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niuhelen 2011-03-12 18:40

ansys分析后面型数据如何进行zernike多项式拟合?

小弟不是学光学的,所以想请各位大侠指点啊!谢谢啦 'f 3HKn<L  
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! XM$GQn]B  
phility 2011-03-12 22:31
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
phility 2011-03-12 22:41
泽尼克多项式的前9项对应象差的
niuhelen 2011-03-12 23:00
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 ;d4 y{  
function z = zernfun(n,m,r,theta,nflag) VUp. j  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7{-@}j`  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N  =^Th[B  
%   and angular frequency M, evaluated at positions (R,THETA) on the CJp-Y}fGEA  
%   unit circle.  N is a vector of positive integers (including 0), and )!A 2>  
%   M is a vector with the same number of elements as N.  Each element D i+4Eb  
%   k of M must be a positive integer, with possible values M(k) = -N(k) hcyn  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, v;Es^ YI  
%   and THETA is a vector of angles.  R and THETA must have the same AP0|z  
%   length.  The output Z is a matrix with one column for every (N,M) Xtkw Z3  
%   pair, and one row for every (R,THETA) pair. u#FXW_-TK  
% &3I$8v|!?  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ilv_D~|  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'j}g  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral G]-%AO{K  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, MI\]IQU  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized : W~f;k  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9\AS@SH{^T  
% X'@'/[?  
%   The Zernike functions are an orthogonal basis on the unit circle. Oxv+1Ub<Dv  
%   They are used in disciplines such as astronomy, optics, and =5ug\S  
%   optometry to describe functions on a circular domain. .Vmtx  
% ;, rnk-  
%   The following table lists the first 15 Zernike functions. &Pq\cNYzW  
% =:gjz4}_8  
%       n    m    Zernike function           Normalization ?I[h~vr6.  
%       -------------------------------------------------- _dr*`yXi  
%       0    0    1                                 1 !lhFKb;  
%       1    1    r * cos(theta)                    2 ra]:$XJ5=a  
%       1   -1    r * sin(theta)                    2 ,Aj }]h\L  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) \!<"7=(J{4  
%       2    0    (2*r^2 - 1)                    sqrt(3) aM$=|%9/  
%       2    2    r^2 * sin(2*theta)             sqrt(6) &hI>L  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) f*<ps o  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) B'p5M.6d#:  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) ^wJEfac  
%       3    3    r^3 * sin(3*theta)             sqrt(8) -2 x E#r  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) &2{]hRM  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hd 0Xx}3&  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 3D[=b%2\  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) < Y>3  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ]G*$W+G]  
%       -------------------------------------------------- skR,-:"8  
% ]_u`EvEx6  
%   Example 1: %K zbO0  
% _R74/|  
%       % Display the Zernike function Z(n=5,m=1) s@~/x5jwCs  
%       x = -1:0.01:1; <Oa9oM},d  
%       [X,Y] = meshgrid(x,x); t#5:\U5r.  
%       [theta,r] = cart2pol(X,Y); G3dh M#!  
%       idx = r<=1; 6vobta^w  
%       z = nan(size(X)); Dx3%K S  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); &$#99\ /  
%       figure W 2<3C  
%       pcolor(x,x,z), shading interp JYV\oV{  
%       axis square, colorbar v9rVpYc"  
%       title('Zernike function Z_5^1(r,\theta)') 3ji:O T  
% x: ~d@  
%   Example 2: H&bh<KPMh  
% o/1JO_41  
%       % Display the first 10 Zernike functions h0Jl_f#Y  
%       x = -1:0.01:1; a#y{pT2 b  
%       [X,Y] = meshgrid(x,x); 0`n 5x0R  
%       [theta,r] = cart2pol(X,Y); nY0sb8lZJ  
%       idx = r<=1; E >}q2  
%       z = nan(size(X)); "PzP; Br  
%       n = [0  1  1  2  2  2  3  3  3  3]; iBoEZEHjw  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; %[ Zz0|A  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; oZ:{@ =  
%       y = zernfun(n,m,r(idx),theta(idx)); e$|VG* d  
%       figure('Units','normalized') ,I`_F,  
%       for k = 1:10 .zS D`v@[  
%           z(idx) = y(:,k); |I^y0Q:K  
%           subplot(4,7,Nplot(k)) a$m_D!b~_  
%           pcolor(x,x,z), shading interp _- %d9@x  
%           set(gca,'XTick',[],'YTick',[]) _z8;lt   
%           axis square ~`R1sSr"  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7'OPjt M  
%       end Rd%0\ B  
% /DO'IHC.o  
%   See also ZERNPOL, ZERNFUN2. 4ht\&2&:  
x=,8[W#XT  
%   Paul Fricker 11/13/2006 >a=d;  
\r;F2C0*i  
l>7r2;  
% Check and prepare the inputs: l^r' $;<m  
% ----------------------------- GwQn;gkF  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) GMm'of#  
    error('zernfun:NMvectors','N and M must be vectors.') "HC)/)Mv@  
end gs`> C(  
*]x_,:R6Ow  
if length(n)~=length(m)  YqU/\f+  
    error('zernfun:NMlength','N and M must be the same length.') D9-Lg%  
end Km*<Kfcz  
cNj*E =~;  
n = n(:); &N\[V-GP2G  
m = m(:); bk3Unreh  
if any(mod(n-m,2)) e<5Y94YE  
    error('zernfun:NMmultiplesof2', ... o.^y1mH'  
          'All N and M must differ by multiples of 2 (including 0).') yr{B5z,  
end 0M8.U  
|+NuYz?  
