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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 \9j +ejGf  
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! 'c7C*6;a  
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 O(x1Ja,&  
function z = zernfun(n,m,r,theta,nflag) Q*4{2oQ  
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y;2WY 0eq  
%   Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )!z4LE  
%   and angular frequency M, evaluated at positions (R,THETA) on the yp}J+/PX}  
%   unit circle.  N is a vector of positive integers (including 0), and 3v\69s  
%   M is a vector with the same number of elements as N.  Each element Qw>~] d,Z  
%   k of M must be a positive integer, with possible values M(k) = -N(k) O0^m_  
%   to +N(k) in steps of 2.  R is a vector of numbers between 0 and 1, o%3i(H  
%   and THETA is a vector of angles.  R and THETA must have the same e}lF#$  
%   length.  The output Z is a matrix with one column for every (N,M) Ckd j|  
%   pair, and one row for every (R,THETA) pair. ^UU@7cSi|G  
% kU :ge  
%   Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ` 1Ui  
%   functions.  The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zF: :?L~  
%   with delta(m,0) the Kronecker delta, is chosen so that the integral g0({$2Q7R  
%   of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, J9aqmQj('  
%   and theta=0 to theta=2*pi) is unity.  For the non-normalized "x1?T+j4  
%   polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 1 S<E=7  
% 3CL1Z\8To  
%   The Zernike functions are an orthogonal basis on the unit circle. ~mBY_[_s=  
%   They are used in disciplines such as astronomy, optics, and we:P_\6  
%   optometry to describe functions on a circular domain. wrP3:!=  
% arK(dg~S  
%   The following table lists the first 15 Zernike functions. HxUJ 0Q  
% z)%Ke~)<\@  
%       n    m    Zernike function           Normalization } H#C<:A  
%       -------------------------------------------------- _oz1'}=  
%       0    0    1                                 1 /]U),LbN  
%       1    1    r * cos(theta)                    2 %f)%FN . S  
%       1   -1    r * sin(theta)                    2 GJs{t1 E  
%       2   -2    r^2 * cos(2*theta)             sqrt(6) wjtFZGx&  
%       2    0    (2*r^2 - 1)                    sqrt(3) pyUzHF0  
%       2    2    r^2 * sin(2*theta)             sqrt(6) &/m0N\n?  
%       3   -3    r^3 * cos(3*theta)             sqrt(8) #W$6[#7=I  
%       3   -1    (3*r^3 - 2*r) * cos(theta)     sqrt(8) d+qeZGg^A  
%       3    1    (3*r^3 - 2*r) * sin(theta)     sqrt(8) Kz~E"?  
%       3    3    r^3 * sin(3*theta)             sqrt(8) 8I8{xt4   
%       4   -4    r^4 * cos(4*theta)             sqrt(10) KWS\iu  
%       4   -2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Six2{b)p  
%       4    0    6*r^4 - 6*r^2 + 1              sqrt(5) |@W|nbAfX  
%       4    2    (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U8S<wf&  
%       4    4    r^4 * sin(4*theta)             sqrt(10) xiv8q/  
%       -------------------------------------------------- `y3*\l  
% cqYMzS t  
%   Example 1: :3N6Ej  
% _ <Ip0?N  
%       % Display the Zernike function Z(n=5,m=1) n  +v(t  
%       x = -1:0.01:1; n\GN}?4  
%       [X,Y] = meshgrid(x,x); ^*G UcQ$  
%       [theta,r] = cart2pol(X,Y); t5CJG'!ql  
%       idx = r<=1; c( _R xLJ  
%       z = nan(size(X)); t/lQSUip  
%       z(idx) = zernfun(5,1,r(idx),theta(idx)); V= g u'~  
%       figure g& ou[_A  
%       pcolor(x,x,z), shading interp 63`5A3rii  
%       axis square, colorbar T O&^%d  
%       title('Zernike function Z_5^1(r,\theta)') }wB!Bx2  
% '2qbIYanh  
%   Example 2: r}:D g fn  
% vs^)=  
%       % Display the first 10 Zernike functions !k<k]^Z\  
%       x = -1:0.01:1; q*K[?  
