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 f#7=N{wm
function z = zernfun(n,m,r,theta,nflag) bR:hu}YS
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K:Z(jF!j
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N IGlyx'\_
% and angular frequency M, evaluated at positions (R,THETA) on the B[#n,ay
% unit circle. N is a vector of positive integers (including 0), and ;kR=vv
% M is a vector with the same number of elements as N. Each element tGbx/$Y
% k of M must be a positive integer, with possible values M(k) = -N(k) BJ'pe[Xa5
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zKaj<Og
% and THETA is a vector of angles. R and THETA must have the same 5j0 Ib>\
% length. The output Z is a matrix with one column for every (N,M) ?4aW^l6/
% pair, and one row for every (R,THETA) pair. tTubW=H
% OQKc_z'"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ^|hVFM2
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), >LH}A6dUC
% with delta(m,0) the Kronecker delta, is chosen so that the integral f|F=)tJO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =*zde0T?l
% and theta=0 to theta=2*pi) is unity. For the non-normalized JR&yaOws
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -XK;B--c
% p&)d]oV>
% The Zernike functions are an orthogonal basis on the unit circle. R?tjobk!
% They are used in disciplines such as astronomy, optics, and yx*<c#Uf
% optometry to describe functions on a circular domain. Of$R+n.
% \IudS{
.?;
% The following table lists the first 15 Zernike functions. \j BA4?(S
% >El]5M7h7
% n m Zernike function Normalization gSj0+|
% -------------------------------------------------- &@BAVc z
% 0 0 1 1
ylS6D
% 1 1 r * cos(theta) 2 Q"c/]Sk)
% 1 -1 r * sin(theta) 2 UWK|_RT6SA
% 2 -2 r^2 * cos(2*theta) sqrt(6) +9pock
% 2 0 (2*r^2 - 1) sqrt(3) 0M&~;`W}
% 2 2 r^2 * sin(2*theta) sqrt(6) W2zG"Q
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^Oeixi@f
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) kUT^o
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) C@zG(?X
% 3 3 r^3 * sin(3*theta) sqrt(8) ._<,
Eodv
% 4 -4 r^4 * cos(4*theta) sqrt(10) sX3qrRY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;_|4c7
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Uq{$j5p8
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) :xbj&
l
% 4 4 r^4 * sin(4*theta) sqrt(10) |-S+ x]9
% -------------------------------------------------- :*DWL!a
% lFSvHs5
% Example 1: _'X
% b?lRada{I
% % Display the Zernike function Z(n=5,m=1) E`hR(UL
?
% x = -1:0.01:1; X Z3fWcw[
% [X,Y] = meshgrid(x,x); jAv3qMQA
% [theta,r] = cart2pol(X,Y); bhbTloCR
% idx = r<=1; 2mMi=pv9
% z = nan(size(X)); ?~.:C'
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0E,QOF{o
% figure {.[EX MX
% pcolor(x,x,z), shading interp JRZp'Ln
% axis square, colorbar gu~R4@3
% title('Zernike function Z_5^1(r,\theta)') zxH<~2
% 4sRg+mMI
% Example 2: "USzk7=&.
% R$A%Zh6
% % Display the first 10 Zernike functions 4<)*a]\c5M
% x = -1:0.01:1; z 0zB&}
% [X,Y] = meshgrid(x,x); suW|hh1/Ya
% [theta,r] = cart2pol(X,Y); .X"&kO>G
% idx = r<=1; v6[VdWOx5
% z = nan(size(X)); 8O60pB;4
% n = [0 1 1 2 2 2 3 3 3 3]; h(J$-SUs
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; e>.^RtDF
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ],~[ ^0
% y = zernfun(n,m,r(idx),theta(idx)); J=(i0A
% figure('Units','normalized') zxD=q5in
% for k = 1:10 2Ub-ufkU
% z(idx) = y(:,k); 5} ur,0{
% subplot(4,7,Nplot(k)) #CAZ}];Qx
% pcolor(x,x,z), shading interp j6$@vA)
% set(gca,'XTick',[],'YTick',[]) }$qrNbLJ
% axis square JKO*bbj
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) yJO Jw o^
% end *O @Zn
% j!oX\Y-: &
% See also ZERNPOL, ZERNFUN2. S')DAx
D^P0X:T]
% Paul Fricker 11/13/2006 YWD gRb
5L~lF8
(: kn)
% Check and prepare the inputs: 0dS (g&ZR
% ----------------------------- N#)Klq87z
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) S1@r.z2L
error('zernfun:NMvectors','N and M must be vectors.') Nq\)o{<1
end <SOG?Lh~
IR:{ { (
if length(n)~=length(m) 2@pEiq3
error('zernfun:NMlength','N and M must be the same length.') P$N5j~*
end Mqk|H~l5c
* a1q M?
