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 (c{<JYEC
function z = zernfun(n,m,r,theta,nflag) tkN5|95
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ypoJ4EZ(
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N B&tU~
% and angular frequency M, evaluated at positions (R,THETA) on the z}Qt6na]-
% unit circle. N is a vector of positive integers (including 0), and ;NyX9&@
% M is a vector with the same number of elements as N. Each element {V> >a
% k of M must be a positive integer, with possible values M(k) = -N(k) `%8by y@$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Y#'?3
% and THETA is a vector of angles. R and THETA must have the same f}4bnu3
% length. The output Z is a matrix with one column for every (N,M) CC(At.dd
% pair, and one row for every (R,THETA) pair. |@}Yady@C
% zi^T?<t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6[-N})
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H#/}FoBiS
% with delta(m,0) the Kronecker delta, is chosen so that the integral Z3ucJH/)V
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^|z
% and theta=0 to theta=2*pi) is unity. For the non-normalized SA5
g~{"
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. p8%/T>hK
% ZDmBuf
q
% The Zernike functions are an orthogonal basis on the unit circle. :{iS0qJ
% They are used in disciplines such as astronomy, optics, and ?m)3n0Uh
% optometry to describe functions on a circular domain. Q%.V\8#|V
% XO*|P\#^
% The following table lists the first 15 Zernike functions. RHV&m()Q
% G0Q8"]
% n m Zernike function Normalization 2#sJ`pdQ
% -------------------------------------------------- <X7x
% 0 0 1 1 &^R0kCF`
% 1 1 r * cos(theta) 2 "V|1w>s
% 1 -1 r * sin(theta) 2 [LwmzmV+F
% 2 -2 r^2 * cos(2*theta) sqrt(6) IF<?TYy=3B
% 2 0 (2*r^2 - 1) sqrt(3) ;C1]gJZ,
% 2 2 r^2 * sin(2*theta) sqrt(6) *vx!twu1o
% 3 -3 r^3 * cos(3*theta) sqrt(8) 8vhg{L..
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) TFX*kk&R
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ])dq4\Bw
% 3 3 r^3 * sin(3*theta) sqrt(8) 99'e)[\
% 4 -4 r^4 * cos(4*theta) sqrt(10) gm**9]k ^{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u$7od$&S
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) n'<FH<x
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <&n\)R4C1
% 4 4 r^4 * sin(4*theta) sqrt(10) Vb0((c%&
% -------------------------------------------------- eq0&8/=
% p[E}:kak_-
% Example 1: uG1)cm
B}
% D^(Nijl9U
% % Display the Zernike function Z(n=5,m=1) }L.xt88
% x = -1:0.01:1; gO0X-fN8
% [X,Y] = meshgrid(x,x); beLT4~Z=
% [theta,r] = cart2pol(X,Y); :iWW2fY
% idx = r<=1; JXG%Cx!2}
% z = nan(size(X)); jhd&\z-
% z(idx) = zernfun(5,1,r(idx),theta(idx)); C_SJ4Sh
% figure HZp}<7NR(7
% pcolor(x,x,z), shading interp 2}Ga
% axis square, colorbar aCu 8
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% title('Zernike function Z_5^1(r,\theta)') K{eq'F5M
% Ga5O&`h
% Example 2: IMaa#8,
% <cQ)*~hN
% % Display the first 10 Zernike functions #0K122oY
% x = -1:0.01:1; sdk%~RN0T
% [X,Y] = meshgrid(x,x);
.;ptgX
% [theta,r] = cart2pol(X,Y); <:[P&Y
% idx = r<=1; L : hEt
% z = nan(size(X)); |7$Fr[2d
% n = [0 1 1 2 2 2 3 3 3 3]; ZT*RD2,
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !(:R=J_h
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *v+xKy#M
% y = zernfun(n,m,r(idx),theta(idx)); AE1EZ#
% figure('Units','normalized') RR,gC"cTi
% for k = 1:10 #r\,oXTm
% z(idx) = y(:,k); Ns ?8N":
% subplot(4,7,Nplot(k)) ^Ht!~So
% pcolor(x,x,z), shading interp Gqe?CM
% set(gca,'XTick',[],'YTick',[]) ?`wO
\>y
% axis square 2Zf}t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) dso6ZRx
% end V)[ta`9
% PQ6.1}
% See also ZERNPOL, ZERNFUN2. [)K?e!c8
q)Qd+:a7{
% Paul Fricker 11/13/2006 V`F]L^m=L
PL;PId<9w
wR)U&da`@
% Check and prepare the inputs: 6Fp}U
% ----------------------------- QWqEe|}6
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i98>=y~
error('zernfun:NMvectors','N and M must be vectors.') B=E<</i
end mmE!!J`B
Q-scL>IkCb
if length(n)~=length(m) Lye^G%{
error('zernfun:NMlength','N and M must be the same length.') [sx J<
end R#D>m8&}3
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n = n(:); xa7~{ E,
m = m(:); k!9LJ%Xh
if any(mod(n-m,2)) "eqN d"~
error('zernfun:NMmultiplesof2', ... j2@19YXe@
'All N and M must differ by multiples of 2 (including 0).') ]yc&ffe%
end t0^chlJP$
jc%
if any(m>n) u"WqI[IV
error('zernfun:MlessthanN', ... 9$]I3k
'Each M must be less than or equal to its corresponding N.') 0?x9.]
