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 Dl%?OG<
function z = zernfun(n,m,r,theta,nflag) %[wTz$S"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Xo Y7/&&
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R<_?W#$j
% and angular frequency M, evaluated at positions (R,THETA) on the XaW4C-D&
% unit circle. N is a vector of positive integers (including 0), and .Gh%p`<
% M is a vector with the same number of elements as N. Each element &5u BNpH
% k of M must be a positive integer, with possible values M(k) = -N(k) dK.R[aQ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, !.EcP=S
% and THETA is a vector of angles. R and THETA must have the same {I{3 (M#"
% length. The output Z is a matrix with one column for every (N,M) '[nmFCG%m*
% pair, and one row for every (R,THETA) pair. XLm@etf
% JA`H@qE
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >AG^fUArH
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (/K5! qh
% with delta(m,0) the Kronecker delta, is chosen so that the integral @EHIp{0.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ,/&Z3e
% and theta=0 to theta=2*pi) is unity. For the non-normalized ?;
[ T
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ]>D)#
% vZ@g@zB4o0
% The Zernike functions are an orthogonal basis on the unit circle. *69c-`o
% They are used in disciplines such as astronomy, optics, and uEx9-,!
% optometry to describe functions on a circular domain. xc;DdK=1X
% zDDK
% The following table lists the first 15 Zernike functions. G2]^F Y
% s qpGrW.
% n m Zernike function Normalization V^n0GJNo
% -------------------------------------------------- (#o t^
% 0 0 1 1 0|XKd24BN
% 1 1 r * cos(theta) 2 LkBZlh_
% 1 -1 r * sin(theta) 2 tPU-1by$
% 2 -2 r^2 * cos(2*theta) sqrt(6) ^s{hs(8%R
% 2 0 (2*r^2 - 1) sqrt(3) Ox qguT,
% 2 2 r^2 * sin(2*theta) sqrt(6) vXdZmYrC
% 3 -3 r^3 * cos(3*theta) sqrt(8) S`iR9{+&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ES}. xZ#~
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) A
WHU'
% 3 3 r^3 * sin(3*theta) sqrt(8) )KY:m |Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) - $JO8'TP
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,Ff n)+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) sDC*J\X
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VFj(M
j`}G
% 4 4 r^4 * sin(4*theta) sqrt(10) !][F
% -------------------------------------------------- {)@D`{$
% gnLn7?
% Example 1: Jdj?I'XtY
% dz%EM8
% % Display the Zernike function Z(n=5,m=1) 8IGt4UF&?
% x = -1:0.01:1; XE rUS80
% [X,Y] = meshgrid(x,x); ;YyXT"6/p
% [theta,r] = cart2pol(X,Y); -M4p\6)Ge
% idx = r<=1; +E5=$`
% z = nan(size(X)); =X1?_~}
% z(idx) = zernfun(5,1,r(idx),theta(idx)); xAhxD|4_
% figure +e P.s_t
% pcolor(x,x,z), shading interp G[Tl%w
% axis square, colorbar Qi9-z'
% title('Zernike function Z_5^1(r,\theta)') DlTR|(AL
% rzeLx Wt
% Example 2: A\$
>>Z
% 4(cJ^]wb ^
% % Display the first 10 Zernike functions S8vV!xO
% x = -1:0.01:1; Vz%OV}\
% [X,Y] = meshgrid(x,x); >t <pFh
% [theta,r] = cart2pol(X,Y); ~/-eyxLTm
% idx = r<=1; {0v*xL_O^
% z = nan(size(X)); 9V
0}d2d
% n = [0 1 1 2 2 2 3 3 3 3]; UBZ9A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; L}%dCe
% Nplot = [4 10 12 16 18 20 22 24 26 28]; M \D]ml~
% y = zernfun(n,m,r(idx),theta(idx)); |<qs
% figure('Units','normalized') jz'<
% for k = 1:10 u\1>gDI )|
% z(idx) = y(:,k); 60}! LmL
% subplot(4,7,Nplot(k)) Y`GOER
% pcolor(x,x,z), shading interp ^,8R,S\}$
% set(gca,'XTick',[],'YTick',[]) ,EpH4*e
% axis square @;Opx."