if any(m>n) -0 0}if7  
    error('zernfun:MlessthanN', ... te'*<HM  
          'Each M must be less than or equal to its corresponding N.') X/+OF'po  
end ;fGx;D  
*IZf^-=Q  
if any( r>1 | r<0 ) (X}@^]lpa  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') J&6:d  
end wFL3& *  
8R xc&`_X  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )#`H."Z  
    error('zernfun:RTHvector','R and THETA must be vectors.') Hr }k5'  
end n )K6i7]xk  
jvs[ /  
r = r(:); f0oek{  
theta = theta(:); 7>-yaL{  
length_r = length(r); >n!ni(  
if length_r~=length(theta) .Z%G@X*  
    error('zernfun:RTHlength', ... %^.P~s6  
          'The number of R- and THETA-values must be equal.') HomN/wKh  
end >V!LitdJ  
&1Fply7(Ay  
% Check normalization: _N'75  
% -------------------- arh@`'Q  
if nargin==5 && ischar(nflag) qY# d+F,t  
    isnorm = strcmpi(nflag,'norm'); 7ZFJexN]  
    if ~isnorm n`L,]dco  
        error('zernfun:normalization','Unrecognized normalization flag.') i2`0|8mw'  
    end +R[4\ hC0Y  
else W9R`A  
    isnorm = false; 0"4@;e_)>  
end QnKC#   
qY(:8yC36  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r4eUZ .8R  
% Compute the Zernike Polynomials }<[Db}?9  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,{{SI  
6/2v  
% Determine the required powers of r: xl] ;*&  
% ----------------------------------- <NB41/  
m_abs = abs(m); Oif,|:  
rpowers = []; I/p]DT  
for j = 1:length(n) 59!)j>f  
    rpowers = [rpowers m_abs(j):2:n(j)]; h&'=F)5  
end vJC f~'  
rpowers = unique(rpowers); o3h-=t  
=! m JG  
% Pre-compute the values of r raised to the required powers, S,vu]?-8  
% and compile them in a matrix: |}S1o0v{(a  
% ----------------------------- 8wIK:   
if rpowers(1)==0 1dv=xe.  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h<3p8eB  
    rpowern = cat(2,rpowern{:}); $qm~c[x%  
    rpowern = [ones(length_r,1) rpowern]; } uQ${]&D  
else 3g'+0tEl  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); >q(6,Mmb  
    rpowern = cat(2,rpowern{:}); U e*$&VlT  
end jA`a/v Wu  
Hed$ytMaGz  
% Compute the values of the polynomials: Lq0 4T0  
% -------------------------------------- Q}P-$X+/ n  
y = zeros(length_r,length(n)); /V^sJ($V$~  
for j = 1:length(n) e@jfIF0=}  
    s = 0:(n(j)-m_abs(j))/2; <abKiXA"  
    pows = n(j):-2:m_abs(j); !N~*EI$  
    for k = length(s):-1:1 =`p&h}h-L  
        p = (1-2*mod(s(k),2))* ... dAxp ,):&J  
                   prod(2:(n(j)-s(k)))/              ... wkikD  
                   prod(2:s(k))/                     ... IZ~.{UQ  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mk=#\>  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); GS%b=kc  
        idx = (pows(k)==rpowers); 3{3/: 7  
        y(:,j) = y(:,j) + p*rpowern(:,idx); fIyPFqf7w)  
    end y8?t-Pp]1  
     yGEb7I$h  
    if isnorm `-O= >U5nH  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4vqNule  
    end -P#nT 2  
end 4<}A]BQVkJ  
% END: Compute the Zernike Polynomials ~ hm`uP  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?}sOG?{  
8p =>?wG  
% Compute the Zernike functions: oVkr3K Z  
% ------------------------------ rJ(OAKnY  
idx_pos = m>0; d8:C3R  
idx_neg = m<0; 3qo e^e  
(,LL[&;:  
z = y; #:{6b *}  
if any(idx_pos) O5;-Om  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]kS7n @8  
end w3bIb$12  
if any(idx_neg) ou6j*eSN  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c]v +  
end }W}G X(?P  
{!=2<-Aq  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) z%};X$V`J  
%ZERNFUN2 Single-index Zernike functions on the unit circle. F5IZ"Itu(  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated A&l7d0Z^j5  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive Wz{%"o  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, bp%S62Dj  
%   and THETA is a vector of angles.  R and THETA must have the same t w!.%_1^  
%   length.  The output Z is a matrix with one column for every P-value, ) `I=oB  
%   and one row for every (R,THETA) pair. ,}<RrUfD  
% /*P7<5n0  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike qLRE}$P  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) F< |c4  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) DV,DB\P$  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 (1 yGg==W.  
