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 4adCMfP7.
function z = zernfun(n,m,r,theta,nflag) DGC-`z
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ci+Pg9sS
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j^1T3 +
% and angular frequency M, evaluated at positions (R,THETA) on the r4gkSwy
% unit circle. N is a vector of positive integers (including 0), and `V w9j,G
% M is a vector with the same number of elements as N. Each element 'P)xY-15
% k of M must be a positive integer, with possible values M(k) = -N(k) w(J-[t118
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +IuV8XT2(
% and THETA is a vector of angles. R and THETA must have the same 9|v
% length. The output Z is a matrix with one column for every (N,M) )"WImf:*
% pair, and one row for every (R,THETA) pair. (u]ft]z,-B
% .Y&_k
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .Ap[C? mV
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7\"-<z;kK
% with delta(m,0) the Kronecker delta, is chosen so that the integral `kwyF27v]
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vPi\ vU{
% and theta=0 to theta=2*pi) is unity. For the non-normalized lBR6O!sBP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nOkX:5
% +;C|5y
% The Zernike functions are an orthogonal basis on the unit circle. zf$OC}|\w
% They are used in disciplines such as astronomy, optics, and ;G0~f9
% optometry to describe functions on a circular domain. ~`#.ZMO
% a,d\<mx
% The following table lists the first 15 Zernike functions. 56G5JSB=\
% R=i$*6}a
% n m Zernike function Normalization MQQiQ 2
% -------------------------------------------------- YM 7P!8Gc
% 0 0 1 1 Aw4Qm2Kf
% 1 1 r * cos(theta) 2 z Rz#0
% 1 -1 r * sin(theta) 2 dDi 1{s
% 2 -2 r^2 * cos(2*theta) sqrt(6) kX'1.<[
% 2 0 (2*r^2 - 1) sqrt(3) B6 x5E
% 2 2 r^2 * sin(2*theta) sqrt(6) -njxc{b
% 3 -3 r^3 * cos(3*theta) sqrt(8) 9=rYzA?)+
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 18}L89S>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ~NpnRIt
% 3 3 r^3 * sin(3*theta) sqrt(8) E-*udQ
% 4 -4 r^4 * cos(4*theta) sqrt(10) #E^ %h
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sG}}a}U1
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) `*KS`
z?
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >/6v`
8F
% 4 4 r^4 * sin(4*theta) sqrt(10) E"#<I*b
% --------------------------------------------------
*X*D,
VY
% qI<*Cze
% Example 1: k,X)PQc
% aMm`G}9n
% % Display the Zernike function Z(n=5,m=1)
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% x = -1:0.01:1; s<E_74q1
% [X,Y] = meshgrid(x,x); )09_CC!a
% [theta,r] = cart2pol(X,Y); [mw#a9
% idx = r<=1; 5Lum$C
c}
% z = nan(size(X)); VY=~cVkzS
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p&Nw:S
% figure ]K XknEaxl
% pcolor(x,x,z), shading interp sFSrMI#R
% axis square, colorbar @faf
% title('Zernike function Z_5^1(r,\theta)') RZOk.~[v
% d>@{!c-
% Example 2: e Yyl=YW
% (niZN_qv
% % Display the first 10 Zernike functions }mu8fm'
% x = -1:0.01:1; BAzc'x&<
% [X,Y] = meshgrid(x,x); -/#3U{O
% [theta,r] = cart2pol(X,Y); [<wy@W
% idx = r<=1; p"q-sMYl
% z = nan(size(X)); ai#EFo+#
% n = [0 1 1 2 2 2 3 3 3 3]; #g ~~zwx/N
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; #0!C3it6c
% Nplot = [4 10 12 16 18 20 22 24 26 28]; +8Peh9"
% y = zernfun(n,m,r(idx),theta(idx)); .IF dJ
% figure('Units','normalized') lba*&j]w=
% for k = 1:10 gxU(&
% z(idx) = y(:,k); k^R>x V
% subplot(4,7,Nplot(k)) 1 68U-<
% pcolor(x,x,z), shading interp -6+HA9zz@C
% set(gca,'XTick',[],'YTick',[]) DgClN:Hw
% axis square Q{>9Dg
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Bw{@YDO{
% end t:m
t9}$d
% SB$~Btr
% See also ZERNPOL, ZERNFUN2. BOt\"N
`q$DNOrS
% Paul Fricker 11/13/2006 AuO%F
YKY
xU@Z<d,k
}pTw$B
% Check and prepare the inputs: &hV;3";
% ----------------------------- _QXo4z!a8
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ta9;;B?$
error('zernfun:NMvectors','N and M must be vectors.') 7yQ r
end YI%S)$
;R2(Gb
if length(n)~=length(m) >z[d~
error('zernfun:NMlength','N and M must be the same length.') b#82G`6r
end :^l*_v{
"T~Ps$
n = n(:); Rw$ @%o%
m = m(:); qIb(uF@l"
if any(mod(n-m,2)) &<tji8Dj
error('zernfun:NMmultiplesof2', ... /Zm@.%.
