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 g7@.Fa.u'!
function z = zernfun(n,m,r,theta,nflag) "^&Te%x_b
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. _<m yM2z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N a"bael
% and angular frequency M, evaluated at positions (R,THETA) on the >4iVVs
% unit circle. N is a vector of positive integers (including 0), and aYrbB#
% M is a vector with the same number of elements as N. Each element /pYp,ak
% k of M must be a positive integer, with possible values M(k) = -N(k) ipH'}~=ID
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;tG@ 6
% and THETA is a vector of angles. R and THETA must have the same S<Od`I
% length. The output Z is a matrix with one column for every (N,M) 1 Q6~O2a
% pair, and one row for every (R,THETA) pair. nz_1Fu>g|
% kpLx?zW--q
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o|bm=&f
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), vH)V\V
% with delta(m,0) the Kronecker delta, is chosen so that the integral \I+#M-V
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, }+dDGFk
% and theta=0 to theta=2*pi) is unity. For the non-normalized 6!$2nK+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. pZV=Co3!I
% k#DMd9
% The Zernike functions are an orthogonal basis on the unit circle. kS1?%E,)q
% They are used in disciplines such as astronomy, optics, and !63]t?QXMG
% optometry to describe functions on a circular domain. G-Dc(QhU&
% r"bV{v
% The following table lists the first 15 Zernike functions. MR}h}JEx0
% %pBc]n@_
% n m Zernike function Normalization #CTeZ/g
% -------------------------------------------------- y41,T&ja
% 0 0 1 1 gvCQ![
% 1 1 r * cos(theta) 2 ~Hb2-V
% 1 -1 r * sin(theta) 2 7x//4G
% 2 -2 r^2 * cos(2*theta) sqrt(6) ck\TTNA
% 2 0 (2*r^2 - 1) sqrt(3) BVe c
% 2 2 r^2 * sin(2*theta) sqrt(6) .
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% 3 -3 r^3 * cos(3*theta) sqrt(8) A|
s\5"??
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) |$G|M=*LN
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 4"d'iY
% 3 3 r^3 * sin(3*theta) sqrt(8) "fOxS\er
% 4 -4 r^4 * cos(4*theta) sqrt(10) d$#DXLA\P
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 3fd?xhWbN
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Cd'`rs}3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E:ti]$$
% 4 4 r^4 * sin(4*theta) sqrt(10) qj1Fj
% -------------------------------------------------- v0u, :eZ4
% B'8T+qvA
% Example 1: v&r\Z @%
% 2f0qfF
% % Display the Zernike function Z(n=5,m=1) r O-=):2
% x = -1:0.01:1; [iUy_ C=qp
% [X,Y] = meshgrid(x,x); PS'SI X
% [theta,r] = cart2pol(X,Y); ^
RIWW0
% idx = r<=1; 6S&OE k
% z = nan(size(X)); )JXy>q#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !~fy".|x
% figure 0@/C5 v
% pcolor(x,x,z), shading interp (g3@3.Kk)
% axis square, colorbar ,?(U4pzX
% title('Zernike function Z_5^1(r,\theta)') g66x;2Q
% fx*Q,}t
% Example 2: @~C
C$Y$
% MwTouEGGgA
% % Display the first 10 Zernike functions $5N\sdyZxg
% x = -1:0.01:1; X1FKcWv
% [X,Y] = meshgrid(x,x); {VT**o
% [theta,r] = cart2pol(X,Y); 6oy[0hj
% idx = r<=1; 3S{3AmKj?
% z = nan(size(X)); NN]8T
% n = [0 1 1 2 2 2 3 3 3 3]; ZYs?65.
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 7_CX6:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; DyM<aT
% y = zernfun(n,m,r(idx),theta(idx)); 0s.X
% figure('Units','normalized') +O!4~k^
% for k = 1:10 pIl[)%F
% z(idx) = y(:,k); 6ac_AsFK
% subplot(4,7,Nplot(k))
a
Ju v{
% pcolor(x,x,z), shading interp vpz l{
% set(gca,'XTick',[],'YTick',[]) c_Jcy
% axis square nQ08(8
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >Y=qSg>Ik
% end .tA=5QY,
% {-1N@*K
% See also ZERNPOL, ZERNFUN2. 04#<qd&ob@
SlI
wLv^
% Paul Fricker 11/13/2006 `i)Pf WdBN
NfND@m{/
J6gn!
