[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 e[Ul"pMvS`  
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！ lVKF^-i  

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)  s~'C'B?  

z+`)|c4-  
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。 =*G'.D /*  
Tp.iRFFkP  
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) 2J;CiEB  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. Mb!^_cS(  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 1MSu]) W  
%   order N and frequency M, evaluated at R.  N is a vector of p$<qT^]&  
%   positive integers (including 0), and M is a vector with the TD9`SSpP  
%   same number of elements as N.  Each element k of M must be a z#/*LP#oY  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) |0mI3r  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is )T_#X!  
%   a vector of numbers between 0 and 1.  The output Z is a matrix 1=.?KAXR  
%   with one column for every (N,M) pair, and one row for every ,:{+ H  
%   element in R. *RM'0[1F4  
% 3!W&J  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- '+wTrW m~j  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is z w9r0bG  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to _Y=yR2O  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 xx8na8  
%   for all [n,m]. EUqG"h5#A{  
% E]Q)pZ{Jb  
%   The radial Zernike polynomials are the radial portion of the 0rUf'S ?K  
%   Zernike functions, which are an orthogonal basis on the unit * vD<6qf  
%   circle.  The series representation of the radial Zernike aoS1Yt'@  
%   polynomials is G.T1rUh=  
% .EwK>ro4  
%          (n-m)/2 7a net  
%            __ ?CDq^)T[  
%    m      \       s                                          n-2s ],|B4\b;  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r PMcyQ2R->  
%    n      s=0 5c8x: e@  
% (#qVtN`t  
%   The following table shows the first 12 polynomials. " cg>g/  
% r MlNp?{_  
%       n    m    Zernike polynomial    Normalization 7(Kc9sJC%%  
%       --------------------------------------------- WTx;,TNG  
%       0    0    1                        sqrt(2) 1\uS~RR  
%       1    1    r                           2 5JXLfYTUI  
%       2    0    2*r^2 - 1                sqrt(6) 7j8_O@_  
%       2    2    r^2                      sqrt(6) =UY@,*q:c  
%       3    1    3*r^3 - 2*r              sqrt(8) dGe  
%       3    3    r^3                      sqrt(8) ;U&VPIX$  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) X*Zv,Wm  
%       4    2    4*r^4 - 3*r^2            sqrt(10) =B1!em|  
%       4    4    r^4                      sqrt(10) ix;8S=eP~{  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) ?%(*bRV -  
%       5    3    5*r^5 - 4*r^3            sqrt(12) f`e.c_n(  
%       5    5    r^5                      sqrt(12) g:yK/1@Hk}  
%       --------------------------------------------- z?xd\x  
% ;f Gi5=-  
%   Example: VbjW$?  
% 2Z~ofrj  
%       % Display three example Zernike radial polynomials 9tO_hhEQ@  
%       r = 0:0.01:1; 26A#X  
%       n = [3 2 5]; ZUycJ-[  
%       m = [1 2 1]; Yi`.zm  
%       z = zernpol(n,m,r); [Wc 73-  
%       figure Nsq%b?#  
%       plot(r,z) 4~4Hst#^  
%       grid on *
O~D lf  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') uY,FugWbl  
% mwxJ#  
%   See also ZERNFUN, ZERNFUN2. vq}V0- <  
aF:
LL>H  
% A note on the algorithm. novZ<?7 5;  
% ------------------------ DVd/OU  
% The radial Zernike polynomials are computed using the series Dts:$PlCk  
% representation shown in the Help section above. For many special W2RS G~|  
% functions, direct evaluation using the series representation can P\JpE  
% produce poor numerical results (floating point errors), because PLD!BD  
% the summation often involves computing small differences between CJ_B.  
% large successive terms in the series. (In such cases, the functions N@}U;x}  
% are often evaluated using alternative methods such as recurrence >qCT#TY  
% relations: see the Legendre functions, for example). For the Zernike SDkN  
% polynomials, however, this problem does not arise, because the 4.8,&{w<m  
% polynomials are evaluated over the finite domain r = (0,1), and Rjf|  
% because the coefficients for a given polynomial are generally all 7_RU*U^  
% of similar magnitude. PA E)3  
% r"+ WUU  
% ZERNPOL has been written using a vectorized implementation: multiple 'py k  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] 4?Io@[7A)  
% values can be passed as inputs) for a vector of points R.  To achieve vzL>ZBe
Z  
% this vectorization most efficiently, the algorithm in ZERNPOL x{m)I<.:  
% involves pre-determining all the powers p of R that are required to ,3wo  
% compute the outputs, and then compiling the {R^p} into a single f]Vz!hM~  
% matrix.  This avoids any redundant computation of the R^p, and 99 ["I:  
% minimizes the sizes of certain intermediate variables. z,)Fvs4U.  
% HwHI$IB  
%   Paul Fricker 11/13/2006 v[x`I;  
Hl#o& *Ui"  
&IcDUr]L  
% Check and prepare the inputs: XP'KgTF  
% ----------------------------- -Jhf]  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I ?1E}bv  
    error('zernpol:NMvectors','N and M must be vectors.') $hL0/T-m  
end 0t) IWD  
X_h+\ 7N>  
if length(n)~=length(m) L@/+u+j0  
    error('zernpol:NMlength','N and M must be the same length.') >,DR{A2hSB  
end )YZ41K5N  
2dC)%]aLme  
n = n(:); L2 I/h`n"  
m = m(:); '&"7(8E} *  
length_n = length(n); vjHbg#0%  
k
z#DBh!&  
if any(mod(n-m,2)) BjA|H  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') txi m|)  
end 8w{V[@QLn  
k=LY 6  
if any(m<0) ?B-aj  
    error('zernpol:Mpositive','All M must be positive.') {S|uQgs6j  
end eN/Jb;W  
4EOu)#  
if any(m>n) PgVM>_nHk  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') !
l'Zar  
end C. Sb4i*  