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Y[
zZw~yx
% end {i [y9
% \7v)iG|#G&
% See also ZERNPOL, ZERNFUN2. q]% T:A=
#8h;Bj
% Paul Fricker 11/13/2006 S*
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NHQF^2 \\
Di5(9]o2
% Check and prepare the inputs: OJO!FH)
% ----------------------------- HU ;#XU1
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !>$4]FkV
error('zernfun:NMvectors','N and M must be vectors.') 5|8^9Oe5
end ,h]o>
1Sz A3c
if length(n)~=length(m) 0CExY9@Wq
error('zernfun:NMlength','N and M must be the same length.') Shr,#wwM`B
end zaimGMJ ,
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n = n(:); NjuiD].
m = m(:); YT#3n
if any(mod(n-m,2)) 3gZ8.8q3
error('zernfun:NMmultiplesof2', ... M8&}j
'All N and M must differ by multiples of 2 (including 0).') ,e722wz
end IE2"rQ T
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if any(m>n) YB( Gk;]
error('zernfun:MlessthanN', ... J^#:qk
'Each M must be less than or equal to its corresponding N.') t=
#&fSR
end Z.PBu|Kx
K2)!h.W
if any( r>1 | r<0 ) hqvE!Of
error('zernfun:Rlessthan1','All R must be between 0 and 1.') cre;P5^E
end d3Mva,bw<
W_|0y4QOo
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4u;9J*r4
error('zernfun:RTHvector','R and THETA must be vectors.') J:*-gwv9*m
end `fNpY#QsN
13k
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r = r(:); g|X ;ahTT
theta = theta(:); 1{x.xi"A/
length_r = length(r); Sl2iz?
if length_r~=length(theta) dUrElXbXd
error('zernfun:RTHlength', ... uN*KHE+h
'The number of R- and THETA-values must be equal.') LpbsYl
end
df}r% i
_gj&$zP
% Check normalization: G3P&{.v
% -------------------- * |.0Myjo
if nargin==5 && ischar(nflag) >SF Uy\3
isnorm = strcmpi(nflag,'norm'); I=)hWC/
if ~isnorm (IqZ@->nw
error('zernfun:normalization','Unrecognized normalization flag.') B(g_Gm<
end u7%D6W~m0
else |077Sf|
isnorm = false; 4S"\~><
end CvSIV7zYo
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .T<=z
% Compute the Zernike Polynomials "Mw[P [w*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BF*kb2"GZ6
8H,4kY?Z
% Determine the required powers of r: ?lGG|9J\
% ----------------------------------- 1J=.N|(@Q
m_abs = abs(m); aimarU
rpowers = []; LsEXM-
for j = 1:length(n) }0#U;_;D
rpowers = [rpowers m_abs(j):2:n(j)]; bK"SKV
end :o-,SrORM
rpowers = unique(rpowers); v,-{Z1N%m
EC2+`HJ"
% Pre-compute the values of r raised to the required powers, n9w9JXp;!
% and compile them in a matrix: G@FI0\t
% ----------------------------- 6oaazB^L
if rpowers(1)==0 omO
S=d!o
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ZRxZume<f
rpowern = cat(2,rpowern{:}); ptatzp]c#
rpowern = [ones(length_r,1) rpowern]; b5$JfjI
else T{wpJ"F5<]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jUv!9Y}F
rpowern = cat(2,rpowern{:}); >^q7c8]~g
end f0<hE2
)[H{yQ
% Compute the values of the polynomials: MObt,[^W
% -------------------------------------- rk+#GO{
y = zeros(length_r,length(n)); WV3|?,y]qm
for j = 1:length(n) \P} p5k[
s = 0:(n(j)-m_abs(j))/2; /kL$4CA
pows = n(j):-2:m_abs(j); qPB8O1fyU
for k = length(s):-1:1 E J$36
p = (1-2*mod(s(k),2))* ... q{s(.Uq$&
prod(2:(n(j)-s(k)))/ ... C{sLz9
prod(2:s(k))/ ... )vmA^nU>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... j?y LDLj
prod(2:((n(j)+m_abs(j))/2-s(k))); ~!s-o|N_\
idx = (pows(k)==rpowers); ur
:i)~wXn
y(:,j) = y(:,j) + p*rpowern(:,idx); t*@2OW`!
end ~$'\L
tQZs.1=z
if isnorm rG#Z=*b%
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); D3|oOOoG
end A(?\>X
9g
end JdIlWJY
% END: Compute the Zernike Polynomials 4h@Z/G!T3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .s#;s'>g
jV.g}F+1m
% Compute the Zernike functions: k(zsm"<q
% ------------------------------ `D9]*c
!mO
idx_pos = m>0; `cPywn@uGZ
idx_neg = m<0; S3L~~X/=
[:xiZ
z = y; 5H=ko8fZ=
if any(idx_pos) KD/V aN
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ??n*2s@t
end DI!V^M[~u
if any(idx_neg) e[sK@jX6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); N`)$[&NG]
end y5Tlpi`g
+?p.?I
% EOF zernfun