%       [X,Y] = meshgrid(x,x); zr ~4@JTS  
%       [theta,r] = cart2pol(X,Y); :x,dYJm  
%       idx = r<=1; u g_c}Nv=Y  
%       z = nan(size(X)); *5u3d`bW  
%       n = [0  1  1  2  2  2  3  3  3  3]; }#q0K  
%       m = [0 -1  1 -2  0  2 -3 -1  1  3]; ')T*cLQ><  
%       Nplot = [4 10 12 16 18 20 22 24 26 28]; vL#I+_ 2  
%       y = zernfun(n,m,r(idx),theta(idx)); mGpBj9jr1  
%       figure('Units','normalized') mg< v9#  
%       for k = 1:10 \WqC^Di  
%           z(idx) = y(:,k); N(e>]ui  
%           subplot(4,7,Nplot(k)) n5 <B*  
%           pcolor(x,x,z), shading interp QYj*|p^x  
%           set(gca,'XTick',[],'YTick',[]) e6>[ZC  
%           axis square >E7s}bL"  
%           title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0h kZ  
%       end aA -j  
% A4*D3\>%u  
%   See also ZERNPOL, ZERNFUN2. \=[38?QOY  
+W/{UddeKU  
%   Paul Fricker 11/13/2006 zjTCq; G  
)xL_jSyh  
)8taMC:H^  
% Check and prepare the inputs: 9e7):ZupO  
% ----------------------------- _&N:%;9uD  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }R~C<3u\2  
    error('zernfun:NMvectors','N and M must be vectors.') I* P xQ  
end T2A74>Nw  
8PqlbLo1  
if length(n)~=length(m) o4^#W;%w  
    error('zernfun:NMlength','N and M must be the same length.') .zy2_3:  
end 7H4\AG\>  
VVEJE$  
n = n(:); YkQ=rurE  
m = m(:); L*P*^I^1  
if any(mod(n-m,2)) <'j ygZ(  
    error('zernfun:NMmultiplesof2', ... gk}.L E  
          'All N and M must differ by multiples of 2 (including 0).') mqBX1D`e2  
end XM3~]  
Abpzf\F  
if any(m>n) K#N5S]2yb  
    error('zernfun:MlessthanN', ... p`S~UBcL.  
          'Each M must be less than or equal to its corresponding N.') Gx|/ Jq  
end J! "m{ 8-  
x}f)P  
if any( r>1 | r<0 ) vos-[$  
    error('zernfun:Rlessthan1','All R must be between 0 and 1.') % C)|fDwN  
end { )4@rM  
rW)}$|-Z  
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F)50 6  
    error('zernfun:RTHvector','R and THETA must be vectors.') CHdYY7\{  
end #UnGU,J  
 "2 }n(8  
r = r(:); dCWq~[[  
theta = theta(:); &!*p>Ns)e  
length_r = length(r); .4!wp&  
if length_r~=length(theta) orEb+  
    error('zernfun:RTHlength', ... }cIj1:  
          'The number of R- and THETA-values must be equal.') "VeNc,-nfQ  
end "^t;V+Io  
W,%qL6qV  
% Check normalization: 1 y7$"N8Xo  
% -------------------- b:&= W>r  
if nargin==5 && ischar(nflag) '1lz`CAB+  
    isnorm = strcmpi(nflag,'norm'); <2\Q Y  
    if ~isnorm I^O`#SA(  
        error('zernfun:normalization','Unrecognized normalization flag.') ?YM0VB,y  
    end Iy2AJ|d.  
else 8WwLKZ}  
    isnorm = false; 5?TjuGc  
end ?o(Y\YJf  
,27=i>>  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7D4I>N'T  
% Compute the Zernike Polynomials /j:-GJb*!u  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UQ2;Dg G%  
#~6X9,x=  
% Determine the required powers of r: ^p~3H  
% ----------------------------------- C 2?p>S/q  
m_abs = abs(m); peU1 t:k?  
rpowers = []; &^ =Y76  
for j = 1:length(n) L_AQS9a^D  
    rpowers = [rpowers m_abs(j):2:n(j)]; f q*V76F  
end (P nrY~9  
rpowers = unique(rpowers); HTP~5J  
M5B?`mTl  
% Pre-compute the values of r raised to the required powers, T) cbpkH4  
% and compile them in a matrix: 3]/Y= A  
% ----------------------------- YifTC-Q;  
if rpowers(1)==0 m6 a @Y<  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [u8JqX  
    rpowern = cat(2,rpowern{:}); GnW_^$Fs  
    rpowern = [ones(length_r,1) rpowern]; Y.o-e)zX  
else E2|c;{ c  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;<v9i#K5  
    rpowern = cat(2,rpowern{:}); bhT:MW!  
