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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 ~> |o3&G{  
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! Ce_E S.  
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 %oqC5O6  
function z = zernfun(n,m,r,theta,nflag) P|4qbm4%O,  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. gN/6%,H}  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lq\/E`fc`  
%   and angular frequency M, evaluated at positions (R,THETA) on the OMVK\_oXo  
%   unit circle.  N is a vector of positive integers (including 0), and @XFy^?  
%   M is a vector with the same number of elements as N.  Each element DZ~qk+,I  
%   k of M must be a positive integer, with possible values M(k) = -N(k) 7!"OF  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, [agp06 $D?  
%   and THETA is a vector of angles.  R and THETA must have the same JVeb$_0k  
%   length.  The output Z is a matrix with one column for every (N,M) x*2'I  
%   pair, and one row for every (R,THETA) pair. COk;z.Kn  
% |] 8Hh>  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike RkuPMs Hw;  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DKxzk~sOM  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral ^&6'FE  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ffqz :6  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized _MC\\u/C/  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KZ;Q71  
% N++jI(  
%   The Zernike functions are an orthogonal basis on the unit circle. }+Ne)B E  
%   They are used in disciplines such as astronomy, optics, and bI?YNt,  
%   optometry to describe functions on a circular domain. W bW@V_rr  
% Ot#O];3  
%   The following table lists the first 15 Zernike functions. =UW! 7OzC  
% T,eP&IN  
%       n    m    Zernike function           Normalization 4#^?-6  
%       -------------------------------------------------- amY\1quD|  
%       0    0    1                                 1 <Fa]k'<^)  
%       1    1    r * cos(theta)                    2 Vx6/Rehj  
%       1   -1    r * sin(theta)                    2 ``p( )^zT  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) 42wa9UL<Ka  
%       2    0    (2*r^2 - 1)                    sqrt(3) Zw`vPvb!  
%       2    2    r^2 * sin(2*theta)             sqrt(6) v2uyn  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) g:sn/Zug]  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) z]Dbca1a`  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) w[S!U<9/  
%       3    3    r^3 * sin(3*theta)             sqrt(8) xbv  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) - s,M+Q(<  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a*Oc:$  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) 0[qU k(=}[  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6cV -iDOH  
%       4    4    r^4 * sin(4*theta)             sqrt(10) ~Yw`w 2  
%       -------------------------------------------------- N5%zbfKM  
% RN3-:Zd_X  
%   Example 1: D< h+r?  
% :DlgNR`bq  
%       % Display the Zernike function Z(n=5,m=1) 3 0fsVwE2  
%       x = -1:0.01:1; !F_BLHig  
%       [X,Y] = meshgrid(x,x); 9$u'2TV  
%       [theta,r] = cart2pol(X,Y); Gx]J6Z8  
%       idx = r<=1; i,Q{Z@,  
%       z = nan(size(X)); jeM/8~^4-  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); EGZ F@#N  
%       figure YGj3W.eH  
%       pcolor(x,x,z), shading interp nf7l}^/UE  
%       axis square, colorbar UE[5Bw?4X  
%       title('Zernike function Z_5^1(r,\theta)') lo%:$2*'p  
% Xo{|m[,  
%   Example 2: < c}cgD4  
% vIi#M0@N  
%       % Display the first 10 Zernike functions JToc("V  
%       x = -1:0.01:1; J4-64t nZ  
%       [X,Y] = meshgrid(x,x); $H9+>Z0(  
%       [theta,r] = cart2pol(X,Y); KfO$bmwmx  
%       idx = r<=1; %$)[qa3  
%       z = nan(size(X)); FOFZ/q  
%       n = [0  1  1  2  2  2  3  3  3  3]; d&dp#)._8  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; TfYXF`d  
%       Nplot = [4 10 12 16 18 20 22 24 26 28];  4|9c+^%^  
%       y = zernfun(n,m,r(idx),theta(idx)); ~e ,D`Lv  
%       figure('Units','normalized') 8KQ]3Z9p  
%       for k = 1:10 9D2}heTN  
