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 bB;5s`-
function z = zernfun(n,m,r,theta,nflag) h@]XBv
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. JOim3(5?s
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Sw^u3
% and angular frequency M, evaluated at positions (R,THETA) on the ">jj
% unit circle. N is a vector of positive integers (including 0), and 84 pFc;<
% M is a vector with the same number of elements as N. Each element wtV#l4
% k of M must be a positive integer, with possible values M(k) = -N(k) c>~*/%+
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3%;a)c;D
% and THETA is a vector of angles. R and THETA must have the same R=
o2K
% length. The output Z is a matrix with one column for every (N,M) ;H.^i|_/
% pair, and one row for every (R,THETA) pair. WPG(@zD
% PO7Lf#9]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike @\P;W(m.i
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), pDCeQ6?
% with delta(m,0) the Kronecker delta, is chosen so that the integral @)&=%
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5vZ^0yFQ
% and theta=0 to theta=2*pi) is unity. For the non-normalized :s6o"VkW
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. U,- 39mr
% >:!X.TG$
% The Zernike functions are an orthogonal basis on the unit circle. pKrN:ExB"\
% They are used in disciplines such as astronomy, optics, and s)Cjc.Qs
% optometry to describe functions on a circular domain. TNh1hhJ$b
% E5lBdM>2
% The following table lists the first 15 Zernike functions. !*. -`$x
% 6Yxh9*N~]
% n m Zernike function Normalization f|lU6EkU
% -------------------------------------------------- `eCo~(Fy
% 0 0 1 1 7 uKY24
% 1 1 r * cos(theta) 2 !pdb'*,n
% 1 -1 r * sin(theta) 2 RnI&8
% 2 -2 r^2 * cos(2*theta) sqrt(6) o;R2p $
% 2 0 (2*r^2 - 1) sqrt(3) JU5C}%Q6
% 2 2 r^2 * sin(2*theta) sqrt(6) Nyj( 0W
% 3 -3 r^3 * cos(3*theta) sqrt(8) Mz~D#6=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) iBgx
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .KUv(-
% 3 3 r^3 * sin(3*theta) sqrt(8) l
+OFw)8od
% 4 -4 r^4 * cos(4*theta) sqrt(10) +sUFv)!4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ApV~(k)W
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r^a7MHY1
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) os={PQRD
% 4 4 r^4 * sin(4*theta) sqrt(10) iv;Is[<o
% -------------------------------------------------- scou%K
% m~d]a$KQ5-
% Example 1: EbE-}>7OO
% B1C-J/J
% % Display the Zernike function Z(n=5,m=1) usCt#eZK
% x = -1:0.01:1; s<eb;Z2D
% [X,Y] = meshgrid(x,x); {Um)15K
% [theta,r] = cart2pol(X,Y); 4f'V8|QM{
% idx = r<=1; lqZ 5?BD1
% z = nan(size(X)); 5}]"OXQ
% z(idx) = zernfun(5,1,r(idx),theta(idx)); '*w00
% figure EYEnN
% pcolor(x,x,z), shading interp ~W+kiTsD?
% axis square, colorbar /%TI??PGu
% title('Zernike function Z_5^1(r,\theta)') FZ,#0ZYJGP
% W=vP]x
>J
% Example 2: QpA/SmJ
% C3],n
% % Display the first 10 Zernike functions J| bd)0
% x = -1:0.01:1; $#S&QHyEe
% [X,Y] = meshgrid(x,x); Sf7\;^
% [theta,r] = cart2pol(X,Y); ,>-< (Qi
% idx = r<=1; Dq5j1m.
% z = nan(size(X)); )~] (&
% n = [0 1 1 2 2 2 3 3 3 3]; .=;3d~.]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; f@DYN!Z_m
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8b-Q F
% y = zernfun(n,m,r(idx),theta(idx)); F,dx2ZPIs?
