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 rvd$4l^
function z = zernfun(n,m,r,theta,nflag) %|(c?`2|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ~SQxFAto
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N F,p0OL.
% and angular frequency M, evaluated at positions (R,THETA) on the 6I@j$edZ
% unit circle. N is a vector of positive integers (including 0), and P{n#^4
% M is a vector with the same number of elements as N. Each element ?x #K:a?
% k of M must be a positive integer, with possible values M(k) = -N(k) dz9U.:C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }<A.zwB<i
% and THETA is a vector of angles. R and THETA must have the same Re8x!e'>
% length. The output Z is a matrix with one column for every (N,M) c("|xe
% pair, and one row for every (R,THETA) pair. El<*)
% *tF~CG$r
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike b/z-W`gw
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `sCaGCp
% with delta(m,0) the Kronecker delta, is chosen so that the integral 4Lt9Dx1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, N2}SR|.
% and theta=0 to theta=2*pi) is unity. For the non-normalized S"Cz.
bv
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. QE(.w
dHP
% :'Qiwf&
% The Zernike functions are an orthogonal basis on the unit circle. _ Ncbo#G
% They are used in disciplines such as astronomy, optics, and [v"Z2F<.=
% optometry to describe functions on a circular domain. j1K3|E
% {'O><4
% The following table lists the first 15 Zernike functions. }UW7py!TN
% %vJHr!x
% n m Zernike function Normalization }%jF!d
% -------------------------------------------------- :jl*Y-mM
% 0 0 1 1 +{I_%SsG
% 1 1 r * cos(theta) 2 .Ix3wR9
% 1 -1 r * sin(theta) 2 'V:Q :
% 2 -2 r^2 * cos(2*theta) sqrt(6) V^2_]VFj
% 2 0 (2*r^2 - 1) sqrt(3) n(F!t,S1i
% 2 2 r^2 * sin(2*theta) sqrt(6) FbE/x$;~O
% 3 -3 r^3 * cos(3*theta) sqrt(8) m;OvOc,
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) d+JK")$9C
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 2!/Kt
O)i^
% 3 3 r^3 * sin(3*theta) sqrt(8) N6y9'LGG`
% 4 -4 r^4 * cos(4*theta) sqrt(10) F<* / J]
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !,Uo{@E)Y
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5)
7=6:ZSI
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ff#7}9_mh
% 4 4 r^4 * sin(4*theta) sqrt(10) ]<f)Rf">:`
% -------------------------------------------------- ANhtz1Fl
% .{1$;K @
% Example 1: ]ZY2\'
% 2zBk#c+
% % Display the Zernike function Z(n=5,m=1) Js,! G
% x = -1:0.01:1; NfgXOLthM
% [X,Y] = meshgrid(x,x);
r6m^~Wq!}
% [theta,r] = cart2pol(X,Y); F(G..XJQ
% idx = r<=1; Bs~~C8+
% z = nan(size(X)); O sgPNy0
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?*fa5=ql
% figure <&5z0rDKWw
% pcolor(x,x,z), shading interp }T?X6LA$I8
% axis square, colorbar G$<(>"Yr~$
% title('Zernike function Z_5^1(r,\theta)') >f]/VaMH{
% AjVC{\Ik
% Example 2: CY1WT
% E=s h^Q(A
% % Display the first 10 Zernike functions %6m/ve
% x = -1:0.01:1; Mg2+H+C~:
% [X,Y] = meshgrid(x,x); |p|Zv H
% [theta,r] = cart2pol(X,Y); 8 1,N92T5
% idx = r<=1; G]K1X"W?
% z = nan(size(X)); iiPVqU%
% n = [0 1 1 2 2 2 3 3 3 3]; ;sB=f
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; l;; 2\mL?
% Nplot = [4 10 12 16 18 20 22 24 26 28]; E'AR.!
% y = zernfun(n,m,r(idx),theta(idx)); *QC6zJ
% figure('Units','normalized') my'nDi
% for k = 1:10 -c`xeuzK'
% z(idx) = y(:,k); %F*9D3^h
% subplot(4,7,Nplot(k)) mxv?PP
% pcolor(x,x,z), shading interp (Z),gxt
% set(gca,'XTick',[],'YTick',[]) BhJ>G%
% axis square E)v~kC}7.
