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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 x(EwHg>;  
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! S`  U,  
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 }L|B@fW  
function z = zernfun(n,m,r,theta,nflag) @?]>4+Oa0  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y$,~"$su|  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {Z <`@\K3  
%   and angular frequency M, evaluated at positions (R,THETA) on the X)RgXl{  
%   unit circle.  N is a vector of positive integers (including 0), and Io IhQ  
%   M is a vector with the same number of elements as N.  Each element +U ziO#D  
%   k of M must be a positive integer, with possible values M(k) = -N(k) +$>aT (q  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, alzdYiGf  
%   and THETA is a vector of angles.  R and THETA must have the same 7uw-1F5x7  
%   length.  The output Z is a matrix with one column for every (N,M) |/xA5_-N  
%   pair, and one row for every (R,THETA) pair. $i<+O,@-  
% b5%<},ySq  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike sx7zRw >X  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), vc3r [mT  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral L;?h)8  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ex]Ku  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized ~$^ >Vo  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?ZC!E0]  
%  <{ v %2  
%   The Zernike functions are an orthogonal basis on the unit circle. I~~":~&  
%   They are used in disciplines such as astronomy, optics, and WB'1_a  
%   optometry to describe functions on a circular domain. JURu>-i  
% rrgOp5aV"  
%   The following table lists the first 15 Zernike functions. $A,YQH+  
% [h B$%i]\<  
%       n    m    Zernike function           Normalization /L(}VJg-  
%       -------------------------------------------------- ()Wu_Q  
%       0    0    1                                 1 $Q'LDmot  
%       1    1    r * cos(theta)                    2 "B +F6  
%       1   -1    r * sin(theta)                    2 o>+mw|{  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) ct,;V/Dx  
%       2    0    (2*r^2 - 1)                    sqrt(3) Oop6o $k  
%       2    2    r^2 * sin(2*theta)             sqrt(6) .C+(E@eyA  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) NB^Al/V@  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) ,1CmB@  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) N5K2Hv<"  
%       3    3    r^3 * sin(3*theta)             sqrt(8) <?DI!~  
%       4   -4    r^4 * cos(4*theta)             sqrt(10) t(6i4c>  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) QH7 GEj]  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) @aFk|.6  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 47{5{/B-  
%       4    4    r^4 * sin(4*theta)             sqrt(10) }#&[[}@th  
%       -------------------------------------------------- rqBoUS4  
% EAWBgOO8iC  
%   Example 1: sEfT#$ a^8  
% !or_CJ8%  
%       % Display the Zernike function Z(n=5,m=1) %c]N-  
%       x = -1:0.01:1; ~W4SFp  
%       [X,Y] = meshgrid(x,x); 6v%ePFul  
%       [theta,r] = cart2pol(X,Y); Us# /#-hJ  
%       idx = r<=1; Jwj=a1I 53  
%       z = nan(size(X)); mv,a>Cvs[  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); (igB'S5wf  
%       figure y.8nzlkE{  
%       pcolor(x,x,z), shading interp aYc<C$:NC"  
%       axis square, colorbar hHDLrr  
%       title('Zernike function Z_5^1(r,\theta)') a!u5}[{  
% ,|z zq@fk  
%   Example 2: hG<[F@d  
% LH_ U#P`E  
%       % Display the first 10 Zernike functions c 8Q2H  
