[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 "xMnD(p  
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！ ,a@jg&Mb]  

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)  MY"8!  

]B3\IT  
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。 N@r`+(_t  
aX{i   
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) |QcE5UC  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. LOX}  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 3yp?|>e  
%   order N and frequency M, evaluated at R.  N is a vector of ,ctm;T1H+  
%   positive integers (including 0), and M is a vector with the 5KIlU78  
%   same number of elements as N.  Each element k of M must be a j8#xNA  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) ! uX0G4  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is FQW{c3%qZ  
%   a vector of numbers between 0 and 1.  The output Z is a matrix vnOl-`Z ~  
%   with one column for every (N,M) pair, and one row for every O/1:2G/`  
%   element in R. d:SLyFD$q  
% 5nSi2
9C  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- q9iHJ'lMD*  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is (HD8Mm  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to Tw+V$:$$  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 $$f89, h  
%   for all [n,m]. 2SV}mK U  
% b^q8s4(  
%   The radial Zernike polynomials are the radial portion of the {sb2r%U!+  
%   Zernike functions, which are an orthogonal basis on the unit ngI3.v/R  
%   circle.  The series representation of the radial Zernike ZW)_dg9  
%   polynomials is |(77ao3  
% 7wB*@a-  
%          (n-m)/2 L2<IG)oXU  
%            __ eb#p-=^KP  
%    m      \       s                                          n-2s tS:/:0HnA)  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r k=M_2T'  
%    n      s=0 }K'gjs/N;  
% y13Y,cz~B  
%   The following table shows the first 12 polynomials. @:%p#$V  
% :HW\awv  
%       n    m    Zernike polynomial    Normalization J_eu(d[9  
%       --------------------------------------------- #WqpU.
  
%       0    0    1                        sqrt(2) $z48~nu@j  
%       1    1    r                           2 _6!@>`u~  
%       2    0    2*r^2 - 1                sqrt(6) 9^<Y~rkm  
%       2    2    r^2                      sqrt(6) t$zeBOI)  
%       3    1    3*r^3 - 2*r              sqrt(8) Y7_2pGvZ  
%       3    3    r^3                      sqrt(8) Ehw2o-s^  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) "HwSW4a]  
%       4    2    4*r^4 - 3*r^2            sqrt(10) -.!+i8d>  
%       4    4    r^4                      sqrt(10) J_`a}ox  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) W:hg*0z-*  
%       5    3    5*r^5 - 4*r^3            sqrt(12) Ygs:Ox"[-G  
%       5    5    r^5                      sqrt(12) \qZ>WCp>r  
%       --------------------------------------------- ?4%@"49n X  
% Uy98lv  
%   Example: 2m{d>  
% T:=ST3#m  
%       % Display three example Zernike radial polynomials e]DuV)k&  
%       r = 0:0.01:1; G<:gNWXd\  
%       n = [3 2 5]; a>8&B  
%       m = [1 2 1]; cf+EQY  
%       z = zernpol(n,m,r); [M/0Qx[,  
%       figure ,+GS.]8<  
%       plot(r,z) v9 *WM3  
%       grid on ;>*Pwz`~jT  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') $e
iW2@  
% !=bGU=^  
%   See also ZERNFUN, ZERNFUN2. nI7v:h4  
G(;R+%pu  
% A note on the algorithm. ?d' vIpzO!  
% ------------------------ 1EAQ ~S!2  
% The radial Zernike polynomials are computed using the series WG]`Sy  
% representation shown in the Help section above. For many special /CXQ&nwY9=  
% functions, direct evaluation using the series representation can 2|\WaH9P  
% produce poor numerical results (floating point errors), because Iq@&?,W  
% the summation often involves computing small differences between )o`[wq  
% large successive terms in the series. (In such cases, the functions Y. Uca<{.[  
% are often evaluated using alternative methods such as recurrence ~+S,`8-P  
% relations: see the Legendre functions, for example). For the Zernike dJ.up*aR  
% polynomials, however, this problem does not arise, because the zGaqYbQD  
% polynomials are evaluated over the finite domain r = (0,1), and Oj8xc!d'  
% because the coefficients for a given polynomial are generally all r)|6H"n#]S  
% of similar magnitude. ;Z.sK-NJ4  
% j.kv!;Rj=  
% ZERNPOL has been written using a vectorized implementation: multiple wJF(&P  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] jp880}  
% values can be passed as inputs) for a vector of points R.  To achieve k@P?,r  
% this vectorization most efficiently, the algorithm in ZERNPOL ~B\O{5W  
% involves pre-determining all the powers p of R that are required to $bFH%EA.
 
