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光行天下 -> ZEMAX,OpticStudio -> ansys分析后面型数据如何进行zernike多项式拟合? [点此返回论坛查看本帖完整版本] [打印本页]

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 x!@3.$  
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! ";>>{lYA.  
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 n#.~XNbxv  
function z = zernfun(n,m,r,theta,nflag) Uiz#QGt  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. "c'K8,+?  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N dM^1O-K:  
%   and angular frequency M, evaluated at positions (R,THETA) on the ruf*-&Kr7  
%   unit circle.  N is a vector of positive integers (including 0), and gPA), NrN  
%   M is a vector with the same number of elements as N.  Each element Z: e|~#  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 8'n xc#&  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, s/" l ?d  
%   and THETA is a vector of angles.  R and THETA must have the same piy_9nk  
%   length.  The output Z is a matrix with one column for every (N,M) |Nfi y  
%   pair, and one row for every (R,THETA) pair. ar!`8"  
% 9pPohR*#V  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~Wox"h}(  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~\ C.Nm  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral /2&:sHWW  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {dk%j~w8  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized )q>mt/,  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $ M8ZF(W  
% AD=qB5:  
%   The Zernike functions are an orthogonal basis on the unit circle. P%nN#Qm  
%   They are used in disciplines such as astronomy, optics, and F^xhhz&e  
%   optometry to describe functions on a circular domain. ([+u U!  
% wQnW2)9!  
%   The following table lists the first 15 Zernike functions. .8I\=+Zi  
% c8"9Lv  
%       n    m    Zernike function           Normalization >0kZ-M5  
%       -------------------------------------------------- }CoR$K   
%       0    0    1                                 1 1lf 5xm.  
%       1    1    r * cos(theta)                    2 5VXI/Lw#  
%       1   -1    r * sin(theta)                    2 x9NcIa9  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) OZ'=Xtbn  
%       2    0    (2*r^2 - 1)                    sqrt(3) 4)zHkN+  
%       2    2    r^2 * sin(2*theta)             sqrt(6) h\@\*Xz<v  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) &.ENcEic  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) "G+g(?N]j  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) K< ;I*cAX  
%       3    3    r^3 * sin(3*theta)             sqrt(8) Xc!0'P0T  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) !MNnau%O  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0j-- X?-  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) Tw{}Ht_Qq  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X,Q'Xe /  
%       4    4    r^4 * sin(4*theta)             sqrt(10) >&mNC \PA  
%       -------------------------------------------------- Y<"BhE  
% YaWZOuxm  
%   Example 1: Z/p>>SCak  
% 04u^Q  
%       % Display the Zernike function Z(n=5,m=1) "; PW#VHC  
%       x = -1:0.01:1; zlw+=NX  
%       [X,Y] = meshgrid(x,x); ^Qx qv  
%       [theta,r] = cart2pol(X,Y); @5i m*ubzM  
%       idx = r<=1; zK5/0zMZ  
%       z = nan(size(X)); KO~_  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); >3v j<v}m  
%       figure sFvu@Wm'7W  
%       pcolor(x,x,z), shading interp d5hYOhO[  
%       axis square, colorbar Tf|?j=f  
%       title('Zernike function Z_5^1(r,\theta)') G(n e8L8  
% dE`a1H%  
%   Example 2: O :'ENoQ:&  
% =FB[<%  
%       % Display the first 10 Zernike functions s\CZ os&  
%       x = -1:0.01:1;  ./iC  
%       [X,Y] = meshgrid(x,x); Vg>\@ C .s  
%       [theta,r] = cart2pol(X,Y); ;AjY-w  
%       idx = r<=1; )yOdRRP  
%       z = nan(size(X)); e?V7<7$  
%       n = [0  1  1  2  2  2  3  3  3  3]; T@S\:P  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; b!h*I>`  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; 'UW(0 PXw  