%       x = -1:0.01:1; km^ZF<.@  
%       [X,Y] = meshgrid(x,x); -U_,RMw~  
%       [theta,r] = cart2pol(X,Y); G*%U0OTi  
%       idx = r<=1; IW@phKz  
%       z = nan(size(X)); <:nyRy}  
%       n = [0  1  1  2  2  2  3  3  3  3]; `YZl2c<w*  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; !.pcldx  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; b *0uxvLu  
%       y = zernfun(n,m,r(idx),theta(idx)); v,~f G>Y}  
%       figure('Units','normalized') "szJ[ _B  
%       for k = 1:10 ",Mrdxn7  
%           z(idx) = y(:,k); G^VOA4  
%           subplot(4,7,Nplot(k)) EX, {1^h  
%           pcolor(x,x,z), shading interp {0/2Hw n  
%           set(gca,'XTick',[],'YTick',[]) ;0?OBUDO  
%           axis square Nq9M$Nt]  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])  ZpBP#Y*  
%       end f TK84v"7_  
% z`Nss o=  
%   See also ZERNPOL, ZERNFUN2. 3q@JhB  
c (5XT[Tw  
%   Paul Fricker 11/13/2006 1#+|RL4o  
:1bDkoK  
[C;Neslo  
% Check and prepare the inputs: l1L8a I,8  
% ----------------------------- AkO);4A;Jd  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) H*f2fyC1\  
    error('zernfun:NMvectors','N and M must be vectors.') 9CN'2 9c  
end v7#|%  
=_@) KWeX$  
if length(n)~=length(m) cuy9QBB :  
    error('zernfun:NMlength','N and M must be the same length.') tW-[.Y -M,  
end Tj<B;f!u  
tgl 4pAc  
n = n(:); >O~V#1 H  
m = m(:); CS-jDok  
if any(mod(n-m,2)) _]D 6m2R  
    error('zernfun:NMmultiplesof2', ... >mEfd=p  
          'All N and M must differ by multiples of 2 (including 0).') MI:%Eq  
end i -@V  
<1* \ ~CX  
if any(m>n) P-8QXDdr  
    error('zernfun:MlessthanN', ... 1_c%p#?K  
          'Each M must be less than or equal to its corresponding N.') KPjAk  
end /<k 5"C% z  
Y"s8j=1m  
if any( r>1 | r<0 ) 31e O2|7  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') )z4eRs F|  
end [>3dhj[;  
5e7\tBab  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) S}]B|Q  
    error('zernfun:RTHvector','R and THETA must be vectors.') ?q2Yk/P  
end +$2`"%nBG  
Zv-1*hhHf  
r = r(:); mDD96y  
theta = theta(:); o>Dd1 j  
length_r = length(r); Y(?SE< 4R  
if length_r~=length(theta) xpwy%uo  
    error('zernfun:RTHlength', ... e:.?T\  
          'The number of R- and THETA-values must be equal.') .ns=jp  
end SK 5]7C2  
MpJ<.|h  
% Check normalization: IX<9_q  
% -------------------- DvOvtd  
if nargin==5 && ischar(nflag) w8J8III\~  
    isnorm = strcmpi(nflag,'norm'); A2A_F|f  
    if ~isnorm 'Yc^9;C(  
        error('zernfun:normalization','Unrecognized normalization flag.') p T z]8[^  
    end ! R3P@,j  
else n'JS-  
    isnorm = false; @'?gan#(  
end BB(v,W  
3=} P l,  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dZb;`DjTH  
% Compute the Zernike Polynomials UTN[! 0[  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k9:|CEP  
h/8p2Mrqi  
% Determine the required powers of r: 9e vQQN6D|  
% ----------------------------------- C-h?#/#?y  
m_abs = abs(m); nXI8`7D  
rpowers = []; +/]*ChrS  
for j = 1:length(n) A:yql`&s  
    rpowers = [rpowers m_abs(j):2:n(j)]; $\~cWpv  
end ;#0$iE  
rpowers = unique(rpowers); SB.=x  
S'NLj(  
% Pre-compute the values of r raised to the required powers, S{f,EBE  
% and compile them in a matrix: k#8`996P  
% ----------------------------- 4*5e0:O  
if rpowers(1)==0 {9x>@p/  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r ) _*MPY  
    rpowern = cat(2,rpowern{:}); qK9A /Mc  
    rpowern = [ones(length_r,1) rpowern]; hdSP#Y'-  
else de.f?y  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); t imY0fx #  
    rpowern = cat(2,rpowern{:}); `ah|BV  
end 6PS[OB{3  
oayu*a.  
