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 ZtlF]k:MV
function z = zernfun(n,m,r,theta,nflag) gs_"H
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. pR4{}=g,
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N T#DJQ"$
% and angular frequency M, evaluated at positions (R,THETA) on the 'C]zB'H=
% unit circle. N is a vector of positive integers (including 0), and 4 C/
% M is a vector with the same number of elements as N. Each element yyPkjUy[
% k of M must be a positive integer, with possible values M(k) = -N(k) br.jj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, T?N' k=
% and THETA is a vector of angles. R and THETA must have the same puG$\D-[
% length. The output Z is a matrix with one column for every (N,M) ^DS9D:oE
% pair, and one row for every (R,THETA) pair. ,+3l9FuQ
% #*BcO-N
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike W @Y$!V<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), {# ;e{v
% with delta(m,0) the Kronecker delta, is chosen so that the integral rtS(iD@B"
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, hzg&OW=:
% and theta=0 to theta=2*pi) is unity. For the non-normalized dB ?+-aE
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9`f]Rf"
% ;eI,1
[_
% The Zernike functions are an orthogonal basis on the unit circle. Vl5}m
% They are used in disciplines such as astronomy, optics, and ,@tYD(Z
% optometry to describe functions on a circular domain. 8c`g{
*z
% %a<N[H3NV@
% The following table lists the first 15 Zernike functions. *>n<7T0
% !lG5BOJM
% n m Zernike function Normalization .e!dEF)D
% -------------------------------------------------- ^*#5iT8/
% 0 0 1 1 5?kJ]:
% 1 1 r * cos(theta) 2 (Gf1#,/3~
% 1 -1 r * sin(theta) 2 +yiGZV/X
% 2 -2 r^2 * cos(2*theta) sqrt(6) \`;FL\1+W
% 2 0 (2*r^2 - 1) sqrt(3) B_i@D?bTD
% 2 2 r^2 * sin(2*theta) sqrt(6) <_=a1x
% 3 -3 r^3 * cos(3*theta) sqrt(8) sn
'#]yM
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0V`s 3,k
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) DDq*#;dP
% 3 3 r^3 * sin(3*theta) sqrt(8) 5&D)W>{d
% 4 -4 r^4 * cos(4*theta) sqrt(10) ~'m
GGH2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *.K+"WS%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Pni
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) p 2It/O
% 4 4 r^4 * sin(4*theta) sqrt(10) G&)A7WaC
% -------------------------------------------------- cW4:eh
% cod__.
% Example 1:
ZE.nB- H
% -'QvUHL|
% % Display the Zernike function Z(n=5,m=1) \< <u
% x = -1:0.01:1; v:
cO+dQ
% [X,Y] = meshgrid(x,x); 5, R\tJCK
% [theta,r] = cart2pol(X,Y); \-a^8{.^E
% idx = r<=1; vz#VW
% z = nan(size(X)); N%v}$58Z
% z(idx) = zernfun(5,1,r(idx),theta(idx)); f]L`^WU
% figure v] &
)+0
% pcolor(x,x,z), shading interp 9G_bM(q'^2
% axis square, colorbar !4\`g?
% title('Zernike function Z_5^1(r,\theta)') {P"$;_Y"<
% 5+].$
% Example 2: G7yCGT)vQ
% [tGAo/
% % Display the first 10 Zernike functions Vz6p^kMB
% x = -1:0.01:1; Gl}[1<~o
% [X,Y] = meshgrid(x,x); Q*&>Ui[&
% [theta,r] = cart2pol(X,Y); |s`j=<rNQI
% idx = r<=1; VC5LxA0{
% z = nan(size(X)); ,X25 -OFZ
% n = [0 1 1 2 2 2 3 3 3 3]; ivYHq#b59
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; @GDe{GG+
% Nplot = [4 10 12 16 18 20 22 24 26 28]; B38_1X7
% y = zernfun(n,m,r(idx),theta(idx)); 6Qnerd%Ec
% figure('Units','normalized') CG*eo!Nw
% for k = 1:10 kW0|\
% z(idx) = y(:,k); 92!1I$zi
% subplot(4,7,Nplot(k)) Kmc*z (Q
% pcolor(x,x,z), shading interp 7nM]E_
% set(gca,'XTick',[],'YTick',[]) va F^[/
(g
% axis square u>o<ua
p
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0h/gqlTK1
% end `T7gfb%1-3
% R_ymTB}<t(
% See also ZERNPOL, ZERNFUN2. A:PQIcR;V
^ZV1Ev8T6
% Paul Fricker 11/13/2006 H^z6.!$m
JJ`RF
d2`m0U
% Check and prepare the inputs: Oya:{d&=
% ----------------------------- C"}CD{<H]M
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 1Z%^U ?
error('zernfun:NMvectors','N and M must be vectors.') s ;EwAd(
end j3 ,6UjlU
soh)IfZ
if length(n)~=length(m) p vone,y2
error('zernfun:NMlength','N and M must be the same length.') Z^ynw8k"
end uJ<nW%}
jkCa2!WQ'i
n = n(:); hr3RC+ y
m = m(:); f'&30lF
if any(mod(n-m,2)) (3a]#`Q
error('zernfun:NMmultiplesof2', ... u`?MV2jU2
'All N and M must differ by multiples of 2 (including 0).') nAIV]9RAZ%
end D=Ia$O0.
