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 \ a(ce?C
function z = zernfun(n,m,r,theta,nflag) iy]?j$B$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @_#\qGY
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rcY &n^:
% and angular frequency M, evaluated at positions (R,THETA) on the <l5m\A
% unit circle. N is a vector of positive integers (including 0), and &%t&[Se_~
% M is a vector with the same number of elements as N. Each element Nv6"c<(L=
% k of M must be a positive integer, with possible values M(k) = -N(k) MHye!T6fO\
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u3pFH(
% and THETA is a vector of angles. R and THETA must have the same IvI..#EzG
% length. The output Z is a matrix with one column for every (N,M) %:;g|PC
% pair, and one row for every (R,THETA) pair. !H9^j6|
% $b53~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike YgS,5::SU
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DL!%Np?`
% with delta(m,0) the Kronecker delta, is chosen so that the integral =]/<Kd}A.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /4+(e I7
% and theta=0 to theta=2*pi) is unity. For the non-normalized !=a]Awr\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. K'
<[kh:cl
% b`^Q ':^A
% The Zernike functions are an orthogonal basis on the unit circle. uI%7jA~@
% They are used in disciplines such as astronomy, optics, and Zzz94`
% optometry to describe functions on a circular domain. Z,Us<du
% 7v0AG:
% The following table lists the first 15 Zernike functions. j:/Z_v'
% u*,>$(-u
% n m Zernike function Normalization $&KkZ
% -------------------------------------------------- \[^!
ys
% 0 0 1 1 N#t`ZC&m'
% 1 1 r * cos(theta) 2 s;*
UP
% 1 -1 r * sin(theta) 2 ;DR5?N/a
% 2 -2 r^2 * cos(2*theta) sqrt(6) Pt/]Z<VL
% 2 0 (2*r^2 - 1) sqrt(3) AYNdV(
% 2 2 r^2 * sin(2*theta) sqrt(6) FoH1O+e
% 3 -3 r^3 * cos(3*theta) sqrt(8) mZPvG
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0\B{~1(^
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 9n;6zVV%`
% 3 3 r^3 * sin(3*theta) sqrt(8) BzO,(bd!PI
% 4 -4 r^4 * cos(4*theta) sqrt(10) \hBzP^*"n
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ; D/6e6
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) N2duhI6
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Vp|?R65S*
% 4 4 r^4 * sin(4*theta) sqrt(10)
[)~1Lu
% -------------------------------------------------- ?h%Jb^#9
% 5I^;v;F
% Example 1: 3JBXGT0gJ
% ar}-~~h 5
% % Display the Zernike function Z(n=5,m=1) NMf#0Nz-
% x = -1:0.01:1; U,;796h
% [X,Y] = meshgrid(x,x); \]5I atli
% [theta,r] = cart2pol(X,Y); $j<KXR
% idx = r<=1; m_@XoS
yxI
% z = nan(size(X)); 0H_uxkB~
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0`-b57lF&
% figure 9!W$S[ABRB
% pcolor(x,x,z), shading interp $/;K<*O$
% axis square, colorbar &r4|WM/ec
% title('Zernike function Z_5^1(r,\theta)') 9q_{_%G%
% $[,4Ib_|
% Example 2: 4"(rZWv
% ; teM^zyI
% % Display the first 10 Zernike functions GJrmK
% x = -1:0.01:1; -`* 'p i
% [X,Y] = meshgrid(x,x); iRlZWgj4^
% [theta,r] = cart2pol(X,Y); X~D[CwA|`
% idx = r<=1; <<A#4!f
% z = nan(size(X)); U$& '> %#
% n = [0 1 1 2 2 2 3 3 3 3]; e(|Z<6
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 83t/\x,Q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; P~=yTW
% y = zernfun(n,m,r(idx),theta(idx)); /:(A9b-B
% figure('Units','normalized') 7H< IO`
% for k = 1:10 .O5V;&,
% z(idx) = y(:,k); -9,~b9$
% subplot(4,7,Nplot(k)) s_VcC_A
% pcolor(x,x,z), shading interp AguE)I&m
% set(gca,'XTick',[],'YTick',[]) vJ^~J2#5
% axis square }P.Z}n;Uj
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) A`Y^qXFb`
% end PDuBf&/e
% D_czUM
% See also ZERNPOL, ZERNFUN2. SM4`Hys;p
w3);ZQ|
% Paul Fricker 11/13/2006 4d PTrBQ?