% Compute the values of the polynomials: ki/Cpfq40*  
% -------------------------------------- 8c_X`0jy  
y = zeros(length_r,length(n)); 3G2iRr.o  
for j = 1:length(n) ^Qn:#O9  
    s = 0:(n(j)-m_abs(j))/2; #M+_Lk3  
    pows = n(j):-2:m_abs(j); t*A[v  
    for k = length(s):-1:1 IA[:-2_  
        p = (1-2*mod(s(k),2))* ... n~}[/ly  
                   prod(2:(n(j)-s(k)))/              ... 9&`";dg  
                   prod(2:s(k))/                     ... g;nLR<]  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cs9h\]ZA  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); .cw)Y#;IG  
        idx = (pows(k)==rpowers); 1,Mm+_)B  
        y(:,j) = y(:,j) + p*rpowern(:,idx); 2k^rZ^^"  
    end )|k#cT{=M  
     ~w|h;*Bj  
    if isnorm -"i $^Q`  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); v-q-CI? B#  
    end 3/yt  
end Yh fQ pe  
% END: Compute the Zernike Polynomials 4# ]g852  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @ F $}/  
llWY7u"  
% Compute the Zernike functions: /93z3o7D>  
% ------------------------------ -38"S;M8  
idx_pos = m>0; tY!l}:E[  
idx_neg = m<0; H)"]I3  
ZX1/6|_  
z = y; .s!0S-RkC  
if any(idx_pos) Ak kF6d+  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X"r.*fb;N  
end WWZ<[[ >  
if any(idx_neg) F'|e:h  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -1v9  
end Nq8 3 6HL  
XBkaum4j  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) ?E|=eO"I1  
%ZERNFUN2 Single-index Zernike functions on the unit circle. 8AY;WL:;  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated ZeU){CB  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive V\{@c%xW  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, ?< teHFj  
%   and THETA is a vector of angles.  R and THETA must have the same H@$K /  
%   length.  The output Z is a matrix with one column for every P-value, !t"/w6X1I  
%   and one row for every (R,THETA) pair. yI#qkl-  
% 0a8\{(w  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike .KC V|x;QW  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) Qj_)^3`e  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) izaqEz  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 KW[y+c u.#  
%   for all p. 8Q<Nl=g>'  
% !&g_hmnIF  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 Ax;?~v4Z  
%   Zernike functions (order N<=7).  In some disciplines it is 26nwUNak  
%   traditional to label the first 36 functions using a single mode 1Td`S1'#yg  
%   number P instead of separate numbers for the order N and azimuthal &dmIv[LU  
%   frequency M. 8~R.iqLoX  
% RH _b  
%   Example: (@xC-*  
% oST)E5X;7  
%       % Display the first 16 Zernike functions )V[j~uOU)]  
%       x = -1:0.01:1; A^X\  
%       [X,Y] = meshgrid(x,x); <Z3C&BM  
%       [theta,r] = cart2pol(X,Y); _B erHoQd  
%       idx = r<=1; stUv!   
%       p = 0:15; D})/2O p   
%       z = nan(size(X)); ]wEI *c(  
%       y = zernfun2(p,r(idx),theta(idx)); |gP)lR  
%       figure('Units','normalized') #<9'{i3  
%       for k = 1:length(p) \i//Aq  
%           z(idx) = y(:,k); 9ck"JMla  
%           subplot(4,4,k) gL&w:_  
%           pcolor(x,x,z), shading interp 3))R91I  
%           set(gca,'XTick',[],'YTick',[]) yx-"&K=`  
%           axis square e.DN,rhqI  
%           title(['Z_{' num2str(p(k)) '}']) wZ\93W-}  
%       end  =5B5  
% $3>Rw/,  
%   See also ZERNPOL, ZERNFUN. \:1$E[3v  
Zn r4^i&(  
%   Paul Fricker 11/13/2006 "i/GzD7`n  
p19(>|$J  
}(Fmr7%m  
% Check and prepare the inputs: BV>9U5  
% ----------------------------- Tr:@Dv.O  
if min(size(p))~=1 !J#P 'x0  
    error('zernfun2:Pvector','Input P must be vector.') _[-MyUs  
end jFe8s@7  
ri59LYy=  
if any(p)>35 >"<s7$g  
    error('zernfun2:P36', ... 5~DKx7P!Z  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... !w%c= V]tV  
           '(P = 0 to 35).']) KNQj U-A  
end fcF|m5  
K\xM%O?  
% Get the order and frequency corresonding to the function number: SUMfebW5  
% ---------------------------------------------------------------- Oi:Hs  
p = p(:); 0R\.G1f%  
n = ceil((-3+sqrt(9+8*p))/2); >\pF5a`  
m = 2*p - n.*(n+2); 9M9Fif.  