%   for all p. C1 {ZW~"YI  
% XnY"oDg^>  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 A& =pw#  
%   Zernike functions (order N<=7).  In some disciplines it is o)!m$Q~v  
%   traditional to label the first 36 functions using a single mode  _8S4Q!  
%   number P instead of separate numbers for the order N and azimuthal o2}N=|&  
%   frequency M. wlVvxX3%  
% $e1:Q#den2  
%   Example: pqq?*\W&[v  
% ]xrD<  
%       % Display the first 16 Zernike functions >0V0i%inmF  
%       x = -1:0.01:1; ohplj`X[21  
%       [X,Y] = meshgrid(x,x); 4\1;A`2%0  
%       [theta,r] = cart2pol(X,Y); [B,p,Q"  
%       idx = r<=1; d h?dO`  
%       p = 0:15; w, 7Cr  
%       z = nan(size(X)); TkT-$=i  
%       y = zernfun2(p,r(idx),theta(idx)); yoW> BX  
%       figure('Units','normalized') @+t (xCv  
%       for k = 1:length(p) |."G?*  
%           z(idx) = y(:,k); -yA3 RP  
%           subplot(4,4,k) k_O-5{  
%           pcolor(x,x,z), shading interp m&cvU>lC  
%           set(gca,'XTick',[],'YTick',[]) 0BFz7  
%           axis square GBr,LN  
%           title(['Z_{' num2str(p(k)) '}']) >{F!ntEj  
%       end QQ~23TlA  
% ;NG1{]|Z  
%   See also ZERNPOL, ZERNFUN. @<5Tba>SC  
N(BiOLZL6  
%   Paul Fricker 11/13/2006 cIZ[[(Db  
EW9b*r7./  
,`"K  
% Check and prepare the inputs: ]|y}\7Aa  
% ----------------------------- )-#i8?y3C  
if min(size(p))~=1 @Wz%KdXA  
    error('zernfun2:Pvector','Input P must be vector.') QES^^PQe:  
end U1kh-8  :  
yDuq6`R*  
if any(p)>35 7@@<5&mN  
    error('zernfun2:P36', ...  z \^  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... uAT/6@  
           '(P = 0 to 35).']) |Q6h /"2  
end ()B7(Y  
o"4E+1qwM  
% Get the order and frequency corresonding to the function number: 6Wb!J>93  
% ---------------------------------------------------------------- Tplg2p% k  
p = p(:); {s9<ej~<R  
n = ceil((-3+sqrt(9+8*p))/2); <b'1#Pd>0  
m = 2*p - n.*(n+2); N5ci};?  
 RY9. n  
% Pass the inputs to the function ZERNFUN: ( mt*y]p?  
% ---------------------------------------- UI_v3c3b  
switch nargin 4`6< {  
    case 3 d^y86pq.  