'All N and M must differ by multiples of 2 (including 0).') ~x4Y57
end D+ jk0*bJ
0;k3
if any(m>n) (\WePOy&
error('zernfun:MlessthanN', ... .`hlw'20
'Each M must be less than or equal to its corresponding N.') h[XGFz
end NiyAAw
[FC%_R&&
if any( r>1 | r<0 ) WZFV8'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7[u&%
end 4~o\Os+8
NugJjd56x
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) `-w;=_Bm
error('zernfun:RTHvector','R and THETA must be vectors.') (8H^{2K~
end '}!dRpx
uQ8]j .0
r = r(:); 8,['q~z
theta = theta(:); BA-n+WCWJ
length_r = length(r); g|n Pr)<
if length_r~=length(theta) ja9y
error('zernfun:RTHlength', ... /iukiWeW
'The number of R- and THETA-values must be equal.') 2t(E+^~
end cDAO5^
W?6RUyMC$T
% Check normalization: $6Ty~.RP5H
% -------------------- nY~CAo/:
if nargin==5 && ischar(nflag) cFH,fj
isnorm = strcmpi(nflag,'norm'); [9>1e
if ~isnorm xNm<` Y?
error('zernfun:normalization','Unrecognized normalization flag.') yq&]>ox
end H:~LL0Md%
else +, PBhB
isnorm = false; ){ wE)NN
end 1miTE4;?
;OVJM
qg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nR
,j1IUF
% Compute the Zernike Polynomials Ad`;O+/;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w>m/c1
H"n"Q:Yp
% Determine the required powers of r: A4SM@ry
% ----------------------------------- Yoaz|7LS
m_abs = abs(m); hd^?svID
rpowers = []; Sc*p7o: A
for j = 1:length(n) IS8ppu&E
rpowers = [rpowers m_abs(j):2:n(j)]; ea B-u
end f+F /`P%
rpowers = unique(rpowers); R%5\1!Fl=G
Bj\0RmVa1
% Pre-compute the values of r raised to the required powers, <k^h&1J#g
% and compile them in a matrix: J6f;dF^
% ----------------------------- #_Tceq5
if rpowers(1)==0 ZA:YoiaC#
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b$M? _<G
rpowern = cat(2,rpowern{:}); /@%
rpowern = [ones(length_r,1) rpowern]; Ai/ay# E
else y]@_DL#J=
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GJH6b7I
rpowern = cat(2,rpowern{:}); r,0> 40^
end *t*yozN
|\,e9U>
% Compute the values of the polynomials: \:O5, wf2
% -------------------------------------- U?@UIhtM|
y = zeros(length_r,length(n)); sLB{R#Pt
for j = 1:length(n) Q=>@:1=
s = 0:(n(j)-m_abs(j))/2; ,.F,]m=
pows = n(j):-2:m_abs(j); JLs7[W)O
for k = length(s):-1:1 FK+jfr [
p = (1-2*mod(s(k),2))* ... O </<
prod(2:(n(j)-s(k)))/ ... YSeXCJ:Iy
prod(2:s(k))/ ... cMtkdIO
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +[D=2&tmk
prod(2:((n(j)+m_abs(j))/2-s(k))); f<y""0L9
idx = (pows(k)==rpowers); j!jZJD
y(:,j) = y(:,j) + p*rpowern(:,idx); dNbN]gHC
end .F> cZ,
N?R1;|Z]
if isnorm R$cg\DD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); P\w.:.2
end iF<VbQP=X^
end %tmK6cY4Y
% END: Compute the Zernike Polynomials PcJ,Y\"[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q.rn ZU
> \KBXS}
% Compute the Zernike functions: !U*i13
% ------------------------------ VNA VdP
idx_pos = m>0; nh,N(t9
idx_neg = m<0; :)%Vahu
']4sx_)S
z = y; gK`6NUj
if any(idx_pos) X}g!Lp
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ~Kt.%K5lgt
end ;|}6\=(
if any(idx_neg) ^Cpvh}1#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E!jM&\Z j
end RqH"+/wR
K4A=lD+
% EOF zernfun