end !%YV0O0  
H{*R(S<I  
% Compute the values of the polynomials: >c@1UEwkm  
% -------------------------------------- p:qj.ukw  
y = zeros(length_r,length(n)); 9/50+2F  
for j = 1:length(n) a~;`&Uj  
    s = 0:(n(j)-m_abs(j))/2; aEqDxr6  
    pows = n(j):-2:m_abs(j); $g)X,iQu  
    for k = length(s):-1:1 >l!DW i6  
        p = (1-2*mod(s(k),2))* ... +DP{_x)t  
                   prod(2:(n(j)-s(k)))/              ... rxAb]~MMp  
                   prod(2:s(k))/                     ... " ZFK-jn/  
                   prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 24/ ^_Td  
                   prod(2:((n(j)+m_abs(j))/2-s(k))); .JL?RH2@8  
        idx = (pows(k)==rpowers); (yi{<$ U*  
        y(:,j) = y(:,j) + p*rpowern(:,idx); ' |K408i   
    end WUqfY?5  
     0Bhf(5  
    if isnorm TfqQh!Y  
        y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 97(*-e=e  
    end $F86Dwd  
end . xdSUe  
% END: Compute the Zernike Polynomials 8Dy;'BtT  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~@bh[o~rF  
2M+'9 +k~  
% Compute the Zernike functions: ~m.@{Do0p  
% ------------------------------ DU-&bm  
idx_pos = m>0; ]Syr{|  
idx_neg = m<0; v}\Nx[}  
xA2 "i2k9  
z = y; TwXqk>J  
if any(idx_pos) Q#rj>+?  
    z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); S-k:+4  
end .`K<Iug1  
if any(idx_neg) S&YC"  
    z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Do5)ilt  
end ]J7.d$7T  
(-U6woB6o  
% EOF zernfun
niuhelen 2011-03-12 23:01
function z = zernfun2(p,r,theta,nflag) <)\  
%ZERNFUN2 Single-index Zernike functions on the unit circle. G\1\L*+0  
%   Z = ZERNFUN2(P,R,THETA) returns the Pth Zernike functions evaluated 3R sbi  
%   at positions (R,THETA) on the unit circle.  P is a vector of positive na1*^S`[  
%   integers between 0 and 35, R is a vector of numbers between 0 and 1, fSFb)+  
%   and THETA is a vector of angles.  R and THETA must have the same Q-}yZ  
%   length.  The output Z is a matrix with one column for every P-value, Akbt%&  
%   and one row for every (R,THETA) pair. 69$[yt>KYz  
% %Z=%E!*  
%   Z = ZERNFUN2(P,R,THETA,'norm') returns the normalized Zernike e1 j3X\ \  
%   functions, defined such that the integral of (r * [Zp(r,theta)]^2) @H^Yf  
%   over the unit circle (from r=0 to r=1, and theta=0 to theta=2*pi) a.yCd/  
%   is unity.  For the non-normalized polynomials, max(Zp(r=1,theta))=1 _c 4kj  
%   for all p. $Dm2>:Dmt  
% 'dstAlt?  
%   NOTE: ZERNFUN2 returns the same output as ZERNFUN, for the first 36 !w8t`Z['  
%   Zernike functions (order N<=7).  In some disciplines it is R6A{u(  
%   traditional to label the first 36 functions using a single mode M3U*'A\  
%   number P instead of separate numbers for the order N and azimuthal ~S,R`wo  
%   frequency M. j%m9y_rg}  
% #DI%l`B  
%   Example: z"n7du}v  
% K*_5M  
%       % Display the first 16 Zernike functions :mwJJIjUW  
%       x = -1:0.01:1; nM1F4G  
%       [X,Y] = meshgrid(x,x); zMf .  