8}U/fQ~  
if any( r>1 | r<0 ) 7B'0(70  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') }RD,JgmV  
end R)#"Ab Z'  
S`NH6?/uH  
if ~any(size(r)==1) 5vS'Qhc  
    error('zernpol:Rvector','R must be a vector.') !XK p_v  
end 
=14pEe  
3mx7[Q  
r = r(:); FCg,p2  
length_r = length(r); y_\d[  
[9w8oNg0  
if nargin==4 3c6<JW  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); 2^|*M@3r  
    if ~isnorm A0'Yfuie  
        error('zernpol:normalization','Unrecognized normalization flag.') k(T/ydrw  
    end ^f
4qs  
else vwP83b0ov"  
    isnorm = false; akaQ6DIdG  
end AR&u9Y)I  
,#s}nJ4  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z{%h6""  
% Compute the Zernike Polynomials %R}}1
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P,,@&* :  
_v_ak4m>  
% Determine the required powers of r: XrYz[h*)!  
% ----------------------------------- /H}83 C  
rpowers = []; S]k<Ixvf  
for j = 1:length(n) La9dFe-uu{  
    rpowers = [rpowers m(j):2:n(j)]; nL\BB&  
end ).xQ~A\.  
rpowers = unique(rpowers); KpSHf9!&[  
'DVP
x%p  
% Pre-compute the values of r raised to the required powers, !sUo+Y  
% and compile them in a matrix: ^ng?+X>mP  
% ----------------------------- $LKn
iK  
if rpowers(1)==0 q4Q1Ib-<2  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 75gE>:f  
    rpowern = cat(2,rpowern{:}); P
.LMu  
    rpowern = [ones(length_r,1) rpowern]; ao Y"uT+  
else 0&Zm3(}  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ]Rz]"JZ\S  
    rpowern = cat(2,rpowern{:}); $n!saPpxS  
end _8kZ>w(L  
GBN^ *I  
% Compute the values of the polynomials: 1H%LUA  
% -------------------------------------- Fj|C+;Q.  
z = zeros(length_r,length_n); 7)z^*;x  
for j = 1:length_n EZa
o\,t  
    s = 0:(n(j)-m(j))/2; jeC3}BL}  
    pows = n(j):-2:m(j); CsXIq.9  
    for k = length(s):-1:1 {Dqf.w>t  
        p = (1-2*mod(s(k),2))* ... 8IbHDDS
  
                   prod(2:(n(j)-s(k)))/          ... ,LX]  
                   prod(2:s(k))/                 ... _z~|*7@  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... tyNT1F{  
                   prod(2:((n(j)+m(j))/2-s(k))); a*!wiTGf  
        idx = (pows(k)==rpowers); ^lf{IM-Y  
        z(:,j) = z(:,j) + p*rpowern(:,idx); BG/M3  
    end ;i>|5tEy  
     dFK/  
    if isnorm ~[t%g9  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); yY[N\*P  
    end =rGjOb3+  
end ]^p6dbzWe  
YR^J7b\  
% EOF zernpol

