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 _Vt9ckaA
function z = zernfun(n,m,r,theta,nflag) MAX?,-x
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1E4`&?
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <1x u&Z7
% and angular frequency M, evaluated at positions (R,THETA) on the /Zx"BSu
% unit circle. N is a vector of positive integers (including 0), and B !rb*"[
% M is a vector with the same number of elements as N. Each element V}Q`dEk2r
% k of M must be a positive integer, with possible values M(k) = -N(k) 8)Vl2z
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, P9S)7&+DL
% and THETA is a vector of angles. R and THETA must have the same GlJOb|WOX
% length. The output Z is a matrix with one column for every (N,M) Su
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% pair, and one row for every (R,THETA) pair. 5UK}AkEe&x
% KRP6b:+4L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .]<gm9l
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), UxMei
% with delta(m,0) the Kronecker delta, is chosen so that the integral H3iYE~^#
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, XGYsTquSe
% and theta=0 to theta=2*pi) is unity. For the non-normalized oGbh*
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. fmLDufx
% =t~]@?]1D
% The Zernike functions are an orthogonal basis on the unit circle. [IHG9Xg
% They are used in disciplines such as astronomy, optics, and 5dX0C
% optometry to describe functions on a circular domain. w=ufJRj
% *`Ge8?qC
% The following table lists the first 15 Zernike functions. hX-^h2eV
% 'fzJw
% n m Zernike function Normalization 'cK{FiIT
% -------------------------------------------------- $t5>1G1j7
% 0 0 1 1 ox";%|PP1
% 1 1 r * cos(theta) 2 oJE<}~_k
% 1 -1 r * sin(theta) 2 #a]\3X
% 2 -2 r^2 * cos(2*theta) sqrt(6) `J7@G]X;2
% 2 0 (2*r^2 - 1) sqrt(3) kaECjZ_&+
% 2 2 r^2 * sin(2*theta) sqrt(6) "/taatcH
% 3 -3 r^3 * cos(3*theta) sqrt(8) HuN_$aP
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,Vz-w;oDn
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) tpgD{BY^wJ
% 3 3 r^3 * sin(3*theta) sqrt(8) <p`
F/p-
% 4 -4 r^4 * cos(4*theta) sqrt(10) Z`%^?My
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C8(0|XX
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2q9$5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NKVLd_f k
% 4 4 r^4 * sin(4*theta) sqrt(10) c2Y\bKeN
% -------------------------------------------------- ybIqn0&[
% #??[;xjs!
% Example 1: ,,S 2>X*L
% UZ:z|a3
% % Display the Zernike function Z(n=5,m=1) 4O{,oN~7
% x = -1:0.01:1; d@Wze[M?0
% [X,Y] = meshgrid(x,x); Y%zWaH
% [theta,r] = cart2pol(X,Y); Y|KT3
% idx = r<=1; Tx'anP
% z = nan(size(X)); .^ba*qb`{
% z(idx) = zernfun(5,1,r(idx),theta(idx)); md/h\o&
% figure -BwZ
% pcolor(x,x,z), shading interp !rZZ/M"i
% axis square, colorbar OU?.}qc<wE
% title('Zernike function Z_5^1(r,\theta)') wRX#^;O9?>
% h`p=~u +
% Example 2: @v\8+0
%
-f<}lhmQ
% % Display the first 10 Zernike functions p@@*F+
% x = -1:0.01:1; .GCJA`0h
% [X,Y] = meshgrid(x,x); ? a/\5`gnN
% [theta,r] = cart2pol(X,Y); |`AJP
% idx = r<=1;
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% z = nan(size(X)); @)ls+}=Y
% n = [0 1 1 2 2 2 3 3 3 3]; $L'[_J
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 2f rwU~y
% Nplot = [4 10 12 16 18 20 22 24 26 28]; !_iv~Q zv
% y = zernfun(n,m,r(idx),theta(idx)); Jgq#m~M6
% figure('Units','normalized') ~svea>Fmr
% for k = 1:10 )]zsAw`/
% z(idx) = y(:,k); [[ll4|
% subplot(4,7,Nplot(k)) mWMtz]M}
% pcolor(x,x,z), shading interp "|E'E"_1
% set(gca,'XTick',[],'YTick',[]) +'[/eW
% axis square iBY16_q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m:9|5W
% end xd+aO=)Td
% Xhpcu1nA
% See also ZERNPOL, ZERNFUN2. 8 9maN
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% Paul Fricker 11/13/2006 3/,}&SX
"9NWsy}<c
$OzVo&P;
% Check and prepare the inputs: jK{qw
% ----------------------------- M>{*PHze0
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4(`U]dNcs
error('zernfun:NMvectors','N and M must be vectors.') jq_ i&~S
end 2r@9|}La
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if length(n)~=length(m) rG%8ugap
error('zernfun:NMlength','N and M must be the same length.') .OlPVMFt
end \
# la8,+9
c1
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n = n(:); }V 4u`=
m = m(:); A,?6|g`q'
if any(mod(n-m,2)) 4)p ID`
error('zernfun:NMmultiplesof2', ... R}D[ z7
'All N and M must differ by multiples of 2 (including 0).') ]\/"-Y#4Q
end /^WOrMR
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if any(m>n) 8zh o\'
error('zernfun:MlessthanN', ... ~1nKL0C6u
'Each M must be less than or equal to its corresponding N.') 64Tb,AL_
end :OA;vp~$x
-U|Z9sia
if any( r>1 | r<0 ) 5+qdn|9%T
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'oUTY *
end FRsp?i
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ONFx -U]
error('zernfun:RTHvector','R and THETA must be vectors.') [i_evsUj?
