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 9}K(Q=
function z = zernfun(n,m,r,theta,nflag) |G`4"``]k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Y yQf
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7I2a*4}
% and angular frequency M, evaluated at positions (R,THETA) on the MEdIw#P.}{
% unit circle. N is a vector of positive integers (including 0), and M"$jpBN*
% M is a vector with the same number of elements as N. Each element 7Va#{Y;Zy
% k of M must be a positive integer, with possible values M(k) = -N(k) N"q+UCRC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, J4Q)`Y\~
% and THETA is a vector of angles. R and THETA must have the same ~:P8g<w
% length. The output Z is a matrix with one column for every (N,M) 2n-Tpay0
% pair, and one row for every (R,THETA) pair. :IP;FrcMP
% !`O_VV`/@
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Nqo#sBS
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *@$($<pY&
% with delta(m,0) the Kronecker delta, is chosen so that the integral |k['wqn"
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, } kh/mq
% and theta=0 to theta=2*pi) is unity. For the non-normalized }iiG$?|.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. h%CEb<
% 9H
!B)
% The Zernike functions are an orthogonal basis on the unit circle. _{2Fx[m%
% They are used in disciplines such as astronomy, optics, and ,q'gG`M
N
% optometry to describe functions on a circular domain. IGF37';;
% NIWI6qCw
% The following table lists the first 15 Zernike functions. e"v[)b++Y
% LX(iuf+l
% n m Zernike function Normalization ~vjr;a(B
% -------------------------------------------------- clR?< LO
% 0 0 1 1 k#IS,NKE
% 1 1 r * cos(theta) 2 M<M#<kD
% 1 -1 r * sin(theta) 2 HwVgT"
% 2 -2 r^2 * cos(2*theta) sqrt(6) :?&WKW
% 2 0 (2*r^2 - 1) sqrt(3) 7(+OsE
% 2 2 r^2 * sin(2*theta) sqrt(6) a@S4IoBg%
% 3 -3 r^3 * cos(3*theta) sqrt(8) $Z(g=nS>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) &bS"N)je
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) BRSgB-Rr7
% 3 3 r^3 * sin(3*theta) sqrt(8) b.%B;qB
% 4 -4 r^4 * cos(4*theta) sqrt(10) vP87{J*DE1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) mvL0F%\.\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) P"~qio-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U4^p({\|-
% 4 4 r^4 * sin(4*theta) sqrt(10) 8 /RfNGY
% -------------------------------------------------- -!bLMLIg
% c9ov;Bw6S
% Example 1: 5u
u2 _B_L
% yG4LQE
% % Display the Zernike function Z(n=5,m=1) !e#I4,f n
% x = -1:0.01:1; YjIED,eRv
% [X,Y] = meshgrid(x,x); &)"7am(S`
% [theta,r] = cart2pol(X,Y); _]?Dt%MkD
% idx = r<=1; p.TiTFu/
% z = nan(size(X)); "[".3V
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Fy(nu-W
% figure [-:<z?(n4
% pcolor(x,x,z), shading interp ^*?B)D =,
% axis square, colorbar .olPm3MC
% title('Zernike function Z_5^1(r,\theta)') }Nd`;d
% 0imqj7L
% Example 2: ~d#;r5>
% 8H%I|fm
% % Display the first 10 Zernike functions u{{xnyl?
% x = -1:0.01:1; N`|Ab(.
