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 LoOyqJ,
function z = zernfun(n,m,r,theta,nflag) =ADAMP
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ZgtW
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N $Pzvv`f*
% and angular frequency M, evaluated at positions (R,THETA) on the ]SBv3Q0D7
% unit circle. N is a vector of positive integers (including 0), and
&
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% M is a vector with the same number of elements as N. Each element miuJ!Kr'
% k of M must be a positive integer, with possible values M(k) = -N(k) V?Lf&X?
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u):z1b3*?
% and THETA is a vector of angles. R and THETA must have the same 1k2Ck
% length. The output Z is a matrix with one column for every (N,M) j!mI9*hP
% pair, and one row for every (R,THETA) pair. < t>N(e
% hz Vpv,|G
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 1kio.9NIp
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), H4k`wWOk
% with delta(m,0) the Kronecker delta, is chosen so that the integral uP|AP
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, VOG DD@
% and theta=0 to theta=2*pi) is unity. For the non-normalized T
fzad2}^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~W5fJd0
% J2aA"BhdC"
% The Zernike functions are an orthogonal basis on the unit circle. akm) X0!-}
% They are used in disciplines such as astronomy, optics, and :b=`sUn<X+
% optometry to describe functions on a circular domain. m f4@g05
% J9/9k
% The following table lists the first 15 Zernike functions. ]_d(YHYf
% kC|tv{g#>
% n m Zernike function Normalization K_]LK
% -------------------------------------------------- 3(^9K2.s}
% 0 0 1 1 kt[#@M!}
% 1 1 r * cos(theta) 2 F!pUfF,&
% 1 -1 r * sin(theta) 2 b44H2A.
% 2 -2 r^2 * cos(2*theta) sqrt(6) o"Ef>5N
% 2 0 (2*r^2 - 1) sqrt(3) kG?tgO?*
% 2 2 r^2 * sin(2*theta) sqrt(6)
*}ay
% 3 -3 r^3 * cos(3*theta) sqrt(8) tjDVU7um
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) L2{to f
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) v
bb mmv
% 3 3 r^3 * sin(3*theta) sqrt(8) !!2~lG<]
% 4 -4 r^4 * cos(4*theta) sqrt(10) e{=7,DRH<
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CFul_qZ/e
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (d#?\
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9!2KpuWji
% 4 4 r^4 * sin(4*theta) sqrt(10) OMKEn!Wq
% -------------------------------------------------- UY}lJHp0
% hJFQ/(
% Example 1: jq.@<<j|$
% YI%7#L7C
% % Display the Zernike function Z(n=5,m=1) YLPiK
% x = -1:0.01:1; $23="Jcl
% [X,Y] = meshgrid(x,x); c0Q`S"o+
% [theta,r] = cart2pol(X,Y); ucoBeNsHx
% idx = r<=1; ik&loM_
% z = nan(size(X)); 3XL0Pm
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Aba6/
% figure "ajZ&{Z
% pcolor(x,x,z), shading interp #\`6ZHW
% axis square, colorbar Yv"uIj+']
% title('Zernike function Z_5^1(r,\theta)') +"'h?7'C
% <LBMth
% Example 2: v]VIUVd
% tp 5]n`3rD
% % Display the first 10 Zernike functions c%xxsq2n
% x = -1:0.01:1; rB=1*.}FLc
% [X,Y] = meshgrid(x,x); lV]l`$XI
% [theta,r] = cart2pol(X,Y); tQ`tHe
% idx = r<=1; w?Q@"^IL
% z = nan(size(X)); SvI
% n = [0 1 1 2 2 2 3 3 3 3]; ^gb2=gWZ<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;yHA.}
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {tWfLfzU
% y = zernfun(n,m,r(idx),theta(idx)); kx'6FkZPIr
% figure('Units','normalized') &p=~=&g=
% for k = 1:10 c:=Z<0S;
% z(idx) = y(:,k); pMX7Rl
% subplot(4,7,Nplot(k)) q/4PX
% pcolor(x,x,z), shading interp g@nE7H1V
% set(gca,'XTick',[],'YTick',[]) W9eR3q
% axis square &mY<e4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) X_%78$N-a`
% end E"V|Plf
c
% anl?4q3;9
% See also ZERNPOL, ZERNFUN2. {?5EOp~
-Ep-v4}
% Paul Fricker 11/13/2006 -O(.J'=8
!3HMGzt
(5Cm+Sy
% Check and prepare the inputs: Yt|{l
% ----------------------------- j4G,Z4
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >aa-ix
&
error('zernfun:NMvectors','N and M must be vectors.') ky!'.3yoI
end [dt1%DD`M
/]+t$K\cBq
if length(n)~=length(m) hP9+|am%
error('zernfun:NMlength','N and M must be the same length.') :+[q`
end \f
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n = n(:); gVZ~OcB!W
m = m(:); )0UQy#r
if any(mod(n-m,2)) $9hOWti
error('zernfun:NMmultiplesof2', ... Cu/w><h)
'All N and M must differ by multiples of 2 (including 0).')