        z = zernfun(n,m,r,theta); _1\poAy  
    case 4 =t HD 4I  
        z = zernfun(n,m,r,theta,nflag); S!<"Swf:  
    otherwise 1Df, a#,y"  
        error('zernfun2:nargin','Incorrect number of inputs.') IE}Sdeqi)  
end @x*.5:[  
p$XnOh  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) ]An_5J  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. <ipWMZae0F  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of k3#wLJ  
%   order N and frequency M, evaluated at R.  N is a vector of .}`V I`z*  
%   positive integers (including 0), and M is a vector with the }\ EL;sT  
%   same number of elements as N.  Each element k of M must be a oLd:3,p}  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) C,(j$Id  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is !;jgzi?z  
%   a vector of numbers between 0 and 1.  The output Z is a matrix EFeAr@nj  
%   with one column for every (N,M) pair, and one row for every N?zV*ngBS  
%   element in R. _=6vW^ s  
% b70AJe=  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- 4':MI|/my_  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is 9V.+U7\w  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to ZDfS0]0F  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 >6w@{p2B  
%   for all [n,m]. K(' 9l& A  
% X&/(x  
%   The radial Zernike polynomials are the radial portion of the 2G H)iUmc  
%   Zernike functions, which are an orthogonal basis on the unit $Q=$?>4U  
%   circle.  The series representation of the radial Zernike YN$`y1V  
%   polynomials is o16d`}/<  
% *XHj)DC;  
%          (n-m)/2 :bI,rEW#_  
%            __ TX&[;jsj  
%    m      \       s                                          n-2s BL7>dZOa  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r =#jTo|~u4o  
%    n      s=0 )G mb? !/^  
% X"wF Qa  
%   The following table shows the first 12 polynomials. \;i G{}(  
% 1R*1BStc  
%       n    m    Zernike polynomial    Normalization N=.}h\{0  
%       ---------------------------------------------  GsI[N%  
%       0    0    1                        sqrt(2) &:-GI)[o  
%       1    1    r                           2 ]Kof sU_{  
%       2    0    2*r^2 - 1                sqrt(6) aD0Q0C+  
%       2    2    r^2                      sqrt(6) ~ 0av3G  
%       3    1    3*r^3 - 2*r              sqrt(8) .iN-4"_j1  
%       3    3    r^3                      sqrt(8) F|9a}(-7  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) 6V+V zDo  
%       4    2    4*r^4 - 3*r^2            sqrt(10) y(V&z"wk[  
%       4    4    r^4                      sqrt(10) `Yg7,{A\J  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12)  MK<  
%       5    3    5*r^5 - 4*r^3            sqrt(12) /@6E3lh S  
%       5    5    r^5                      sqrt(12) <1TlW ~q<  
%       --------------------------------------------- p!C_:Z5i  
% SlR7h$r'  
%   Example: b!0'Qidh0  
% Y!bpOa&  
%       % Display three example Zernike radial polynomials cPV5^9\T  
%       r = 0:0.01:1; "4KkKi  
%       n = [3 2 5]; ,"x23=]  
%       m = [1 2 1]; 6Wf*>G*h  
%       z = zernpol(n,m,r); :P HUsy  
%       figure ys:1Z\$P  
%       plot(r,z) ,xm;JXJ  
%       grid on ]r"31.w(  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') %g!yccD9  
% 0TpBSyx.  
%   See also ZERNFUN, ZERNFUN2. lO%MyP  
3| GNi~  
% A note on the algorithm. #8P#^v]H  
% ------------------------ oH!$eAU?  
% The radial Zernike polynomials are computed using the series cdd P T  
% representation shown in the Help section above. For many special ZD$-V 3e`  
% functions, direct evaluation using the series representation can "bv,I-\  
% produce poor numerical results (floating point errors), because p7%0hLW  
% the summation often involves computing small differences between Xw&QrTDS`  
% large successive terms in the series. (In such cases, the functions SD)5?{6<  
% are often evaluated using alternative methods such as recurrence F<6KaZ|  
% relations: see the Legendre functions, for example). For the Zernike ;D%$Eh&oma  
% polynomials, however, this problem does not arise, because the ?j &V:kF  
% polynomials are evaluated over the finite domain r = (0,1), and @m*&c*r  
% because the coefficients for a given polynomial are generally all 2tm~QL  
% of similar magnitude. M!;`(_2  
% vo:52tCk}m  
% ZERNPOL has been written using a vectorized implementation: multiple Ukk-(gjX  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] )$2%&9b  
% values can be passed as inputs) for a vector of points R.  To achieve `I]1l MJ)o  
% this vectorization most efficiently, the algorithm in ZERNPOL AAIyr703cQ  
% involves pre-determining all the powers p of R that are required to }CB=c]p  
% compute the outputs, and then compiling the {R^p} into a single o=mq$Z:}  
% matrix.  This avoids any redundant computation of the R^p, and fvAh?<Ul  
% minimizes the sizes of certain intermediate variables. j1'xp`jgv  
% DDqC}l_  
%   Paul Fricker 11/13/2006 )R [@G.  
