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 @'FE2^~Jj
function z = zernfun(n,m,r,theta,nflag) Z9`TwS@x[
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. wD\ZOn_J
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j f~wBmd7
% and angular frequency M, evaluated at positions (R,THETA) on the sp9W?IJ 6c
% unit circle. N is a vector of positive integers (including 0), and OEhHR
% M is a vector with the same number of elements as N. Each element s<QkDERMX
% k of M must be a positive integer, with possible values M(k) = -N(k) + =$
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u0s8yPA
% and THETA is a vector of angles. R and THETA must have the same rVSZ.+n
% length. The output Z is a matrix with one column for every (N,M) @I3eK^#|P
% pair, and one row for every (R,THETA) pair.
=Ufr^naA
% Q\Kx"Y3i
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike T3%C%BcX
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |9K<-yD
% with delta(m,0) the Kronecker delta, is chosen so that the integral "h"NW[R
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3)Ac"nuyqH
% and theta=0 to theta=2*pi) is unity. For the non-normalized dE`-\J
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. yx{3J
% dR^"X3$
% The Zernike functions are an orthogonal basis on the unit circle. D1s4`V -
% They are used in disciplines such as astronomy, optics, and H U+ I
% optometry to describe functions on a circular domain. _RkuBOv@e
% Z=S>0|`R
% The following table lists the first 15 Zernike functions. s 0u{dqP
% Y'VBz{brf
% n m Zernike function Normalization JC?N_kP%W
% -------------------------------------------------- ?
zDa=7 J
% 0 0 1 1 2{,n_w?Wy
% 1 1 r * cos(theta) 2 A
Io|TD5{~
% 1 -1 r * sin(theta) 2 n'FwM\
% 2 -2 r^2 * cos(2*theta) sqrt(6) sq /]wzT:
% 2 0 (2*r^2 - 1) sqrt(3) ?`_jFj+<\S
% 2 2 r^2 * sin(2*theta) sqrt(6) c:!z O\P#
% 3 -3 r^3 * cos(3*theta) sqrt(8) ~ Hy,7
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) _Xcn
N:Rt
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .4y>QN#VL
% 3 3 r^3 * sin(3*theta) sqrt(8) #uCB)n&.
% 4 -4 r^4 * cos(4*theta) sqrt(10) E-5_{sc
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xw^.bz|
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P$GjF-!:
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) &[mZD,
% 4 4 r^4 * sin(4*theta) sqrt(10) `Nh"
% -------------------------------------------------- cE'L% Z
% ~p0c3*
% Example 1: K0pac6]
% >gll-&;t
% % Display the Zernike function Z(n=5,m=1) !9iGg*0dx
% x = -1:0.01:1; &;TJ~r#K
% [X,Y] = meshgrid(x,x); UYP9c}_,4
% [theta,r] = cart2pol(X,Y); `6Qdfmk=
% idx = r<=1; K5t0L!6<+
% z = nan(size(X)); "6ECgyD+E!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); G9P!_72
% figure T GB_~Bqe
% pcolor(x,x,z), shading interp D('2p8;2"7
% axis square, colorbar mog[pu:!,
% title('Zernike function Z_5^1(r,\theta)') >O9o,o/6R
% t`'iU$:1f
% Example 2: 5+Mdh`
% t>)45<PEw
% % Display the first 10 Zernike functions BI?@1q}:
% x = -1:0.01:1; y&[y=0!
% [X,Y] = meshgrid(x,x); t+r:"bb
% [theta,r] = cart2pol(X,Y); ~ I}9;XT
% idx = r<=1; ~tFqb<n
% z = nan(size(X)); /e}#'
H
% n = [0 1 1 2 2 2 3 3 3 3]; DaH Z{T8>d
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; wd@aw /
% Nplot = [4 10 12 16 18 20 22 24 26 28]; m(iR|Zx
% y = zernfun(n,m,r(idx),theta(idx)); 69y;`15
% figure('Units','normalized') A=zPLq{Sb
% for k = 1:10 >kZ57,
% z(idx) = y(:,k); lS^(&<{
% subplot(4,7,Nplot(k)) ?YM4b5!3T
% pcolor(x,x,z), shading interp 1_'? JfY-
% set(gca,'XTick',[],'YTick',[]) Mp$@`8X`
% axis square w@\vHH.;V
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !}+tdT(y
% end XZNY4/25G
% :q<Z'EnW
% See also ZERNPOL, ZERNFUN2. 8N%Bn&
}V;+l8
% Paul Fricker 11/13/2006 :1q4"tv|
'uDjFQX
jDM
w2#<
% Check and prepare the inputs: bOp54WI-g
% ----------------------------- R
#]jSiS
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qH,l#I\CG
error('zernfun:NMvectors','N and M must be vectors.') u}bf-;R
end >gKh
# {fTgq
if length(n)~=length(m) gnp~OVDqfL
error('zernfun:NMlength','N and M must be the same length.') <mMTD8Sx]
end V}o n|A
XNMa0
n = n(:); kU-t7'?4
m = m(:); Z4$cyL'$P
if any(mod(n-m,2)) d1@%W;qX!