GiuE\J9i  
% Pass the inputs to the function ZERNFUN: i>h 3UIx\  
% ---------------------------------------- KP0(w(q  
switch nargin ZZ^A&%E(a  
    case 3 yQj J-g(.  
        z = zernfun(n,m,r,theta); FO_}9<s  
    case 4 y4p"LD5%^  
        z = zernfun(n,m,r,theta,nflag); c)^A|{,G  
    otherwise 0Q7<;'m  
        error('zernfun2:nargin','Incorrect number of inputs.') {%>~ ]9E  
end @aUNyyVP  
N5;z5E  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) ,{q#U3  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. Sft vN-  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of Jf7H;ZM<  
%   order N and frequency M, evaluated at R.  N is a vector of |iBf6smF  
%   positive integers (including 0), and M is a vector with the L7rr/D  
%   same number of elements as N.  Each element k of M must be a X4hz\={  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) 6q>iPK Jt  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is 420K6[  
%   a vector of numbers between 0 and 1.  The output Z is a matrix oP56f"BE(  
%   with one column for every (N,M) pair, and one row for every |#cqxr"  
%   element in R. P(8Yz W  
% ~s'}_5;VY  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- *: FS/ir  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is %{r3"Q=;W  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to b&h'>(  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1  bV(BwWm  
%   for all [n,m]. 6%6dzZ  
% ,R-k]^O  
%   The radial Zernike polynomials are the radial portion of the ]Ri=*KZa  
%   Zernike functions, which are an orthogonal basis on the unit HX| p4-L  
%   circle.  The series representation of the radial Zernike m2(}$z3e  
%   polynomials is [W$Mn.5<s  
% o ?05bv  
%          (n-m)/2 ( EJ1g^|"  
%            __ Il@K8?H@  
%    m      \       s                                          n-2s lj&\F|-i  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r 4pXY7+e2'  
%    n      s=0 Q/Z>w+zh#  
% b+whZtNk7  
%   The following table shows the first 12 polynomials. _IU5HT}2  
% z@19gD#8  
%       n    m    Zernike polynomial    Normalization R7z @y o  
%       --------------------------------------------- &I'J4gk[  
%       0    0    1                        sqrt(2) 8irTGA  
%       1    1    r                           2 bg0ix"  
%       2    0    2*r^2 - 1                sqrt(6) ;DZj.| Sj+  
%       2    2    r^2                      sqrt(6) z$m(@Q  
%       3    1    3*r^3 - 2*r              sqrt(8) M+ <SSi"  
%       3    3    r^3                      sqrt(8) p</t##]3ks  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) ~0-)S@  
%       4    2    4*r^4 - 3*r^2            sqrt(10) 0)K~pV0aT  
%       4    4    r^4                      sqrt(10) NqKeQezX  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) 9&^5!R8  
%       5    3    5*r^5 - 4*r^3            sqrt(12) V:5aq.o!  
%       5    5    r^5                      sqrt(12) P;gd!Yl<-  
%       --------------------------------------------- p8.JJt^  
% 9 {SzE /[  
%   Example: OC,yLQ  
% Z)(#D($-  
%       % Display three example Zernike radial polynomials _[&.`jTFn  
%       r = 0:0.01:1; >HwVP.~HN  
%       n = [3 2 5]; o%lxEd r  
%       m = [1 2 1]; _f>)G3p  
%       z = zernpol(n,m,r); Xs$a^zZ  
%       figure @m`H~]AU  
%       plot(r,z) UL~~J[1r  
%       grid on w-km qh  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') :si&A;k  
% 8VG~n?y  
%   See also ZERNFUN, ZERNFUN2. [\Aws^fD_  
G^L9[c= ,  
% A note on the algorithm. A`(Cuw-o  
% ------------------------ ;cL+= !  
