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 =x7ODBYW^
function z = zernfun(n,m,r,theta,nflag) [w{ZP4d>
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y\op9Fw
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N `Mjm/9+18
% and angular frequency M, evaluated at positions (R,THETA) on the "Y%\qw/wq
% unit circle. N is a vector of positive integers (including 0), and l w%fY{
% M is a vector with the same number of elements as N. Each element R(zsn;
% k of M must be a positive integer, with possible values M(k) = -N(k) 2sU"p5 j
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, IcQ?^9%{
% and THETA is a vector of angles. R and THETA must have the same KDXo9FzF
% length. The output Z is a matrix with one column for every (N,M) {xH
\!!"T
% pair, and one row for every (R,THETA) pair. %kc g#p+tE
% {^\-%3$
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bTiw?i+6Dv
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B"qG-ci
% with delta(m,0) the Kronecker delta, is chosen so that the integral {'b8;x8h
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SHGO;
% and theta=0 to theta=2*pi) is unity. For the non-normalized K[ \z'9Q
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. kqyMrZ#
% TgUQD(d^
% The Zernike functions are an orthogonal basis on the unit circle. {[s<\<~B*
% They are used in disciplines such as astronomy, optics, and ScTqnY$v
% optometry to describe functions on a circular domain. 9V"j=1B}
% {$EXI]f
% The following table lists the first 15 Zernike functions. b~Ruhi[E
% 5sE^MS1
% n m Zernike function Normalization G{"1I
% -------------------------------------------------- y&CUT:M6
% 0 0 1 1 MO D4O4z&
% 1 1 r * cos(theta) 2 I#Bz
UF
% 1 -1 r * sin(theta) 2 cr/|dc'
% 2 -2 r^2 * cos(2*theta) sqrt(6) T+[e6/|
% 2 0 (2*r^2 - 1) sqrt(3) <N*>9S,}
% 2 2 r^2 * sin(2*theta) sqrt(6) >ciq4H43Q|
% 3 -3 r^3 * cos(3*theta) sqrt(8) \
bhok
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) c !;wp,c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) m!2Dk#t
% 3 3 r^3 * sin(3*theta) sqrt(8) B.WJ6.DkS
% 4 -4 r^4 * cos(4*theta) sqrt(10) {c1qC zM4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +/X'QB$R
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5{5ABV
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Yn#8uaU
% 4 4 r^4 * sin(4*theta) sqrt(10) H|!s.
% -------------------------------------------------- 6,7omYof
% 7*5ctc!dG
% Example 1: Stc\P]%d
% 4tC_W!?$t
% % Display the Zernike function Z(n=5,m=1) Qnw$=L:
% x = -1:0.01:1; =I5XG"",
% [X,Y] = meshgrid(x,x); esHiWHAC
% [theta,r] = cart2pol(X,Y); _qg6(
X
% idx = r<=1; joA+
% z = nan(size(X)); \1Bgs^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 35>}$1?-6
% figure 6a@~;!GlI
% pcolor(x,x,z), shading interp gP<_DEd^`
% axis square, colorbar s6D-?G*u%8
% title('Zernike function Z_5^1(r,\theta)') wY95|QS
% S3_4i;K\
% Example 2: l+6\U6_)B
% ]/bE${W*]
% % Display the first 10 Zernike functions 'l:2R,cP
% x = -1:0.01:1; y#0w\/<
% [X,Y] = meshgrid(x,x); ]R@G5d
% [theta,r] = cart2pol(X,Y);
p4t)Z#0
% idx = r<=1; 9PJDT]
% z = nan(size(X)); </X"*G't
% n = [0 1 1 2 2 2 3 3 3 3]; SSXS
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @5wg' mM
% Nplot = [4 10 12 16 18 20 22 24 26 28]; r83~o/T@
% y = zernfun(n,m,r(idx),theta(idx)); hkJZqUA
% figure('Units','normalized') )
b10%n^
% for k = 1:10 2X*<Fma3C
% z(idx) = y(:,k); k)s 7Ev*
% subplot(4,7,Nplot(k)) @"!SU'*
% pcolor(x,x,z), shading interp p5l$On
% set(gca,'XTick',[],'YTick',[]) gp)ds^
% axis square @9h#o5y q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =M5M;
% end ='0!B]<G
% _);Kb/
% See also ZERNPOL, ZERNFUN2. }/spo3,6
+][P*/ Ek
% Paul Fricker 11/13/2006 { 9 ".o,
ra>`J_
,7P^]V1
% Check and prepare the inputs: ~-`02
% ----------------------------- d*$<%J
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %B*dj9n^q
error('zernfun:NMvectors','N and M must be vectors.') =LxmzQO#
end uw=Ube(
<gLtX[v!CL
if length(n)~=length(m) $0}bi:7
error('zernfun:NMlength','N and M must be the same length.') r6JkoPMh
end ts<dUO
YSo7~^1W"
n = n(:); fZ}Y(TG/
m = m(:); 5V~p@vCx
if any(mod(n-m,2)) Zk
UuniO
error('zernfun:NMmultiplesof2', ... ok[=1gA#h
'All N and M must differ by multiples of 2 (including 0).') 9M]"%E!s
end suFOc
n-3j$x1Ne
if any(m>n) ,,@`l\Pgd
error('zernfun:MlessthanN', ... `HG19_Z
'Each M must be less than or equal to its corresponding N.') =jc8=h[F<
end Lc<xgN+cJ
K9Xd?