% [X,Y] = meshgrid(x,x); @L>NN>?SGQ
% [theta,r] = cart2pol(X,Y); }JpslY*aS
% idx = r<=1; (fk, 80
% z = nan(size(X)); yZ(Nv $[5
% n = [0 1 1 2 2 2 3 3 3 3]; 9^
*ZH1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; eM1;Nl
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ncw?;
% y = zernfun(n,m,r(idx),theta(idx)); meM.?kk(
% figure('Units','normalized') \Zz= 4
j
% for k = 1:10 2cX"#."5p
% z(idx) = y(:,k); M:1F@\<
% subplot(4,7,Nplot(k)) Zh~Lm
% pcolor(x,x,z), shading interp <*(UvOQuX
% set(gca,'XTick',[],'YTick',[]) /YugQ.>| l
% axis square G}?P
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) GZhfA ;O,
% end W1vAK
% Bg+]_:<U
% See also ZERNPOL, ZERNFUN2. !Bd*
L~D
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% Paul Fricker 11/13/2006 ZNH*[[Pf
5 dNf$a0E
|>o0d~s
% Check and prepare the inputs: s*~jvL
% ----------------------------- <}Wy;!L
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'B<qG<>
error('zernfun:NMvectors','N and M must be vectors.') + x;ML
end g7}z
&S;_
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if length(n)~=length(m) a_iQlsU
error('zernfun:NMlength','N and M must be the same length.') Qpv}N*v^
end s3E~X
^B6i6]Pd=9
n = n(:); /HJ(Wt
q
m = m(:); =*>4Gh
i
if any(mod(n-m,2)) 7%"\DLA
error('zernfun:NMmultiplesof2', ... :_YG/0%I
'All N and M must differ by multiples of 2 (including 0).') gc8PA_bFz
end Y /ac}q
g
/ @yK
if any(m>n) qL;T&h
error('zernfun:MlessthanN', ... G$kwc
F'C
'Each M must be less than or equal to its corresponding N.') $I6eHjYT
end 46?F+,Rzl
{7~ $$AR(
if any( r>1 | r<0 ) Jx
;"a\KD
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Md?bAMnG+}
end 'St= izhd
,vdP
#:
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3w:Z4]J
error('zernfun:RTHvector','R and THETA must be vectors.') tDLk ZCP
end @G$<6CG\
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r = r(:); ut5!2t$c
theta = theta(:); W*DIW;8p
length_r = length(r); tD0>(41K
if length_r~=length(theta) oY6|h3T=Q$
error('zernfun:RTHlength', ... }:D~yEP
'The number of R- and THETA-values must be equal.') |%cO"d^ri
end MbF e1U]B
<C96]}/ ?
% Check normalization: ]XafFr6pe
% -------------------- WKJL<
D ]:
if nargin==5 && ischar(nflag) |{LaZXU &
isnorm = strcmpi(nflag,'norm'); L(n~@gq
if ~isnorm R6$F<;nw
error('zernfun:normalization','Unrecognized normalization flag.') E!~2\qKT
end <W%Z_d&Xv
else CU`Oc>;*T
isnorm = false; GGL4<P7
end t7+Ic
l}-`E@w
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~)8i5p;P/k
% Compute the Zernike Polynomials jv=f@:[`I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IS4K$Ac.
v4##(~Tu
% Determine the required powers of r: wJR i;fvi
% ----------------------------------- N3c)ce7[
m_abs = abs(m); s]8J+8
<uO
rpowers = []; rJQ|Oi&1i
for j = 1:length(n) mS&\m#s<
rpowers = [rpowers m_abs(j):2:n(j)]; 2xdJ(\JWM
end Wk6&TrWlY
rpowers = unique(rpowers); x&/Syb
+Y]*>afG
% Pre-compute the values of r raised to the required powers, |{IU<o
x
% and compile them in a matrix: .-~%w
% ----------------------------- Z*aU2Kr`;
if rpowers(1)==0 BOQV X&g%
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); d?y\~<
rpowern = cat(2,rpowern{:}); =LY^3TlDj
rpowern = [ones(length_r,1) rpowern]; AbI*/|sY
else m1o65FsY08
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `/ReJj&~
rpowern = cat(2,rpowern{:});
x Bw.M{
end &`Z)5Ww
&Wz:-G7<n
% Compute the values of the polynomials: $<%
nt
% -------------------------------------- (C|V-}/*m
y = zeros(length_r,length(n)); 7^ {hn_%;
for j = 1:length(n) 35kbE'
s = 0:(n(j)-m_abs(j))/2; s ^R2jueR
pows = n(j):-2:m_abs(j); 5f@YrTO[@
for k = length(s):-1:1 4m!3P"$
p = (1-2*mod(s(k),2))* ... H08YMP>dc
prod(2:(n(j)-s(k)))/ ... PxD}j
2Kd
prod(2:s(k))/ ... 1gej$G@
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... >t2)Z|1
prod(2:((n(j)+m_abs(j))/2-s(k))); N_[ Q.HD"
idx = (pows(k)==rpowers); 7{F9b0zwk
y(:,j) = y(:,j) + p*rpowern(:,idx); c
O>:n
end Sz@?%PnU|
kR?n%`&k
if isnorm a(T4WDl^
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Y8'_5?+ 0
end 3^yWpSC
end 6Q.whV%y
% END: Compute the Zernike Polynomials ?o5#Ve$-X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% O|zmDp8a+
^l9
*h
% Compute the Zernike functions: TFNU+
% ------------------------------ > 0)`uJ
idx_pos = m>0; zGz'2,o3
idx_neg = m<0; ;OqLNfU3y
@7 HBXP
z = y; 8&hn$~ate
if any(idx_pos) Cy'W!qH
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @$}\S
end MtTHKp
if any(idx_neg) [z@RgDXv
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)');
VZ@@j[F(
end %-po6Vf
}U1shG[
% EOF zernfun