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) voa)V1A/]
% end
0,Ds1y^
% -^@FZR^Y
% See also ZERNPOL, ZERNFUN2. x5lVb$!G
r&u1-%%9[
% Paul Fricker 11/13/2006
|Xso}Y{
m eF7[>!U
C;BO6$*_e
% Check and prepare the inputs: 5aQ)qUgAW
% ----------------------------- $S6(V}yh
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LRJX>+@
error('zernfun:NMvectors','N and M must be vectors.') `Skvqo(5:
end QQJGqM3a2
Aiq Kf=
if length(n)~=length(m)
?8>a;0
error('zernfun:NMlength','N and M must be the same length.') PR{ubMn
end #7uH>\r
7e<=(\(yl
n = n(:); ti5fsc
m = m(:); BtJkvg(2]
if any(mod(n-m,2)) /J`}o}
error('zernfun:NMmultiplesof2', ... lu#a.41
'All N and M must differ by multiples of 2 (including 0).') CsR[@&n'
end )vtbA=RH?
-laH^<jm5
if any(m>n) HSruue8
error('zernfun:MlessthanN', ... {cdICWy(F3
'Each M must be less than or equal to its corresponding N.') uLdHE5vr
end (hc!!:N~q
>tg)F|@
if any( r>1 | r<0 ) }8O9WS
error('zernfun:Rlessthan1','All R must be between 0 and 1.') NEBhVh
end 6i/unwe!`)
H1N@E}> |
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e~vO
error('zernfun:RTHvector','R and THETA must be vectors.') g@H<Q('fJ
end vn.5X
R@\fqNq
r = r(:); 1hbQ30
theta = theta(:); 0:{W
t
length_r = length(r); 6~dAK3v5
if length_r~=length(theta) rJ/HIda
error('zernfun:RTHlength', ... 0akJv^^D
'The number of R- and THETA-values must be equal.') _`2%)#^o
end [if(B\&
V9[_aP;
% Check normalization: 1d<?K7%^
% -------------------- tB;PGk_6
if nargin==5 && ischar(nflag) h7]+#U]mi
isnorm = strcmpi(nflag,'norm'); 4"?`p;{Z
if ~isnorm _a&gbSQv
error('zernfun:normalization','Unrecognized normalization flag.') |gkNhxzB
end +*.*bo
else g$Tsht(rHD
isnorm = false; ,ei9 ?9J1
end ~&:-c v
fw %p_Cm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Q<>u)%92@
% Compute the Zernike Polynomials 7(/yyZQnZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nOC\ =<Nsg
L|[i<s;
% Determine the required powers of r: 3Ei^WDJ
% ----------------------------------- 9fp"r,aHN&
m_abs = abs(m); -zECxHjx
rpowers = []; &>-'|(m+2
for j = 1:length(n) 1c,#`\Iikd
rpowers = [rpowers m_abs(j):2:n(j)]; /l`zZ>
end mxqZj8VuH
rpowers = unique(rpowers); V@0T&#
t__f=QB/
% Pre-compute the values of r raised to the required powers, kQI'kL8>
% and compile them in a matrix: $mxG-'x%K
% ----------------------------- WvU[9ME^)
if rpowers(1)==0 b GSj?t9/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); aPJTH0u
rpowern = cat(2,rpowern{:}); Xau%v5r
rpowern = [ones(length_r,1) rpowern]; YusmMsN?
else |X{j^JP5
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); U*nB=
=
rpowern = cat(2,rpowern{:}); )d[n-Si
end Jk{SlH3'
)pI( <
% Compute the values of the polynomials: 3MX#}_7A
% -------------------------------------- @zGF9O<3,@
y = zeros(length_r,length(n)); 5CnNp?.t^
for j = 1:length(n) S^R dj ]
s = 0:(n(j)-m_abs(j))/2; T6y~iNd<
pows = n(j):-2:m_abs(j); R1JD{
for k = length(s):-1:1 \=({T_j4
p = (1-2*mod(s(k),2))* ... t<Sa;[+
prod(2:(n(j)-s(k)))/ ... o4: e1
prod(2:s(k))/ ... _"*vj-{-y
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &SIf