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 461p 4)
function z = zernfun(n,m,r,theta,nflag) 1V]j8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. y)7;"3Q<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N &
5'cN
% and angular frequency M, evaluated at positions (R,THETA) on the I=k`VI d:
% unit circle. N is a vector of positive integers (including 0), and cdg&)
% M is a vector with the same number of elements as N. Each element Qs 'dwc
% k of M must be a positive integer, with possible values M(k) = -N(k) WmblY2
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, j^Ln\N]^
% and THETA is a vector of angles. R and THETA must have the same $
\ I|6[P
% length. The output Z is a matrix with one column for every (N,M) Ft @ZK!'@
% pair, and one row for every (R,THETA) pair. c}2"X,
% :ZXaJ!
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p0pA|
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FCChB7c`
% with delta(m,0) the Kronecker delta, is chosen so that the integral Emv9l~mIu
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WwLV^m]
% and theta=0 to theta=2*pi) is unity. For the non-normalized wNl "y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. TEbE-h0)]
% g7K<"Z {M
% The Zernike functions are an orthogonal basis on the unit circle. D Z=OZ.v
% They are used in disciplines such as astronomy, optics, and l YjPrA]TC
% optometry to describe functions on a circular domain. >UV=k :Q
% t k+t3+
% The following table lists the first 15 Zernike functions. (2/i1)Cq
% p8z"Jn2P
% n m Zernike function Normalization B,A\/%<
% -------------------------------------------------- #/WjKr n
% 0 0 1 1 oXGP6#
% 1 1 r * cos(theta) 2 J*qo3aJjE
% 1 -1 r * sin(theta) 2 #3-hE
% 2 -2 r^2 * cos(2*theta) sqrt(6) JL?|NV-
% 2 0 (2*r^2 - 1) sqrt(3) 21~~ =+)X
% 2 2 r^2 * sin(2*theta) sqrt(6) i]0$7s9!
% 3 -3 r^3 * cos(3*theta) sqrt(8) ZRC7j?ui8`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q/3co86c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) )umW-A
% 3 3 r^3 * sin(3*theta) sqrt(8) P}D5 j
% 4 -4 r^4 * cos(4*theta) sqrt(10) ^NO;A=9b[
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :LD+B1$y
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P~@I`r567
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R)9FXz$).
% 4 4 r^4 * sin(4*theta) sqrt(10) 4$4n9`odE
% -------------------------------------------------- Q0TKM>
% dkOERVRe
% Example 1: /gE9 W
% KI5099 _/
% % Display the Zernike function Z(n=5,m=1) +/Vzw
% x = -1:0.01:1; bpfSe
% [X,Y] = meshgrid(x,x); Oz.Zxw
% [theta,r] = cart2pol(X,Y); g)iw.M2
% idx = r<=1; }-paGM@'Nd
% z = nan(size(X)); :$oi P
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Y1 6pT
% figure `aaT
#r
% pcolor(x,x,z), shading interp q<A,S8'm
% axis square, colorbar _P{v=`]Eu
% title('Zernike function Z_5^1(r,\theta)') |r53>,oR<:
% \MtdT[*
% Example 2: b'4r5@GO
% f8L3+u
% % Display the first 10 Zernike functions ^Kh>La:>O
% x = -1:0.01:1; .t{?doOT
% [X,Y] = meshgrid(x,x); SwmX_F#_
% [theta,r] = cart2pol(X,Y); aB4L$M8x
% idx = r<=1; c]:@y"W5$
% z = nan(size(X)); 3hNb
?
% n = [0 1 1 2 2 2 3 3 3 3]; (OHd} YQ
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; g?!;04
% Nplot = [4 10 12 16 18 20 22 24 26 28]; JT 5+d ,
% y = zernfun(n,m,r(idx),theta(idx)); JLV?n,nF
% figure('Units','normalized') 8\8%FSrc
% for k = 1:10 `jCq`-.
