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 0r1GGEW`s
function z = zernfun(n,m,r,theta,nflag) FrXP"U}Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. )-Hs]D:
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5 k3m"*
% and angular frequency M, evaluated at positions (R,THETA) on the gI;"P kN
% unit circle. N is a vector of positive integers (including 0), and :#^qn|{e
% M is a vector with the same number of elements as N. Each element 8$\j| mN
% k of M must be a positive integer, with possible values M(k) = -N(k) {Fw"y %a^
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, zH}3J}
% and THETA is a vector of angles. R and THETA must have the same _Ik?WA_;
% length. The output Z is a matrix with one column for every (N,M) kP&I}RY
% pair, and one row for every (R,THETA) pair. 7UMZs7L$
% >U{iof<
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6Qt(Yu*s
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), |di(hY|
% with delta(m,0) the Kronecker delta, is chosen so that the integral .F 6US<]
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, NknS:r&2
% and theta=0 to theta=2*pi) is unity. For the non-normalized (is' ,4^b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. WT<}3(S'?
% CE`]X;#y
% The Zernike functions are an orthogonal basis on the unit circle. nXLz<wE
% They are used in disciplines such as astronomy, optics, and 7b>_vtrt
% optometry to describe functions on a circular domain. xj>P5\mW#
% 2MRd
% The following table lists the first 15 Zernike functions. b},2A'X
% 9efey? z
% n m Zernike function Normalization jL\j$'KC
% -------------------------------------------------- Qq`S=:}~x
% 0 0 1 1 <}{<FXk[
% 1 1 r * cos(theta) 2 iv~R4;;)
% 1 -1 r * sin(theta) 2 j*?8w(!
% 2 -2 r^2 * cos(2*theta) sqrt(6) T:@6(_Z
% 2 0 (2*r^2 - 1) sqrt(3) >^jBE''
% 2 2 r^2 * sin(2*theta) sqrt(6) 1Z< ^8L<
% 3 -3 r^3 * cos(3*theta) sqrt(8) ;um)JCXz
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) A[+)PkR
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) mufGv%U2
% 3 3 r^3 * sin(3*theta) sqrt(8) qhxMO[f
% 4 -4 r^4 * cos(4*theta) sqrt(10) Unb2D4&'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s`bGW1#io
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +Pl)E5W!=`
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H_RfIX)X
% 4 4 r^4 * sin(4*theta) sqrt(10) \s*UUODWK
% -------------------------------------------------- HXKM<E{j
% SPb+H19;
% Example 1: dXh[Ea^
% aKriO
% % Display the Zernike function Z(n=5,m=1) )hrsA&1w
% x = -1:0.01:1; M/p9 I
gp
% [X,Y] = meshgrid(x,x); ,yGbMOV
% [theta,r] = cart2pol(X,Y); ~ps,U
% idx = r<=1; 0Gs\x
% z = nan(size(X)); uMw6b=/U
% z(idx) = zernfun(5,1,r(idx),theta(idx)); P!+Gwm{
% figure nKm#
kb
% pcolor(x,x,z), shading interp 'M~`IN`
% axis square, colorbar D5c
8sB
% title('Zernike function Z_5^1(r,\theta)') ~6t!)QATnp
% w UxFE=ia
% Example 2: -13}]Gls7Q
% %@vF%
% % Display the first 10 Zernike functions OK80-/8HI
% x = -1:0.01:1; 'z8FU~oU
% [X,Y] = meshgrid(x,x); NF8<9
% [theta,r] = cart2pol(X,Y); >g{&Qx`&
% idx = r<=1; N4+Cg t(
% z = nan(size(X)); JI.=y5I
% n = [0 1 1 2 2 2 3 3 3 3]; ~ZVz
sNrx
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; F9o7=5WAb
% Nplot = [4 10 12 16 18 20 22 24 26 28]; C~pas~
% y = zernfun(n,m,r(idx),theta(idx)); bIiuna\
% figure('Units','normalized') Q[#}Oh6$
% for k = 1:10 \:J=tAC
% z(idx) = y(:,k); -r sbSt ?_
% subplot(4,7,Nplot(k)) dHIk3j-!