function z = zernfun2(p,r,theta,nflag) y}Oc^Fc  
%ZERNFUN2 Single-index Zernike functions on the unit circle. sFuB[ JJ}  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 6=0"3%jn@  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive $i;%n1VBg  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, Z.ky=vCt  
%   and THETA is a vector of angles.  R and THETA must have the same }w}2'P'T  
%   length.  The output Z is a matrix with one column for every P-value, |VQ17*4ff1  
%   and one row for every (R,THETA) pair. HN]roSt~  
% wsYvbI!  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike ~7IXJeon  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) tN&4t xB  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) w9Bbvr6  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 YzNSZJPD  
%   for all p. ,4M7:=gf  
% XvETys@d  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 '@i0~  
%   Zernike functions (order N<=7).  In some disciplines it is B+
:/!_  
%   traditional to label the first 36 functions using a single mode Ii FeO  
%   number P instead of separate numbers for the order N and azimuthal dZK/v  
%   frequency M. 7&;M"?m&  
% fP# !ywgr%  
%   Example: LX2rg\a+%  
% F!(Vg  
%       % Display the first 16 Zernike functions ^YiGvZJ  
%       x = -1:0.01:1; K[r<-6TS  
%       [X,Y] = meshgrid(x,x); 3}~.#`QeY  
%       [theta,r] = cart2pol(X,Y); %? -E)n[  
%       idx = r<=1; cNOtfn6?F  
%       p = 0:15; j
whc;y  
%       z = nan(size(X)); d5jZ?
  
