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 6x^$W ]R
function z = zernfun(n,m,r,theta,nflag) s| p I`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. b`X''6
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N oPi>]#X
% and angular frequency M, evaluated at positions (R,THETA) on the BwT[SI<Sg
% unit circle. N is a vector of positive integers (including 0), and >._d2.Q'
% M is a vector with the same number of elements as N. Each element n^nE&'[?0g
% k of M must be a positive integer, with possible values M(k) = -N(k) l@);U%\pS
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, oz&`3`
% and THETA is a vector of angles. R and THETA must have the same 9JFN8Gf*)
% length. The output Z is a matrix with one column for every (N,M) BpIyw
% pair, and one row for every (R,THETA) pair. 'dwW~4|B
% ~
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike `n$Ak5f
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }xsO^K
% with delta(m,0) the Kronecker delta, is chosen so that the integral {<yapBMw
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, CY o
m
% and theta=0 to theta=2*pi) is unity. For the non-normalized HAn{^8"@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Vg'R=+Wb
% LwB1~fF
% The Zernike functions are an orthogonal basis on the unit circle. iTHwH{!
% They are used in disciplines such as astronomy, optics, and ~A>fB2.pM
% optometry to describe functions on a circular domain. necY/&Ld-
% `/sNX<mp
% The following table lists the first 15 Zernike functions. HJ&P[zV^
% i >3`V6
% n m Zernike function Normalization -m@c{&r
% -------------------------------------------------- c~hH
7/v
% 0 0 1 1 FW-I|kK.
% 1 1 r * cos(theta) 2 `N\ ^JAGW
% 1 -1 r * sin(theta) 2 P}4&J ^
% 2 -2 r^2 * cos(2*theta) sqrt(6)
^xHKoOTj[
% 2 0 (2*r^2 - 1) sqrt(3) ZxvH1qx8
% 2 2 r^2 * sin(2*theta) sqrt(6) l\Ozy
% 3 -3 r^3 * cos(3*theta) sqrt(8) ( eKgc
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) JX0M3|I=
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) :UdW4N-
% 3 3 r^3 * sin(3*theta) sqrt(8) W'4/cO
% 4 -4 r^4 * cos(4*theta) sqrt(10) [-\ Y?3
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4o#]hB';ni
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) k3bQ32()
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WX4sTxJK
% 4 4 r^4 * sin(4*theta) sqrt(10) k'iiRRM
% -------------------------------------------------- _UVpQ5pN
% _9>,9aL
% Example 1: jq
H)o2"/
% _%Z.Re
% % Display the Zernike function Z(n=5,m=1) <);q,|eh2
% x = -1:0.01:1; CtY-Gs
% [X,Y] = meshgrid(x,x); o^epXIrIPi
% [theta,r] = cart2pol(X,Y); g}%ODa !H
% idx = r<=1; QYbB\Y
% z = nan(size(X)); (m3hD)!+y
% z(idx) = zernfun(5,1,r(idx),theta(idx)); F?6kkLS/
% figure {\5(aQ)Vi5
% pcolor(x,x,z), shading interp e_b,{l#
% axis square, colorbar 9p8ajlYg,
% title('Zernike function Z_5^1(r,\theta)') N|i>|2EB
% y11^q*}
% Example 2: UIEvwQ
% 7RT{RE
% % Display the first 10 Zernike functions O: :FB.k
% x = -1:0.01:1; !l*A3qA
% [X,Y] = meshgrid(x,x); 3uYLA4[-B
% [theta,r] = cart2pol(X,Y); 2BC!,e$Z
% idx = r<=1; Ubu&$4a
% z = nan(size(X)); [R4#bl
% n = [0 1 1 2 2 2 3 3 3 3]; x/<ow4C
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; VVQ~;{L
% Nplot = [4 10 12 16 18 20 22 24 26 28];
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% y = zernfun(n,m,r(idx),theta(idx)); i$y=tJehi
% figure('Units','normalized') {jD?obs
% for k = 1:10 |V5BL<4
% z(idx) = y(:,k); _YX% M|#
% subplot(4,7,Nplot(k)) (GRW(Zd4
% pcolor(x,x,z), shading interp 2xN7lfu1RB
% set(gca,'XTick',[],'YTick',[]) Vs5 &X+k
% axis square h.tj8O1
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %uo8z~+
% end a>GA=r
% nC3+Zka
% See also ZERNPOL, ZERNFUN2. L9/'zhiZBx
ZJ{DW4#t
% Paul Fricker 11/13/2006 O
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}!^h2)'7
b_Y+XXb<
% Check and prepare the inputs: a >fA-@
% ----------------------------- KJFQ)#SW!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !po,Z&
error('zernfun:NMvectors','N and M must be vectors.') S+06pj4Ie
end wA{)9.
