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 h28")c.pH=
function z = zernfun(n,m,r,theta,nflag) rToZN!q\S
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. pmm?Fq!s=
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :%[=v(G[
% and angular frequency M, evaluated at positions (R,THETA) on the 'H"wu
/#
% unit circle. N is a vector of positive integers (including 0), and en"]u,!
% M is a vector with the same number of elements as N. Each element \8Mn[G9TL
% k of M must be a positive integer, with possible values M(k) = -N(k) R+'$V$g\X
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, %+\ PN
% and THETA is a vector of angles. R and THETA must have the same hu?Q,[+o
% length. The output Z is a matrix with one column for every (N,M) tDWW
4H
% pair, and one row for every (R,THETA) pair. &`#k1t'
% S6k
R o^2
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike (0L7Ivg<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), RrFq"
% with delta(m,0) the Kronecker delta, is chosen so that the integral G,tJ\xMw8
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \Wdl1 =`
% and theta=0 to theta=2*pi) is unity. For the non-normalized $uw[X
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *&WkorByW
%
]/l"
% The Zernike functions are an orthogonal basis on the unit circle. PUt\^ke
% They are used in disciplines such as astronomy, optics, and c$Vu/dgx
% optometry to describe functions on a circular domain. 4*k>M+o/C4
% O$Wi=5
% The following table lists the first 15 Zernike functions. ;yfKYN[
% bW"bkA80
% n m Zernike function Normalization
bsfYz
% -------------------------------------------------- ZXCq>
% 0 0 1 1 w_c)iJ
% 1 1 r * cos(theta) 2 `pMI@"m
% 1 -1 r * sin(theta) 2 ;^XF;zpg
% 2 -2 r^2 * cos(2*theta) sqrt(6) t=,ZR}M1`
% 2 0 (2*r^2 - 1) sqrt(3) 26SXuFJ@
% 2 2 r^2 * sin(2*theta) sqrt(6) xJG&vOf;?
% 3 -3 r^3 * cos(3*theta) sqrt(8) UQ0Sfu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) fL0dy[Ch@
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) w}8
,ICL
% 3 3 r^3 * sin(3*theta) sqrt(8) AcZ{B<
% 4 -4 r^4 * cos(4*theta) sqrt(10) lk.]!K$}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0P{^aSxTP
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1NHiW
v
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) noSkKqP
% 4 4 r^4 * sin(4*theta) sqrt(10) ^Rr!YnEN
% -------------------------------------------------- <WXGDCj
% JD`IPQb~E
% Example 1: qPI\Y3ZU
% d#-scv}s5
% % Display the Zernike function Z(n=5,m=1) {Ad4H[]|]
% x = -1:0.01:1; sj9j47y
% [X,Y] = meshgrid(x,x); l*r8.qp
% [theta,r] = cart2pol(X,Y); _Y{8FN(4
% idx = r<=1; /"(`oe<
% z = nan(size(X)); Mi>!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ae%Bl[
% figure 6o5NeKZ
% pcolor(x,x,z), shading interp
kM:Z(Z7$
% axis square, colorbar x;^DlyyYU
% title('Zernike function Z_5^1(r,\theta)') -yP|CZM
% {l
E\y9
% Example 2: /)%$xi
% C VXz>oM
% % Display the first 10 Zernike functions gGaA;YW1
% x = -1:0.01:1; _i3?;Fds
% [X,Y] = meshgrid(x,x); |wxAdPe
% [theta,r] = cart2pol(X,Y); H{)DI(,Y^P
% idx = r<=1; c
-sc*.&
% z = nan(size(X)); 3_DwqZ 'O
% n = [0 1 1 2 2 2 3 3 3 3]; 8"'Z0
Ey
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; p*NKM}
]I
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Sg &0a$
% y = zernfun(n,m,r(idx),theta(idx)); Y)O88C
% figure('Units','normalized') 009[`Z
% for k = 1:10 Ub,5~I+`
% z(idx) = y(:,k); dguN<yS-E
% subplot(4,7,Nplot(k)) T'ko =k
% pcolor(x,x,z), shading interp mm
dQ\\
% set(gca,'XTick',[],'YTick',[])
rSg OQ
% axis square ngt?9i;N
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) V}Ok>6(~
% end
vE~>9
% 3>T2k }
% See also ZERNPOL, ZERNFUN2. 3wYhDxY1
[g/ &%n0^
% Paul Fricker 11/13/2006 5cF7w
H2jF=U"=
`o4%UkBpM
% Check and prepare the inputs: rq#\x{l
% ----------------------------- v:IpZ;^
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) qo*%S
error('zernfun:NMvectors','N and M must be vectors.') eqY8;/
end .)g7s? K
NiSyb yR$
if length(n)~=length(m) @$7'{*
error('zernfun:NMlength','N and M must be the same length.') Z1~`S!(}
end cU|tG!Ij?