%           z(idx) = y(:,k); 8e`'Ox_5a  
%           subplot(4,7,Nplot(k)) Y 7a<3>  
%           pcolor(x,x,z), shading interp |,&5.|E 7  
%           set(gca,'XTick',[],'YTick',[]) $R'  
%           axis square =lzRx%tm  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZZ<uiN$  
%       end dQ5_=( 9  
% 39|4)1e  
%   See also ZERNPOL, ZERNFUN2. meHnT9a^  
!f\q0Gnl  
%   Paul Fricker 11/13/2006 "gcHcboU5$  
8J P{`)  
0'giAA  
% Check and prepare the inputs: cH&-/|N  
% ----------------------------- G\y:O9(  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) pohA??t2:  
    error('zernfun:NMvectors','N and M must be vectors.') ~VRt 6C  
end n( |~z   
CLb~6LD  
if length(n)~=length(m) 1e 8J-Nkj  
    error('zernfun:NMlength','N and M must be the same length.') G d".zsn  
end [7Yfv Xp  
k* ayzg3F>  
n = n(:); %6\e_y%  
m = m(:); {Lex((  
if any(mod(n-m,2)) JF%eC}[d  
    error('zernfun:NMmultiplesof2', ... O>Vb7`z0<  
          'All N and M must differ by multiples of 2 (including 0).') U4J9b p|  
end 5Av bKT  
ZeUA  e  
if any(m>n) \GL!x 7s1A  
    error('zernfun:MlessthanN', ... p7UdZOi2  
          'Each M must be less than or equal to its corresponding N.') `CW I%V  
end %_rdO(   
:u%Jrc (W  
if any( r>1 | r<0 ) dE<}X7J%  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^ |k 7g  
end X=i^[?C  
?onaJ=mT  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2yu\f u  
    error('zernfun:RTHvector','R and THETA must be vectors.') %SG**7  
end tOJK~%'  
rOt`5_2f  
r = r(:); 7oj ^(R,  
theta = theta(:); sD|P*ir  
length_r = length(r); #J 1vN]g  
if length_r~=length(theta) <oweLRt  
    error('zernfun:RTHlength', ... ~ .}  
          'The number of R- and THETA-values must be equal.') O F$0]V  
end 5pF4{Jd1  
tE i-0J  
% Check normalization: 9~bl  
% -------------------- 0y>]6 8D  
if nargin==5 && ischar(nflag) hJr cy!P<a  
    isnorm = strcmpi(nflag,'norm'); 1o&] =(  
    if ~isnorm RTPxAp+\5  
        error('zernfun:normalization','Unrecognized normalization flag.') *Nv!Kuk  
    end ^6tcB* #A  
else HgHhc&-  
    isnorm = false; sFd"VRAV~E  
end L/2{}l>D  
IO,ddVO  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {s=n "*Qp)  
% Compute the Zernike Polynomials o5!"dxR  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AOT +4*)%  
mlIX>ss|7B  
% Determine the required powers of r: .T*K4m{b0  
% ----------------------------------- N! 7r~B   
m_abs = abs(m); Who7{|M\'  
rpowers = []; X67.%>#3  
for j = 1:length(n) wv.FL$f[@  
    rpowers = [rpowers m_abs(j):2:n(j)]; l>l)m-;O  
end  3kAmRU  
rpowers = unique(rpowers); m!{}Y]FZn  
5|&:l8=  
% Pre-compute the values of r raised to the required powers, <ej Wl%4  
% and compile them in a matrix: S >E|A %  
% ----------------------------- JfJUOaL  
if rpowers(1)==0 4)'8fi  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @,Je*5$o"  
    rpowern = cat(2,rpowern{:}); 6XVr-ef  
    rpowern = [ones(length_r,1) rpowern]; 1!u}~E_   
else r"yA=d'c  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )_*<uSl  
    rpowern = cat(2,rpowern{:}); aE[>^~Lv}  
end 0_Gi1)  
Mx ?{[zT"  
% Compute the values of the polynomials:  2C9wOO  
% -------------------------------------- qT`sPEs;V  
y = zeros(length_r,length(n)); B;SN}I  
for j = 1:length(n) $"P9I-\m  
    s = 0:(n(j)-m_abs(j))/2; ,@+ 7(W  
    pows = n(j):-2:m_abs(j); ]Lc:M'V#  
    for k = length(s):-1:1 g+5{&YD  
        p = (1-2*mod(s(k),2))* ... E)eRi"a46  
                   prod(2:(n(j)-s(k)))/              ... <+MNv#1:w  
                   prod(2:s(k))/                     ... wzX 1!?  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D`nW9i7  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); Ri$wt.b  
        idx = (pows(k)==rpowers); UjmBLXz@T  
        y(:,j) = y(:,j) + p*rpowern(:,idx); kF:4 [d  
    end 6S-1Wc4  
     ]LFY2w<  
    if isnorm C4Pi6.wf  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); F_8nxQ-  
    end n^8LF9r  
end i^c  
% END: Compute the Zernike Polynomials 5):2;hk  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + y!B`'J  
sOc<'):TK  
% Compute the Zernike functions: 8 *@knkJ  
% ------------------------------ V K/;ohTTP  