]a
if any( r>1 | r<0 ) HFuaoS+b*
error('zernfun:Rlessthan1','All R must be between 0 and 1.') "GI&S% F
end WgJAr73
l
Us,[x Q
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (V.,~t@
error('zernfun:RTHvector','R and THETA must be vectors.') 7/_ VE
end 5jV97x)BGx
>JPJ%~y
r = r(:); 5w)^~#'
theta = theta(:); ~e77w\Q0
length_r = length(r); Sn2Ds)Pfx3
if length_r~=length(theta) *}ee"eHs
error('zernfun:RTHlength', ... "P5bYq%0v
'The number of R- and THETA-values must be equal.') A}bHfn|
end ^>8]3@ Nh
U?fN3
% Check normalization: F[D0x26^
% -------------------- QYfAf3te
if nargin==5 && ischar(nflag) nX\]i~
isnorm = strcmpi(nflag,'norm'); BrH;(*H)8
if ~isnorm CKt|c!3 7
error('zernfun:normalization','Unrecognized normalization flag.') ht3T{4qCS
end P!+nZXo
else !Vr45l
isnorm = false; )^f9[5ee
end 9LO.8Jy
%C`'>,t>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `3y!XET
% Compute the Zernike Polynomials cbCE
$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M=[q+A
`x$}~rP&)!
% Determine the required powers of r: e*2&s5 #RT
% ----------------------------------- .\~P -{Hd
m_abs = abs(m); 8#]7`o
rpowers = []; NnLhJPh
for j = 1:length(n) )yNw2+ ~5
rpowers = [rpowers m_abs(j):2:n(j)]; T]#,R|)d
end FK@ f'
rpowers = unique(rpowers); R_>TEYZ
Q;XHHk
% Pre-compute the values of r raised to the required powers, nK jeH@
% and compile them in a matrix: 1%hM8:)i_
% ----------------------------- `@$"L/AJ
if rpowers(1)==0 85|95P.<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $}^\=p}X
rpowern = cat(2,rpowern{:}); MeI2i
rpowern = [ones(length_r,1) rpowern]; NB+$ym
else \'??
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7"n1it[RJ8
rpowern = cat(2,rpowern{:}); c. TB8Ol
end !q-:rW?c
? gA=39[j
% Compute the values of the polynomials: )-.Cne;n
% -------------------------------------- -.b
I o
y = zeros(length_r,length(n)); ^\vfos
for j = 1:length(n) 20/P M9
s = 0:(n(j)-m_abs(j))/2; =tS[&6/
pows = n(j):-2:m_abs(j); 9*=@/1
for k = length(s):-1:1 }+{*, z
p = (1-2*mod(s(k),2))* ... hINnb7o
prod(2:(n(j)-s(k)))/ ... Q"OV>kl k
prod(2:s(k))/ ... q: Bt]2x
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... x0u?*5-t
prod(2:((n(j)+m_abs(j))/2-s(k))); Qh|-a@
idx = (pows(k)==rpowers); V#zhGAMy.
y(:,j) = y(:,j) + p*rpowern(:,idx); -B *<Q[_
end ''(fH$pY
vn0cKz@
if isnorm hi {2h04
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); vLnq%@x
end 6+Wr6'kuH
end mmrW`~-
% END: Compute the Zernike Polynomials ZVdsxo<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^_*jp[!`b$
x2j/8]'o
% Compute the Zernike functions: -7-Fd_F8
% ------------------------------ 5W[3_P+
idx_pos = m>0; j8[`~pb
idx_neg = m<0; ]cF1c90%
W(uP`M%][0
z = y; VY+(,\)U
if any(idx_pos) x{NNx:T1
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ><;l:RGK|
end A*7Io4e!
if any(idx_neg) qJ{r!NJJ
8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f?=r3/AO
end Kk!6B
="3a%\
% EOF zernfun