% pcolor(x,x,z), shading interp T<0 r,
% set(gca,'XTick',[],'YTick',[]) Li6|c*K'
% axis square z='%NZY
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) U)8yd,qG[%
% end Mm@G{J\\
% j2Dw7"f3
% See also ZERNPOL, ZERNFUN2. p Run5 )7
yIKpyyC9H
% Paul Fricker 11/13/2006 33DP?nI}
csW\Q][
:*KTpTa
% Check and prepare the inputs: u$R5Q{H_
% ----------------------------- )7*'r@
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ni2#20L
error('zernfun:NMvectors','N and M must be vectors.') /8e}c`
end "M5
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if length(n)~=length(m) {d[Nc,AMb
error('zernfun:NMlength','N and M must be the same length.') [cnuK
end eY :"\c3
.+1I>L
n = n(:); ~QbHp|g
m = m(:); [<53_2]~
if any(mod(n-m,2)) {ze69 h
error('zernfun:NMmultiplesof2', ... |2l-s 1|y
'All N and M must differ by multiples of 2 (including 0).') L4Jm8sy{
end \eKXsO"d
f8lyH'z0
@
if any(m>n) Hq}g1?b
error('zernfun:MlessthanN', ... SvSO?H!-
'Each M must be less than or equal to its corresponding N.') [gBf1,bK
end /iO"4%v
"BSY1?k{
if any( r>1 | r<0 ) Y|LL]@Lv
error('zernfun:Rlessthan1','All R must be between 0 and 1.') yDqwz[v b
end <5E'`T
^!S4?<v
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {*O%A
error('zernfun:RTHvector','R and THETA must be vectors.') 0E26J@jcZ7
end i) e6U(H
bBs{PI2(p1
r = r(:); )58O9b
theta = theta(:); zU!{_Ao9
length_r = length(r); |V\{U j
if length_r~=length(theta) m
.(ja
error('zernfun:RTHlength', ... PFX,X
'The number of R- and THETA-values must be equal.') Xq$-&~
end twr{jdY9
~Yd[&vpQ
% Check normalization: XDCm
% -------------------- )r46I$]>
if nargin==5 && ischar(nflag) KhWy
isnorm = strcmpi(nflag,'norm'); E'\gd7t ;
if ~isnorm 9 wR D=a
error('zernfun:normalization','Unrecognized normalization flag.') !d()'N
end YxM\qy{Vr
else R7Qj<,
isnorm = false; 6 EqN>.
end _5 SvZ;4
KuwhA-IL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }SWfP5D@
% Compute the Zernike Polynomials vy~6]hH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5Yv*f:
G@DNV3Cc
% Determine the required powers of r: ZOfv\(iJ;
% ----------------------------------- AHs%?5YTY;
m_abs = abs(m); 4|_xz;i
rpowers = []; HVA:|Z19
for j = 1:length(n) @\F7nhSfa
rpowers = [rpowers m_abs(j):2:n(j)]; o`n8Fk}i
end 0\!Bh^++1
rpowers = unique(rpowers); I?D=Q$s
5b rM..
% Pre-compute the values of r raised to the required powers, liYsUmjZ=
% and compile them in a matrix: 3Y#
% ----------------------------- H&ek"nP_
if rpowers(1)==0 'G65zz
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !X7z y9
rpowern = cat(2,rpowern{:}); =*'yGB[x)
rpowern = [ones(length_r,1) rpowern]; wm#(\dj
else #"6l+}
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )*}\fmOv{
rpowern = cat(2,rpowern{:}); EC$F|T0f
end &]a(5
(QIU 3EN
% Compute the values of the polynomials: }BS
EK<W
% -------------------------------------- e-`9-U%6
y = zeros(length_r,length(n)); $DH/
for j = 1:length(n) Fw[1Aa#
s = 0:(n(j)-m_abs(j))/2; iyCH)MA
pows = n(j):-2:m_abs(j); x(u.(:V
for k = length(s):-1:1 BsXF'x<U*
p = (1-2*mod(s(k),2))* ... {G=> WAXo
prod(2:(n(j)-s(k)))/ ... 7Q{&L#;
prod(2:s(k))/ ... 3q/"4D
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0(c,J$I]Z!
prod(2:((n(j)+m_abs(j))/2-s(k))); =55)|$hgD
idx = (pows(k)==rpowers); a`yCPnB(
y(:,j) = y(:,j) + p*rpowern(:,idx); qDGx(d
end M#2<|VUW,
P}AwE,&Q
if isnorm H8"RdKwg?
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2Ax(q&`9
end w$$pTk|&n
end a?Fz&BE
% END: Compute the Zernike Polynomials JT}"CuC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }6LcimQyK
)X#$G?|Hn
% Compute the Zernike functions: ^'N!k{x
% ------------------------------ qK;J:GT>
idx_pos = m>0; M GC=L .
idx_neg = m<0; ^Mm%`B7W
=Cf@!wZ^
z = y; w`boQ_Ir
if any(idx_pos) 6@0?~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); m6
M/G
end zLr:zf l
if any(idx_neg) l{r HXST|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); uUaDesz~=
end dn~k_J=p
T:'<:*pD
% EOF zernfun