[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 aoy Be|H~=  
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！ Vh-8pFt  

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)  #*bmwb*i  

yzN[%/  
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。 uZ6krI  
+W4}&S  
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) NQqw|3  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. 5t:Zp\$+`  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of .{ v$;g  
%   order N and frequency M, evaluated at R.  N is a vector of wlNL;W@w  
%   positive integers (including 0), and M is a vector with the t/[lA=0 )2  
%   same number of elements as N.  Each element k of M must be a 5&8E{YXr  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) %DSr@IX  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is }T[@G6#  
%   a vector of numbers between 0 and 1.  The output Z is a matrix |jIHgm  
%   with one column for every (N,M) pair, and one row for every \9[vi +T  
%   element in R. 2
}&ERW  
% Aits<0  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- b d 1^  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is `%Fp'`ZM$8  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to <ww D*t  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ZSu.0|0#  
%   for all [n,m]. ;VLDXvGd  
% yx8G9SO?  
%   The radial Zernike polynomials are the radial portion of the #R5\k-I  
%   Zernike functions, which are an orthogonal basis on the unit Kxr{Nx  
%   circle.  The series representation of the radial Zernike *}vvS^c0  
%   polynomials is !` 1h *}  
% +,spC`M6h  
%          (n-m)/2 s*GZOz  
%            __ wNi%u{T  
%    m      \       s                                          n-2s S#GxKMO%  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r _&, A  
%    n      s=0 Iynks,ikA  
% k1,k 9BK  
%   The following table shows the first 12 polynomials. jgE{JK\n4  
% Lc~m`=B  
%       n    m    Zernike polynomial    Normalization kZF]BPh.  
%       --------------------------------------------- v:SHaUS  
%       0    0    1                        sqrt(2) ) k/&,J3  
%       1    1    r                           2 br=e+]C Y)  
%       2    0    2*r^2 - 1                sqrt(6) i6paNHi*  
%       2    2    r^2                      sqrt(6) ]-t)wGr  
%       3    1    3*r^3 - 2*r              sqrt(8) +jE)kaV%  
%       3    3    r^3                      sqrt(8) &m--}  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 3}}/,pGSc  
%       4    2    4*r^4 - 3*r^2            sqrt(10) j#f/M3  
%       4    4    r^4                      sqrt(10) )#AYb   
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) L9/'zhiZBx  
%       5    3    5*r^5 - 4*r^3            sqrt(12) ZJ{DW4#t  
%       5    5    r^5                      sqrt(12) "22./vWV|i  
%       --------------------------------------------- <l1/lm<#  
% ])?dqgwa  
%   Example: 
"5eD >!  
% k$v7@|Aw  
%       % Display three example Zernike radial polynomials F,_cci`p  
%       r = 0:0.01:1; DAq H  
%       n = [3 2 5]; |Kd6.Mx  
%       m = [1 2 1]; ai?uJ}  
%       z = zernpol(n,m,r); Q3>qT84  
%       figure "dCIg{j  
%       plot(r,z) 4AhFE@  
%       grid on $MasYi
 
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') q<\r}1Dm  
% @Xoh@:j\  
%   See also ZERNFUN, ZERNFUN2. .U(6])%;@  
-v9(43  
% A note on the algorithm. >> cW0I/`  
% ------------------------ xLIyh7$t  
% The radial Zernike polynomials are computed using the series eQQVfEvS  
% representation shown in the Help section above. For many special Oi?Q^ISxP  
% functions, direct evaluation using the series representation can <@`K^g;W  
% produce poor numerical results (floating point errors), because I&|8 qx#  
% the summation often involves computing small differences between u&p8S#e  
% large successive terms in the series. (In such cases, the functions =3,<(F5Y[  
% are often evaluated using alternative methods such as recurrence _$*-?*V&  
% relations: see the Legendre functions, for example). For the Zernike jEKa9