%       y = zernfun2(p,r(idx),theta(idx)); /enlkZx=8  
%       figure('Units','normalized') BQTZt'p  
%       for k = 1:length(p) 3Z/_}5%"  
%           z(idx) = y(:,k); RC?gozBFJ  
%           subplot(4,4,k) :+#$=4  
%           pcolor(x,x,z), shading interp W>W b|W  
%           set(gca,'XTick',[],'YTick',[]) v,]-;V~<  
%           axis square AH-B/c5  
%           title(['Z_{' num2str(p(k)) '}']) In13crr4!  
%       end y``[CBj  
% U1nObA  
%   See also ZERNPOL, ZERNFUN. c[VVCN8dA  
t@r>GHO  
%   Paul Fricker 11/13/2006 F/p/&9  
x9\z^GU%H  
3ScOJo  
% Check and prepare the inputs: pY.R?\  
% ----------------------------- +;,65j+n   
if min(size(p))~=1 .Nk'y
ow  
    error('zernfun2:Pvector','Input P must be vector.') 4Ys\<\~d  
end WAq!_xE  
+q*WY*gX  
if any(p)>35 vo(riHH  
    error('zernfun2:P36', ... Z;/QB6|%  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !U::kr=t  
           '(P = 0 to 35).']) '_ZiZ4O  
end +%Z#!1u  
NW]zMU{c  
% Get the order and frequency corresonding to the function number: nMM:Tr  
% ---------------------------------------------------------------- *? V boyU  
p = p(:); (>49SOu;$\  
n = ceil((-3+sqrt(9+8*p))/2); >G9YYt~  
m = 2*p - n.*(n+2); &ci;0P#Q  
!#y_
vz9  
% Pass the inputs to the function ZERNFUN: 5]f6YlJZ  
% ---------------------------------------- b I"+b\K  
switch nargin CH9Psr78  
    case 3 Tfq7<<0$N  
        z = zernfun(n,m,r,theta); k%D|17I  
    case 4 :MaP58dhh  
        z = zernfun(n,m,r,theta,nflag); w`YN#G  
    otherwise M"\Iw'5$  
        error('zernfun2:nargin','Incorrect number of inputs.') q!;u4J  
end :_8Nf1B+T  
F:7d}Jx  
% 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 7!F -.kG  
function z = zernfun(n,m,r,theta,nflag) cY^'Cj  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. icK$W2<8mg  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ^ 0
.`1$  
%   and angular frequency M, evaluated at positions (R,THETA) on the 6Vgxfic  
%   unit circle.  N is a vector of positive integers (including 0), and :i3 W U%  
%   M is a vector with the same number of elements as N.  Each element 8kLHQ0pmu  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 7#&e0fw/I  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1,  "F=ta  
%   and THETA is a vector of angles.  R and THETA must have the same }U'VVPh_  
%   length.  The output Z is a matrix with one column for every (N,M) +!Q*ie+q  
%   pair, and one row for every (R,THETA) pair. vRh)o1u)  
% cJE4uL<  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XL7||9,(h  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), SM8f"H28  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral + )n}n5  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !bIE%cq  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized Mt4*`CxtH;  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k4PXH  
% I5@8=rFk  
%   The Zernike functions are an orthogonal basis on the unit circle. "m%EFWUOl  
%   They are used in disciplines such as astronomy, optics, and d#HlO}  
%   optometry to describe functions on a circular domain. G<-<>)zO!  
% X[!S7[d-y  
%   The following table lists the first 15 Zernike functions. GG`j9"t4  
% 3bRW]mP8  
%       n    m    Zernike function           Normalization Cg(&WJw(ep  
%       -------------------------------------------------- sXm
P<c  
%       0    0    1                                 1 ?bPW*A82{q  
%       1    1    r * cos(theta)                    2 &5[B\yv  
%       1   -1    r * sin(theta)                    2 '#C5m#v  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) .}5qi;CA  
%       2    0    (2*r^2 - 1)                    sqrt(3) D*>#]0X  
%       2    2    r^2 * sin(2*theta)             sqrt(6) 6zi 5#23  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) Z,tHyyF?j  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) 1Va=.#<  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) 34QW^{dgE  
%       3    3    r^3 * sin(3*theta)             sqrt(8) ^T*!~K8A  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) Vr@tSc&  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Qz89=#W  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) ^/VnRpU  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u* G+=aV.6  
%       4    4    r^4 * sin(4*theta)             sqrt(10) *aJO5&w<T  
%       -------------------------------------------------- \a4X},h\  
% JZK93R  
%   Example 1: S['cX ~  
% /ykc`E?f  
%       % Display the Zernike function Z(n=5,m=1) 1?yj<^"  
%       x = -1:0.01:1; z%1e>`\E  
%       [X,Y] = meshgrid(x,x); h@z0 x4_])  
%       [theta,r] = cart2pol(X,Y); q65]bs4M  
%       idx = r<=1; Ms
Zx 0]  
%       z = nan(size(X)); CG95ScrX  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); ~%2yDhdQ  
%       figure i&8|@CACb  
%       pcolor(x,x,z), shading interp l,~`o$_  
%       axis square, colorbar :+ mULUi  
%       title('Zernike function Z_5^1(r,\theta)') }'?qUy3x  
% eY-h<K)y  
%   Example 2: d"@ /{O^1  
% {kBsiSvsA;  
%       % Display the first 10 Zernike functions tJ7F.}\;C  
%       x = -1:0.01:1; `!spi=f  
%       [X,Y] = meshgrid(x,x); VR .t  
%       [theta,r] = cart2pol(X,Y); 4AKr.a0q  
%       idx = r<=1; as'yYn8  
%       z = nan(size(X)); ?"^{:~\N  
%       n = [0  1  1  2  2  2  3  3  3  3]; Mna yiJl  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; [Y~~C J  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 4"H*hKp  
%       y = zernfun(n,m,r(idx),theta(idx)); g*(z.  
%       figure('Units','normalized') ZyDNtX%  
%       for k = 1:10 a]Pw:lT  
%           z(idx) = y(:,k); a#
{"3Z2|  
%           subplot(4,7,Nplot(k)) Zk/ejhy0  
%           pcolor(x,x,z), shading interp F

,A+O+  
%           set(gca,'XTick',[],'YTick',[]) qpMcVJL  
%           axis square Bz <I7h  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =36fS/Gb  
%       end 7{(UiQbf  
% L N Fe7<y  
%   See also ZERNPOL, ZERNFUN2. ; o Y|~  
U[|5:qWs
 
