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 D(X:dB50@
function z = zernfun(n,m,r,theta,nflag) V7S[rI<<r
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. f*%Y]XL;%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &eA!h
% and angular frequency M, evaluated at positions (R,THETA) on the )(/Bw&$
% unit circle. N is a vector of positive integers (including 0), and /s~(? =qYH
% M is a vector with the same number of elements as N. Each element 4{v?<x8
% k of M must be a positive integer, with possible values M(k) = -N(k) GEs5@EH
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, XI5TVxo(q
% and THETA is a vector of angles. R and THETA must have the same Jc=~BT_G
% length. The output Z is a matrix with one column for every (N,M) O)FkpZc@9c
% pair, and one row for every (R,THETA) pair. >2^|r8l5
%
8MZ:=
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (ah^</
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &_1x-@oI2:
% with delta(m,0) the Kronecker delta, is chosen so that the integral -J&
b~t@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7*MjQzg-P
% and theta=0 to theta=2*pi) is unity. For the non-normalized eaWK2%v
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hy}n&h
% L> \/%x>Wx
% The Zernike functions are an orthogonal basis on the unit circle. ^[=1J
% They are used in disciplines such as astronomy, optics, and /EvnwYQy
% optometry to describe functions on a circular domain. hpBn_
% $/)0iL{0
% The following table lists the first 15 Zernike functions. XS_Ib\-50
% (>,}C/-UG
% n m Zernike function Normalization 4#Rq}/h
% -------------------------------------------------- 8mnzxtk
% 0 0 1 1 zI&).
% 1 1 r * cos(theta) 2 X[E!q$ag
% 1 -1 r * sin(theta) 2 ?y|8bw<
% 2 -2 r^2 * cos(2*theta) sqrt(6) 3E$h
W
% 2 0 (2*r^2 - 1) sqrt(3) FdE9k\E#/)
% 2 2 r^2 * sin(2*theta) sqrt(6) +\GuZ5`
% 3 -3 r^3 * cos(3*theta) sqrt(8) gk^`-`P
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @|;XDO`k;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) EJm*L6>@R&
% 3 3 r^3 * sin(3*theta) sqrt(8) ;kLp}CqV
% 4 -4 r^4 * cos(4*theta) sqrt(10) 8eDKN9kq
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y{`hRz`
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5)
W*Gp0pX
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VD0U]~CWR
% 4 4 r^4 * sin(4*theta) sqrt(10)
B@K =^77
% -------------------------------------------------- JfVGs;_,
% _OY<Hb3%M
% Example 1: Aw,#oG {N
% dMDSyd<(
% % Display the Zernike function Z(n=5,m=1) FV>xAU$
% x = -1:0.01:1; $1.l|
% [X,Y] = meshgrid(x,x); JrJTIUf_
% [theta,r] = cart2pol(X,Y); @D2KDV3'
% idx = r<=1; p}MH LM
% z = nan(size(X)); #(dERET*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); I`KBj6n
% figure G&,2>qxKR
% pcolor(x,x,z), shading interp `\Hs{t]
% axis square, colorbar )A*Sl2ew
% title('Zernike function Z_5^1(r,\theta)') jx-8%dxtZ
% K/D,sH!
% Example 2: Y^ti;:
% _/RP3" #
% % Display the first 10 Zernike functions q,fk@GI'2
% x = -1:0.01:1; :qxd
s>Xm
% [X,Y] = meshgrid(x,x); kOLS<>.
% [theta,r] = cart2pol(X,Y); Yvxp(
% idx = r<=1; 1+NmiGKg
% z = nan(size(X)); fud Lm
% n = [0 1 1 2 2 2 3 3 3 3]; gt:Ot0\7
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Xb5$ijH
% Nplot = [4 10 12 16 18 20 22 24 26 28]; SX6P>:`
% y = zernfun(n,m,r(idx),theta(idx)); d
A' h7D
% figure('Units','normalized') OJ4-p&1
% for k = 1:10 ~glFB`?[
% z(idx) = y(:,k); BGZvgMxLJ
% subplot(4,7,Nplot(k)) -"X}
)N2
% pcolor(x,x,z), shading interp n 7m!
% set(gca,'XTick',[],'YTick',[]) SPY4l*kX
% axis square d){Al(/
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) }RY&f4&GV,
% end x|IG'R1:Y
% CJ
9tO#R
% See also ZERNPOL, ZERNFUN2. Bl8&g]dk
wA>bL PTw
% Paul Fricker 11/13/2006 bcy(
?(
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o]@g%_3X
% Check and prepare the inputs: :fE*fU@
% ----------------------------- h|
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O-K!Bv^
Q
error('zernfun:NMvectors','N and M must be vectors.') +gsk}>"
end 8L}N,6gC4_
s7#|'jhZt
if length(n)~=length(m) rXR}]|;>
error('zernfun:NMlength','N and M must be the same length.') R@H}n3,
end )gq(
Y2Y!^A89
n = n(:); )B'U_*
m = m(:); ;o0o6pF
if any(mod(n-m,2)) *tZ#^YG{(
error('zernfun:NMmultiplesof2', ... -?Aa RwZ,
'All N and M must differ by multiples of 2 (including 0).') m%?b"kxL[
end tXIre-. 2}
CJNz J(
if any(m>n) 4D\+_Ic3
error('zernfun:MlessthanN', ... P!)k 4n
'Each M must be less than or equal to its corresponding N.') %C8fv|@:f
end D3emO'`gQ
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if any( r>1 | r<0 ) {\HE'C/?
error('zernfun:Rlessthan1','All R must be between 0 and 1.') FE/2.!]&o
end ^D0BGC&&
NR)[,b\v
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :4D#hOI
error('zernfun:RTHvector','R and THETA must be vectors.') !jDqRXi(
end ?ixzlDto\
UVD::
r = r(:); 9/k?Lv
theta = theta(:); !u#o"e<qh
length_r = length(r); IBzHXa>75
if length_r~=length(theta) kty,hAXe
error('zernfun:RTHlength', ... }PY?
ZG
'The number of R- and THETA-values must be equal.') K,IPVjS
end ]41G!'E=
V8xv@G{;
% Check normalization:
ka&-tGg
% -------------------- \g}FoN&