end 6!([Hu#= *
XI,= W
r = r(:); lWUQkS
theta = theta(:); .dwbJT
length_r = length(r); VeOM `jy
if length_r~=length(theta) B)dG:~
error('zernfun:RTHlength', ... 8=g~+<A
'The number of R- and THETA-values must be equal.') &
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end Y3SV6""y/
$v5 >6+-n
% Check normalization: S#T u/2<}
% -------------------- ^4et;
F%
if nargin==5 && ischar(nflag) |+qsO;
isnorm = strcmpi(nflag,'norm'); bEmzigN[
if ~isnorm .0MY$ 0s
error('zernfun:normalization','Unrecognized normalization flag.') #8y"1I=i&
end JkKbw&65
else gLK0L%"5
isnorm = false; L XTtV0F
end n3$u9!|P
;s8\F]K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?A-f_0<0
% Compute the Zernike Polynomials pwV~[+SS_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s|X_:3\x
_9?v?mL5;
% Determine the required powers of r: FU;a
{irB
% ----------------------------------- 'lOQb)
m_abs = abs(m); n Q{~D5y,,
rpowers = []; bH!_0+$P
for j = 1:length(n) mE&SAm5#d
rpowers = [rpowers m_abs(j):2:n(j)]; J|VDZ# c7
end >:BgatyPH
rpowers = unique(rpowers); Iz>\qC}
s+E4AG1r
% Pre-compute the values of r raised to the required powers, ;Eh"]V,e
% and compile them in a matrix: ?8;WP&
% ----------------------------- ?yu@eo
if rpowers(1)==0 fUPYCw6F
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Dn#UcMO>W
rpowern = cat(2,rpowern{:}); -#R63f&
rpowern = [ones(length_r,1) rpowern]; ;vn0b"Fi3
else 12: Q`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `YO&
rpowern = cat(2,rpowern{:}); @q{.
end Qh*}v!3Jo
5xU}}[|~-
% Compute the values of the polynomials: fA=Lb^,M
% -------------------------------------- `'gcF});
y = zeros(length_r,length(n)); Dj 6^|R$z&
for j = 1:length(n) _qh\
s = 0:(n(j)-m_abs(j))/2; =5uhIU0O
pows = n(j):-2:m_abs(j); 12Fnv/[n'K
for k = length(s):-1:1 k L4 #
p = (1-2*mod(s(k),2))* ... ngk:q5Tp
prod(2:(n(j)-s(k)))/ ... @g*[}`8]y
prod(2:s(k))/ ... Y@qugQM>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2EO9IxIf
prod(2:((n(j)+m_abs(j))/2-s(k))); u#Bj#y!
idx = (pows(k)==rpowers); Ak$9\Sl
y(:,j) = y(:,j) + p*rpowern(:,idx); ;";>7k/}
end 0T0I<t
gADqIPu]
if isnorm MJa`4[/
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); o,xy'
end _ozg=n2(
end x@:98P
% END: Compute the Zernike Polynomials tCGA3t
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }r"E\~E
NGEE'4!i7T
% Compute the Zernike functions: >
kwhZ/x
% ------------------------------ )QmmI[,tq
idx_pos = m>0; (&, E}{p9
idx_neg = m<0; V9%9nR!'
$"#M:V@
z = y; {}=5uU 2Tu
if any(idx_pos) Ki%)LQAg
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dkSd
Y+Q
end A>(EM}\,
if any(idx_neg) "j.Q*Hazg
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); auM1k]
end C[;7i!Dv
.'2"83f
% EOF zernfun