% figure('Units','normalized') cy3B({PLy
% for k = 1:10 L3 --r
% z(idx) = y(:,k); _Khc3Jo
% subplot(4,7,Nplot(k)) F,MO@&ue"
% pcolor(x,x,z), shading interp S.m{eur!,E
% set(gca,'XTick',[],'YTick',[]) ruzspS
% axis square `t9?=h!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) O_DtvjI'
% end x+x40!+\
% 0#&5.Gr)
% See also ZERNPOL, ZERNFUN2. fb8g7H|
*ikc]wQr$
% Paul Fricker 11/13/2006 -}=%/|\FG
lq&wXi
FCuB\Q
% Check and prepare the inputs: e5B Qr$j
% ----------------------------- ReI/]#Us
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5>j)kx=J9
error('zernfun:NMvectors','N and M must be vectors.') #+5pgD2C
end Jjv=u
"a1n_>#Fb
if length(n)~=length(m) dhr3,&+T2
error('zernfun:NMlength','N and M must be the same length.') @I/]D6
~"
end 3]UUG
^!z[t\$
n = n(:); H77"
m = m(:); yo)%J
if any(mod(n-m,2)) ;@Z#b8aM}
error('zernfun:NMmultiplesof2', ... Vq;A>
'All N and M must differ by multiples of 2 (including 0).') G *;a^]-
end "WK{ >T
? 1$fJ3
if any(m>n) M9@ri ^x
error('zernfun:MlessthanN', ... ;b(p=\i
'Each M must be less than or equal to its corresponding N.') oifv+oY
end :^x?2%
~K.
~-m "
if any( r>1 | r<0 ) ^__Dd)(
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ICkp$u^
end J@X'PG<
6B
lh D,\3/O
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) oDYRQozo>
error('zernfun:RTHvector','R and THETA must be vectors.') BWuqo
end QC;^xG+W
KiOcu=F
r = r(:); iN0nw]_*
theta = theta(:); .0O2Qqdg
length_r = length(r); {0^&SI"5`E
if length_r~=length(theta) 3?Pn6J{O
error('zernfun:RTHlength', ... ,gOOiB
}
'The number of R- and THETA-values must be equal.') !M]\I &
end [$"n^5_~
I=9!Rs(QF
% Check normalization: g[7#w,o
% -------------------- 16i"Yg!*
if nargin==5 && ischar(nflag) mAW,?h
isnorm = strcmpi(nflag,'norm'); )R
2.
if ~isnorm
$g+[yb7@
error('zernfun:normalization','Unrecognized normalization flag.') Xo*%/0q'
end '@CR\5 @
else iVTGF<
isnorm = false; ?Wt$6{)
end `8>Py~
deixy.
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JPWOPB'H
% Compute the Zernike Polynomials &F5@6nJ`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (S`2[.j
&0(
% Determine the required powers of r: 9>rPe1iv
% ----------------------------------- T%n2$
m_abs = abs(m); ZwerDkd
rpowers = [];
pzgSg[|
for j = 1:length(n) $aPfGZ<i
rpowers = [rpowers m_abs(j):2:n(j)]; _#}n~}d
end F.=Bnw/-
rpowers = unique(rpowers); 9Xo[(h)5d
*[R
eb%
% Pre-compute the values of r raised to the required powers, V{&rQ@{W
% and compile them in a matrix: Cssl{B
% ----------------------------- dVo.Czyd
if rpowers(1)==0 U*P. :BvG
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yxq}QSb \3
rpowern = cat(2,rpowern{:}); lP!;3iJ B
rpowern = [ones(length_r,1) rpowern]; "a/ Q%.P
else FwZ>{~?3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); P7f,OY<@%o
rpowern = cat(2,rpowern{:}); .eO?Z^
end wL^%w9q-
NwR}yb6
% Compute the values of the polynomials: t"YNgC ^
% -------------------------------------- d/e|'MPX
y = zeros(length_r,length(n)); LW:LFzp
for j = 1:length(n) `\6?WXk3T
s = 0:(n(j)-m_abs(j))/2; I]y.8~xs
pows = n(j):-2:m_abs(j); mTEx,
for k = length(s):-1:1 }Lw>I94e
p = (1-2*mod(s(k),2))* ... !'*csg
prod(2:(n(j)-s(k)))/ ... O8W7<Wc|z
prod(2:s(k))/ ... {?}*1,I
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... fQ=MJ7l
prod(2:((n(j)+m_abs(j))/2-s(k))); e<#DdpX!H~
idx = (pows(k)==rpowers); !!nuAQ"E[
y(:,j) = y(:,j) + p*rpowern(:,idx); +/;*|
end "A)("
?}Lg)EFH
if isnorm GzTq5uU&
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }O4se"xK
end 08m;{+|vY
end K!mOr
% END: Compute the Zernike Polynomials nPgeLG"00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :g\rQazxO
oq_6L\
~
% Compute the Zernike functions: 35x 0T/8
% ------------------------------ leiW4Fj
idx_pos = m>0; %&\ jOq~
idx_neg = m<0; @MK"X}3
=_8Tp~j
z = y; @i3bgx>_o
if any(idx_pos) vkRi5!bR
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); R,
8s_jN
end <p?&udqD
if any(idx_neg) lRP1&FH0
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ?n\*,{9
end y9|K|xO[
*X38{rj
% EOF zernfun