jKY Aid{-  
g=8|z#S  
% Check and prepare the inputs: ]be 0I)  
% ----------------------------- @lb=-oR!~  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) SX1Fyy6 w  
    error('zernpol:NMvectors','N and M must be vectors.') \NvC   
end pfJVE  
Im0+`9Jw  
if length(n)~=length(m) +X2 i/}  
    error('zernpol:NMlength','N and M must be the same length.') pq/ FLYiv  
end 2n-Tpay0  
:IP;Frc MP  
n = n(:); S0Rf>Eo4  
m = m(:); 1 $KLMW  
length_n = length(n); H{Y5YTg]  
($ [r>)TG  
if any(mod(n-m,2)) U5@TaGbx  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') ne !j%9Ar  
end Knw'h;,[  
thlpj*|  
if any(m<0) MA/"UV&M(  
    error('zernpol:Mpositive','All M must be positive.') 7Ap~7)z[  
end NIWI6qCw  
e"v[)b++Y  
if any(m>n) q>(I*=7  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') %>I?'y^  
end $BR=IYby  
_.Z&<.lJ  
if any( r>1 | r<0 ) _!$Up  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') T}b( M*E  
end Xn ZX *Y]"  
eaYkYuS/  
if ~any(size(r)==1) &4[#_(pk  
    error('zernpol:Rvector','R must be a vector.') lD;,I^Lt6  
end k^'d@1z;C  
<x>k3bD  
r = r(:); N18diP[C  
length_r = length(r); .JD4gF2N  
3-_U-:2"  
if nargin==4 N,sqrk]  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); yuy+}]uB@  
    if ~isnorm **w*hd]  
        error('zernpol:normalization','Unrecognized normalization flag.') cc2oFn  
    end ?-.Ep0/  
else cciAMQhA  
    isnorm = false; C9z~)aL}7  
end mKf>6/s{c  
-UD~>s  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M|Lw`?T  
% Compute the Zernike Polynomials 28UL  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g qJEJ~  
Fy(nu-W  
% Determine the required powers of r: [-:<z?(n4  
% ----------------------------------- iuU3*yyn  
rpowers = []; QE{;M  
for j = 1:length(n) ffoo^1}1  
    rpowers = [rpowers m(j):2:n(j)]; W:+2We@  
end MLv.v&@S  
rpowers = unique(rpowers); o,''f_tRQ|  
qeK  
% Pre-compute the values of r raised to the required powers, Z 9 q{r s  
% and compile them in a matrix: $E9daUt8"J  
% ----------------------------- x4HMT/@AG2  
if rpowers(1)==0 !+|N<`  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); G(|(y=ck  
    rpowern = cat(2,rpowern{:}); p$b= r+1f  
    rpowern = [ones(length_r,1) rpowern]; K^cWj_a"  
else OL ]T+6X  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); I$6 f.W  
    rpowern = cat(2,rpowern{:}); |>/&EElD  
end \Q?|gfJH  
X-ki%jp3  
% Compute the values of the polynomials: sWZtbW;)  
% -------------------------------------- f'i8Mm4IL  
z = zeros(length_r,length_n); `6S=KRv  
for j = 1:length_n G}?P r4Gj  
    s = 0:(n(j)-m(j))/2; GZhfA ;O,  
    pows = n(j):-2:m(j); vA7jZw  
    for k = length(s):-1:1 Z564K7IV  
        p = (1-2*mod(s(k),2))* ... L ugk`NUvF  
                   prod(2:(n(j)-s(k)))/          ... '__3[D  
                   prod(2:s(k))/                 ... _]~ht H  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... 1KIq$lG{ E  
                   prod(2:((n(j)+m(j))/2-s(k))); zs]/Y2  
        idx = (pows(k)==rpowers); :Z]+Z_9p  
        z(:,j) = z(:,j) + p*rpowern(:,idx); E[Ws} n.  
    end MCrO]N($b  
     tU2to V  
    if isnorm vL=--#  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); 8,H5G`  
    end KutgW#+40  
end )Fr;'JYC1S  
A7,$y!D  
% EOF zernpol
niuhelen 2011-03-12 23:03
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!
li_xin_feng 2012-09-28 10:52
我也正在找啊
guapiqlh 2014-03-04 11:35
我也一直想了解这个多项式的应用,还没用过呢
phoenixzqy 2014-04-22 23:39
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)   aC }1]7  
eF;1l<<   
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。 dQ|Ht[ s=  
C<@1H>S4_  
07年就写过这方面的计算程序了。
查看本帖完整版本: [-- ansys分析后面型数据如何进行zernike多项式拟合? --] [-- top --]

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