%       y = zernfun(n,m,r(idx),theta(idx)); hI^Hqv  
%       figure('Units','normalized') ;aY.CgX  
%       for k = 1:10 _gGI&0(VM  
%           z(idx) = y(:,k); *i=+["A  
%           subplot(4,7,Nplot(k)) Q7~9~  
%           pcolor(x,x,z), shading interp [b=l'e/  
%           set(gca,'XTick',[],'YTick',[]) ;`{PA !>  
%           axis square I|`/#BYbW  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) -Tzp;o  
%       end xYgG  
% l$1z%|I  
%   See also ZERNPOL, ZERNFUN2. j.b7<Vr4;  
QXQ'QEG  
%   Paul Fricker 11/13/2006 sM4Qu./  
ib3 u:  
U:a-Wi+  
% Check and prepare the inputs: {DI`HB[  
% ----------------------------- "<e<0::  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ez= Q{g  
    error('zernfun:NMvectors','N and M must be vectors.') iPD5 KsAOA  
end 9L"Z ~CUL  
sy ]k  
if length(n)~=length(m) N`G* h^YQ  
    error('zernfun:NMlength','N and M must be the same length.') :3uCW1  
end nO ^m  
M<AjtDF%  
n = n(:); WeqE 9@V  
m = m(:); 7jj.maK  
if any(mod(n-m,2)) ({R-JkW: ;  
    error('zernfun:NMmultiplesof2', ... 5`!Bj0Uf  
          'All N and M must differ by multiples of 2 (including 0).') HB>&}z0  
end HP$GI  
')bas#=uP  
if any(m>n)  c"pI+Q  
    error('zernfun:MlessthanN', ... (.CEEWj%{  
          'Each M must be less than or equal to its corresponding N.') J$]-)`[G&  
end fAW(  
x34 4}\  
if any( r>1 | r<0 ) :jJ;&t^^  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') -w[j`}([P9  
end !mM`+XH  
n]15 ~GO.  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oCCTRLb02  
    error('zernfun:RTHvector','R and THETA must be vectors.') .nN>Ipv  
end Wk0"U V  
jHU5>Gt-}  
r = r(:); N=JZtf/i  
theta = theta(:); [SJ)4e|)  
length_r = length(r); E`"<t:RzF  
if length_r~=length(theta) ~36)3W[4  
    error('zernfun:RTHlength', ... 6> fQe8Y  
          'The number of R- and THETA-values must be equal.') \V1geSoE  
end tK|jh  
by:"aDGK.  
% Check normalization: 65P*Gu?  
% -------------------- >Q`\|m}x)Q  
if nargin==5 && ischar(nflag) dN8@ 0AMSf  
    isnorm = strcmpi(nflag,'norm'); | 4slG   
    if ~isnorm b3zxiq x  
        error('zernfun:normalization','Unrecognized normalization flag.') -$x5[6bN  
    end rryC^Vma  
else 3eg)O34  
    isnorm = false; [110[i^  
end "%mu~&Ga  
; qQ* p  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VbwB<nQl  
% Compute the Zernike Polynomials Fm|h3.`V  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8iB}gHe9  
$*KM%M6  
% Determine the required powers of r: (>LJv |wn  
% ----------------------------------- ^L,Uz:[J  
m_abs = abs(m); vi4lmkyh^  
rpowers = []; A#&,S4Wi|  
for j = 1:length(n) S260h,(,  
    rpowers = [rpowers m_abs(j):2:n(j)]; `veq/!  
end si!jB%^  
rpowers = unique(rpowers); f3p)Q<H>`(  
2i4&*& A  
% Pre-compute the values of r raised to the required powers, jQV.U~25Q  
% and compile them in a matrix: ~8j4IO(  
% ----------------------------- %VSjMZ  
if rpowers(1)==0 ~+HZQv3Y  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6>hW.aq}  
    rpowern = cat(2,rpowern{:}); >k)zd-  
    rpowern = [ones(length_r,1) rpowern]; <Rno ;  
else a%IJ8t+mn  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )J"*[[e  
    rpowern = cat(2,rpowern{:}); 4D4Y.g_x  
end QMI6l'"s  
pE~>k:  
% Compute the values of the polynomials: ZJod=^T  
% -------------------------------------- &|LP>'H;  
y = zeros(length_r,length(n)); J/{!_M-  
for j = 1:length(n) )[Bl3+'  
    s = 0:(n(j)-m_abs(j))/2; 4(hHp6}b  
    pows = n(j):-2:m_abs(j); 5LF#w_x  
    for k = length(s):-1:1 \nKpJ9!  
        p = (1-2*mod(s(k),2))* ... hE9UWa.Q>  
                   prod(2:(n(j)-s(k)))/              ... ,~TV/l<  
                   prod(2:s(k))/                     ... ^T.icSxP  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xc&&UKd  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); (c'kZ9&  
        idx = (pows(k)==rpowers); v=Y) A?  
        y(:,j) = y(:,j) + p*rpowern(:,idx); Xh[02iL-  
    end mT3'kUZ}]  
     Z2t r?]  