Rl6E
end Gc
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I.(/j
if any(m>n) _-^KqNyy
error('zernfun:MlessthanN', ... 4;&(
'Each M must be less than or equal to its corresponding N.') D $ `yxc
end a&y%|Gs^f
RJd55+h
if any( r>1 | r<0 ) hg\$>W~2
error('zernfun:Rlessthan1','All R must be between 0 and 1.') JsiJ=zo<
end FQ O6w'
tWc!!Hf2j
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) w/Q'T&>b/
error('zernfun:RTHvector','R and THETA must be vectors.') 5ue{&z
@T
end uFECfh
{){i
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r = r(:); eOLS
theta = theta(:); }0f[x ?V
length_r = length(r); &|gn%<^
if length_r~=length(theta) .O lq_wuH
error('zernfun:RTHlength', ... \9D
'7/$I,
'The number of R- and THETA-values must be equal.') gv<9XYByt
end 0!!pNK%(
iyj&O"
% Check normalization: v?Y9z!M
% --------------------
neOR/]
if nargin==5 && ischar(nflag) 4pA(.<#A
isnorm = strcmpi(nflag,'norm'); bh_i*DJ]
if ~isnorm =zI
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error('zernfun:normalization','Unrecognized normalization flag.') 5N '
QG<jE
end odj|"ZK
else m2VF}%
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isnorm = false; IURi90Ir
end rF
7EO%,
}HXNhv-K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% L!/USh:IP
% Compute the Zernike Polynomials cty.)e=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \.Q"fd?a_D
{) jQbAr(G
% Determine the required powers of r: oIbd+6>f
% ----------------------------------- 6)DYQ^4y
m_abs = abs(m); yjN|PqtSV
rpowers = []; }R.cqk\qa^
for j = 1:length(n) \Fc"Q@.u
rpowers = [rpowers m_abs(j):2:n(j)]; J}<k`af
end [\.
ho9
rpowers = unique(rpowers); %'EOFv]
~f){`ZJc
% Pre-compute the values of r raised to the required powers, O2A Z|[*I
% and compile them in a matrix: %:((S]vAi
% ----------------------------- g^8bY=*
.
if rpowers(1)==0 :9K5zD
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Q{mls
rpowern = cat(2,rpowern{:}); qTiX;e\W
rpowern = [ones(length_r,1) rpowern]; U2+CL)al^
else W^al`lg+y
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); <W\~A$
rpowern = cat(2,rpowern{:}); b6oPnP_3P
end N6yqA)z?;
J;'?(xO3\
% Compute the values of the polynomials: `<+D<x)(3
% -------------------------------------- _.wLQL~y
y = zeros(length_r,length(n)); O/l|\n
for j = 1:length(n) j s7J#b7
s = 0:(n(j)-m_abs(j))/2; lty`7(\
pows = n(j):-2:m_abs(j); ^K&&O{
for k = length(s):-1:1 mKWA-h+f
p = (1-2*mod(s(k),2))* ... U3%!#E{
prod(2:(n(j)-s(k)))/ ... uVOOw&q_
prod(2:s(k))/ ... [4(TG<I
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D='/-3f!F]
prod(2:((n(j)+m_abs(j))/2-s(k))); RH>b,
idx = (pows(k)==rpowers); c9iCH~
y(:,j) = y(:,j) + p*rpowern(:,idx); r~TiJ?8I
end lHz:Iibt
Lj({
T'f(
if isnorm 4d9iAN
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Qn<J@%
end PS(9?rX#+
end [*8wv^
% END: Compute the Zernike Polynomials )#i]exZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CI$F#j
g:e|
% Compute the Zernike functions: ;STO!^9~
% ------------------------------ N;RZIg(x
idx_pos = m>0; kw|bEL9!u
idx_neg = m<0;
<k/'mBDk
7f[nNng
z = y; @T]gwJ
if any(idx_pos) !tHqF
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kzgHp,;R{
end H>-,1/IY
if any(idx_neg) *sB=Ys?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); tkV:kh< L~
end \f0I:%-
8~\Fpz|Og
% EOF zernfun