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 X2C&q$8
function z = zernfun(n,m,r,theta,nflag) a.G;s2>
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5tu 4uYp;
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N CDDOm8
% and angular frequency M, evaluated at positions (R,THETA) on the {edjvPlk
% unit circle. N is a vector of positive integers (including 0), and l 1Ns~
% M is a vector with the same number of elements as N. Each element #s]` jdc
% k of M must be a positive integer, with possible values M(k) = -N(k) ,wH]|`w
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Xp_G9I,+
% and THETA is a vector of angles. R and THETA must have the same MN. $a9m
% length. The output Z is a matrix with one column for every (N,M) Jbqm?Fy4X
% pair, and one row for every (R,THETA) pair. ^yVKW5x
% \m3ca-Y
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {-e|x&-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !:<n]-U
% with delta(m,0) the Kronecker delta, is chosen so that the integral 6#Afj0
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ]c$)0O\O
% and theta=0 to theta=2*pi) is unity. For the non-normalized kmF@u@5M
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~BD 80s:f
% 20k@!BNq
% The Zernike functions are an orthogonal basis on the unit circle. ^@n?&
% They are used in disciplines such as astronomy, optics, and bZzB\FB~
% optometry to describe functions on a circular domain. D eM/B5qw
% xe!6Pgcb
% The following table lists the first 15 Zernike functions. C:@JLZB
% `l`)Cs;a
% n m Zernike function Normalization tU>?j1
% -------------------------------------------------- {Z{!tR?+
% 0 0 1 1 rIZ^ix-N
% 1 1 r * cos(theta) 2 :]k`;;vh
% 1 -1 r * sin(theta) 2 $Z{Xt*
% 2 -2 r^2 * cos(2*theta) sqrt(6) EnnE@BJ"
% 2 0 (2*r^2 - 1) sqrt(3) ^+'\
u;\
% 2 2 r^2 * sin(2*theta) sqrt(6) L<MH:
% 3 -3 r^3 * cos(3*theta) sqrt(8) |$a!Zx94^
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q, XRb
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) jxNnrIA
% 3 3 r^3 * sin(3*theta) sqrt(8) E [b6k&A
% 4 -4 r^4 * cos(4*theta) sqrt(10) w{5v*SHl}`
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x72T5.
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tg'2v/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) a!Ht81gj
% 4 4 r^4 * sin(4*theta) sqrt(10) !JWZ}uM6
% --------------------------------------------------
]pP:
% !;s5\91
% Example 1: ] B3\IT
% U
*']7-
% % Display the Zernike function Z(n=5,m=1) ^
woCwW8n
% x = -1:0.01:1; l#k&&rI5x.
% [X,Y] = meshgrid(x,x); |?/,ED+|>D
% [theta,r] = cart2pol(X,Y); LyWgaf#/d
% idx = r<=1; t}q\.
% z = nan(size(X)); [$AOu0J
% z(idx) = zernfun(5,1,r(idx),theta(idx)); pu?D^h9/
% figure TIre,s)_
% pcolor(x,x,z), shading interp N=@Nn)
% axis square, colorbar kcN#g-0
% title('Zernike function Z_5^1(r,\theta)') QC^#ns&
% >%{H>?Hn
% Example 2: p`2w\P3;)
% ^L"ENsOs
% % Display the first 10 Zernike functions yV/A%y-P
% x = -1:0.01:1; 5x/LHsr=m
% [X,Y] = meshgrid(x,x); 6A,-?W'\
% [theta,r] = cart2pol(X,Y); MclW!CmJ
% idx = r<=1; o+I'nFtnI
% z = nan(size(X)); Qvl3=[S
% n = [0 1 1 2 2 2 3 3 3 3]; =#|K-X0d=
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1aBQ.-E-
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <Gkmk?x`A
% y = zernfun(n,m,r(idx),theta(idx)); 0\2#(^
% figure('Units','normalized') -K*&I!
% for k = 1:10 O[O[E}8#
% z(idx) = y(:,k); bL9vjD'}
% subplot(4,7,Nplot(k)) 0G}]d17ho
% pcolor(x,x,z), shading interp '|^<|S_+K
% set(gca,'XTick',[],'YTick',[]) 1]% ]"JbV
% axis square Dj(!i1eQNZ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $:D-dUr1
% end (Y>|P
% $>=?'wr
% See also ZERNPOL, ZERNFUN2. B A(PWX`H
O{w'i|
% Paul Fricker 11/13/2006 "Q<
,3~[cE<4
PG*:3![2
% Check and prepare the inputs: (&^k''f
% ----------------------------- .R5(k'g?
