!<F%t9R  
theta = theta(:); h<.[U $,  
length_r = length(r); gNd J=r4  
if length_r~=length(theta) 8TPm[r]  
    error('zernfun:RTHlength', ... ^-!HbbVv  
          'The number of R- and THETA-values must be equal.') |7$h@KF=S  
end ;" *`  
d"UW38K{  
% Check normalization: ,]mwk~HeF  
% -------------------- | dwxe
a  
if nargin==5 && ischar(nflag) U;GoC$b}|  
    isnorm = strcmpi(nflag,'norm'); }$1;<  
    if ~isnorm 2>k)=hl:  
        error('zernfun:normalization','Unrecognized normalization flag.') eeZysCy+DY  
    end vWH>k+9&X  
else Cpcd`y=IN  
    isnorm = false; ([-=NT}Aq  
end `W n5 .V  
u&XkbPZ%4c  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q4iD59yd)S  
% Compute the Zernike Polynomials QP%Fz#u`  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )
^Pvm  
J\'5CG  
% Determine the required powers of r: l%(`<a]VIB  
% ----------------------------------- Xh"iP%  
m_abs = abs(m); })lT fy  
rpowers = []; %UQB?dkf$  
for j = 1:length(n) }%ThnFFBw  
    rpowers = [rpowers m_abs(j):2:n(j)]; ON0+:`3\  
end k)V%.Eobf  
rpowers = unique(rpowers); 5]l7Z35  
O + & xb  
% Pre-compute the values of r raised to the required powers, As
LjU#jn  
% and compile them in a matrix: c/Yi0Rl)  
% ----------------------------- '5/}MMT  
if rpowers(1)==0 B
kxhF  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); DS}rFU  
    rpowern = cat(2,rpowern{:}); u^zitW!X$  
    rpowern = [ones(length_r,1) rpowern]; V55J[s*6!  
else c dbSv=r  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N%A`rY}u  
    rpowern = cat(2,rpowern{:}); 7&1~O#  
end aSkx#mV  
Cw&D}  
% Compute the values of the polynomials: i:M*L< +  
% -------------------------------------- #pQ"+X  
y = zeros(length_r,length(n)); FP'lEp  
for j = 1:length(n) pEj^x[b`^  
    s = 0:(n(j)-m_abs(j))/2; Z/= %J3f  
    pows = n(j):-2:m_abs(j); rHgdvDc  
    for k = length(s):-1:1 .*~u  
        p = (1-2*mod(s(k),2))* ... }K80G~O2<  
                   prod(2:(n(j)-s(k)))/              ... Y\e]2  
                   prod(2:s(k))/                     ... SWjQ.aM  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <yI,cM<c  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); r`R~{;oT  
        idx = (pows(k)==rpowers);
&^n>ZY,  
        y(:,j) = y(:,j) + p*rpowern(:,idx); p?$G>nkdq  
    end PT#eXS9_  
     ~]W[ {3 ;  
    if isnorm Dbdzb m7  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cia-OVX  
    end Kq 4<l  
end :~3{oZGX&  
% END: Compute the Zernike Polynomials H<Kkj  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EKeh>3;?  
d&T6p&V$  
% Compute the Zernike functions: n R\n\
  
% ------------------------------ dH2]ZE0V  
idx_pos = m>0; fb"J Bc}X  
idx_neg = m<0; ::OFW@dS  
xR|eyeR  
z = y; 3> \fP#oQ  
if any(idx_pos) >=~Fo)V!(V  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); M_!u@\  
end =Etwa  
if any(idx_neg) 0^}'+t,lc  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PM-PP8h  
end XK%W^a*x
 
EARfbb"SG7  
% 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