% Check and prepare the inputs: V<Co!2S
% ----------------------------- Mw|lEctN0
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) JQ9+kZ
error('zernfun:NMvectors','N and M must be vectors.') TTD#ovo'
end He1~27+99
=4
NKXP~C
if length(n)~=length(m) Xa_:B\ic
error('zernfun:NMlength','N and M must be the same length.') ?G 'sb}.
end mNKcaM?h
+zZ]Txb(
n = n(:); S~U5xM^s
m = m(:); O:Wd
,3_
if any(mod(n-m,2)) 2Ws'3Jz
error('zernfun:NMmultiplesof2', ... rm4t
'All N and M must differ by multiples of 2 (including 0).') lw_@(E]E
end iz3Hoj
: eFyd`Syw
if any(m>n) %J+k.UrM
error('zernfun:MlessthanN', ... j+[oZfH
'Each M must be less than or equal to its corresponding N.') &(h@]F!
end xtK}XEhG!
>OKc\m2%Q
if any( r>1 | r<0 ) 4@=[rZb9
error('zernfun:Rlessthan1','All R must be between 0 and 1.') y(X^wC
end J3hhh(
?N]G;%3/
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &'u%|A@
error('zernfun:RTHvector','R and THETA must be vectors.') Z_s]2y1
end C:z7R" yj
+>Pq]{Uf1j
r = r(:); F&HvSt}l5
theta = theta(:); WF-^pfRq~
length_r = length(r); R'qBG(?i
if length_r~=length(theta) \jr-^n]
error('zernfun:RTHlength', ... jQ['f\R
'The number of R- and THETA-values must be equal.') DI[
end HG^~7oMf
wlpcuz@
% Check normalization: .J?RaH{i
% -------------------- 7pM&))R
if nargin==5 && ischar(nflag) Iv/h1j> H
isnorm = strcmpi(nflag,'norm'); 7%W@Hr,%F
if ~isnorm f{U,kCv
error('zernfun:normalization','Unrecognized normalization flag.') p+V::O&&r
end k#G+<7c<
else ;}'Z2gZB
isnorm = false;
j]m|}n
end ~*L@|?
KN~Rep cz@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]W7&ZpF
% Compute the Zernike Polynomials jF-0 fK;)*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3
<Zo{;
A_+*b
[P
% Determine the required powers of r: S*%:ID|/C2
% ----------------------------------- syk,e4:oA
m_abs = abs(m); u zL|yxt
rpowers = []; \wV ?QH
for j = 1:length(n) GK&R.R]
rpowers = [rpowers m_abs(j):2:n(j)]; ~zDFL15w
end u?KG%
rpowers = unique(rpowers); .jl^"{@6
LG'1^W{a
% Pre-compute the values of r raised to the required powers, ^+Njz{rpG
% and compile them in a matrix: -v=tM6
% ----------------------------- qot{#tk
d
if rpowers(1)==0 xLw[
aYy4
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -l{ wB"
rpowern = cat(2,rpowern{:}); ZK8DziO
rpowern = [ones(length_r,1) rpowern]; 9g7Ok9dF
else 1~[GGl
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l#a*w
rpowern = cat(2,rpowern{:}); *-gmWATC6
end yn04[PN2
`3.bux~
% Compute the values of the polynomials: =<U'Jtu6'
% -------------------------------------- \>+BvF
y = zeros(length_r,length(n)); `!.c_%m2
for j = 1:length(n) \ $
:)Ka
s = 0:(n(j)-m_abs(j))/2; t}gK)"g
pows = n(j):-2:m_abs(j); 4}Hf"L[ l
for k = length(s):-1:1 EI@ep~
p = (1-2*mod(s(k),2))* ... RMa#z [{0
prod(2:(n(j)-s(k)))/ ... uN6xOq/
prod(2:s(k))/ ... \p\rPfY{>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 94.M8
prod(2:((n(j)+m_abs(j))/2-s(k))); BF
U#FE)s
idx = (pows(k)==rpowers); h|ja67VG
y(:,j) = y(:,j) + p*rpowern(:,idx); D 66!C{
end `;&=m,
W'
hYh~[Kr^@^
if isnorm ]v.Yt/&C{
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); D$SO 6X~
end b<KKF '
end ? \NT'CG
% END: Compute the Zernike Polynomials VqeW;8&*iv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CxVrnb[`q
6/Z 8/PL
% Compute the Zernike functions: qGie~S ##
% ------------------------------ <@=w4\5j9
idx_pos = m>0; c1StA
idx_neg = m<0; < !]7Gt
kYkck]|
z = y; UbSD?Ew@35
if any(idx_pos) G_?qY#"(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6
i]B8Ziq{
end =Lr#
*ep[
if any(idx_neg) "`5BAv;u
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); .,SWa;[iB
end `Dv&.
]BBjFs4#
% EOF zernfun