rt  
% polynomials, however, this problem does not arise, because the 07^.Z[(pCt  
% polynomials are evaluated over the finite domain r = (0,1), and T\wOGaCW  
% because the coefficients for a given polynomial are generally all _x5-!gK  
% of similar magnitude. B#."cg4VR  
% (a!E3y5,  
% ZERNPOL has been written using a vectorized implementation: multiple F@/syX;bb5  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] 8;=?F>]xn  
% values can be passed as inputs) for a vector of points R.  To achieve &h[)nD  
% this vectorization most efficiently, the algorithm in ZERNPOL Ew}G
PJ  
% involves pre-determining all the powers p of R that are required to |QzJHP @  
% compute the outputs, and then compiling the {R^p} into a single aJm5`az)  
% matrix.  This avoids any redundant computation of the R^p, and sUF5Yq:9  
% minimizes the sizes of certain intermediate variables. _BG`!3U+  
% _6FDuCVD-  
%   Paul Fricker 11/13/2006 dY?l oFz  
&\?{%xj  
jM*wm~4>@  
% Check and prepare the inputs: Ct/6<  
% ----------------------------- @W+8z#xr'  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ^?%ThPo_  
    error('zernpol:NMvectors','N and M must be vectors.') JKmd'ZGw  
end "~C\Z} ;  
a
[^dK-  
if length(n)~=length(m) ?{Xp'D\z  
    error('zernpol:NMlength','N and M must be the same length.') unL1/JY z  
end \i~5H]?d  
9LQy0Gx  
n = n(:); gKU*@`6G  
m = m(:); g'L$m|  
length_n = length(n); #tPy0QH  
'iYaA-9j  
if any(mod(n-m,2)) K6<1&  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') r'}#usB(  
end b(ryk./ogx  
[-[|4|CnOm  
if any(m<0) `).;W
 
    error('zernpol:Mpositive','All M must be positive.') $AFiPH9  
end (K"t</]  
'9wD+'c=A  
if any(m>n) qfY=!|O  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') _ )^n[_E  
end Qe/=(P<  
;+qP
V7Z  
if any( r>1 | r<0 ) Dc>)js|"  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') I(WND/&  
end A%M&{S'+|X  
qpc
2;3*7  
if ~any(size(r)==1) P3XP=
G`E  
    error('zernpol:Rvector','R must be a vector.') ~w%Z Bp  
end Q
^V`%+  
y+iuA@WCv  
r = r(:); **YNR:#Y  
length_r = length(r); w
1A&
p  
 K[TMTn  
if nargin==4 aEZn6k1  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); e;}5~dSi  
    if ~isnorm d0Kg,HB  
        error('zernpol:normalization','Unrecognized normalization flag.') z
T+yZA.L  
    end Zr2QeLQC(  
else zck |jhJ6  
    isnorm = false; W%Zyt:H`  
end <^ )0M  
-+I! (?  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vDOeBw=  
% Compute the Zernike Polynomials dl$l5z\  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &akMj@4;R  
U14dQ=~b/  
% Determine the required powers of r: VZlvmN  
% ----------------------------------- 2Nl("e^kJr  
rpowers = []; S(-=I!.G{  
for j = 1:length(n) ^lMnwqx<  
    rpowers = [rpowers m(j):2:n(j)]; +#H8d1^5  
end 01q7n`o#zf  
rpowers = unique(rpowers); J2[QHr&tn  
+[}]a3)  
% Pre-compute the values of r raised to the required powers, .y2<2eW  
% and compile them in a matrix: >qUO_>  
% ----------------------------- '}YXpB  
if rpowers(1)==0 (1}Ndo^;w  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); YL=k&QG  
    rpowern = cat(2,rpowern{:}); /t
v;W  
    rpowern = [ones(length_r,1) rpowern]; hA\8&pI;  
else $xZk{ rK  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); QBn>@jq  
    rpowern = cat(2,rpowern{:}); =nL*/  
end _0j}(Q>|H#  
Zz&i0r  
% Compute the values of the polynomials: ]D-4
8o0  
% -------------------------------------- O}D
8  
z = zeros(length_r,length_n); CC-:dN
b  
for j = 1:length_n =K>Z{%i  
    s = 0:(n(j)-m(j))/2; -5 W0K}  
    pows = n(j):-2:m(j); TgB;R5  
    for k = length(s):-1:1 \j;uN#)28  
        p = (1-2*mod(s(k),2))* ... c_
a$g  
                   prod(2:(n(j)-s(k)))/          ... Y-c~"#  
                   prod(2:s(k))/                 ... ;VFr5.*x  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... o%QQ7S3P  
                   prod(2:((n(j)+m(j))/2-s(k))); yK7>^p}V  
        idx = (pows(k)==rpowers); *f#4S_ws`  
        z(:,j) = z(:,j) + p*rpowern(:,idx); {n{}Y.  
    end G\p; bUF  
     +r!h*4  
    if isnorm l9ihW^  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); ">QY'r  
    end (}}8
DB  
end r"[T9  
)IhY&?jk?  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) _D4}
[`  
%ZERNFUN2 Single-index Zernike functions on the unit circle. L>b,}w  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated %$xFnGb  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive `@D4?8_  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, (B-9M)  
%   and THETA is a vector of angles.  R and THETA must have the same R4(8]oUW  
%   length.  The output Z is a matrix with one column for every P-value, 9p4U\hx  
%   and one row for every (R,THETA) pair. Z :+#3.4$3  
% 1F,_L}=o1s  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike B^?XE(.  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) (`+Z'Y  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) ACYn87tq  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 1W[(+TZ&s  
%   for all p. w0Y%}7  
% ;/T-rVND  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 5,ahKB8  
%   Zernike functions (order N<=7).  In some disciplines it is  vB*oI~<  
%   traditional to label the first 36 functions using a single mode K:@=W1  
%   number P instead of separate numbers for the order N and azimuthal Rk[ * p  
%   frequency M. 3raA^d3!?  
% >NA7,Z2.  
%   Example: [1^wy#  
% Vb9N~v  
%       % Display the first 16 Zernike functions
Ds_ "m,  
%       x = -1:0.01:1; G9inNz*Cx  
%       [X,Y] = meshgrid(x,x); ji -1yX  
%       [theta,r] = cart2pol(X,Y); # :w2Hf6Q  
%       idx = r<=1; =+S3S{\CK  
%       p = 0:15; 9lJj/  
%       z = nan(size(X)); ]/Qy1,  
%       y = zernfun2(p,r(idx),theta(idx));
xN8JrZE&  
%       figure('Units','normalized') )N6[rw<  
%       for k = 1:length(p) D#1~]d  
%           z(idx) = y(:,k); QS*cd|7J;  
%           subplot(4,4,k) Wb)l8[=  
%           pcolor(x,x,z), shading interp i(iP}:3  
%           set(gca,'XTick',[],'YTick',[]) c@7hLUaE2  
%           axis square jsd]7C  
%           title(['Z_{' num2str(p(k)) '}']) p30&JJ!~"  
%       end Z[S+L"0  
% B~zg"
 