    if isnorm W,53|9b@  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kuZs30^  
    end v< Ozr:lL  
end ;LhNz()b  
% END: Compute the Zernike Polynomials U~}cib5W5  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEFh&n  
y?}R,5k  
% Compute the Zernike functions: sT;:V  
% ------------------------------ SBbPO5^](  
idx_pos = m>0; br[n5  
idx_neg = m<0; 0\X'a}8Bu  
'y? HF@NJ  
z = y; pn._u`xMV  
if any(idx_pos) o(|fapK.  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); >+3tOv3:  
end %ylpn7I\6  
if any(idx_neg) g:&V9~FR  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); fN-y8  
end q]}1/JZS  
"x"y3v'  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) @XSxoUF\  
%ZERNFUN2 Single-index Zernike functions on the unit circle. wfJ[" q   
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated /qeSR3WC  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive `(dRb  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, %CaUC'  
%   and THETA is a vector of angles.  R and THETA must have the same `B GU  
%   length.  The output Z is a matrix with one column for every P-value,  M .J  
%   and one row for every (R,THETA) pair. km[ PbC  
%  (r!d4  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike 6j{O/  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) }aJK^>^>A  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) @?gN &Z)I  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 ;xl_9Ht/  
%   for all p. eCI0o5U  
% (d@ =   
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 ` g5S  
%   Zernike functions (order N<=7).  In some disciplines it is ,TdL-a5  
%   traditional to label the first 36 functions using a single mode gL-\@4\wc  
%   number P instead of separate numbers for the order N and azimuthal P-*=e8z{  
%   frequency M. HXg4 T  
% `J}-U\4F{  
%   Example: A[mm_+D>  
% Pqc +pE  
%       % Display the first 16 Zernike functions R8K ?! Z  
%       x = -1:0.01:1; ~ ""?:  
%       [X,Y] = meshgrid(x,x); Dqu][~oQ  
%       [theta,r] = cart2pol(X,Y); ?4 &C)[^  
%       idx = r<=1; .^} vDA  
%       p = 0:15; RgQ;fYS  
%       z = nan(size(X));  #VA8a=t  
%       y = zernfun2(p,r(idx),theta(idx)); zs_^m1t1s  
%       figure('Units','normalized') (H*d">`mz  
%       for k = 1:length(p) ly)b=ph&  
%           z(idx) = y(:,k); HR> X@g<c  
%           subplot(4,4,k) Wz^M*=,  
%           pcolor(x,x,z), shading interp ,7{}}l  
%           set(gca,'XTick',[],'YTick',[]) bR*/d-v^  
%           axis square 6x _tX  
%           title(['Z_{' num2str(p(k)) '}']) H |K}m,g  
%       end e#uk+]  
% Q:+cLl&;hB  
%   See also ZERNPOL, ZERNFUN. IRxFcLk  
J|Xu]fg0  
%   Paul Fricker 11/13/2006 " ~X;u8m  
>[U.P)7;  
V L&5TZtz  
% Check and prepare the inputs: p 7YfOUo k  
% ----------------------------- mAFVjSa2  
if min(size(p))~=1 3X,SCG  
    error('zernfun2:Pvector','Input P must be vector.') yye( ^  
end :<}.3Q?&  
`PY>p!E  
if any(p)>35 L< gp "e  
    error('zernfun2:P36', ...  xlH?J;$  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... w?+v+k\  
           '(P = 0 to 35).']) \AT]$`8@_  
end M]Y72K^  
bLHj<AX#>|  
% Get the order and frequency corresonding to the function number: 9b0M'x'W5  
% ---------------------------------------------------------------- kr_!AW<.tz  
p = p(:); gmY*}d` 'f  
n = ceil((-3+sqrt(9+8*p))/2); N!`8-ap\^  
m = 2*p - n.*(n+2); ;8G( l   
=>0+BD  
% Pass the inputs to the function ZERNFUN: S4jt*]w5b  
% ---------------------------------------- 0F\ e*{gc  
switch nargin -UPlQL  
    case 3 ^rKA=siz  
        z = zernfun(n,m,r,theta); wK7wu.  
