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 ve+bR
function z = zernfun(n,m,r,theta,nflag) S:XsO9:{
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. SpImd IpD
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >(-A"jf
% and angular frequency M, evaluated at positions (R,THETA) on the ]}kw'&
% unit circle. N is a vector of positive integers (including 0), and =Oq*9=v|
% M is a vector with the same number of elements as N. Each element 16>D?;2o(
% k of M must be a positive integer, with possible values M(k) = -N(k) d@p#{ -
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, vz~Oi
% and THETA is a vector of angles. R and THETA must have the same y Vp,)T9
% length. The output Z is a matrix with one column for every (N,M) 7{]dh+)
% pair, and one row for every (R,THETA) pair. Ia<V\$ #
% ;?k<L\zaw
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike !Sw=ns7
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /wax5FS'I,
% with delta(m,0) the Kronecker delta, is chosen so that the integral DJ DQH \&
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, tXqX[Td`0g
% and theta=0 to theta=2*pi) is unity. For the non-normalized m8;w7S7,j~
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $_iE^zZaU^
% ]B UirJ,2
% The Zernike functions are an orthogonal basis on the unit circle. O,9^R
% They are used in disciplines such as astronomy, optics, and @({=~
W^
% optometry to describe functions on a circular domain. m^0vux
% %ioVNbrR7
% The following table lists the first 15 Zernike functions. lKB9n}P
% co~NXpqg
% n m Zernike function Normalization W7Y@]QMX
% -------------------------------------------------- S2e3d
% 0 0 1 1 =kfa1kD&{
% 1 1 r * cos(theta) 2 33EF/k3vW
% 1 -1 r * sin(theta) 2 x+j@YWDpG"
% 2 -2 r^2 * cos(2*theta) sqrt(6) x1?mE)n]
% 2 0 (2*r^2 - 1) sqrt(3) w|6/ i/X
% 2 2 r^2 * sin(2*theta) sqrt(6) )AxD|A
% 3 -3 r^3 * cos(3*theta) sqrt(8) p_g`f9q6D
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) BvsSrse
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 1*yxSU@uY
% 3 3 r^3 * sin(3*theta) sqrt(8) :SS \2
% 4 -4 r^4 * cos(4*theta) sqrt(10) #-\5O
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5 ty2e`~K
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) F3EAjO)ch
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *!ng)3#
% 4 4 r^4 * sin(4*theta) sqrt(10) [8a(4]4
% -------------------------------------------------- v\5O\ I ^
% }%8ZN :
% Example 1: vX\9#Hj
% QM#Vl19>j(
% % Display the Zernike function Z(n=5,m=1) '9Z`y_~)G
% x = -1:0.01:1; pa1<=w
% [X,Y] = meshgrid(x,x); xa@$cxt
% [theta,r] = cart2pol(X,Y); NJQ)Ttt
% idx = r<=1; 8W{M}>;[9
% z = nan(size(X)); O-X(8<~H=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); |~e"i<G#
% figure OemY'M?ZQ
% pcolor(x,x,z), shading interp W`_JERo
% axis square, colorbar -R]0cefC<f
% title('Zernike function Z_5^1(r,\theta)') ewU*5|*[
% jkx>o?s)z
% Example 2: Lo%vG{yTr
% R\5,H!V9n
% % Display the first 10 Zernike functions fwv^dEe
% x = -1:0.01:1; Vf&U`K
% [X,Y] = meshgrid(x,x); tg@61V?>
% [theta,r] = cart2pol(X,Y); ["<Xh0_
% idx = r<=1; hqvhnqQk
% z = nan(size(X)); 0#9H;j<Op
% n = [0 1 1 2 2 2 3 3 3 3]; u"=]cBRWL6
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ?E"192,z@
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 6[3Xe_
% y = zernfun(n,m,r(idx),theta(idx)); $G`CXhbl
% figure('Units','normalized') qC> tni%
% for k = 1:10 O hk\P;}
% z(idx) = y(:,k); Q?i_Nl/|
% subplot(4,7,Nplot(k)) } +}nrJv
% pcolor(x,x,z), shading interp %-!%n=P
% set(gca,'XTick',[],'YTick',[]) ~tA ^[tK
% axis square 1~c\J0h)d
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ng3ZK
% end "00j]e.
% PGJh>[s
% See also ZERNPOL, ZERNFUN2. SYYx>1;8`
+PjTT6
% Paul Fricker 11/13/2006 e'.BTt58Y
94+^K=lAX
;[}OZt
% Check and prepare the inputs: &T,|?0>~=J
% ----------------------------- 4{YA['
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?Ts]zO%%Z
error('zernfun:NMvectors','N and M must be vectors.') b!]O]dk#
end (<eLj Q
yMz#e0k
if length(n)~=length(m) |@X^_L.!
error('zernfun:NMlength','N and M must be the same length.') (N9-YP?qm
end wuH*a3(
+ +}!Gfc?s
n = n(:); R.rch2
m = m(:); jg710.v:
if any(mod(n-m,2)) 'Gn>~m
error('zernfun:NMmultiplesof2', ... <{kPa_`'
'All N and M must differ by multiples of 2 (including 0).') >J7slDRo
end }ssV"5M
m[}k]PB>
if any(m>n) -i`jS_-Cv-
error('zernfun:MlessthanN', ... _ p\L,No
'Each M must be less than or equal to its corresponding N.') ]eKuR"ob0
end uCDe>Q4@/
;d6Dm)/(
if any( r>1 | r<0 ) BYq80Vk%@
error('zernfun:Rlessthan1','All R must be between 0 and 1.') UH!(`Z\C
end r@4A%ql<
y(J~:"}7)
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?}KRAtJ8
error('zernfun:RTHvector','R and THETA must be vectors.') a<<4gXx
end xJCxzJ
kkq1:\pZ]a
r = r(:); `j>5W<5q\
theta = theta(:); SY+0~5E
length_r = length(r); #%"G[B
if length_r~=length(theta) EB[T 5{
error('zernfun:RTHlength', ... u}iuf_
'The number of R- and THETA-values must be equal.') ..}P$
end 9r? Z'~,Za
spV7\Gs.@
% Check normalization: j L|6i-?!
% -------------------- .g8*K "
if nargin==5 && ischar(nflag) 4-yK!LR
isnorm = strcmpi(nflag,'norm'); L!cOg8Z
if ~isnorm KR>)Ek
error('zernfun:normalization','Unrecognized normalization flag.') PQ4mNjXN
end S~Gse+*
else ?@ oF@AEx=
isnorm = false; X<%D@$
end /pj[c;aO
v&d1ACctJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '#+&?6 p
% Compute the Zernike Polynomials Z mJ<h&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p / ITg
[Z$H<m{c-
% Determine the required powers of r: iJzBd7
% ----------------------------------- TPN+jK
m_abs = abs(m); cyCh^- <l@
rpowers = []; } k2Q
for j = 1:length(n) Pu3oQDldV
rpowers = [rpowers m_abs(j):2:n(j)]; %hVR|K|J
end &*v\t\]
rpowers = unique(rpowers); sM-,95H
Wlc&QOfF
% Pre-compute the values of r raised to the required powers, /.SG? 5t4
% and compile them in a matrix: ["3dr@T9Z
% ----------------------------- w)EYj+L
if rpowers(1)==0 AQ'%}(#0
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fp [gKRSF
rpowern = cat(2,rpowern{:}); ]}v]j`9m%
rpowern = [ones(length_r,1) rpowern]; <