error('zernfun:NMmultiplesof2', ... ;;$# )b
'All N and M must differ by multiples of 2 (including 0).') Wjh/M&,
end (}r|yE
0Z<I%<8bK
if any(m>n) {K{EOB_u
error('zernfun:MlessthanN', ... Lj\/Ji_
'Each M must be less than or equal to its corresponding N.') gG%V 9eOQ
end Ch()P.n?
$GQ`clj<
if any( r>1 | r<0 ) F;lI+^}}
error('zernfun:Rlessthan1','All R must be between 0 and 1.') /WV7gO&L1
end R:JX<Ba
l&VjUPz_
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _{vkX<s
error('zernfun:RTHvector','R and THETA must be vectors.') plu$h-$d
end m\>a,oZH
!J*,)kRN
r = r(:); `u!l3VZ/4
theta = theta(:); 49Df?sx
length_r = length(r); wfL-oi'5
if length_r~=length(theta) b?4/#&z]
error('zernfun:RTHlength', ... e6X[vc|Y}
'The number of R- and THETA-values must be equal.') thO ~=RB
end ]u-]'P
LIU}a5
% Check normalization: KD1=Y80P
% -------------------- v]%WH~>
if nargin==5 && ischar(nflag) S|rgCh!h
isnorm = strcmpi(nflag,'norm'); 6ZgU"!|r
if ~isnorm {u!)y?}I-
error('zernfun:normalization','Unrecognized normalization flag.') 1Kvx1p
end 8;y&Pb~)
else >3:?)
isnorm = false; UY2X
end e}@)z3Q<l
KV|}# <dD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -S,ln
% Compute the Zernike Polynomials o]{uc,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hqk}akXt
{
74mf'IW
% Determine the required powers of r: )5%C3/Dl!
% ----------------------------------- [U#72+K
m_abs = abs(m); ,y9iKkg
rpowers = []; 8,O33qwH
for j = 1:length(n) !|2VWI}
rpowers = [rpowers m_abs(j):2:n(j)]; ]Ni$.@Hu$
end e&MC|US=\
rpowers = unique(rpowers); obK*rdg,
<]C$xp<2
% Pre-compute the values of r raised to the required powers, k{tMzx]F__
% and compile them in a matrix: T9 <2A1
% ----------------------------- wOQ#N++C
if rpowers(1)==0 s{ V*1$e~
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); wn>edn
rpowern = cat(2,rpowern{:}); Fg$3N5*
rpowern = [ones(length_r,1) rpowern]; xX0-]Y h:
else =S[yE]v^
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); sfr(/mp(
rpowern = cat(2,rpowern{:}); iFSJL,QZ3
end KucV3-I
d1!i(MaV!
% Compute the values of the polynomials: EzW)'Zzw~
% -------------------------------------- ,1q_pep~?%
y = zeros(length_r,length(n)); P+MA*:
for j = 1:length(n) m6eZ_&+u
s = 0:(n(j)-m_abs(j))/2; %2'A
pp
pows = n(j):-2:m_abs(j); >$gG/WD?KR
for k = length(s):-1:1 O_$dI*RK
p = (1-2*mod(s(k),2))* ... U%7i=Z{^Ks
prod(2:(n(j)-s(k)))/ ... O 2{)WWOT
prod(2:s(k))/ ... yix'rA -T
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... B)$c|dUV
prod(2:((n(j)+m_abs(j))/2-s(k))); I O%6 O
idx = (pows(k)==rpowers); cN! uV-e
y(:,j) = y(:,j) + p*rpowern(:,idx); %CZ-r"A
end 7;.xc{
N_4eM,7t
if isnorm 53 QfTP
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); sGY_{CZ:
end %I!:ITa
end QU{Ech'
% END: Compute the Zernike Polynomials ggtDN{t
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C0.'_
gw+9x<e
% Compute the Zernike functions: "Th$#3
% ------------------------------ |6J ?8y
idx_pos = m>0; q,<[hBri-
idx_neg = m<0; d;tkJ2@NO
Zn:R
PMk*
z = y; kH*P n'
if any(idx_pos) Jxf~&!zR
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8T;IZ(s
end Gy1xG.yM~
if any(idx_neg) ^/wfXm
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tC8(XMVx
end O<9~Kgd8h
/|{,sWf2
% EOF zernfun