1*dN. v:5
%gAT\R_f
% Check and prepare the inputs: NGl
8*Af
% ----------------------------- k)S1Z s~G
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3J
&Ros
error('zernfun:NMvectors','N and M must be vectors.') DlE, aYB
end I@/
G#3Zr
pQ:^ ziwa3
if length(n)~=length(m) .G!xcQ`?
error('zernfun:NMlength','N and M must be the same length.') S,AxrQc
end "}*D,[C5e
{eaR,d~X
n = n(:); f/#Id]B
m = m(:); ?1JY6v]h4
if any(mod(n-m,2)) D48e30
error('zernfun:NMmultiplesof2', ... 4i)5=H
'All N and M must differ by multiples of 2 (including 0).') s!/lQo5/
end CMW4Zqau*
n*wQgC'vw
if any(m>n) EpMxq7*
error('zernfun:MlessthanN', ... 9Sxr9FLW~
'Each M must be less than or equal to its corresponding N.') :) lG}c
end xBTx`+%WS
nJN-U+)u
if any( r>1 | r<0 ) W{"sB:E
error('zernfun:Rlessthan1','All R must be between 0 and 1.') \~E?;q!
end $e7%>*?m
_)
x{TnK
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) P|$n
error('zernfun:RTHvector','R and THETA must be vectors.') U`qC.s(L
end g&xj(SMj-$
nwKp8mfP
r = r(:); *O~y6|U?
theta = theta(:); <.n,:ir
length_r = length(r); OA&'T*)-A6
if length_r~=length(theta) F~
5,-atDM
error('zernfun:RTHlength', ... vu*e*b$}
'The number of R- and THETA-values must be equal.') x:MwM?
end 5:IDl1f5
F%|P#CaB
% Check normalization: vF$(
Y/
% -------------------- Gg'!(]v
if nargin==5 && ischar(nflag) h8`On/Ur_8
isnorm = strcmpi(nflag,'norm'); rwLKY.J]
if ~isnorm {wz)^A
sy
error('zernfun:normalization','Unrecognized normalization flag.') );d 07\V
end agx8 *x
else IAH"vHM
isnorm = false; qKfUm:7Q_
end {q)d
%@Gy<t,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% bQautRW
% Compute the Zernike Polynomials U*=E(l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =\%ER/
g D6S%O
% Determine the required powers of r: t8-Nli*O
% ----------------------------------- ),p0V
m_abs = abs(m); #("M4}~
rpowers = []; RBrb7D{
for j = 1:length(n) /&Oo)OB;
rpowers = [rpowers m_abs(j):2:n(j)]; O]PM L`
end (uvQ/!
rpowers = unique(rpowers); c1k[)O~
(2#Xa,pb
% Pre-compute the values of r raised to the required powers, ]M*`Y[5"
% and compile them in a matrix: 5VTVx1P[8
% ----------------------------- LsWD^JE.
if rpowers(1)==0 W9%v#;2
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e&z@yy$
rpowern = cat(2,rpowern{:}); \.mVLLtG
rpowern = [ones(length_r,1) rpowern]; Pb'(Y
else BwWSztJ+B
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); w&L~+Z<
rpowern = cat(2,rpowern{:}); dBd7#V:}yV
end 4}m9,
3LET zsJ
% Compute the values of the polynomials: v
^h:E
% -------------------------------------- g9" wX?*
y = zeros(length_r,length(n)); [ *Dj:A)V^
for j = 1:length(n) \lQ3j8U
s = 0:(n(j)-m_abs(j))/2; !ddyJJ^a
pows = n(j):-2:m_abs(j); 3UUdJh<~
for k = length(s):-1:1 k 3m_L-
p = (1-2*mod(s(k),2))* ... rgVRF44X{
prod(2:(n(j)-s(k)))/ ... 3Tu]-.
prod(2:s(k))/ ... `CVkjLiy
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e El)wZ,A
prod(2:((n(j)+m_abs(j))/2-s(k))); MMFg{8
idx = (pows(k)==rpowers); b~vV++ou_
y(:,j) = y(:,j) + p*rpowern(:,idx); pZ>yBY?R8>
end .3C::~:
\+V"JIStUj
if isnorm !vf:mMo
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); CK n2ZL
end "HJ^>%ia
end |qMG@
% END: Compute the Zernike Polynomials Bn]=T
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i=#`7pt%'a
B$2b=\
% Compute the Zernike functions: vT EqT
% ------------------------------ C})Dvh
idx_pos = m>0; ,)[9RgsE
idx_neg = m<0; ,5$G0
U}jGr=tu
z = y; 9\.0v{&v
if any(idx_pos) T]wI)
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); SQ,-45@W
end (O+d6oT=Z2
if any(idx_neg)
$L= Dky7
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _+By=B.'
end ^Q`5+
"/6#Z>y
% EOF zernfun