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 mITNx^p4f
function z = zernfun(n,m,r,theta,nflag) ):_@i
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5rmU9L
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :}yT?LIyP
% and angular frequency M, evaluated at positions (R,THETA) on the Ta[\BWR2
% unit circle. N is a vector of positive integers (including 0), and 9;'#,b*(
% M is a vector with the same number of elements as N. Each element Xo:Mar
% k of M must be a positive integer, with possible values M(k) = -N(k) hbg$u$1`,
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, l2kGFgc
% and THETA is a vector of angles. R and THETA must have the same ~8yh,U
% length. The output Z is a matrix with one column for every (N,M) sQJGwZ7
% pair, and one row for every (R,THETA) pair. |j-ng;
% T9I$6HAi
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike S43JaSw)
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B]H8^
% with delta(m,0) the Kronecker delta, is chosen so that the integral 7nPcm;Er
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, x9AFN
% and theta=0 to theta=2*pi) is unity. For the non-normalized S@Rd>4
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. l^d' 8n
% 0>8w On
% The Zernike functions are an orthogonal basis on the unit circle. /`l;u7RD
% They are used in disciplines such as astronomy, optics, and YVwpqOE.=
% optometry to describe functions on a circular domain. )|v y}Jf7
% vJaWHC$q
% The following table lists the first 15 Zernike functions. +ZwoA_k{
% l=b!O
% n m Zernike function Normalization 0ki- /{;
% -------------------------------------------------- &y}7AV
% 0 0 1 1 ,0a_ou"P=_
% 1 1 r * cos(theta) 2 xnt) 1Q
% 1 -1 r * sin(theta) 2 'Y#'ozSQv
% 2 -2 r^2 * cos(2*theta) sqrt(6) :SS \2
% 2 0 (2*r^2 - 1) sqrt(3) E]rXp~AZm
% 2 2 r^2 * sin(2*theta) sqrt(6) -iS^VzI|I
% 3 -3 r^3 * cos(3*theta) sqrt(8) N<8\.z5:<
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Y+UJV6
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 6^WNwe\
% 3 3 r^3 * sin(3*theta) sqrt(8) _
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% 4 -4 r^4 * cos(4*theta) sqrt(10) L2,2Sn*4i
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4"k &9+>
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) '9Z`y_~)G
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pa1<=w
% 4 4 r^4 * sin(4*theta) sqrt(10) xa@$cxt
% -------------------------------------------------- NJQ)Ttt
% 8W{M}>;[9
% Example 1: O-X(8<~H=
% |~e"i<G#
% % Display the Zernike function Z(n=5,m=1) OemY'M?ZQ
% x = -1:0.01:1; W`_JERo
% [X,Y] = meshgrid(x,x); -R]0cefC<f
% [theta,r] = cart2pol(X,Y); ewU*5|*[
% idx = r<=1; jkx>o?s)z
% z = nan(size(X)); Lo%vG{yTr
% z(idx) = zernfun(5,1,r(idx),theta(idx)); YD'gyP4
% figure ?a3wBy
% pcolor(x,x,z), shading interp \^*:1=|7u]
% axis square, colorbar &J&'J~N
% title('Zernike function Z_5^1(r,\theta)') *:@KpYWx"
% o 2Nu@^+
% Example 2: :31_WJ^
% "t&=~eOe3
% % Display the first 10 Zernike functions G;}WZy
% x = -1:0.01:1; 1hY| XZ%qd
% [X,Y] = meshgrid(x,x); iR nj N
% [theta,r] = cart2pol(X,Y); s|e.mZk/
% idx = r<=1; q* p
% z = nan(size(X)); =^rt?F4
% n = [0 1 1 2 2 2 3 3 3 3]; !xfDWbvHV
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; SK\@w9#&$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; M; wKTTQy
% y = zernfun(n,m,r(idx),theta(idx)); U
L
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% figure('Units','normalized') B18BwY
% for k = 1:10 SG)Fk *1
% z(idx) = y(:,k); j|[rT^b@
% subplot(4,7,Nplot(k)) q$:7j5E
% pcolor(x,x,z), shading interp {6v.(Zlh$
% set(gca,'XTick',[],'YTick',[]) `!vqT 3p,
% axis square YWK0.F,8a
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) b^$`2m-?@f
% end bW6| &P}X
% \Nt
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% See also ZERNPOL, ZERNFUN2. *'-4%7C`1
dn#I,xa`
% Paul Fricker 11/13/2006 uaF-3
%=UD~5!G0
YCD|lL#
% Check and prepare the inputs: TRGpE9i
% ----------------------------- v`Jt+?I
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o~~;I
error('zernfun:NMvectors','N and M must be vectors.') .D
4G;=Q
end jg710.v:
'Gn>~m
if length(n)~=length(m) ojy^A
error('zernfun:NMlength','N and M must be the same length.') r >'tE7W9
end O`~#X w
lV$JCNe
n = n(:); -wXeue},>
m = m(:); rE+B}O
if any(mod(n-m,2)) .t_t)'L
error('zernfun:NMmultiplesof2', ... GQtNk<?$I
'All N and M must differ by multiples of 2 (including 0).') 4=^_VDlpd
end T)\}V#iA*
]3O&8,
if any(m>n) eTa_RO,x
error('zernfun:MlessthanN', ... i<"lXu
'Each M must be less than or equal to its corresponding N.') 2t\0vV2)/O
end ='h2z"}\Bn
4/b.;$
if any( r>1 | r<0 ) \_`qon$9
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 61S;M8tNv
end e'K~WNT
5skN'*oG
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) G4@r_VP \
error('zernfun:RTHvector','R and THETA must be vectors.') }3{eVct#|
end A9.TRKb=8
1p}H,\o
r = r(:); SboHo({5VA
theta = theta(:); 1C<cwd;9
length_r = length(r); c<_%KL&R
if length_r~=length(theta) |{ N{VK
error('zernfun:RTHlength', ... x(Bt[=,K3
'The number of R- and THETA-values must be equal.') 6$R9Y.s>Z
end /f#b;qa,
;ek*2Lh
% Check normalization: CPOHqK`k
% -------------------- 3+ 6Ed;P
if nargin==5 && ischar(nflag) (Mk7"FC7
isnorm = strcmpi(nflag,'norm'); ~m6=s~Vn
if ~isnorm $,}jz.R@
error('zernfun:normalization','Unrecognized normalization flag.') }p~2lOI
end nQiZ6[L
else <o%T]
isnorm = false; WQ9e~D"
end `dZ|Ko%k
[|Qzx w9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <8:h%%$?
% Compute the Zernike Polynomials LCB-ewy#E
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% RCWmdR#}V
q^aDZzx,z
% Determine the required powers of r: : "85w#r
% ----------------------------------- Wlc&QOfF
m_abs = abs(m); /.SG? 5t4
rpowers = []; JIyS e:p3
for j = 1:length(n) w)EYj+L
rpowers = [rpowers m_abs(j):2:n(j)]; AQ'%}(#0
end fp [gKRSF
rpowers = unique(rpowers); ]}v]j`9m%
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