j5:/Gl8
n = n(:); 1F'x$~ZI
m = m(:); T;M4NGmvd
if any(mod(n-m,2)) vWH)W?2
error('zernfun:NMmultiplesof2', ... :|HCUZ*H(T
'All N and M must differ by multiples of 2 (including 0).') :!QT ,
end X:>,3[hx|
jmBsPSGIC
if any(m>n) 0woLB#v9
error('zernfun:MlessthanN', ... J$^"cCMr
'Each M must be less than or equal to its corresponding N.') hnnVp_<]
end Ln$= 8x^T
adn2&7H
if any( r>1 | r<0 ) X|'[\v2ld
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Vv&GyqoO]
end 1>=]lMW
j,79G^/YG
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) h:=W`(n5u
error('zernfun:RTHvector','R and THETA must be vectors.') M\A6;dz'
end ZK4d;oa",
L2Fi/UWM
r = r(:); sh/4ui{
theta = theta(:); Tg@:mw5
length_r = length(r); {nj`>
if length_r~=length(theta) C <d]0)
error('zernfun:RTHlength', ... @:/H)F^x
'The number of R- and THETA-values must be equal.') ++!'6!l
end Ou] !@s
~,Kx"VK
% Check normalization: V`4/oM`
% -------------------- &9ERlZ(A
if nargin==5 && ischar(nflag) {%D4%X<
isnorm = strcmpi(nflag,'norm'); G.:QA}FE'
if ~isnorm aeE~[m
error('zernfun:normalization','Unrecognized normalization flag.') ew&"n2r
end w\1K.j=>|N
else 6(/*E=bOKV
isnorm = false; 5
)z'=
end 6J<R;g23R]
gn:&akg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8^puC
% Compute the Zernike Polynomials E/hO0Ox6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !8*7 {7
C ~Doj
% Determine the required powers of r: avY<~-44B
% ----------------------------------- e3k58
m_abs = abs(m); &<EixDi4q
rpowers = []; /], 9N
for j = 1:length(n) y`Zn{mQ@[
rpowers = [rpowers m_abs(j):2:n(j)]; mq+x=
end l^2m7 7)
rpowers = unique(rpowers); !>:]k?$b
*{(tg~2'(
% Pre-compute the values of r raised to the required powers, L5wR4Ue)
% and compile them in a matrix: ZKJhmk
% ----------------------------- nP0rg
if rpowers(1)==0 ~{ucr#]C
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @!*I
mNMI
rpowern = cat(2,rpowern{:}); Z3f}'vr
rpowern = [ones(length_r,1) rpowern]; ZU;nXqjc
else [$@EQ]tt/
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); GO3KKuQ=
rpowern = cat(2,rpowern{:}); $lg{J$
h8
end qb$M.-\ne
h\4enu9[RL
% Compute the values of the polynomials: T%yGSk
% -------------------------------------- CQs,G8\/
y = zeros(length_r,length(n)); Q[9W{l+
for j = 1:length(n)
= Atyy
s = 0:(n(j)-m_abs(j))/2; eMtQa;Lc9o
pows = n(j):-2:m_abs(j); x$z>.4
for k = length(s):-1:1 _adW>-wQ!d
p = (1-2*mod(s(k),2))* ... |Es,$
prod(2:(n(j)-s(k)))/ ... y;fnC5Q
prod(2:s(k))/ ... ~En]sj
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $ve*j=p
prod(2:((n(j)+m_abs(j))/2-s(k))); -0+h&CO
idx = (pows(k)==rpowers); !`dMTW
y(:,j) = y(:,j) + p*rpowern(:,idx); aWY#gI{
end $XcuU
sG
Y+gNi_dE
if isnorm A#gy[.Bb
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 6('CB|ga
end !O4)YM
end fs2y$HN
% END: Compute the Zernike Polynomials kR<\iT0j
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zd=N.
mOJ-M@ME
% Compute the Zernike functions: tlgg~MViS
% ------------------------------ #Eqx Eo;
idx_pos = m>0; _sQhD i
idx_neg = m<0; ;Q<2Y#
t\O#5mo
z = y; f%yNq6l
if any(idx_pos) QwLSL<.
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Ej<`HbJ'Q
end sW&h?jdf
if any(idx_neg) MAD t$_
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j2oU1' b
end (Ft#6oK"
n`D-?]*
% EOF zernfun