%   Paul Fricker 11/13/2006 <R+?>kz6  
kz1#"8Zd!  
"\O7_od-  
% Check and prepare the inputs: o[}Dj6e\t  
% ----------------------------- Jfk
#E^1  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @0s' (  
    error('zernfun:NMvectors','N and M must be vectors.') 934j5D  
end jpO0dtn3=  
j}tM0U
g.U  
if length(n)~=length(m) IG# wY  
    error('zernfun:NMlength','N and M must be the same length.') hRRxOr#*
$  
end cc*?4C/t  
8'L:D  
n = n(:); K#N9N@WjR  
m = m(:); bhGRD{=  
if any(mod(n-m,2)) RRPPojKZ  
    error('zernfun:NMmultiplesof2', ... >Oj$Dn=  
          'All N and M must differ by multiples of 2 (including 0).') 9 " t;6  
end 4r`I)  
6)ibXbH  
if any(m>n) VBQAkl?(}4  
    error('zernfun:MlessthanN', ... Xz^k.4 Y{4  
          'Each M must be less than or equal to its corresponding N.') -(F}=o'  
end Q,JH/X  
E0Q6Ryn  
if any( r>1 | r<0 ) #^r-D[/m  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') wM4{\  f\  
end }~|`h1JF  
v@
OELJX  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _AFje  
    error('zernfun:RTHvector','R and THETA must be vectors.') 4K'U}W  
end Dka8[z7  
kmC0.\  
r = r(:); eOiH7{OA,  
theta = theta(:); -&`_bf%M  
length_r = length(r); :d9GkC  
if length_r~=length(theta) p0 X%^A,4  
    error('zernfun:RTHlength', ... pP1DR'  
          'The number of R- and THETA-values must be equal.') iAQ[;M3p  
end Iy49o!  
Y @'do)  
% Check normalization: oA[`| ji  
% -------------------- yQUrHxm  
if nargin==5 && ischar(nflag) s`H|o'0  
    isnorm = strcmpi(nflag,'norm'); n]Yz<#  
    if ~isnorm
3))CD,|  
        error('zernfun:normalization','Unrecognized normalization flag.') &_-=(rK  
    end p?>J86%[  
else fcEm:jEZ*  
    isnorm = false; v~Dobk/n  
end |v%$Q/zp&  
-rI7ihr*  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fsPNxy"_  
% Compute the Zernike Polynomials 8v2Wi.4T  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Cip|eM&l  
DJgM>&Y6,  
% Determine the required powers of r: B=K<k+{6"  
% ----------------------------------- #*qV kPX  
m_abs = abs(m); zO\_^A|8H  
rpowers = []; z+;$cfN  
for j = 1:length(n) }v2p]D5n.  
    rpowers = [rpowers m_abs(j):2:n(j)]; X
e\}(O  
end ~&p]kmwXSX  
rpowers = unique(rpowers); AZhI~QWo  
9C,gJp}P  
% Pre-compute the values of r raised to the required powers, }NwmZw>_  
% and compile them in a matrix: mfI[9G  
% ----------------------------- guYP|  
if rpowers(1)==0 _ps4-<ugC  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); sj&(O@~R  
    rpowern = cat(2,rpowern{:}); ]kmAN65c  
    rpowern = [ones(length_r,1) rpowern]; #e-7LmO~  
else uKXU.u*C  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9NVtvBA  
    rpowern = cat(2,rpowern{:}); 89D`!`Ah]  
end ym6Emf]  
/];N1  
% Compute the values of the polynomials: T+P{,,a/]  
% -------------------------------------- )E=B;.FH  
y = zeros(length_r,length(n)); ,Aq, f$5V  
for j = 1:length(n) um]*nXIr  
    s = 0:(n(j)-m_abs(j))/2; jWxa [>  
    pows = n(j):-2:m_abs(j); ld(_+<e  
    for k = length(s):-1:1 2BOH8Mp9  
        p = (1-2*mod(s(k),2))* ... Q$.CtECo  
                   prod(2:(n(j)-s(k)))/              ... `_Iyr3HAf  
                   prod(2:s(k))/                     ... ~oSA&v4V  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i=b'_SZ'  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); 7YTO{E6]d\  
        idx = (pows(k)==rpowers); E5P.x^  
        y(:,j) = y(:,j) + p*rpowern(:,idx); t"%~r3{  
    end -M]/Xv]  
     ZT&[:>upR  
    if isnorm j^8Hjg  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Y(rQ032s  
    end x?{l<m
c  
end =u9e5n  
% END: Compute the Zernike Polynomials S?v;+3TG  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SP2";,%/9  
~rOvVi&4  
% Compute the Zernike functions: {
yf,:5  
% ------------------------------ 8[^b8^  
idx_pos = m>0; [C 7X#|  
idx_neg = m<0; A;C4>U Y  
Sb?v5  
z = y; ?=iy 6q  
if any(idx_pos) i
0x[w>\-  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); E(S$
Q^  
end 0\ j)!b  
if any(idx_neg) fH
,h\0  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @d3yqA  
end yyVJb3n5:!  
bsc b
 
% 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