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 ]:Ocu--
function z = zernfun(n,m,r,theta,nflag) {Km|SG[-q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `L7 cS
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N XOVZ'V
% and angular frequency M, evaluated at positions (R,THETA) on the "kVN|Do
% unit circle. N is a vector of positive integers (including 0), and 5qR76iH)/
% M is a vector with the same number of elements as N. Each element Z9 }qds6 y
% k of M must be a positive integer, with possible values M(k) = -N(k) =}u;>[3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }a-ikFQ]
% and THETA is a vector of angles. R and THETA must have the same I)O%D3wfMW
% length. The output Z is a matrix with one column for every (N,M) IcI y
% pair, and one row for every (R,THETA) pair. v #IC
% cSoZq4
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike el5F>)
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9F ).i
% with delta(m,0) the Kronecker delta, is chosen so that the integral OA&N WAm4
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Cf2rRH
% and theta=0 to theta=2*pi) is unity. For the non-normalized Nbuaw[[iz
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5"]PwC
% : :e=6i
% The Zernike functions are an orthogonal basis on the unit circle. _nOio ?
% They are used in disciplines such as astronomy, optics, and $bD 3
% optometry to describe functions on a circular domain. 82efqzT
% M'R^?Jjb
% The following table lists the first 15 Zernike functions. /Y|9!{.
% ir3iW*5k
% n m Zernike function Normalization C[_{ $j(J
% -------------------------------------------------- ^kt#[N
% 0 0 1 1 VS1gg4tCv
% 1 1 r * cos(theta) 2 C} Ewi-
% 1 -1 r * sin(theta) 2 wF$8#=
% 2 -2 r^2 * cos(2*theta) sqrt(6) NJLU+byU
% 2 0 (2*r^2 - 1) sqrt(3) qA
Jgz7=c
% 2 2 r^2 * sin(2*theta) sqrt(6) E':y3T@."
% 3 -3 r^3 * cos(3*theta) sqrt(8) C
`k^So)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ukzXQe;l1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) >x(^g~i
% 3 3 r^3 * sin(3*theta) sqrt(8) h&;\
% 4 -4 r^4 * cos(4*theta) sqrt(10) H2p1gb#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) S!up2OseW
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) gXc&uR0S
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /,c9&it(M
% 4 4 r^4 * sin(4*theta) sqrt(10) T->O5t c
% -------------------------------------------------- !>
% I]jVnQ>&
% Example 1: -QI1>7sl
% oIQor%z
% % Display the Zernike function Z(n=5,m=1) WVf;uob{
% x = -1:0.01:1; ATPc~f
% [X,Y] = meshgrid(x,x); \E]s]ft;+
% [theta,r] = cart2pol(X,Y); \CX`PZ><
% idx = r<=1; Gk'J'9*
% z = nan(size(X)); w!8h4U.
;
% z(idx) = zernfun(5,1,r(idx),theta(idx)); r6*0H/*
% figure 52{jq18&
% pcolor(x,x,z), shading interp MpGWt#
% axis square, colorbar 8&3+=<U
% title('Zernike function Z_5^1(r,\theta)') V~NS<!+q
% *~:4&$
% Example 2: 3:dQN;=
% - "h
{B
% % Display the first 10 Zernike functions q J@XVN4
% x = -1:0.01:1; & i)p^AmM
% [X,Y] = meshgrid(x,x); Z\4l+.R`
% [theta,r] = cart2pol(X,Y); I>C;$Lp]
% idx = r<=1; | t3_E
% z = nan(size(X)); wvBJ?t,
% n = [0 1 1 2 2 2 3 3 3 3]; C4#'`8E
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; <+
>y GPp
% Nplot = [4 10 12 16 18 20 22 24 26 28]; DLJu%5F
% y = zernfun(n,m,r(idx),theta(idx)); n)^B0DnIk
% figure('Units','normalized') MJ4+|riB
% for k = 1:10 ;_1D-Mf
% z(idx) = y(:,k); ,^`+mP
% subplot(4,7,Nplot(k)) f.,S-1D]h
% pcolor(x,x,z), shading interp GwxfnCKi9
% set(gca,'XTick',[],'YTick',[]) KZsSTB6J
% axis square G0xk @SE
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nhiCV>@y
% end &5~bJ]P
% - YJ7ne]
% See also ZERNPOL, ZERNFUN2. Z
r
gM^ Hs7o,
% Paul Fricker 11/13/2006 }gGcYRT
bTb|@
|r%6;8A]i
% Check and prepare the inputs: g()YP
% ----------------------------- l"*zr ;#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W7_X=>l
error('zernfun:NMvectors','N and M must be vectors.') HT[<~c
end o~*% g.
SB:-zQ5
if length(n)~=length(m) PZ
AyHXY
error('zernfun:NMlength','N and M must be the same length.') |z-A;uL <
end ys u"+J
CM!bD\5
n = n(:); PL%U
m = m(:); Z ZX|MA!
if any(mod(n-m,2)) :-69,e
error('zernfun:NMmultiplesof2', ... -'*B%yy
'All N and M must differ by multiples of 2 (including 0).') Oz-X}eM
end [y W0U:m
OLXG0@
if any(m>n) 8I$>e (
error('zernfun:MlessthanN', ... &?#V*-;^
'Each M must be less than or equal to its corresponding N.') o vvR{MTc
end !w(J]<
g;UB+Y 247
if any( r>1 | r<0 ) es6!p 7p?
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z[[qW
f
end tux/@}I
|p-, B>p!
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) v8I&~_b
error('zernfun:RTHvector','R and THETA must be vectors.') >DP9S@W
end bLhTgss](
H"=%|/1M0
r = r(:); iT227v!s
theta = theta(:); xVfAlN37(
length_r = length(r); AVF(YD<U
if length_r~=length(theta) ; {iX_%
error('zernfun:RTHlength', ... TMpV.iH
'The number of R- and THETA-values must be equal.') .hzzoLI2
end 6c$ so
SDwTGQ/0
% Check normalization: LPc)-t|p"
% -------------------- wqk D
if nargin==5 && ischar(nflag)
{^a"T'+
isnorm = strcmpi(nflag,'norm'); | (JxtQqQg
if ~isnorm G3
rTzMO
error('zernfun:normalization','Unrecognized normalization flag.') iVTC"v
end NjrF":'Y
else l=GcgxD+"d
isnorm = false; u!hY
bCB
end C&