    case 4 DM-8azq $  
        z = zernfun(n,m,r,theta,nflag); 4+V+SD  
    otherwise v%e-vl  
        error('zernfun2:nargin','Incorrect number of inputs.') Au6*hv3:  
end sp0j2<$a  
qX GAlCq@  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) e%v<nGN.-  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. 6!,Am^uXM  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of C^ hHt,&  
%   order N and frequency M, evaluated at R.  N is a vector of fC2   
%   positive integers (including 0), and M is a vector with the aN.t) DG}J  
%   same number of elements as N.  Each element k of M must be a nC3U%*l  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) Z(S=2r.  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is +G!N@O  
%   a vector of numbers between 0 and 1.  The output Z is a matrix R?Dbv'lp>  
%   with one column for every (N,M) pair, and one row for every 9q2 >_Mv  
%   element in R. 0`KB|=>  
% \;?=h  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- N F[v/S  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is *ay>MlcV2=  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to n.{+\M6k  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 UXXN\D  
%   for all [n,m]. ?Jlz{msI  
%  0x}8}  
%   The radial Zernike polynomials are the radial portion of the Fr}e-a  
%   Zernike functions, which are an orthogonal basis on the unit <"uT=]wZ=  
%   circle.  The series representation of the radial Zernike S^@I4Z  
%   polynomials is sOJH$G3O  
% _jw A_  
%          (n-m)/2 9E (VU.  
%            __ }LH>0v_<Y  
%    m      \       s                                          n-2s G\\zk  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r Phsdn`,  
%    n      s=0 ~-,P1 u!  
% wX6VapFboI  
%   The following table shows the first 12 polynomials. u"K-mr#$[o  
% 1n! Jfs U  
%       n    m    Zernike polynomial    Normalization ZhKYoPIq  
%       ---------------------------------------------  H}NW?  
%       0    0    1                        sqrt(2) Mj&f7IUO  
%       1    1    r                           2 @L%9NqE`O  
%       2    0    2*r^2 - 1                sqrt(6) rl^_RI  
%       2    2    r^2                      sqrt(6) D?rQQxb  
%       3    1    3*r^3 - 2*r              sqrt(8) (Al.hEs'  
%       3    3    r^3                      sqrt(8) DRD%pm(  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) C#X0Cn0ln  
%       4    2    4*r^4 - 3*r^2            sqrt(10) 0px@3/  
%       4    4    r^4                      sqrt(10) "D7*en  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) Urhh)i  
%       5    3    5*r^5 - 4*r^3            sqrt(12) W'\{8&:!  
%       5    5    r^5                      sqrt(12) =6t)-53  
%       --------------------------------------------- NDhHU#Q9  
% m :ROq  
%   Example: /vFw5KUu  
% tvptaw A.  
%       % Display three example Zernike radial polynomials vN%zk(?T  
%       r = 0:0.01:1; WlVC0&  
%       n = [3 2 5]; S @\Pki+n[  
%       m = [1 2 1]; ZD1UMB0$4  
%       z = zernpol(n,m,r); fmH$ 1C<  
%       figure wobTT1!|  
%       plot(r,z) /<Cl\q2 A  
%       grid on v709#/ cR  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') w .l2  
% ]*pALT6  
%   See also ZERNFUN, ZERNFUN2. TuphCu+Oh  
~130"WQ;  
% A note on the algorithm. g% :Q86u  
% ------------------------ -p`hevRr  
% The radial Zernike polynomials are computed using the series WO*YBH@  
% representation shown in the Help section above. For many special A>^\jIB>  
% functions, direct evaluation using the series representation can i\G@kJNnF  
% produce poor numerical results (floating point errors), because \;p5Pagx0-  
% the summation often involves computing small differences between eKpH|S!x U  
% large successive terms in the series. (In such cases, the functions E/:<9xl  
% are often evaluated using alternative methods such as recurrence ghJ81  
% relations: see the Legendre functions, for example). For the Zernike nH*U  
% polynomials, however, this problem does not arise, because the \`r5tQr  
% polynomials are evaluated over the finite domain r = (0,1), and kRnh20I  
% because the coefficients for a given polynomial are generally all m.-l&@I2/<  
% of similar magnitude. fk`y}#7M  
% b&p*IyJR  
% ZERNPOL has been written using a vectorized implementation: multiple Z~$=V:EA?  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] y[S 5  
% values can be passed as inputs) for a vector of points R.  To achieve ]*U\ gm%  
% this vectorization most efficiently, the algorithm in ZERNPOL I(/W+ o  
% involves pre-determining all the powers p of R that are required to R>[2}R30  
% compute the outputs, and then compiling the {R^p} into a single e{Pgz0sO Q  
% matrix.  This avoids any redundant computation of the R^p, and 1=O Xi!G  
% minimizes the sizes of certain intermediate variables. #B|`F?o  
% z?_c:]D  
%   Paul Fricker 11/13/2006 {FraM,w:  
(w?@qs!  