I0Do%
if length(n)~=length(m) L~ax`i1:"
error('zernfun:NMlength','N and M must be the same length.') k
Fl*Im
end HVvm3qu4
q5g_5^csM{
n = n(:); O5du3[2x7a
m = m(:); #xmiUN,|
if any(mod(n-m,2)) q2
7Ac;y
error('zernfun:NMmultiplesof2', ... ANPG3^w
'All N and M must differ by multiples of 2 (including 0).') ]/!*^;cY(
end GYw/KT~$
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if any(m>n) .:H'9QJg
error('zernfun:MlessthanN', ... O#igH
'Each M must be less than or equal to its corresponding N.') }|h-=T '
end {Q/@ Y.~<
9>+>s ?IgK
if any( r>1 | r<0 ) =x w:@(]{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') g{DOQA
end NH/jkt&F[
leHKBu'd
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h`fZ8|yw
error('zernfun:RTHvector','R and THETA must be vectors.') 5%S5*c6BD
end b5g^{bzwu
ip'v<%,Q3"
r = r(:); _`Kh8G
{e
theta = theta(:); R&s/s`pLW
length_r = length(r); yYOV:3!"
if length_r~=length(theta) h1>.w
pr
error('zernfun:RTHlength', ... Uj 3{c
'The number of R- and THETA-values must be equal.') WL%T nux
end _BG`!3U+
_6FDuCVD-
% Check normalization: e7G>'K
% -------------------- y3*IF2G
if nargin==5 && ischar(nflag) pnz@;+f
isnorm = strcmpi(nflag,'norm'); Ct/6<
if ~isnorm @W+8z#xr'
error('zernfun:normalization','Unrecognized normalization flag.') ^?%ThPo_
end JKmd'ZGw
else "~C\Z} ;
isnorm = false; BvlY\^
end ,_,7cor
Z[+Qf3j}o6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L%9yFg%u
% Compute the Zernike Polynomials #oGvxc7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P)TeF1~T
5}NO~Xd<
% Determine the required powers of r: \l6mXIn=>
% ----------------------------------- @Ng q+uXm
m_abs = abs(m); ku^2K
rpowers = []; hy"p8j7_
for j = 1:length(n) GmGq69]J*
rpowers = [rpowers m_abs(j):2:n(j)]; <.7W:s,f=
end a(o[ bH.|;
rpowers = unique(rpowers); /7*qa G
lSId<v?C>
% Pre-compute the values of r raised to the required powers, AMgvk`<f
% and compile them in a matrix: nDC5/xB
% ----------------------------- BcGQpv&x
if rpowers(1)==0 ]*S_fme
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ,@gDY9Q3r/
rpowern = cat(2,rpowern{:}); /=OSGIJzm
rpowern = [ones(length_r,1) rpowern]; of<>M4/g4y
else Pb D|7IM
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); \v_t:
"
rpowern = cat(2,rpowern{:}); ~?A,GalS
end = &aD!nTx
Y@%6*uTLa
% Compute the values of the polynomials: xcIZ'V
% -------------------------------------- :kI
x?cc
y = zeros(length_r,length(n)); UE\@7
for j = 1:length(n) &4MVk3SLx#
s = 0:(n(j)-m_abs(j))/2; 48%a${Nvvj
pows = n(j):-2:m_abs(j); Ll&5#q
for k = length(s):-1:1 -p!KsU
p = (1-2*mod(s(k),2))* ... p|%Y\!
prod(2:(n(j)-s(k)))/ ... >Q\H1|?
prod(2:s(k))/ ... ?t.?f`(|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cfe[6N
prod(2:((n(j)+m_abs(j))/2-s(k))); FkECY
idx = (pows(k)==rpowers); f<'&_*7,|t
y(:,j) = y(:,j) + p*rpowern(:,idx); Zk;;~ESOU
end uJp}9B60_
/0YNB)
if isnorm k0D&F;a%
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); kAk,:a;P
end s9:2aLZ{
end Z*e7W O.
% END: Compute the Zernike Polynomials "AVj]jR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1aMBCh<}JN
U ._1'pW
% Compute the Zernike functions: 0_y%Qj^e
% ------------------------------ w)8@Tu:Q
idx_pos = m>0; LP)mp cQ
idx_neg = m<0; N$,)vb<
@x J^JcE
z = y; x}>tX
if any(idx_pos) n_ez6{
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ujWHO$uz!
end /7"1\s0 U
if any(idx_neg) tw3d>H`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); z=Vvb
end =L
wX+c
>`\*{]
% EOF zernfun