% compute the outputs, and then compiling the {R^p} into a single hV}C.- 6h  
% matrix.  This avoids any redundant computation of the R^p, and &fe67#0r)  
% minimizes the sizes of certain intermediate variables. 4L/nEZ!Nsu  
% Xmw%f[Xl  
%   Paul Fricker 11/13/2006 {J*|)-eAw  
4 z^7T  
}6"l`$=Ev  
% Check and prepare the inputs:  N<~LgH  
% ----------------------------- Hq aay  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vD76IG jm  
    error('zernpol:NMvectors','N and M must be vectors.') {sW>J0  
end -unQ4G  
w*\JA+  
if length(n)~=length(m) s0m k<>z  
    error('zernpol:NMlength','N and M must be the same length.') %$'Z"njO&  
end a[jNT$8  
#_Z)2ESX  
n = n(:); c)Ne/E{!0  
m = m(:); !.{"Ttn;s  
length_n = length(n); y7vA[us  
>Z>sR0s7  
if any(mod(n-m,2)) :Q ?p^OC  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') L KLLBrm:  
end {~`{bnx^]7  
V3<#_:;  
if any(m<0) C9 j{:&  
    error('zernpol:Mpositive','All M must be positive.') g>QN9v})  
end tuJ{IF  
Ym?VF{e,  
if any(m>n) {wD:!\5  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') S5\KI+;PW  
end xoQ(GrBY  
LKgo(&mY  
if any( r>1 | r<0 ) pP%9MSCi  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') Mtc-  
end S59^$  
 \|C*b<
 
if ~any(size(r)==1) U~w8yMxX  
    error('zernpol:Rvector','R must be a vector.') NInZ~4:  
end p\Fxt1Y@X  
NH4T*R)Vz  
r = r(:); MS\?+8|SV(  
length_r = length(r); @M6F?;  
=1\mLI}@  
if nargin==4 xy4P_  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ,I
H~  
    if ~isnorm +1^L35\@  
        error('zernpol:normalization','Unrecognized normalization flag.') F{Oaxn  
    end HM
hdK  
else |>b;M,`OO  
    isnorm = false; h:i FLSf  
end :r7!HG_  
e?pQuF~  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =s9*=5r8  
% Compute the Zernike Polynomials xT-`dS0u  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% h)^|VM   
x,HD,VQR/  
% Determine the required powers of r: Zr(eH2}0D  
% ----------------------------------- >J#/IjCW  
rpowers = []; sl:1P^b  
for j = 1:length(n) JAy-N bb
\  
    rpowers = [rpowers m(j):2:n(j)]; BS%pS(  
end LtPaTe  
rpowers = unique(rpowers); jp|*kBDq\  
N*+WGsxl$z  
% Pre-compute the values of r raised to the required powers, c]"w0a-`^@  
% and compile them in a matrix: z pDc~ebh  
% ----------------------------- i(kx'ua?  
if rpowers(1)==0 _{n4jdw%(  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
]|u7P{Z"R  
    rpowern = cat(2,rpowern{:}); ~V0 GRPnI  
    rpowern = [ones(length_r,1) rpowern]; 'K\H$<CJ  
else LI<Emez  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Gd$!xN%O  
    rpowern = cat(2,rpowern{:}); JvNd'u)Z<  
end  FL b  
< <F  
% Compute the values of the polynomials: 7=s0Pm  
% -------------------------------------- @d:GtAW  
z = zeros(length_r,length_n); pu_?)U  
for j = 1:length_n @$nh6l>i  
    s = 0:(n(j)-m(j))/2; ^^< C9  
    pows = n(j):-2:m(j); w`v`aw]  
    for k = length(s):-1:1 FAX[|p  
        p = (1-2*mod(s(k),2))* ... y}?PyPz  
                   prod(2:(n(j)-s(k)))/          ... y`z4S,  
                   prod(2:s(k))/                 ... {R"mvB`  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... D5:|CMQ  
                   prod(2:((n(j)+m(j))/2-s(k))); Xka+1c  
        idx = (pows(k)==rpowers); "H=N>=g0E  
        z(:,j) = z(:,j) + p*rpowern(:,idx); F6}YM|  
    end 2:<H)oB  
     ) I(9qt>Y  
    if isnorm JJ'f\f9  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 9|Ylv:sR  
    end 5,-:31(j\  
end ~NLthZ(O  
\.>7w 1p  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) vif)g6,  
%ZERNFUN2 Single-index Zernike functions on the unit circle. 4S4gK   
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /t! 5||G  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive 9XKqsvdS  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, n5;@}Rai  
%   and THETA is a vector of angles.  R and THETA must have the same :{VXDT"  
%   length.  The output Z is a matrix with one column for every P-value, C%{2 sMJz  
%   and one row for every (R,THETA) pair. (nXnP{yb
 
