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 q~qz^E\T
function z = zernfun(n,m,r,theta,nflag) }%KQrlbHJl
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. caEIE0H~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GVT 6cR
% and angular frequency M, evaluated at positions (R,THETA) on the 6Emn@Mn=
% unit circle. N is a vector of positive integers (including 0), and -n:2US<
% M is a vector with the same number of elements as N. Each element Yte*$cJ=
% k of M must be a positive integer, with possible values M(k) = -N(k) ,IiKe_B
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +aL6$
% and THETA is a vector of angles. R and THETA must have the same 9ERdjS
% length. The output Z is a matrix with one column for every (N,M)
4H;g"nWqO
% pair, and one row for every (R,THETA) pair. 2`i&6iz
% ~Dg:siw
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @Hj]yb5
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 6?"Gj}|r
% with delta(m,0) the Kronecker delta, is chosen so that the integral @G&oUhS
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, jzi%[c<G
% and theta=0 to theta=2*pi) is unity. For the non-normalized [?z;'O}y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ufR|V-BWx
% q4:zr
% The Zernike functions are an orthogonal basis on the unit circle. mcwd2)
% They are used in disciplines such as astronomy, optics, and li3X}
% optometry to describe functions on a circular domain. aR6~r^jB
% ,>6mc=p
% The following table lists the first 15 Zernike functions. 65B&>`H~
% dhLd2WSyH
% n m Zernike function Normalization covCa )kf
% -------------------------------------------------- FUI/ A>
% 0 0 1 1 m<Gd 6V5
% 1 1 r * cos(theta) 2 |QrVGm@2
% 1 -1 r * sin(theta) 2 W&A^.% 2l
% 2 -2 r^2 * cos(2*theta) sqrt(6) B{#Fm6
% 2 0 (2*r^2 - 1) sqrt(3) kb-XEJ}L
% 2 2 r^2 * sin(2*theta) sqrt(6) i{g~u<DH)Q
% 3 -3 r^3 * cos(3*theta) sqrt(8) &*Z)[Bl
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) p7},ymQ|YQ
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b#709VHm
% 3 3 r^3 * sin(3*theta) sqrt(8) x+sSmW
% 4 -4 r^4 * cos(4*theta) sqrt(10) NrcV%-+u%
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *"|f!t
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ;&b=>kPlZ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y}vV.q
% 4 4 r^4 * sin(4*theta) sqrt(10) =)#XZ[#F
% -------------------------------------------------- k H06Cb
% Kj"n
Id)
% Example 1: %i&am=
% f`}u9!jVR
% % Display the Zernike function Z(n=5,m=1) ?zo7.R-Vac
% x = -1:0.01:1; |r*y63\T
% [X,Y] = meshgrid(x,x); GWx?RIKF
% [theta,r] = cart2pol(X,Y); 1\jj3Y'i'
% idx = r<=1; 5=s|uuw/
% z = nan(size(X)); MNfc1I_#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Mt4`~`6
% figure #;2kN
&
% pcolor(x,x,z), shading interp 6_EfOD9
% axis square, colorbar IFSIQ
q
% title('Zernike function Z_5^1(r,\theta)') gd)VL}k
% d.sn D)X
% Example 2: N,)rrBD
% y_IF{%i
% % Display the first 10 Zernike functions i;2V
% x = -1:0.01:1; 4YMUkwh
% [X,Y] = meshgrid(x,x); Ud-c+, xX
% [theta,r] = cart2pol(X,Y); Swv
=gu
% idx = r<=1; m,J9:S<5;
% z = nan(size(X)); voN, u>U
% n = [0 1 1 2 2 2 3 3 3 3]; -z/>W+k
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Dk~
JH9#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; `yXHb
% y = zernfun(n,m,r(idx),theta(idx)); K>+c2;t;
% figure('Units','normalized') N8wA">u
% for k = 1:10 o<S(ODOfi
% z(idx) = y(:,k); Xp^71A?>
% subplot(4,7,Nplot(k)) Mc|UD*Z
% pcolor(x,x,z), shading interp :JxuaM8
% set(gca,'XTick',[],'YTick',[]) yV@~B;eW0
% axis square K?wo AuY
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) EU7mP
MxJ
% end U_0"1+jbq
% ~RM_c
% See also ZERNPOL, ZERNFUN2. [MM`#!K%
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q|1
% Paul Fricker 11/13/2006 }
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q-0(
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3 G*
% Check and prepare the inputs: ,|R\ Z,s
% ----------------------------- [{-;cpM\
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) k5Df97\s
error('zernfun:NMvectors','N and M must be vectors.') WGMEZx
end sU?%"q
SR'u*u!