%   See also ZERNPOL, ZERNFUN. cCGXB|9fYR  
*;E+9^:V  
%   Paul Fricker 11/13/2006 @[lc0_b  
]NV ]@*`tO  
+JS/Z5dl+}  
% Check and prepare the inputs: |;m`874  
% ----------------------------- &MZy;Sq  
if min(size(p))~=1 EvQwGt1)P  
    error('zernfun2:Pvector','Input P must be vector.') D8AIVK]  
end `{lAhZ5  
QsJW"4d  
if any(p)>35 DE\bYxJ  
    error('zernfun2:P36', ... q,+kPhHEgy  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... xTFrrmxOf  
           '(P = 0 to 35).']) D>b5Uwt  
end (2bZ]  
6y,P4O*q  
% Get the order and frequency corresonding to the function number: w1@b5-  
% ---------------------------------------------------------------- S50x0$%<W  
p = p(:); /dJ)TW(Ir  
n = ceil((-3+sqrt(9+8*p))/2); F0'A/T'ht  
m = 2*p - n.*(n+2); 66@3$P%1p  
!_-sTZ  
% Pass the inputs to the function ZERNFUN: I,4-  
% ---------------------------------------- R
=9~*9  
switch nargin ~J>gVg%66  
    case 3 ?t0zsq  
        z = zernfun(n,m,r,theta); ~@uY?jr  
    case 4 !H|82:`t+  
        z = zernfun(n,m,r,theta,nflag); H]$=*(aje  
    otherwise AOlt,MNpQ  
        error('zernfun2:nargin','Incorrect number of inputs.') ]ZKt1@4AY  
end =PFR{=F  
CnSX  
% 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 ~S^X"8(U  
function z = zernfun(n,m,r,theta,nflag) +-aU+7tu  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y0PGT5].@'  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N UROj9COv  
%   and angular frequency M, evaluated at positions (R,THETA) on the wAu[pWD'6;  
%   unit circle.  N is a vector of positive integers (including 0), and 6i]
Nr@1C  
%   M is a vector with the same number of elements as N.  Each element oT|P1t.  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 'z-;*!A}j  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, (~ ]g,*+  
%   and THETA is a vector of angles.  R and THETA must have the same S~k 0@  
%   length.  The output Z is a matrix with one column for every (N,M) 9P7xoXJ@y  
%   pair, and one row for every (R,THETA) pair. &N"'7bK6n  
% Ns.3s7&  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike vQKn=  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), BEXQTM3])I  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral #ox9&  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c{
mK
ra  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized 2#AeN6\@  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \6SMn6a4  
% ,pyQP^u-  
%   The Zernike functions are an orthogonal basis on the unit circle. b8N[."~:  
%   They are used in disciplines such as astronomy, optics, and ~5r=FF6
  