% ^Wn+G8n  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike !
aKu9SR^e  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) IP@3R(DS%  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) sKJr34  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 1Kr$JIcd  
%   for all p. =qpGAv_#  
% t0r0{:  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 gsUF\4A(J  
%   Zernike functions (order N<=7).  In some disciplines it is
fK *l?Hr  
%   traditional to label the first 36 functions using a single mode C`.eJF  
%   number P instead of separate numbers for the order N and azimuthal U_;="y  
%   frequency M.
Gt _tL%  
% )l.uj  
%   Example: -~4r6ZcA  
% ew~?&=  
%       % Display the first 16 Zernike functions
|"S#uJW  
%       x = -1:0.01:1; R MOs1<D  
%       [X,Y] = meshgrid(x,x); *|y$z+g/  
%       [theta,r] = cart2pol(X,Y); sINf/mv+  
%       idx = r<=1; ,>za|y<n  
%       p = 0:15; bsr]Z&9rrk  
%       z = nan(size(X)); Vh2uzG  
%       y = zernfun2(p,r(idx),theta(idx)); _0FMwC#DY  
%       figure('Units','normalized') ,zr,>^v  
%       for k = 1:length(p) ZJc{P5a1J  
%           z(idx) = y(:,k); iH@u3[w  
%           subplot(4,4,k) l'@!'  
%           pcolor(x,x,z), shading interp WPAUY<6f  
%           set(gca,'XTick',[],'YTick',[]) %NJ0Y(:9(  
%           axis square *uf)t,%  
%           title(['Z_{' num2str(p(k)) '}']) w!v^6[!  
%       end (6NDY5h~=n  
% |)" y  
%   See also ZERNPOL, ZERNFUN. cruBJZr*  
hdcB*j?4  
%   Paul Fricker 11/13/2006 i+_=7(
e  
6xwjKh:9  
UNwjx7usD  
% Check and prepare the inputs: 1]5k lJ  
% ----------------------------- %<+uJ'pj  
if min(size(p))~=1 '+ZJf&Ox  
    error('zernfun2:Pvector','Input P must be vector.') g|->W]q@;  
end @"A 5yD5
  
^Ifm1$X}  
if any(p)>35 a5s
aN5)H  
    error('zernfun2:P36', ... <DPRQhNW]  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... tm1&OY  
           '(P = 0 to 35).']) e`H>}O/ai  
end
r_T"b  
3e47UquZ  
% Get the order and frequency corresonding to the function number: 9I2&Vx=DSt  
% ---------------------------------------------------------------- rXT?w]4  
p = p(:); %&VI-7+K
 