idx_pos = m>0; sb 3l4(8g  
idx_neg = m<0; w(w%~;\kLP  
TH_Vw,)  
z = y; >QwZt  
if any(idx_pos) kyQUaFG  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p5<2tSD  
end (<ybst6+I  
if any(idx_neg) S~WsGLF s  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); VKtrSY}6T  
end iz'#K?PF_  
4:$?u}9[:[  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) ;6N@raP7  
%ZERNFUN2 Single-index Zernike functions on the unit circle. ny:c&XS  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated A1xY8?#?~c  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive q7}$F]UM"  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, S5eQHef  
%   and THETA is a vector of angles.  R and THETA must have the same 0{ov LzW  
%   length.  The output Z is a matrix with one column for every P-value, V1R=`  
%   and one row for every (R,THETA) pair. F(E3U'G  
% w$J0/eX{A  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike &*TwEN^h  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) ^H'zS3S  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) S5:`fo^5  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 5MJ`B: He+  
%   for all p. `r"euO r\  
% jd;=5(2  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 L]{ 1"`#  
%   Zernike functions (order N<=7).  In some disciplines it is mk JS_6  
%   traditional to label the first 36 functions using a single mode ;I'pC?!y  
%   number P instead of separate numbers for the order N and azimuthal `Q!|/B  
%   frequency M. J~z;sTR  
% ]a|3"DP5  
%   Example: }+U} [G  
% y}jX/Ln  
%       % Display the first 16 Zernike functions t0Q/vp*/  
%       x = -1:0.01:1; XZhX%OT!  
%       [X,Y] = meshgrid(x,x); ?'H);ou-p  
%       [theta,r] = cart2pol(X,Y); )i&9)_ro  
%       idx = r<=1; ehEXC  
%       p = 0:15; A:3bL: ;t  
%       z = nan(size(X)); WXj}gL`  
%       y = zernfun2(p,r(idx),theta(idx)); [0 7N<<  
%       figure('Units','normalized') SJ1 1LF3)  
%       for k = 1:length(p) ['pk/h  
%           z(idx) = y(:,k); 4VE7%.z+  
%           subplot(4,4,k) -d\O{{%>.z  
%           pcolor(x,x,z), shading interp <Vp7G%"'W  
%           set(gca,'XTick',[],'YTick',[]) 3=xb%Upw  
%           axis square F,#)8>O  
%           title(['Z_{' num2str(p(k)) '}']) hvu>P {  
%       end nGA'\+zj L  
% P/._ tQu6  
%   See also ZERNPOL, ZERNFUN. -ecP@,  
>'eOzMBn  
%   Paul Fricker 11/13/2006 yTw0\yiO  
@L|X('i  
(x9d7$2  
% Check and prepare the inputs: #ej^K |Qx  
% ----------------------------- }~dXz?{p8  
if min(size(p))~=1 WAiEINQ^)  
    error('zernfun2:Pvector','Input P must be vector.') P8.tl"q  
end }x4,a6^  
bL 5z%bV  
if any(p)>35 cOra`7L`  
    error('zernfun2:P36', ... h+|3\>/@9{  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... -bSe=09;S|  
           '(P = 0 to 35).']) fAWjk&9  
end ~*D)L'`2M  
\s Fdp!M}2  
% Get the order and frequency corresonding to the function number: vrXmzq  
% ---------------------------------------------------------------- #5O'XH5_  
p = p(:); b3e:F{n ^  
n = ceil((-3+sqrt(9+8*p))/2); [6qa"Ie  
m = 2*p - n.*(n+2); HbF.doXK  
-bo2"*|m  
% Pass the inputs to the function ZERNFUN: zmy94Y5PE  
% ---------------------------------------- fXMVl\ <  
switch nargin E!'H,#"P  
    case 3 [@[!esC  
        z = zernfun(n,m,r,theta); ''!pvxA  
    case 4 9BB<. p  
        z = zernfun(n,m,r,theta,nflag); xbrxh-gV  
    otherwise \/4ipU.  
        error('zernfun2:nargin','Incorrect number of inputs.') ]aq!@rDX  
end iC&=-$vu  
xEaRuH c  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) (r`+q[  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. OlhfBu)~  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of 8 f%@:}H  
%   order N and frequency M, evaluated at R.  N is a vector of vP&dvAUF  
%   positive integers (including 0), and M is a vector with the @~fg[)7M  
%   same number of elements as N.  Each element k of M must be a wXDF7tJh  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) 0"o<( 1  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is sj2v*tFb  
%   a vector of numbers between 0 and 1.  The output Z is a matrix gBHev1^y  
%   with one column for every (N,M) pair, and one row for every W5(t+$L.  