%   optometry to describe functions on a circular domain. cQ(}^KO  
% 6_R\l@a  
%   The following table lists the first 15 Zernike functions. y@o9~?M  
% W!/vm  
%       n    m    Zernike function           Normalization t1e4H=d>  
%       -------------------------------------------------- \}$*}gW[}  
%       0    0    1                                 1 r]k*7PK  
%       1    1    r * cos(theta)                    2 _m
9~*  
%       1   -1    r * sin(theta)                    2 Ky[bX  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 5-8]N>/b!  
%       2    0    (2*r^2 - 1)                    sqrt(3) (
TEo_BW|+  
%       2    2    r^2 * sin(2*theta)             sqrt(6) F8{ldzh  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) M!N` Orz  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) N@Xg5huO  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 81g9ZV(4  
%       3    3    r^3 * sin(3*theta)             sqrt(8) N9i}p^F<_  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) #_.g2 Y  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >vlQ|/C  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |x &Z~y  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V~OUE]]Q  
%       4    4    r^4 * sin(4*theta)             sqrt(10) 0jR){G9+  
%       -------------------------------------------------- sA/,+aM  
% sS, z
zx<  
%   Example 1: {]&R8?%  
% DpA\r_D  
%       % Display the Zernike function Z(n=5,m=1) <fNGhmL  
%       x = -1:0.01:1; DVObrL)znL  
%       [X,Y] = meshgrid(x,x); Ur^YG4(  
%       [theta,r] = cart2pol(X,Y); L)q`D2|'  
%       idx = r<=1; xME(B@j  
%       z = nan(size(X)); 3PsxOb+  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); a*Rz<08  
%       figure -NAmu97V}  
%       pcolor(x,x,z), shading interp ?E%+}P  
%       axis square, colorbar saatU;V  
%       title('Zernike function Z_5^1(r,\theta)') oG!6}5  
% cX2$kIs;  
%   Example 2: Q"A_bdg5  
% ~\2;i]|  
%       % Display the first 10 Zernike functions 1|W
2s\  
%       x = -1:0.01:1; vx'l>@]k  
%       [X,Y] = meshgrid(x,x); _zdNLwE[  
%       [theta,r] = cart2pol(X,Y); 1{^CfamF  
%       idx = r<=1; s~L`53A
 