5-'jYp/
if any(m>n) :U;n?Zu
S
error('zernfun:MlessthanN', ... `/?XvF\
'Each M must be less than or equal to its corresponding N.') %+Hhe]J ld
end sjl(
"Kky|(EQ$$
if any( r>1 | r<0 ) -OV:y],-
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \yt-_W=[
end E57:ap)/
8T"C]
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3ht>eaHi
error('zernfun:RTHvector','R and THETA must be vectors.') OXu*wl(z
end t8SvU
LpRl!\FY$
r = r(:); 3sr>?/>:
theta = theta(:); UQ]WBS\
length_r = length(r); $Ro]]NUz|
if length_r~=length(theta) MI8f(ZJK5
error('zernfun:RTHlength', ... +9mE1$C
'The number of R- and THETA-values must be equal.') =AEl:SY+
end t6-He~
<X@XbM
% Check normalization: 7G6XK
% -------------------- lO^Ly27
if nargin==5 && ischar(nflag) 'Mp8!9=&
isnorm = strcmpi(nflag,'norm'); +c4-7/kE
if ~isnorm bm/pLC6%.
error('zernfun:normalization','Unrecognized normalization flag.') >
mI1wV[
end ~)J]`el,Q
else "rxhS;
R1>
isnorm = false; H}v.0R
end )v\zaz
&n6'r^[D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ek' ~i
% Compute the Zernike Polynomials f@JMDJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% yC%zX}5
,q9nHZG^
% Determine the required powers of r: [/Q .MmnL
% ----------------------------------- FXLY*eRk
m_abs = abs(m); O5rHN;\_
rpowers = []; ai,\'%N
for j = 1:length(n) n*(9:y=l1
rpowers = [rpowers m_abs(j):2:n(j)]; ;/-v4
end I^}q;L![\
rpowers = unique(rpowers); ~H<oqk:O-
=*paa
% Pre-compute the values of r raised to the required powers, d7,ZpHt
% and compile them in a matrix: *[VO03
% ----------------------------- Myj5qh
if rpowers(1)==0 j ?c"BF.
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5hxG\f#}?
rpowern = cat(2,rpowern{:}); o )\\(^ld
rpowern = [ones(length_r,1) rpowern]; \\ZR~f!<
else g5",jTn#
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y4N8B:j%
rpowern = cat(2,rpowern{:}); Rs$fNW@P
end [N@t/^gRC
^nO0/nqz]
% Compute the values of the polynomials: r6,EyCWcCs
% -------------------------------------- X283 . ?
y = zeros(length_r,length(n)); :Xe,=M(l~
for j = 1:length(n) c<k=8P
s = 0:(n(j)-m_abs(j))/2; #|92+
pows = n(j):-2:m_abs(j); ~wejy3|@0
for k = length(s):-1:1 cWp5' e]A
p = (1-2*mod(s(k),2))* ... .Iu8bN(L`
prod(2:(n(j)-s(k)))/ ... !xE/
prod(2:s(k))/ ... ]n \Qa
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... OM>,1;UH]
prod(2:((n(j)+m_abs(j))/2-s(k))); ,(&p"O":
idx = (pows(k)==rpowers); :.VI*X:aQh
y(:,j) = y(:,j) + p*rpowern(:,idx); Ym% $!#
end v|K,
(7X|W<xT
if isnorm Os90fR
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); DPWt=IFU
end "V=IG{.
end 5SB!)F]
% END: Compute the Zernike Polynomials ,H)v+lI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ri
k4C3SI*`4
% Compute the Zernike functions: Mzg zOM
% ------------------------------ $yn7XonS
idx_pos = m>0; *XU2%"Sc
idx_neg = m<0; =%)Y,
)"
S|jE1v"L
z = y; 21T#NYfew
if any(idx_pos) 2@Nt6r
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); VxP cC+
end K]{x0A
if any(idx_neg) +GYO<N7
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !&eKq?P{j
end iJ&jg`"=F
B,5kG{2!
% EOF zernfun