% z(idx) = y(:,k); |b)N;t
% subplot(4,7,Nplot(k)) c#(&\g2H
% pcolor(x,x,z), shading interp `H\NJ,
% set(gca,'XTick',[],'YTick',[]) gPWl# 5P:
% axis square WWWfQ_u2
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {,i='!WIm
% end v7-
d+P=
% .t9zF-jk
% See also ZERNPOL, ZERNFUN2. = DXvt5G
[0hZg
% Paul Fricker 11/13/2006 ]ch=D
0B~Q.tyP
=u]FKY
% Check and prepare the inputs: 2E}^'o
% ----------------------------- *gXm&/2*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~b{j`T
error('zernfun:NMvectors','N and M must be vectors.') ;V3d"@R,
end NbW5a3=
Y{ 2xokJ N
if length(n)~=length(m) G6x 2!Ny
error('zernfun:NMlength','N and M must be the same length.') 9<I;9.1S?^
end ecy41y'~:
S~ 3|
n = n(:); ,@*`2I>`
m = m(:); q CB9z
if any(mod(n-m,2)) f7QX"p&P
error('zernfun:NMmultiplesof2', ... 1_.#'U>
'All N and M must differ by multiples of 2 (including 0).') %uLyL4*L(p
end R|H_F#eVn}
[u2)kH$
if any(m>n) "t"&6\
error('zernfun:MlessthanN', ... q! U'DDEP
'Each M must be less than or equal to its corresponding N.') '$n#~/#}
end uP[:P?,t
H=k*;'
if any( r>1 | r<0 ) 8?7:sfc
error('zernfun:Rlessthan1','All R must be between 0 and 1.') XS/5y(W
end CiGN?1|
_Uz}z#jt
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) f*SAbDE
error('zernfun:RTHvector','R and THETA must be vectors.') c F(]`49(
end L)ry!BuHI
.U !;fJ9
r = r(:); emI]'{_G
theta = theta(:); 1"CbuV
6
length_r = length(r); d\ Z#XzI8
if length_r~=length(theta) oWUDTio#[
error('zernfun:RTHlength', ... @*c) s_
'The number of R- and THETA-values must be equal.') 'u2Qq"d+
end bz?
*#S
\;A\ vQ[
% Check normalization: C&'Y@GE5
% --------------------
" V`MNZ
if nargin==5 && ischar(nflag) Ma3Hn
isnorm = strcmpi(nflag,'norm'); $0zH2W
if ~isnorm XDJQO /qN
error('zernfun:normalization','Unrecognized normalization flag.') cNG6 A4
end PF(P"f.?D
else prY9SQd
isnorm = false; f(E 'i>
end `&U ['_%
Do|`wpR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ? I}T[j
% Compute the Zernike Polynomials ?Y~>H2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pz"!8b-MN
jjm-%W@
% Determine the required powers of r: -j9R%+YW<
% ----------------------------------- F#^ .L|d4
m_abs = abs(m); LV 94i
rpowers = []; ;.h5; `&
for j = 1:length(n) 3;`93TO{
rpowers = [rpowers m_abs(j):2:n(j)]; `#X{.
end hGF(E*
rpowers = unique(rpowers); kc8T@5+I0
XI,F^K
% Pre-compute the values of r raised to the required powers, &w3LMOT
% and compile them in a matrix: P"u* bqk
% ----------------------------- JCZJ\f*EZ
if rpowers(1)==0 p$@=N6)I.k
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <96ih$5D1
rpowern = cat(2,rpowern{:}); 0r=Lilu{q
rpowern = [ones(length_r,1) rpowern]; FO}4~_W{
else k>n^QHM
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); (.!q~G
rpowern = cat(2,rpowern{:}); N[ArwV2O
end (w% hz']
u6jJf@!ws
% Compute the values of the polynomials: U'.>wjO
% -------------------------------------- s$:]$&5
y = zeros(length_r,length(n)); Zk}e?Grc
for j = 1:length(n) ( L RX
s = 0:(n(j)-m_abs(j))/2; !HDk]
pows = n(j):-2:m_abs(j); =W ! m`
for k = length(s):-1:1 ASy7")5
p = (1-2*mod(s(k),2))* ... fC%;|V'Nd
prod(2:(n(j)-s(k)))/ ... rf1nC$Sop
prod(2:s(k))/ ... M7,|+W/RK
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1xq1te)
prod(2:((n(j)+m_abs(j))/2-s(k))); r:Cad0xj;^
idx = (pows(k)==rpowers); xYt{=
y(:,j) = y(:,j) + p*rpowern(:,idx); wQnr*kyza
end =4 JVUu~Z
?67j+)
if isnorm %v~j10e
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); x_Ais&Gc
end iJrscy-
end '}4[m>/
% END: Compute the Zernike Polynomials >cMU<'&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% pdnL~sv
^#^u90I
% Compute the Zernike functions: ^ad>
(W
% ------------------------------ gYzKUX@
idx_pos = m>0; ocgbBE
idx_neg = m<0; 9y]$c1
//Tr=!TQu
z = y; /e{Oqhf[n
if any(idx_pos) R!pV`N
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <O\z`aA'q
end tg8VFH2q.z
if any(idx_neg) XcfTE
m
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); NKd@Kp`,
end }.b[a z\T
`(o1&
% EOF zernfun