%   element in R. (?xGl V`n  
% j11\t  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- /dhx+K~  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is JU:!lyd  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to zB\g'F/  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 ?AP2Opsl  
%   for all [n,m]. b$[_(QUw  
%  m3 ;  
%   The radial Zernike polynomials are the radial portion of the QRdNi 1&M  
%   Zernike functions, which are an orthogonal basis on the unit zc]F  
%   circle.  The series representation of the radial Zernike VP\HPSp  
%   polynomials is M$-4.+G  
% KSVIX!EsX  
%          (n-m)/2 F/ 2@%,2n  
%            __ ~w Dmt  
%    m      \       s                                          n-2s f2_LfbvH  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r z=!$3E ecr  
%    n      s=0 rfYu8-  
% u_@%}zo?5*  
%   The following table shows the first 12 polynomials. Ql%7wrK  
% jM1_+Lm1  
%       n    m    Zernike polynomial    Normalization nSsVONHfa  
%       --------------------------------------------- wc)[r~On(5  
%       0    0    1                        sqrt(2) n)} J<  
%       1    1    r                           2 x `PIJE  
%       2    0    2*r^2 - 1                sqrt(6) :84ja>`c  
%       2    2    r^2                      sqrt(6) {d}-SoxH  
%       3    1    3*r^3 - 2*r              sqrt(8) L;nRI.  
%       3    3    r^3                      sqrt(8) BQSA;;n]  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) >?'q P ]  
%       4    2    4*r^4 - 3*r^2            sqrt(10) &o^wgmS   
%       4    4    r^4                      sqrt(10) Y::fcMJr;Q  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) !W^2?pqN  
%       5    3    5*r^5 - 4*r^3            sqrt(12) GWjKZ1p  
%       5    5    r^5                      sqrt(12) nX5*pTfjL3  
%       --------------------------------------------- #i ?@S$  
% 9/}i6j8Z  
%   Example: (J.(Fl>^  
% 7zr\AgV9  
%       % Display three example Zernike radial polynomials .`Z{ptt>  
%       r = 0:0.01:1; >1pD'UZIy7  
%       n = [3 2 5]; 90sMS]a  
%       m = [1 2 1]; <Ml,H%F  
%       z = zernpol(n,m,r); n8[sR;r5f  
%       figure jm RYL("  
%       plot(r,z) ;DC0LJ  
%       grid on ?MKf=! w  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') :m8ED[9b  
% ^Q!:0D*  
%   See also ZERNFUN, ZERNFUN2. }S*6+4  
{N)\It  
% A note on the algorithm. Ez?vJDd  
% ------------------------ t8i"f L  
% The radial Zernike polynomials are computed using the series [w=x0J&  
% representation shown in the Help section above. For many special HZ 8 j[kO  
% functions, direct evaluation using the series representation can (N;Jw^C@  
% produce poor numerical results (floating point errors), because [7<X&Q  
% the summation often involves computing small differences between Z;%  
% large successive terms in the series. (In such cases, the functions hp-< 8Mf  
% are often evaluated using alternative methods such as recurrence d/Py,  
% relations: see the Legendre functions, for example). For the Zernike cnLC>_hY  
% polynomials, however, this problem does not arise, because the v^@L?{" }8  
% polynomials are evaluated over the finite domain r = (0,1), and ~lDLdUs  
% because the coefficients for a given polynomial are generally all X&wK<  
% of similar magnitude. x ?^c:`.  
% ~|DF-t V  
% ZERNPOL has been written using a vectorized implementation: multiple 15yIPv+5  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] %M}zi'qQ?  