end XTzz/.T;Z
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if any( r>1 | r<0 ) WA~PE` U
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2P&KU%D)0s
end F 7v 1rf]
R^[b
I;
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $2tPqZ>
error('zernfun:RTHvector','R and THETA must be vectors.') L?aaR%6#
end mmN!=mf*
W3AtO
r = r(:); _9y
theta = theta(:); 6p=OM=R
length_r = length(r); u\)2/~<]
if length_r~=length(theta) vKX6@eg"
error('zernfun:RTHlength', ... Kx8>
'The number of R- and THETA-values must be equal.') EbG`q!C
end gb_r <j:w
J5i$D0K[
% Check normalization: #YABbwH
% -------------------- 8`I/\8;H'p
if nargin==5 && ischar(nflag) p\>im+0oh
isnorm = strcmpi(nflag,'norm'); dV~d60jOF
if ~isnorm #kmZS/"
error('zernfun:normalization','Unrecognized normalization flag.') @<n8?"{5S
end ;+86q"&n
else ;%#.d$cU
isnorm = false; ,PmQ}1kGW
end MQ~OG9.
HB/q
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Qp]-:b
% Compute the Zernike Polynomials t<UtSkE1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ym]Dlz,o
y2_^lW%
% Determine the required powers of r: S2^Ckg
% ----------------------------------- cH==OM7&-
m_abs = abs(m); Q!%C:b
rpowers = []; ITUwIpAE
for j = 1:length(n) LTof$4s
rpowers = [rpowers m_abs(j):2:n(j)];
!623;
end P&6hk6#
rpowers = unique(rpowers); 1u%e7
R)[ l3
% Pre-compute the values of r raised to the required powers, o ?9k{
% and compile them in a matrix: *5Mg^}ZC5
% ----------------------------- Qz[4M` M
if rpowers(1)==0 vk^ /[eha
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Q')0 T>F-
rpowern = cat(2,rpowern{:}); $ts%SDM
rpowern = [ones(length_r,1) rpowern]; oo+nqc`,O
else &EZq%Sd
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); | e&v;48
rpowern = cat(2,rpowern{:}); BAJEn6f?
end }mhD2 ' E
BGe&c,feIc
% Compute the values of the polynomials: `S&$y4|Vs
% -------------------------------------- Za5bx,^
y = zeros(length_r,length(n)); CH`_4UAX%
for j = 1:length(n) xs'vd:l.Pp
s = 0:(n(j)-m_abs(j))/2; \W;+@w|c
pows = n(j):-2:m_abs(j); MO1t0My c
for k = length(s):-1:1 7aV(tMzd
p = (1-2*mod(s(k),2))* ... BLno/JK0}
prod(2:(n(j)-s(k)))/ ... .b3cn
prod(2:s(k))/ ... e>GX]tK
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;$0)k(c9
prod(2:((n(j)+m_abs(j))/2-s(k))); nMBKZ
idx = (pows(k)==rpowers); SLj2/B0
y(:,j) = y(:,j) + p*rpowern(:,idx);
Z>O2
end F74^HQ*J
=Nc}XFq
if isnorm 3lZ5N@z69
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Z*M]AvO+#
end 0_A|K>7
end CP%?,\
% END: Compute the Zernike Polynomials 3ZAPcpB2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1TuN
e1
yvvi
% Compute the Zernike functions: szDd!(&pv
% ------------------------------ u> YC4&
idx_pos = m>0; (,i&pgVZ
idx_neg = m<0; $_u9Y!
ZQ0R3=52r
z = y; O%Mi`\W@
if any(idx_pos) j9bn|p$DA
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |k3^
eeLk
end Bq20U:f
if any(idx_neg) R
_c!
,y
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (B0tgg^jj,
end ;QiSz=DyA
RTEzcJ>
% EOF zernfun