if length(n)~=length(m) JLxAk14lc
error('zernfun:NMlength','N and M must be the same length.')
cCy*?P@
end .ktyA+r8v
[tz}H&
n = n(:); [)p>pA2GZj
m = m(:); >]8H@. \
if any(mod(n-m,2)) 2G`tS=Un
error('zernfun:NMmultiplesof2', ... [RUYH5>Ik
'All N and M must differ by multiples of 2 (including 0).') *p%=u>?&
end 6SD9lgF*-
RC]-9gd3Q
if any(m>n) "f`{4p0v
error('zernfun:MlessthanN', ... TzY[-YlvF
'Each M must be less than or equal to its corresponding N.') )1!*N)$
end 7%^/Jm
eN]9=Y~-K
if any( r>1 | r<0 ) k|
,F/:
error('zernfun:Rlessthan1','All R must be between 0 and 1.') g~$cnU
end h>'Mh;+
L Z#SX5N
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ],ISWb
error('zernfun:RTHvector','R and THETA must be vectors.') nx'D&,VX
end .q(1
v|u[BmA)*k
r = r(:); Wi{ jC?2Q
theta = theta(:); io(Rb\#"
length_r = length(r); gflu!C6
if length_r~=length(theta) .qk_m-o
error('zernfun:RTHlength', ... ;V\l,
u
'The number of R- and THETA-values must be equal.') ]eZrb%B.
end
UB1/0o
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% Check normalization: Qc]Ki3ls
% -------------------- BO)Q$*G~JD
if nargin==5 && ischar(nflag) m'.y,@^B
isnorm = strcmpi(nflag,'norm'); J PK(S~
if ~isnorm iqPMCOPZ
error('zernfun:normalization','Unrecognized normalization flag.') "_
i:
end ^8eu+E.{
else E#m|Sq
isnorm = false; #)N}F/Od^
end 8!(09gW'>
-9z!fCu3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =Hwlo!
% Compute the Zernike Polynomials 0'0GAh2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o\;cXuh
Sr?2~R0&
% Determine the required powers of r: Wc
qUF"A
% ----------------------------------- KN-)m ta&
m_abs = abs(m); e$ {Cf
rpowers = []; Ryrvu 1 k
for j = 1:length(n) %d#h<e|,.
rpowers = [rpowers m_abs(j):2:n(j)]; 05gdVa,
end (W4H?u@X0
rpowers = unique(rpowers); lo:{T_ay
Doj>Irj?7
% Pre-compute the values of r raised to the required powers, 6<h
==I
% and compile them in a matrix: Xe7/
% ----------------------------- >Kjl>bq
if rpowers(1)==0 zLda+
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ic(`E v
rpowern = cat(2,rpowern{:}); ;Wu6f"+Y#
rpowern = [ones(length_r,1) rpowern]; 7dbGUbT
else (6#,
$Ze
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7I`8r2H
rpowern = cat(2,rpowern{:}); I@/+=
end 4V9S~^v|
\&Zp/;n
% Compute the values of the polynomials: TtKV5
% -------------------------------------- FLzC kzJ:6
y = zeros(length_r,length(n)); #%$U-ti
for j = 1:length(n) waI:w,
s = 0:(n(j)-m_abs(j))/2; |qFCzK9tD/
pows = n(j):-2:m_abs(j); nA?Ks!9T
for k = length(s):-1:1 nWz7$O
p = (1-2*mod(s(k),2))* ... &K.js
prod(2:(n(j)-s(k)))/ ... vlS+UFH0
prod(2:s(k))/ ... s7>a
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... A5[iFT>
prod(2:((n(j)+m_abs(j))/2-s(k))); /_l$h_{DH
idx = (pows(k)==rpowers); L.tW]43K
y(:,j) = y(:,j) + p*rpowern(:,idx); &;wNJ)Uc
end C_ 4(-OWq
]w;!x7bU(
if isnorm P ")1_!
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +l) [A{
end "vL,c]D
end _(%;O:i
% END: Compute the Zernike Polynomials yJn<S@)VT:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^ 9`O
^
wXKg^%t\
% Compute the Zernike functions: :'0.
% ------------------------------ x@*!MC#
idx_pos = m>0; zz_(*0,Qcr
idx_neg = m<0; tEFbL~n
/fDXO;tN
z = y; JKy06I
if any(idx_pos) O
!
iN
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); nc/F@HCB
end dlJc~|
if any(idx_neg) eWWtMnq
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); F+Q(^Nk
end Sxzt|{
,|G~PC8
% EOF zernfun