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 N%;Q[*d@/
function z = zernfun(n,m,r,theta,nflag) Fp4?/-]
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. P]!$MOt
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $D5[12X
% and angular frequency M, evaluated at positions (R,THETA) on the qyl~*r*
% unit circle. N is a vector of positive integers (including 0), and ?15k~1nA
% M is a vector with the same number of elements as N. Each element y$s}-O]/-
% k of M must be a positive integer, with possible values M(k) = -N(k) n>>hfxv(O!
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $t.N|b`'
% and THETA is a vector of angles. R and THETA must have the same d|TRP,y
% length. The output Z is a matrix with one column for every (N,M) }D
dg
% pair, and one row for every (R,THETA) pair. ;hF >iw
% s=#IoNh
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @dX0gHU[c
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), asP>(Li
% with delta(m,0) the Kronecker delta, is chosen so that the integral RyD2LAf)J
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WhE5u&`
% and theta=0 to theta=2*pi) is unity. For the non-normalized j)Kk:BFFY
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. +A
W6 >yV`
% ^T'+dGU`
% The Zernike functions are an orthogonal basis on the unit circle. FMY
r6/I
% They are used in disciplines such as astronomy, optics, and As@~%0 S
% optometry to describe functions on a circular domain. X^% I 3
% Z]$yuM
% The following table lists the first 15 Zernike functions. :eS7"EG{3
% %_M B-
% n m Zernike function Normalization Fdd$Bl.&XS
% -------------------------------------------------- "A__z|sQ
% 0 0 1 1 V5KAiG<d
% 1 1 r * cos(theta) 2 _jH1Mcq
% 1 -1 r * sin(theta) 2 \|R`wFn^P
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]=9%fA
% 2 0 (2*r^2 - 1) sqrt(3) @^Mn
PM
% 2 2 r^2 * sin(2*theta) sqrt(6) d|on
y
% 3 -3 r^3 * cos(3*theta) sqrt(8) I OF~V)8k=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) vtR<(tOu@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ig; ~
T
% 3 3 r^3 * sin(3*theta) sqrt(8) R.A}tV=j#
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0'^? m$
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9^0 'VRG
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) .)|jBC8|}
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *bn9j>|iv
% 4 4 r^4 * sin(4*theta) sqrt(10) 'Twi
@I
% -------------------------------------------------- 5
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% wX#\\Jgi
% Example 1: dcU|y%k%
% |Y(].G,
% % Display the Zernike function Z(n=5,m=1) 1>a^Q
% x = -1:0.01:1; Uvf-h4^J]:
% [X,Y] = meshgrid(x,x); C'n 9n!hR
% [theta,r] = cart2pol(X,Y); 3I:DL#f
% idx = r<=1; TW3:Y\ p
% z = nan(size(X)); "4g1I<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); RfN5X}&A
% figure XIBw&mWf
% pcolor(x,x,z), shading interp ]*i>KR@G
% axis square, colorbar Tj0eW(<!s
% title('Zernike function Z_5^1(r,\theta)') Uj k``;
% <py~(q
% Example 2: $
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% =AgY8cF!sl
% % Display the first 10 Zernike functions ih+kh7J-
% x = -1:0.01:1; 7azxqa5:
% [X,Y] = meshgrid(x,x); L8bq3Q'p
% [theta,r] = cart2pol(X,Y); z@~1e]%
% idx = r<=1; KN}[N+V>
% z = nan(size(X)); ;i:Uoyi
% n = [0 1 1 2 2 2 3 3 3 3]; ip>dHj
z
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; H:[z#f|t
% Nplot = [4 10 12 16 18 20 22 24 26 28]; cR@z^
% y = zernfun(n,m,r(idx),theta(idx)); 9D<^)ShY
% figure('Units','normalized') 9\Xl3j!
% for k = 1:10 ACyQsmqm:
% z(idx) = y(:,k); t"0~2R6i
% subplot(4,7,Nplot(k)) vZ]gb$
% pcolor(x,x,z), shading interp B]*&lRR
% set(gca,'XTick',[],'YTick',[]) OPKX&)SE-
% axis square
r.K4<ly-N
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) g LpWfT29V
% end _R5^4 -Qe
% ;"Ot\:0
% See also ZERNPOL, ZERNFUN2. ,R^Pk6m>
U4N
S.`V
% Paul Fricker 11/13/2006 Do_L
Z@I%ppd
-\NB*|9m|
% Check and prepare the inputs: dk(-yv'
% ----------------------------- U_VD* F4Bv
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ww\/$ |
error('zernfun:NMvectors','N and M must be vectors.') `Z@wWs
end |LNXu
m
if length(n)~=length(m) ~{5%~8h.0r
error('zernfun:NMlength','N and M must be the same length.') /`s^.Xh
end Nc"h8p?