%       [theta,r] = cart2pol(X,Y); M;\iL?,  
%       idx = r<=1; 5HZt5="+  
%       p = 0:15; /uM;g9 m  
%       z = nan(size(X)); |ZAR!u&0  
%       y = zernfun2(p,r(idx),theta(idx)); ? %9-5"U[  
%       figure('Units','normalized') WDC+Jmlgp  
%       for k = 1:length(p) U$fh ~w<[  
%           z(idx) = y(:,k); ([r4N#lx  
%           subplot(4,4,k) 1yS [;  
%           pcolor(x,x,z), shading interp +M"Fv9  
%           set(gca,'XTick',[],'YTick',[]) -r6cK,WVU  
%           axis square 4Y)rgLFj  
%           title(['Z_{' num2str(p(k)) '}']) G98P<cyD  
%       end "+g9}g  
% XvU^DEfW  
%   See also ZERNPOL, ZERNFUN. 9Q<8DMX^  
%8_bh8g-  
%   Paul Fricker 11/13/2006 v\3:R,|'  
'edd6yTd  
0@K?'6  
% Check and prepare the inputs: &p)]Cl/`  
% ----------------------------- ^]&uMkPN  
if min(size(p))~=1 sO,%Ok1  
    error('zernfun2:Pvector','Input P must be vector.') 5,I|beM  
end $?.0>0 ,<  
i|]Kw9  
if any(p)>35 =ZE]jmD4P  
    error('zernfun2:P36', ... ] rP^  
          ['ZERNFUN2 only computes the first 36 Zernike functions ' ... eD4qh4|u.  
           '(P = 0 to 35).']) -K 7jigac  
end )"W(0M] >  
^usZ&9"@P  
% Get the order and frequency corresonding to the function number: ''{REFjK7  
% ---------------------------------------------------------------- |,3>A@  
p = p(:); kK27hfsw  
n = ceil((-3+sqrt(9+8*p))/2); u?(@hUV.  
m = 2*p - n.*(n+2); jT~PwDSFt3  
M.|cl#  
% Pass the inputs to the function ZERNFUN: 0W92Z@_GY  
% ---------------------------------------- *&f^R}O  
switch nargin gn2*'_V~3  
    case 3 :!SVpCt3  
        z = zernfun(n,m,r,theta); s$cr|p;7#  
    case 4 yqEX0|V%  
        z = zernfun(n,m,r,theta,nflag); B-oQ 9[~  
    otherwise vD=>AAvG  
        error('zernfun2:nargin','Incorrect number of inputs.') O%g Q  
end L}E~CiL0n  
:bh#,]'  
% EOF zernfun2
niuhelen 2011-03-12 23:01
function z = zernpol(n,m,r,nflag) [~3[Tu( C  
%ZERNPOL Radial Zernike polynomials of order N and frequency M. y&ZyThqg  
%   Z = ZERNPOL(N,M,R) returns the radial Zernike polynomials of eP d  
%   order N and frequency M, evaluated at R.  N is a vector of R cZg/{[{  
%   positive integers (including 0), and M is a vector with the Dl a }-A:  
%   same number of elements as N.  Each element k of M must be a pyvH [  
%   positive integer, with possible values M(k) = 0,2,4,...,N(k) WH>=*\  
%   for N(k) even, and M(k) = 1,3,5,...,N(k) for N(k) odd.  R is ~4 ~c+^PF  
%   a vector of numbers between 0 and 1.  The output Z is a matrix I~^t\iujs  
%   with one column for every (N,M) pair, and one row for every dRdI('  
%   element in R. y:Wq;xEiDo  
% z\d{A7  
%   Z = ZERNPOL(N,M,R,'norm') returns the normalized Zernike poly- wbIgZ]o!/;  
%   nomials.  The normalization factor Nnm = sqrt(2*(n+1)) is \# p@ef  
%   chosen so that the integral of (r * [Znm(r)]^2) from r=0 to s+tPHftp  
%   r=1 is unity.  For the non-normalized polynomials, Znm(r=1)=1 @U8}K#  
%   for all [n,m]. |/qwR~  
% 'YKzs;y$  
%   The radial Zernike polynomials are the radial portion of the lOp7rW]$  
%   Zernike functions, which are an orthogonal basis on the unit 1^f7  
%   circle.  The series representation of the radial Zernike PBeBI:  
%   polynomials is S55h}5Y  
% 2)~`.CD?L  
%          (n-m)/2 i;flK*HOZ9  
%            __ ]u]BxMs  
%    m      \       s                                          n-2s syJLcK+e  
%   Z(r) =  /__ (-1)  [(n-s)!/(s!((n-m)/2-s)!((n+m)/2-s)!)] * r XIGz_g;#'w  
%    n      s=0 "{;E+-/ aL  
% t@&U2JaL>W  
%   The following table shows the first 12 polynomials. R@X65o  
% Wbq0K6X  