%       z = nan(size(X)); ZQ|5W6c  
%       n = [0  1  1  2  2  2  3  3  3  3]; a;%I\w;2  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ;:P7}v fz!  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 8Bq-0=E  
%       y = zernfun(n,m,r(idx),theta(idx)); iBucT"d]  
%       figure('Units','normalized') ze&#i6S  
%       for k = 1:10 ^Sw2xT$p{j  
%           z(idx) = y(:,k); 8`}l\ Y  
%           subplot(4,7,Nplot(k)) R6 ;jY/*#  
%           pcolor(x,x,z), shading interp =tq1
ogE  
%           set(gca,'XTick',[],'YTick',[])  Q.yb4  
%           axis square W;qP=DK2  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) jDkm:X}:  
%       end y>`5Kyj3-@  
% DacN{r"3  
%   See also ZERNPOL, ZERNFUN2. OZ=Cp$  
]a M-p@  
%   Paul Fricker 11/13/2006 a~,Kz\Tt  
?b56AE  
8yn4}`Nc@  
% Check and prepare the inputs: #*^e,FF<  
% ----------------------------- n,CD  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +,z)#  
    error('zernfun:NMvectors','N and M must be vectors.') rS
4%$p"  
end "e@n:N!  
+>!V]S  
if length(n)~=length(m) >zQOK-  
    error('zernfun:NMlength','N and M must be the same length.') f~q4{  
end H;QA@tF>5  
fH[Wkif  
n = n(:); zZ:xEc  
m = m(:); zd.'*Dj  
if any(mod(n-m,2)) +Juh:1H  
    error('zernfun:NMmultiplesof2', ... ?pqU3-knH  
          'All N and M must differ by multiples of 2 (including 0).') FI$XSG  
end a$}N
W.  
tF^g<)S;t  
if any(m>n) W!91tzs:  
    error('zernfun:MlessthanN', ... [X<Pk  
          'Each M must be less than or equal to its corresponding N.') myIe_k,F  
end QjJfE<h  
0\Y1}C  
if any( r>1 | r<0 ) ~@D/A/|  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') pw\P<9e=  
end gqfDacDJL  
^&Q<tN7  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <!F3s`7~  
    error('zernfun:RTHvector','R and THETA must be vectors.') ,:UX<6l R  
end )S*1C@  
&?y7I Pp  
r = r(:); x#r<,uNn,  
theta = theta(:); h)cY])tGtK  
length_r = length(r); R&*@@F-dx  
if length_r~=length(theta) dxCPV6 XI  
    error('zernfun:RTHlength', ... n'M>xq_  
          'The number of R- and THETA-values must be equal.') JhP\u3 QE  
end cDIBDC  
'1-maM\r  
% Check normalization: hF>u)%J/S  
% -------------------- mlB~V3M'G  
if nargin==5 && ischar(nflag) G?xJv`"9iC  
    isnorm = strcmpi(nflag,'norm'); 2.nE k  
    if ~isnorm !),t"Ae?>  
        error('zernfun:normalization','Unrecognized normalization flag.') {[W(a<%bXm  
    end 9->q|E4  
else /8c&Axuv  
    isnorm = false; 6pp$-uS  
end nQ-mmY>#  
N=~~EtX  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |>+uw|LtZ  
% Compute the Zernike Polynomials y' [LNp V  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w7 *V^B  
qyb
xXK:  
% Determine the required powers of r: z_&P?+"Df  
% ----------------------------------- $FX,zC<=  
m_abs = abs(m); 4TZ cc|B5  
rpowers = []; o\!qcoE2W  
for j = 1:length(n) tJ'iX>9I  
    rpowers = [rpowers m_abs(j):2:n(j)]; ?lKhzH.T  
end ?\y%]1  
rpowers = unique(rpowers); Y3rt5\!  
9]7u_  
% Pre-compute the values of r raised to the required powers, D(\$i.,b2  
% and compile them in a matrix: `q_<Im%I  
% ----------------------------- gKi{Y1
 
if rpowers(1)==0 i=rH7k  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,b|-rU\  
    rpowern = cat(2,rpowern{:}); zK;XFN#U^  
    rpowern = [ones(length_r,1) rpowern]; *eb-rhCVn  
else
yWb4Ify  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GLUUY0  
    rpowern = cat(2,rpowern{:}); +@:L|uFU  
end @D[`Oj)  
N`$!p9r  
% Compute the values of the polynomials: ZA820A>2!  
% -------------------------------------- Apfnx7Fv  
y = zeros(length_r,length(n)); K{=PQ XSU  
for j = 1:length(n) 75NRCXh.  
    s = 0:(n(j)-m_abs(j))/2; 2?DRLF]  
    pows = n(j):-2:m_abs(j); OH'ea5xq  
    for k = length(s):-1:1 d%ME@6K)  
        p = (1-2*mod(s(k),2))* ... NX,-;v  
                   prod(2:(n(j)-s(k)))/              ... $k%Z$NSN=  
                   prod(2:s(k))/                     ... $[ z y  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L:R<e#kgS  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); \?lz&<  
        idx = (pows(k)==rpowers); rx!=q8=0R  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Yj3I5RG  
    end `JURQ:l)3^  
     46No%cSiG  
    if isnorm 5?u}#zO  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); =R
sXI&&vh  
    end f.xA_Y>  
end "![L#)"s  
% END: Compute the Zernike Polynomials .*5Z"Q['G  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MesRa(  
lpmJLH.F  
% Compute the Zernike functions: \".^K5Pm  
% ------------------------------ zm#nV Y`  
idx_pos = m>0; #Dy?GB08  
idx_neg = m<0; l#qv 5f  
[V}, tO|  
z = y; Da1aI]{I  
if any(idx_pos) Xm!-~n@-m7  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Wf26  
end '7)"  
if any(idx_neg) ^ c%N/V \  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \>Zvev!s  
end zfI}Q
}p  
H9 tXSh  
% 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