n = ceil((-3+sqrt(9+8*p))/2); m@+QC$6S  
m = 2*p - n.*(n+2); 6-tIe_5  
S[yrGX8lu  
% Pass the inputs to the function ZERNFUN: KK1?!7  
% ---------------------------------------- X%znNx  
switch nargin 6}Tftw$0z  
    case 3 7FyE?  
        z = zernfun(n,m,r,theta); lMh>eX  
    case 4 d$x vEm  
        z = zernfun(n,m,r,theta,nflag); ='w 2"4  
    otherwise C4d'z(<  
        error('zernfun2:nargin','Incorrect number of inputs.') zVf79UrK  
end Z<^EZX3N  
1KGf @u%-1  
% 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 !l0"nPM=  
function z = zernfun(n,m,r,theta,nflag) 0A~UuH0.  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cN?/YkW?]  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SiaW; ks  
%   and angular frequency M, evaluated at positions (R,THETA) on the D}X6I#U'/  
%   unit circle.  N is a vector of positive integers (including 0), and sR83e|4I  
%   M is a vector with the same number of elements as N.  Each element H lM7^3(&  
%   k of M must be a positive integer, with possible values M(k) = -N(k)
E@xrn+L>-  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, ezY^T  
%   and THETA is a vector of angles.  R and THETA must have the same Gos#=H  
%   length.  The output Z is a matrix with one column for every (N,M) %xG
<hNw/  
%   pair, and one row for every (R,THETA) pair. yvzH}$!]  
% t2OBVzK  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 0%[IG$u)|  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), EmrkaV-?k  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral 7)[Ve1;/N  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^,
^MW  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ^xNzppz`]C  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [
wm0a4fg  
% M
&29J  
%   The Zernike functions are an orthogonal basis on the unit circle. 7=u Gf$/  
%   They are used in disciplines such as astronomy, optics, and V>Z4gZp5sc  
%   optometry to describe functions on a circular domain. NyRa.hgZ;  
% ~CV.Ci.dG  
%   The following table lists the first 15 Zernike functions. PWx%~U.8~j  
% (BxmV1  
%       n    m    Zernike function           Normalization Zr2T^p5u  
%       -------------------------------------------------- !vJ$$o6#  
%       0    0    1                                 1 :7"Q  
%       1    1    r * cos(theta)                    2 Ly^bP>2i  
%       1   -1    r * sin(theta)                    2 oOvQAW8`  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 0x5Ax=ut  
%       2    0    (2*r^2 - 1)                    sqrt(3) l=l$9H,  
%       2    2    r^2 * sin(2*theta)             sqrt(6) =. \hCgq  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) b-#{O=B  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ,<#Rk'y$  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Keo<#Cc?  
%       3    3    r^3 * sin(3*theta)             sqrt(8) uo2k  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) ilJ`_QN  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n YUFRV$  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ~@l4T_,k  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gYrB@W;2  
%       4    4    r^4 * sin(4*theta)             sqrt(10) BgT ^  
%       -------------------------------------------------- CR9wp]-Vd  
% : Bo
  
%   Example 1: =<Sn&uL  
% =JfwHFHd#  
%       % Display the Zernike function Z(n=5,m=1) h0k?(O  
%       x = -1:0.01:1; }}]Lf3;  
%       [X,Y] = meshgrid(x,x); =:w,wI.  
%       [theta,r] = cart2pol(X,Y); V~/-e- 9u  
%       idx = r<=1; OOXSJE1  
%       z = nan(size(X)); u*=^>LD  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); EZI#CLT[  
%       figure P)f8lU^z  
%       pcolor(x,x,z), shading interp cf"&22TQ+Z  
%       axis square, colorbar aAGV\o{^  
%       title('Zernike function Z_5^1(r,\theta)') in
O;Uwlv  
% -`\^_nVC  
%   Example 2: &Lt$~}*&6  
% JZxA:dg l  
%       % Display the first 10 Zernike functions ?uL-qsU  
%       x = -1:0.01:1; +3-5\t`  
%       [X,Y] = meshgrid(x,x); H9ES|ZJs  
%       [theta,r] = cart2pol(X,Y); bK0(c1*a[e  
%       idx = r<=1; 9^n0<(99b  
%       z = nan(size(X)); e>e${\=,  
%       n = [0  1  1  2  2  2  3  3  3  3]; j?|V
x'  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; j][&o-Ev  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; )mwwceN  
%       y = zernfun(n,m,r(idx),theta(idx)); 1irSI,j%z  
%       figure('Units','normalized') Yu)GV7\2  
%       for k = 1:10 N_B^k8j  
%           z(idx) = y(:,k); G,?a8(  
%           subplot(4,7,Nplot(k)) weu+$Kr  
%           pcolor(x,x,z), shading interp 'R-\6;3E>9  
%           set(gca,'XTick',[],'YTick',[]) j[dZ*Jr_  
%           axis square WZ,k][~  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) aBaiX
v/*  
%       end \ Xh C  
% hO.b?>3NL  
%   See also ZERNPOL, ZERNFUN2. LFi* O&  
U7n#TPet  
%   Paul Fricker 11/13/2006 q\i&ERr  
7"aN7Q+EbI  
g7hI9(8+  
% Check and prepare the inputs: ,|VLOY^  
% ----------------------------- ub>:dNBN  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aLm~.@Q  
    error('zernfun:NMvectors','N and M must be vectors.') Pm2LB<qS  
end ai?J
  