% The radial Zernike polynomials are computed using the series ca0vN^Ji  
% representation shown in the Help section above. For many special {s6#h#U  
% functions, direct evaluation using the series representation can W>{&" 5  
% produce poor numerical results (floating point errors), because %O[1yZh \  
% the summation often involves computing small differences between dn42'(p@G  
% large successive terms in the series. (In such cases, the functions 0?I  
% are often evaluated using alternative methods such as recurrence 16z Wm JH  
% relations: see the Legendre functions, for example). For the Zernike (A}c22qe  
% polynomials, however, this problem does not arise, because the ~pI`_3  
% polynomials are evaluated over the finite domain r = (0,1), and K'"s9b8  
% because the coefficients for a given polynomial are generally all 6b` Jq>v  
% of similar magnitude. >U4bK^/Bp  
% b/#<::D `  
% ZERNPOL has been written using a vectorized implementation: multiple fD_3lbiL(  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] BjjuZN&  
% values can be passed as inputs) for a vector of points R.  To achieve /K+;HAUTn  
% this vectorization most efficiently, the algorithm in ZERNPOL Ft:_6T%  
% involves pre-determining all the powers p of R that are required to dKchQsgCg  
% compute the outputs, and then compiling the {R^p} into a single trLxg H_Y  
% matrix.  This avoids any redundant computation of the R^p, and #t# S(A9)  
% minimizes the sizes of certain intermediate variables. wc}x [cS  
% v=uQ8_0~N  
%   Paul Fricker 11/13/2006 l9y%@7  
slr>6o%W`  
BTA2['  
% Check and prepare the inputs: f R2,NKM@  
% ----------------------------- b*nI0/cbR.  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #:MoZw`rlw  
    error('zernpol:NMvectors','N and M must be vectors.') G[zysxd  
end xA n|OSe  
`q\v~FT  
if length(n)~=length(m) b3GTsX\2|  
    error('zernpol:NMlength','N and M must be the same length.') Xh~oDnP  
end F?y C=  
+@@( C9  
n = n(:); 0r&FH$  
m = m(:); |NjyO>@Pa  
length_n = length(n); VF7H0XR/k5  
<`A!9+  
if any(mod(n-m,2)) ^ilgd  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') LzB*d  
end 6p,}?6^  
N=8CVI  
if any(m<0) IeIv k55  
    error('zernpol:Mpositive','All M must be positive.') dsK ^-e6:5  
end yN#]Q}4  
hH?ke(&=f  
if any(m>n) 4tapQgj24  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') \/y&l\ k)  
end B,4GxoX`  
(i@(ZG]/  
if any( r>1 | r<0 ) {N-*eV9#  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') w;Pe_m7\EO  
end _xP@kN~  
MF::At[4   
if ~any(size(r)==1) I =K<%.  
    error('zernpol:Rvector','R must be a vector.') V5I xZn%  
end 7H[.o~\  
&v r0{]V^  
r = r(:); *b1NVN$  
length_r = length(r); DEzL]1;P  
W5 RZsS]  
if nargin==4 Yd~Tzh  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm');  $)(Zt^  
    if ~isnorm <,nd]a  
        error('zernpol:normalization','Unrecognized normalization flag.') E{}eYU  
    end ".fnx8v,  
else &gvX<X4e  
    isnorm = false; UWBR5  
end 7q^a@5f BG  
T!5g:;~y >  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q +c~Bd  
% Compute the Zernike Polynomials /9C>{29x!  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fz_8m4  
Sj4@pMh4  
% Determine the required powers of r: W4AFa>h  
% ----------------------------------- 'p'nAB''!  
rpowers = []; {E>(%vD  
for j = 1:length(n) *>Zq79TG  
    rpowers = [rpowers m(j):2:n(j)]; FBI^}^#_  
end ^eqq|(<K  
rpowers = unique(rpowers); wQ9fPOm  
`ve5>aw0_Y  
% Pre-compute the values of r raised to the required powers, SL%lY  
% and compile them in a matrix: rEZMX2  
% ----------------------------- r6`KZ TU  
if rpowers(1)==0 zeHs5P8}r  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Y<`uq'V  
    rpowern = cat(2,rpowern{:}); Ob7F39):N  
    rpowern = [ones(length_r,1) rpowern]; >K;C?gHo  
else 2ISnWzq;  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); as@I0e((  
    rpowern = cat(2,rpowern{:}); &#[6a&9#[A  
end fYZ)5xnj  
& Pzr)W(  
% Compute the values of the polynomials: ,O"zz7  
% -------------------------------------- EWoGdH|  
z = zeros(length_r,length_n); &1Idv}@!  
for j = 1:length_n W6uz G  
    s = 0:(n(j)-m(j))/2; \RyW#[(  
    pows = n(j):-2:m(j); Z6r_T  
    for k = length(s):-1:1 C+c;UzbD  
        p = (1-2*mod(s(k),2))* ... '${xZrzmt  
                   prod(2:(n(j)-s(k)))/          ... ^Zw1X6C5~  
                   prod(2:s(k))/                 ... FDO$(&  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... 1C\[n(9  
                   prod(2:((n(j)+m(j))/2-s(k))); 5i1Xumh 4  
        idx = (pows(k)==rpowers); VSa#X |z  
        z(:,j) = z(:,j) + p*rpowern(:,idx); M&K'5G)7  
    end m]:|j[!*M  
     TW?A/GoXI  
    if isnorm gS4@3BOw&.  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); Ucz`^}+  
    end 2q.J1:lW  
end VE5M}kDCZ  
RI jz7ZG  
% 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)  S0 AaJty  
2}P?N  
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。 xM3T7PV9  
e-dpk^-  
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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