n = n(:); "lC>_A
m = m(:); F2_'U' a
if any(mod(n-m,2)) ~)>.%`v&
error('zernfun:NMmultiplesof2', ... s|c}9/Xe)
'All N and M must differ by multiples of 2 (including 0).') cXf/
end tlg}"lY
nhC8Tq[m
if any(m>n) MZcvr 9y
error('zernfun:MlessthanN', ... i O? f&u
'Each M must be less than or equal to its corresponding N.') PNo:vRtsq
end [q_62[-X
qdKqc,R1{
if any( r>1 | r<0 ) Ie=gI+2
error('zernfun:Rlessthan1','All R must be between 0 and 1.') x%goyXK
end %hZX XpuO
vdB2T2F
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) (JnEso-V
error('zernfun:RTHvector','R and THETA must be vectors.') }Y!s:w#
end .m>Qlh
O'#;Ge/,
r = r(:); w'$>E4\
theta = theta(:); n+Conp/
length_r = length(r); "$K]+0ryG<
if length_r~=length(theta) $F X$nY
error('zernfun:RTHlength', ... a_{'I6a*,
'The number of R- and THETA-values must be equal.') *b0z/6
end v,ni9DIu
@|">j#0
% Check normalization: 5rCJIl.
% -------------------- &(Hw:W9
if nargin==5 && ischar(nflag) |wQ3+WN|
isnorm = strcmpi(nflag,'norm'); ~]?EV?T
if ~isnorm u8|CeA
error('zernfun:normalization','Unrecognized normalization flag.') !Y7$cU &
end Cc`-34/%
else r2i]9>w
isnorm = false; \+Y=}P>
end *&_cp]3-WF
cq
gCcO,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4oryTckS
% Compute the Zernike Polynomials ePv`R'#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% T\6,@7
1{d;Ngx
% Determine the required powers of r: N=T}
% ----------------------------------- q *Hg-J}
m_abs = abs(m); 5[)#3vY
rpowers = []; fz|_c*&64
for j = 1:length(n) $dK430_B
rpowers = [rpowers m_abs(j):2:n(j)]; T3SFG]H
end GVn'p
Wg
rpowers = unique(rpowers); #8M^;4N>[
%{:pBt:Z
% Pre-compute the values of r raised to the required powers, 7 H:y=?X6
% and compile them in a matrix: 0YfmAF$/ B
% ----------------------------- 0o6o<ggi
if rpowers(1)==0 +\&6Zbn
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rLmc(-q
rpowern = cat(2,rpowern{:}); ~Jsu"kr
rpowern = [ones(length_r,1) rpowern]; 'o0o.&/=
else 6|3 X*Orn
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '|5o(6u'
rpowern = cat(2,rpowern{:}); `ZM$\Q=:
end 6w
m-uu
$""kZ
% Compute the values of the polynomials: ;XjXv'
% -------------------------------------- #;@I.
y = zeros(length_r,length(n)); bXXX-Xc
for j = 1:length(n) 'X6Y!VDd
s = 0:(n(j)-m_abs(j))/2; S'ms>ZENC
pows = n(j):-2:m_abs(j); \{~CO{II
for k = length(s):-1:1 d=uGB"
p = (1-2*mod(s(k),2))* ... H`URJ8k$Q
prod(2:(n(j)-s(k)))/ ... VGxab;#,:3
prod(2:s(k))/ ... :~srl)|)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... whP5u/857
prod(2:((n(j)+m_abs(j))/2-s(k))); W_ Hoa*~
idx = (pows(k)==rpowers); 1x\k:2U
y(:,j) = y(:,j) + p*rpowern(:,idx); DS7L}]
end 1MnC5[Q
=Bm|9A1
if isnorm \*b
.f
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9b,0_IMHH
end 59W~bWHCP
end ~$j;@4
% END: Compute the Zernike Polynomials l`:u5\ rM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $G }9iV7
Y{#*;p*I
% Compute the Zernike functions: R)*l)bpZ#
% ------------------------------ M3F1O6=4j
idx_pos = m>0; dw5"}-D
idx_neg = m<0; zF{~Md1
/]-yZ0hX0O
z = y; ~!g2+^G7+P
if any(idx_pos) f/IQ2yT-:D
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); +Ig%h[1a
end z#P`m,~t0
if any(idx_neg) .7 LQ l?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); c|aX4 =Z
end WQiRbb X
L+
XAbL)
% EOF zernfun