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) W_%@nm\y
error('zernfun:NMvectors','N and M must be vectors.') K! I]0!:
end g886RhCe
t`oH7)nut
if length(n)~=length(m) ])3(@.
error('zernfun:NMlength','N and M must be the same length.') uk=f /nT
end |fhYft
fNnX{Wq
n = n(:); V4>qR{5
m = m(:); %=EN 3>,
if any(mod(n-m,2)) 1Q>D^yPI[
error('zernfun:NMmultiplesof2', ... |';oIYs|$
'All N and M must differ by multiples of 2 (including 0).') s !XJ
end F\IJim-Rh
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if any(m>n) 09L"~:rg
error('zernfun:MlessthanN', ... QK0-jYG^
'Each M must be less than or equal to its corresponding N.') +fRABY5C
end PRQEk.C
U+2U#v=<
if any( r>1 | r<0 ) o~ J~-$T{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') [,86||^
end @r=v*hu
<2,NWn.
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) +u\kTn
error('zernfun:RTHvector','R and THETA must be vectors.') w+W!dM
end aTU[H~dTU
g;mX {p_@
r = r(:); wpI_yp
theta = theta(:); j Wjp0ii
length_r = length(r); c[<>e#s+;
if length_r~=length(theta) }{y(&Oy3Y
error('zernfun:RTHlength', ... CD:$22*]
'The number of R- and THETA-values must be equal.') YQ$EN>.eO
end XS oHh-
-J'0qN!
% Check normalization: CEHtr90P
% -------------------- QpI\\Zt6
if nargin==5 && ischar(nflag) U *K6FWqiB
isnorm = strcmpi(nflag,'norm'); r~q3nIe/,
if ~isnorm 2PTAIm Rq
error('zernfun:normalization','Unrecognized normalization flag.') ##r9/`A
end 6haw\ *
else Y6:b
isnorm = false; Xdl7'~k
end 3]wV 1<K
&@ [pJ2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /C\tJs
% Compute the Zernike Polynomials E -+t[W
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -|g9__|@
VqL#w<A%
% Determine the required powers of r: e<+$E%"7hS
% ----------------------------------- -,a@bF:
m_abs = abs(m); J5"d|i
rpowers = []; f[fH1cu&`
for j = 1:length(n) NE5H\
rpowers = [rpowers m_abs(j):2:n(j)]; [x8_ax}w
end %Kzu&*9Hb
rpowers = unique(rpowers); Y5z5LG4
20Z=_},
% Pre-compute the values of r raised to the required powers, XmAun
% and compile them in a matrix: ,,=VF(@G
% ----------------------------- B]#^&89wG)
if rpowers(1)==0 E]dc4US
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 1uco{JX<S
rpowern = cat(2,rpowern{:}); ifI0s)Pn
rpowern = [ones(length_r,1) rpowern]; !%Bhg?
else :`B70D8ku
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); D5"Xjo*
rpowern = cat(2,rpowern{:}); LMHiiOs,
end 3-v&ktD&N'
1A}#j
% Compute the values of the polynomials: Bg.
% -------------------------------------- ?*L{xNC#
y = zeros(length_r,length(n)); 4QBPN@~t
for j = 1:length(n) }U ue}VOA
s = 0:(n(j)-m_abs(j))/2; ^y.|KA3[
pows = n(j):-2:m_abs(j); e:+[}I)
for k = length(s):-1:1 9Yhlq$;g
p = (1-2*mod(s(k),2))* ... szUJh9-
prod(2:(n(j)-s(k)))/ ... h!J|4Qa
prod(2:s(k))/ ... Aaug0X
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M3!4,_!~
prod(2:((n(j)+m_abs(j))/2-s(k))); ^GnR1.ux
idx = (pows(k)==rpowers); ?h)T\z
y(:,j) = y(:,j) + p*rpowern(:,idx); Go)}%[@w
end #`@5`;U>#
q+ 2v9K@
if isnorm I(uM`g
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hdDL92JVg
end kgP6'`}E[
end d]vom@iI
% END: Compute the Zernike Polynomials )nlFyWXh.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t~%( Zu>S
*:?XbtIK u
% Compute the Zernike functions: "EBCf.3-
% ------------------------------ BVG.ZZR})
idx_pos = m>0; }poLHS/
idx_neg = m<0; KEjMxOv1
8Om4G]*|,
z = y; s\ e b
if any(idx_pos) 7QdboEa
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 4m!w<c0NL
end xbzO'C
if any(idx_neg) &2r[4
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); A"/|h].
end >02p,W6S>
8&SWQ
% EOF zernfun