% values can be passed as inputs) for a vector of points R.  To achieve )[p8  
% this vectorization most efficiently, the algorithm in ZERNPOL W**=X\"'  
% involves pre-determining all the powers p of R that are required to =7e8N&-nv  
% compute the outputs, and then compiling the {R^p} into a single w-B\AK?}  
% matrix.  This avoids any redundant computation of the R^p, and T"QY@#E  
% minimizes the sizes of certain intermediate variables. 30DpIkf  
% * uEU9fX  
%   Paul Fricker 11/13/2006 L+I[yJY:!  
m9L+|r  
cYbO)?mC_  
% Check and prepare the inputs: .z[+sy_  
% ----------------------------- ,1~Zqprn  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Phb<##OB  
    error('zernpol:NMvectors','N and M must be vectors.') +B8Ut{l  
end HhqqJEp0  
7PQedZ<\  
if length(n)~=length(m) yLDHJ}R  
    error('zernpol:NMlength','N and M must be the same length.') RLKO0 #  
end ise@,[!  
8U;!1!+ 7)  
n = n(:); W/\7m\ B  
m = m(:); Qb(CH  
length_n = length(n); M2s   
BbU&e z8P  
if any(mod(n-m,2)) Rp@u.C <  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') l w%fY{  
end &1Y+ q]  
N$y4>g  
if any(m<0) Iewq?s\Fo  
    error('zernpol:Mpositive','All M must be positive.') /ZzlC#`  
end {^\-%3$  
78.sf{I  
if any(m>n) ,88Y1|:X  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') iF%q 6R  
end ] !A;-m  
^M80 F7  
if any( r>1 | r<0 ) hCDI;'ls  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') VLO>{"{'  
end 8odVdivh  
N!m%~},s//  
if ~any(size(r)==1) djSN{>S  
    error('zernpol:Rvector','R must be a vector.') }`MO}Pz  
end z)0VP QMT  
)a}5\V  
r = r(:); a>kD G <.A  
length_r = length(r); NRl"!FSD;"  
^s?wnEo;j  
if nargin==4 =3T?U_u@  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); ?UxY4m%R;  
    if ~isnorm /Mk)H d  
        error('zernpol:normalization','Unrecognized normalization flag.') {c1qC zM4  
    end [a`i{(!  
else {Q-U=me\  
    isnorm = false; FjydEV  
end D!P?sq_5r  
[D,:=p`  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% roA1= G\Q  
% Compute the Zernike Polynomials dL\8^L  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {L=[1  
~3%3{a a  
% Determine the required powers of r: g\l;>  
% ----------------------------------- *Q>:|F[vM  
rpowers = []; IN"vi|1  
for j = 1:length(n) \1Bgs^  
    rpowers = [rpowers m(j):2:n(j)]; "4`%NA  
end K+}Z6_:  
rpowers = unique(rpowers); toWmm(7v  
;FmSL#]I  
% Pre-compute the values of r raised to the required powers, 4 * OU  
% and compile them in a matrix: kmoJ`W} N  
% ----------------------------- q>[% C5  
if rpowers(1)==0 /'1y`j<  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); l+6\U6_)B  
    rpowern = cat(2,rpowern{:}); ]/bE${W*]  
    rpowern = [ones(length_r,1) rpowern]; VMWg:=~$  
else lkm(3y@']A  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Z]Y4NO;  
    rpowern = cat(2,rpowern{:}); _y9P]@Q7%  
end </X"*G't  
SSXS  
% Compute the values of the polynomials: [z~Nw#  
% -------------------------------------- V\"5<>+O  
z = zeros(length_r,length_n); NM@An2  
for j = 1:length_n ]'Yw#YB  
    s = 0:(n(j)-m(j))/2; ]cr;PRyv  
    pows = n(j):-2:m(j); (HI%C@e9  
    for k = length(s):-1:1  ~Hs{(7   
        p = (1-2*mod(s(k),2))* ... 5avO48;Vc  
                   prod(2:(n(j)-s(k)))/          ... 3~Ln:4[6ID  
                   prod(2:s(k))/                 ... `k&K"jA7$  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... P1wRt5  
                   prod(2:((n(j)+m(j))/2-s(k))); !cb#fl  
        idx = (pows(k)==rpowers); 0I((UA/7Zs  
        z(:,j) = z(:,j) + p*rpowern(:,idx); BcLt95;.\  
    end sQZ8<DpB  
     }L!`K"^O&  
    if isnorm JNaW> X$K  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); "^z=r]<5  
    end md"%S-a_dT  
end 2-0cB$W+  
8~E)gV+v  
% 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)  ^[}0&_L w  
8V|jL?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。 B9IXa;  
]Wd`GI  
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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