%       n    m    Zernike polynomial    Normalization z4qc)- {L  
%       --------------------------------------------- :17Pc\:DS  
%       0    0    1                        sqrt(2) g<E[IR  
%       1    1    r                           2 %,1xOl4l  
%       2    0    2*r^2 - 1                sqrt(6) Dru iiA  
%       2    2    r^2                      sqrt(6) rD\)ndPv  
%       3    1    3*r^3 - 2*r              sqrt(8) &<=?O a  
%       3    3    r^3                      sqrt(8) kqH:H~sgD  
%       4    0    6*r^4 - 6*r^2 + 1        sqrt(10) i_c'E;|  
%       4    2    4*r^4 - 3*r^2            sqrt(10) K7 J RCLA  
%       4    4    r^4                      sqrt(10) Sl;[9l2  
%       5    1    10*r^5 - 12*r^3 + 3*r    sqrt(12) P=[_W;->}  
%       5    3    5*r^5 - 4*r^3            sqrt(12) u|mTF>L  
%       5    5    r^5                      sqrt(12) qkM)zOZ^  
%       --------------------------------------------- }@y(-7t  
% `SH14A*  
%   Example: O"GuVC}B  
% YYN'LF#j  
%       % Display three example Zernike radial polynomials mo?*nO|-  
%       r = 0:0.01:1; 8nu@6)#  
%       n = [3 2 5]; 7D KTd^^M  
%       m = [1 2 1]; DFXHD,o  
%       z = zernpol(n,m,r); [  *~2Ts  
%       figure 2Ij,OIcdBE  
%       plot(r,z) 0H +!v  
%       grid on 9+iz+  
%       legend('Z_3^1(r)','Z_2^2(r)','Z_5^1(r)','Location','NorthWest') q^DQ9B  
% &7XsyDo6  
%   See also ZERNFUN, ZERNFUN2. Vb8Qh601  
N>H#Ew@2U  
% A note on the algorithm. |@1M'  
% ------------------------ Kqz+:E8D  
% The radial Zernike polynomials are computed using the series U2Tw_  
% representation shown in the Help section above. For many special S\SYFXUl  
% functions, direct evaluation using the series representation can l;SXR <EU  
% produce poor numerical results (floating point errors), because I/*^s  
% the summation often involves computing small differences between _P` ^B  
% large successive terms in the series. (In such cases, the functions .k-t5d  
% are often evaluated using alternative methods such as recurrence x[y}{T  
% relations: see the Legendre functions, for example). For the Zernike zIA)se Js  
% polynomials, however, this problem does not arise, because the vdcPpj^d5  
% polynomials are evaluated over the finite domain r = (0,1), and 9sN#l  
% because the coefficients for a given polynomial are generally all ``-pjD(t  
% of similar magnitude. Sy/Z}H  
% JvsL]yRT  
% ZERNPOL has been written using a vectorized implementation: multiple mKqXB\<  
% Zernike polynomials can be computed (i.e., multiple sets of [N,M] aX2N Qq>s  
% values can be passed as inputs) for a vector of points R.  To achieve O I0N(V  
% this vectorization most efficiently, the algorithm in ZERNPOL vqAEF^HYry  
% involves pre-determining all the powers p of R that are required to ~: fSD0  
% compute the outputs, and then compiling the {R^p} into a single AHo}K\O?r  
% matrix.  This avoids any redundant computation of the R^p, and :}R,a=N  
% minimizes the sizes of certain intermediate variables. #N$\d4q9  
% kWacc&*|  
%   Paul Fricker 11/13/2006 t2iQ[`/?~  
gq]@*C  
RrT`]1".  
% Check and prepare the inputs: e"%uOuIYX  
% ----------------------------- ck b(+*+l  
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~.4y* &  
    error('zernpol:NMvectors','N and M must be vectors.') {r~=mQ  
end WH"'Ju5}  
}4T`)  
if length(n)~=length(m) yk'L_M(=  
    error('zernpol:NMlength','N and M must be the same length.') 2acT w#  
end C+t0Zen  
JeN]sK)8x  
n = n(:); 2+/r~LwbK  
m = m(:); J(K/z,4h  
length_n = length(n); 4TI`   
RU=\eD  
if any(mod(n-m,2)) <5"&]! .  
