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 6b:tyQ
function z = zernfun(n,m,r,theta,nflag) v-PXZ'7~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. R;j!}D!4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N kOed ]>H
% and angular frequency M, evaluated at positions (R,THETA) on the *FM Mjz
% unit circle. N is a vector of positive integers (including 0), and }b-g*dn]5
% M is a vector with the same number of elements as N. Each element (_"*NY0
% k of M must be a positive integer, with possible values M(k) = -N(k) og
kD^
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, N|6MP
e
% and THETA is a vector of angles. R and THETA must have the same >M`CVUf
% length. The output Z is a matrix with one column for every (N,M) *"{lMZ+
% pair, and one row for every (R,THETA) pair. `I3r3WyA
% W' s
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !Ze5)g%H
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), GgB,tam{p
% with delta(m,0) the Kronecker delta, is chosen so that the integral HbxL:~:}J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hK_LEwd;
% and theta=0 to theta=2*pi) is unity. For the non-normalized K|Di1)7=/
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *#C+iAF|)'
% ~FN9 [aJF+
% The Zernike functions are an orthogonal basis on the unit circle. {o>j6RS\
% They are used in disciplines such as astronomy, optics, and yI"6Da6|y
% optometry to describe functions on a circular domain. wf:OK[r9
% zlEX+=3
% The following table lists the first 15 Zernike functions. 1
=M ?GDc
% SF>c\eTtx
% n m Zernike function Normalization pNKhc#-w
% -------------------------------------------------- < Pky9o;
% 0 0 1 1 f>N!wgo[
% 1 1 r * cos(theta) 2 3 yB!M
% 1 -1 r * sin(theta) 2 `nZ )>
% 2 -2 r^2 * cos(2*theta) sqrt(6) d%o&+l#
% 2 0 (2*r^2 - 1) sqrt(3) 5.MGaU^Z$
% 2 2 r^2 * sin(2*theta) sqrt(6) zc;|fHW~O
% 3 -3 r^3 * cos(3*theta) sqrt(8) )s%[T-uKi
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) TL}++e
7+
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iT%} $Lu~
% 3 3 r^3 * sin(3*theta) sqrt(8) p{j.KI s7
% 4 -4 r^4 * cos(4*theta) sqrt(10) c1E'$-
K@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :R~MO&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~x]jB
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) bp~g;h*E2
% 4 4 r^4 * sin(4*theta) sqrt(10) 'LE=6{#
% -------------------------------------------------- `pGa~!vl
% /N&CaH\;^$
% Example 1: /\4'ddGU
% z}MP)|aH:
% % Display the Zernike function Z(n=5,m=1) ;e1ku|>$
% x = -1:0.01:1; $d_|NssvU
% [X,Y] = meshgrid(x,x); 5) pj]S!]-
% [theta,r] = cart2pol(X,Y); O4og?h>
% idx = r<=1; Vz= PiMO
% z = nan(size(X)); !Rhlf.x
% z(idx) = zernfun(5,1,r(idx),theta(idx)); XBp? w
% figure ]% IT|/;9Y
% pcolor(x,x,z), shading interp +,#$:fs u
% axis square, colorbar W7 #9jo
% title('Zernike function Z_5^1(r,\theta)') '*"vkgN
% Ljp%CI[i
% Example 2: V 7Ek-2M
% USM4r!x
% % Display the first 10 Zernike functions $w);5o
% x = -1:0.01:1; cT!\{~
% [X,Y] = meshgrid(x,x); F!]lU`z)=
% [theta,r] = cart2pol(X,Y); Q+W1lv8R
% idx = r<=1; q)^Jj?W
% z = nan(size(X)); PqiB\~o@Z
% n = [0 1 1 2 2 2 3 3 3 3]; 9Ru8~R/\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; mjKS{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; r}mbXvn
% y = zernfun(n,m,r(idx),theta(idx)); J
/f
% figure('Units','normalized') .ZJRO>S
% for k = 1:10 }wHW7SJ
% z(idx) = y(:,k); t3&LO~Ye
% subplot(4,7,Nplot(k)) tX>
G,hw
% pcolor(x,x,z), shading interp IHcD*zQ
% set(gca,'XTick',[],'YTick',[]) &3TEfvz
% axis square hKT
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !g(KK|`,m
% end A*}.EClH
%
l%1!a
% See also ZERNPOL, ZERNFUN2. '8{Ne!y
2-C!jAfd
% Paul Fricker 11/13/2006 BA%pY|"Q
(K)] qNH
(4dhuT
% Check and prepare the inputs: }Du}c3
% ----------------------------- >U]C/P[+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dAkgR~
error('zernfun:NMvectors','N and M must be vectors.') >76\nGO
end Q=/</|
zhpt%7So
if length(n)~=length(m) o!{w"K
error('zernfun:NMlength','N and M must be the same length.') #w\~&0
end ^7 &5
z&o
t ]_VG
n = n(:); 32/MkuY^u
m = m(:); 2E)wpgUc?e
if any(mod(n-m,2)) JAQb{KefdO
error('zernfun:NMmultiplesof2', ... S/ODqL|
'All N and M must differ by multiples of 2 (including 0).') %Ntcvp)
end O"c;|zCc>
5`gQ~
if any(m>n) .xH5fMj,"
error('zernfun:MlessthanN', ... /q5v"iX]T
'Each M must be less than or equal to its corresponding N.') RkBb$q9F]
end JQ6zVS2SSS
9&` 2V
if any( r>1 | r<0 ) O0pDd4)"
error('zernfun:Rlessthan1','All R must be between 0 and 1.') NR5oIKP?
end C86J
IC"
i5K[>5
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /eOzXCSws
error('zernfun:RTHvector','R and THETA must be vectors.') ]2\VweV
end ] 0B2#
d
Ft;^g3N
r = r(:); w &(|e <
theta = theta(:); INi]R^-
length_r = length(r); t_qNq{
if length_r~=length(theta) 0^RXGN
error('zernfun:RTHlength', ... {O`w,dMOI
'The number of R- and THETA-values must be equal.') i* NH'o/
end HY%i`]4X
Y#lk6
% Check normalization: ZfgJ.<<
% -------------------- s#tZg
if nargin==5 && ischar(nflag) !=:$lzS^
isnorm = strcmpi(nflag,'norm'); TG+VEL |T
if ~isnorm k+8q{5>A<
error('zernfun:normalization','Unrecognized normalization flag.') yX0dbW~@y
end < VSA
else nEkR1^30
isnorm = false; zOa_X~!@
end :"IE
yRfSJbzaf\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *UmI]E{g3(
% Compute the Zernike Polynomials }t%!9hr5D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }XSfst5-H
kT!9`S\
% Determine the required powers of r: _oUHJ~&,
% ----------------------------------- I{*<