eM9~&{m.
n = n(:); yS3x))
m = m(:); \C<rg|
if any(mod(n-m,2)) D!Gm9Pa}
error('zernfun:NMmultiplesof2', ... Q'|cOQX
'All N and M must differ by multiples of 2 (including 0).') 6B+
@76w H
end lA]u8+gXd
+5({~2Lzvp
if any(m>n) ol[{1KT{
error('zernfun:MlessthanN', ... RK'( {1
'Each M must be less than or equal to its corresponding N.') 8\a)}k~4
end g|+G(~=e|
M?\)&2f[Z
if any( r>1 | r<0 ) hCo&SRC/5
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9J%>2AA
end be764do
!^m5by
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )Z;Y,g
error('zernfun:RTHvector','R and THETA must be vectors.') /60[T@Mz
end C$(t`G
F)%; gzs
r = r(:); {T^'&W>8G8
theta = theta(:); 9
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length_r = length(r); NeK:[Q@je
if length_r~=length(theta) jkdNisq37
error('zernfun:RTHlength', ... m+u>%Ys`
'The number of R- and THETA-values must be equal.') C>03P.s4c
end
RB\WttI
W*s`1O >
% Check normalization: ?"C]h s
% -------------------- oVhw2pKpM
if nargin==5 && ischar(nflag) Zq`bd55~
isnorm = strcmpi(nflag,'norm'); vc!S{4bN
if ~isnorm sZbzY^P
error('zernfun:normalization','Unrecognized normalization flag.') i5wA=K_
end nRo`O
else ~/#?OLj(T
isnorm = false; z`Q5J9_<cV
end JA)gM
7<tqT
@c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BsRas
% Compute the Zernike Polynomials AnyFg)a<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sXydMk`J
H\b5]q%
% Determine the required powers of r: G|MDo|q]
% ----------------------------------- fwnpmuJ
m_abs = abs(m); UMX+h])#N
rpowers = []; q#778
for j = 1:length(n) tFSdi.|G=
rpowers = [rpowers m_abs(j):2:n(j)]; K;97/"
end y$&a(S]
rpowers = unique(rpowers); (Q4_3<G+
[@y=%\%R
% Pre-compute the values of r raised to the required powers, B>]5/!_4
% and compile them in a matrix: QbNv+Eu5
% ----------------------------- e7?W VV,
if rpowers(1)==0 jK=*~I
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =ddx/zN
rpowern = cat(2,rpowern{:}); "''<:K|
rpowern = [ones(length_r,1) rpowern]; %1<p1u'r?#
else f|G7L5-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 87Uv+((H
rpowern = cat(2,rpowern{:}); .;F+ QP0
end I[`2MKh
C&st7.
(k
% Compute the values of the polynomials: \|pAn
% -------------------------------------- 6f>l~$
y = zeros(length_r,length(n)); hHg
gH4T
for j = 1:length(n) rzmk-V
s = 0:(n(j)-m_abs(j))/2; nSow$6T_
pows = n(j):-2:m_abs(j); a"DV`jn
for k = length(s):-1:1 ICTtubjV"
p = (1-2*mod(s(k),2))* ... 9j2I6lGQ
prod(2:(n(j)-s(k)))/ ... StDmJ]
prod(2:s(k))/ ... 2%qn!+.
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 'f}S,i +q
prod(2:((n(j)+m_abs(j))/2-s(k))); 0;H6b=
idx = (pows(k)==rpowers); u20b+c4
y(:,j) = y(:,j) + p*rpowern(:,idx); 6uXW`/lvX
end IX*S:7S[
)eFFtnu5
if isnorm 7, 13g)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); u`'z~N4}
end R>U<8z"i
end 5p|@ )
% END: Compute the Zernike Polynomials /C:'qhY,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5Hm!5:ZB
`eWcp^|
% Compute the Zernike functions: LJ/qF0L!H
% ------------------------------ SN{*:\>,
idx_pos = m>0; IeB6r+4|
idx_neg = m<0; i@CMPz-h&
+.lWck
z = y; 4ufLP DH
if any(idx_pos) 9sCk\`n
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?R]y}6P$
end =.X?LWKY
if any(idx_neg) ^!<7#kX
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); $
tNhwF
end e]
K=Nm
6}T%m?/ }
% EOF zernfun