]q7\  
% Check and prepare the inputs: +"Ek? )?  
% ----------------------------- A/%K=H?  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~R7rIP8Wr  
    error('zernpol:NMvectors','N and M must be vectors.') B+eB=KL  
end }m/aigA[1  
<6U{I '  
if length(n)~=length(m) 3zY"9KUN  
    error('zernpol:NMlength','N and M must be the same length.') MOP %vS   
end -MJ6~4k2  
q .nsGbl  
n = n(:); A1Mr  
m = m(:); `4}zB#3  
length_n = length(n); vLJ<_&6  
]3xnq<  
if any(mod(n-m,2)) a*6x^R;)  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') Fe%Q8RIh_  
end 8Z:NT_Ss  
. [+ObF9=  
if any(m<0) /"tVOv#  
    error('zernpol:Mpositive','All M must be positive.') AnG/A!G  
end J:)Q)MT24:  
(A\qZtnyl  
if any(m>n) fyYT#r  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') 2sjV*\Udf  
end V /$qD  
1Jc-hrN-  
if any( r>1 | r<0 ) BQ,]]}e43z  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') oOBN  
end \=v7'Hp  
LR]P?  
if ~any(size(r)==1) 0XSZ3dY&+  
    error('zernpol:Rvector','R must be a vector.') x&+/da-E/5  
end {$bAs9L  
s{x2RDAt  
r = r(:);  r75,mX  
length_r = length(r); POfvs]  
Cd#[b)d ?^  
if nargin==4 #q9jFW8  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); }I}RqD:`  
    if ~isnorm V9I5/~0c  
        error('zernpol:normalization','Unrecognized normalization flag.') 5'?K(Jdmp  
    end f}Eoc>n  
else Ra&HzK?  
    isnorm = false; zIf/jk  
end Eh#W*Bg  
`]#DdJ_|  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y+_G L=J  
% Compute the Zernike Polynomials =S4_^UY;  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lJpD>\$}@R  
VzA~w` $d  
% Determine the required powers of r: 8TD:~ee  
% ----------------------------------- 4M*UVdJ;  
rpowers = []; m!er "0  
for j = 1:length(n) OvX z+C,  
    rpowers = [rpowers m(j):2:n(j)]; 8h=H\v^f  
end u^s{r`/  
rpowers = unique(rpowers); K!7o#"GM  
15 x~[?!  
% Pre-compute the values of r raised to the required powers, k+ 5:fB)z  
% and compile them in a matrix: |+mhYq|`  
% ----------------------------- ,u-i9`B  
if rpowers(1)==0 5O#CdN-S  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); i{I'+%~R  
    rpowern = cat(2,rpowern{:}); "KcSOjvJ  
    rpowern = [ones(length_r,1) rpowern]; yP :>vFd7  
else q*{i/=~  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MIub^ $<C  
    rpowern = cat(2,rpowern{:}); >7!4o9)c  
end &1ASWllD  
X<Rh-1$8F  
% Compute the values of the polynomials: QzT)PtX  
% -------------------------------------- br .jj  
z = zeros(length_r,length_n); :iD( [V  
for j = 1:length_n sycN  
    s = 0:(n(j)-m(j))/2; /i"vEI  
    pows = n(j):-2:m(j); F[ N{7C3  
    for k = length(s):-1:1 $2A%y14  
        p = (1-2*mod(s(k),2))* ... -_[n2\|we)  
                   prod(2:(n(j)-s(k)))/          ... IxAKIa[HY  
                   prod(2:s(k))/                 ... d!{,[8&  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... /0s1q  
                   prod(2:((n(j)+m(j))/2-s(k))); kG[u$[B  
        idx = (pows(k)==rpowers); 3v :PBmE  
        z(:,j) = z(:,j) + p*rpowern(:,idx); { xi$'r  
    end vRp =L54z  
     3;Kv9i<~LE  
    if isnorm .e!dEF)D  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); R9f*&lj  
    end DPw"UY:  
end )TnxsFC  
vg&Dr  
% 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)  q4'szDYO2  
;wwhW|A  
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。 EtvZk9d6h*  
Pz7{dQqjk#  
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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