&)t
v
4L&  
if length(n)~=length(m) o*7NyiJ@z  
    error('zernfun:NMlength','N and M must be the same length.') P#!gP
3  
end #Ox@[Z1I  
 C&qo$C  
n = n(:); W>+`e]z  
m = m(:); U.~G{H`G,u  
if any(mod(n-m,2)) rWNe&gFM  
    error('zernfun:NMmultiplesof2', ... iVeH\a  
          'All N and M must differ by multiples of 2 (including 0).') <h#W
*a  
end ZoJqJWsd  
GQYn |vm  
if any(m>n) Oj%5FUP~[%  
    error('zernfun:MlessthanN', ... 7z3t
DE[#  
          'Each M must be less than or equal to its corresponding N.') w<!,mL5 N  
end  9Ca0Tu  
?nL
,
Otz  
if any( r>1 | r<0 ) )mN/e+/Lu  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') aizws[C  
end _>`9]6\&  
Yh!k uS#<  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) [6g$;SicT  
    error('zernfun:RTHvector','R and THETA must be vectors.') Dl0{pGK~  
end zq$L[X  
PPG+~.7  
r = r(:); ]CcRI|g}  
theta = theta(:); G+2fmVB*X  
length_r = length(r); V73/q  
if length_r~=length(theta) aLW3Ub{h  
    error('zernfun:RTHlength', ... ^vSSG5 :  
          'The number of R- and THETA-values must be equal.') YGQ/zB^Pj  
end (?(gz#-  
K>~YO~~  
% Check normalization: v8C($<3%  
% -------------------- G!C }ULq  
if nargin==5 && ischar(nflag) 7>MG8pf3a  
    isnorm = strcmpi(nflag,'norm'); |/xA5_-N  
    if ~isnorm $i<+O,@-  
        error('zernfun:normalization','Unrecognized normalization flag.') b5%<},ySq  
    end sx7zRw >X  
else "v0bdaQH3  
    isnorm = false; l SKq  
end fH9"sBiO  
1]0;2THx  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;m.6 ~A  
% Compute the Zernike Polynomials 0'A"]6  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q4!\^HwQ  
V,& O
O  
% Determine the required powers of r: 9v
DOSwU*  
% ----------------------------------- qo\9,<  
m_abs = abs(m); \@h$|nb  
rpowers = []; jzpDKc%  
for j = 1:length(n) jp4-w(  
    rpowers = [rpowers m_abs(j):2:n(j)]; /L(}VJg-  
end ()Wu_Q  
rpowers = unique(rpowers); c]U+6JH  
"B+F6  
% Pre-compute the values of r raised to the required powers, o>+mw|{  
% and compile them in a matrix: +CSv@ />3  
% ----------------------------- Oop6o$k  
if rpowers(1)==0 .C+(E@eyA  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NB^Al/V@  
    rpowern = cat(2,rpowern{:});  yoe@]c=  
    rpowern = [ones(length_r,1) rpowern]; N5K2Hv<"  
else <?DI!~  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); UB8n,+R  
    rpowern = cat(2,rpowern{:}); m&q0 _nay  
end S"
^'ksL\  
_ 3>E+9TQ  
% Compute the values of the polynomials: (s|WmSQ  
% -------------------------------------- Fx1FxwIJ  
y = zeros(length_r,length(n)); ;{R;lF,  
for j = 1:length(n) @}PX:*c  
    s = 0:(n(j)-m_abs(j))/2; f9y+-GhaD  
    pows = n(j):-2:m_abs(j); Dz2Z (EXI~  
    for k = length(s):-1:1 Z'5&N5hx  
        p = (1-2*mod(s(k),2))* ... $7Z-Nn38  
                   prod(2:(n(j)-s(k)))/              ... @\oZ2sB  
                   prod(2:s(k))/                     ... 3gJZlH5IR  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... T <k;^iqR  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); >fT%CGLC0  
        idx = (pows(k)==rpowers); 74 )G.!  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Vep41\g^  
    end M5:*aCN6P  
     e~'z;%O~  
    if isnorm B2LXF3#/  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); g;[t1
~oF  
    end hc0$mit  
end o
F_rC[  
% END: Compute the Zernike Polynomials km^ZF<.@  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >@?mP$;=  
.tHc*Eh  
% Compute the Zernike functions: sy4Nm0m  
% ------------------------------ Tw*p^rU  
idx_pos = m>0; >mMfZvxl%  
idx_neg = m<0; b *0uxvLu  
{^;7
DV:  
z = y; "szJ[ _B  
if any(idx_pos) UpSJ%%.n  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G^VOA4  
end <u# 7K\:  
if any(idx_neg) &IRM<A!8  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;0?OBUDO  
end Ml?KnSb  
'YbE%i}  
% 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