    error('zernpol:NMmultiplesof2','All N and M must differ by multiples of 2 (including 0).') BNF*1JO  
end gj iFpW4  
($!uBF-b  
if any(m<0) lQiw8qD  
    error('zernpol:Mpositive','All M must be positive.') d'&OEGb<  
end Io"3wL)2  
kBLFK3i  
if any(m>n) +!W:gA  
    error('zernpol:MlessthanN','Each M must be less than or equal to its corresponding N.') y@,PTF  
end E=CAWj\  
lXF7)H&T  
if any( r>1 | r<0 ) 8#HnV%|N  
    error('zernpol:Rlessthan1','All R must be between 0 and 1.') 6.a5%:  
end pY[b[ezb  
`K:n=hpF  
if ~any(size(r)==1) IN@o9pUjV  
    error('zernpol:Rvector','R must be a vector.') 9XYm8g'X  
end IdMwpru(  
G'u[0>  
r = r(:); g4Q' Fub+I  
length_r = length(r); NMfHrYHbh  
J<0d"'  
if nargin==4 A'rd1"K  
    isnorm = ischar(nflag) & strcmpi(nflag,'norm'); H3*] }=   
    if ~isnorm zc=G4F01  
        error('zernpol:normalization','Unrecognized normalization flag.') by0K:*C  
    end :4-,Ru1C"  
else .%)uCLZr$  
    isnorm = false; @87Y/_l  
end NP\mzlI~@  
=4'V}p  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KO7&dM  
% Compute the Zernike Polynomials ~Kt1%&3{a?  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NtM ? Jh  
kX+9U"` C  
% Determine the required powers of r: Sgv_YoD?-  
% ----------------------------------- ]{# =WTp]  
rpowers = []; tc/  
for j = 1:length(n) I*^t!+q$  
    rpowers = [rpowers m(j):2:n(j)]; ?>U=bA  
end dt@c,McN|Q  
rpowers = unique(rpowers); [U",yN]d  
Bm,Vu 1]t  
% Pre-compute the values of r raised to the required powers, .D ^~!A  
% and compile them in a matrix: j'U1lEZm2  
% ----------------------------- x>B\2;  
if rpowers(1)==0 Ha|}Oj  
    rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2"?DaX  
    rpowern = cat(2,rpowern{:}); #) :.1Z?  
    rpowern = [ones(length_r,1) rpowern]; g)^s+Y  
else P`{$7ST'Hh  
    rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); vG'#5%,|  
    rpowern = cat(2,rpowern{:}); ~  z3J4s  
end 1P[x.t#  
,dTmI{@O  
% Compute the values of the polynomials: $A9Pi"/*z  
% -------------------------------------- P{UV3ZA%  
z = zeros(length_r,length_n); $l"%o9ICG  
for j = 1:length_n ?RS:I%bL  
    s = 0:(n(j)-m(j))/2; z`t~N  
    pows = n(j):-2:m(j); l*z% Jw  
    for k = length(s):-1:1 [.fh2XrVM  
        p = (1-2*mod(s(k),2))* ... p*,T~(A6  
                   prod(2:(n(j)-s(k)))/          ... ,3 /o7'  
                   prod(2:s(k))/                 ... !CX WoM  
                   prod(2:((n(j)-m(j))/2-s(k)))/ ... ObEz0Rj  
                   prod(2:((n(j)+m(j))/2-s(k))); *l{epum;  
        idx = (pows(k)==rpowers); 7`;f<QNo  
        z(:,j) = z(:,j) + p*rpowern(:,idx); 0&|0l>wy.  
    end  |@'O3KA  
     KS93v9|  
    if isnorm z,ERq,g+L  
        z(:,j) = z(:,j)*sqrt(2*(n(j)+1)); L55 UeP\  
    end "lw|EpQk`  
end )Ln".Bu,  
F/BR#J1  
% EOF zernpol
niuhelen 2011-03-12 23:03
这三个文件,我不知道该怎样把我的面型节点的坐标及轴向位移用起来,还烦请指点一下啊,谢谢啦!
li_xin_feng 2012-09-28 10:52
我也正在找啊
guapiqlh 2014-03-04 11:35
我也一直想了解这个多项式的应用,还没用过呢
phoenixzqy 2014-04-22 23:39
guapiqlh:我也一直想了解这个多项式的应用,还没用过呢 (2014-03-04 11:35)  NAvR^"I~  
*tF~CG$r  
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。 ^_bG{du  
Bxm,?=h  
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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