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 mNnw G);$
function z = zernfun(n,m,r,theta,nflag) Rvu3Qo+
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8~[C'+r
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Qa7S'(
% and angular frequency M, evaluated at positions (R,THETA) on the }n2-*{)x
% unit circle. N is a vector of positive integers (including 0), and /_VRO9R\V
% M is a vector with the same number of elements as N. Each element #<tWYE
% k of M must be a positive integer, with possible values M(k) = -N(k) {xBjEhQm
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pw<q?q%
% and THETA is a vector of angles. R and THETA must have the same rjpafGCp
% length. The output Z is a matrix with one column for every (N,M) a7v[l04
% pair, and one row for every (R,THETA) pair. Hh/
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% Io4:$w
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike rs 1*H
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Y)4Nydq
% with delta(m,0) the Kronecker delta, is chosen so that the integral c~L6fvS
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Dt~}9HrU
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8SCW.;0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $?/Xk%d+
% \_I)loPc8
% The Zernike functions are an orthogonal basis on the unit circle. SJ~I
r#
% They are used in disciplines such as astronomy, optics, and d*\C^:Z
% optometry to describe functions on a circular domain. Lx:N!RDw
% q5\LdI2
% The following table lists the first 15 Zernike functions. D
5 r
% wx"6",M
% n m Zernike function Normalization Er/5 ,
% -------------------------------------------------- @Z=|$*9
% 0 0 1 1 tzW<&^
% 1 1 r * cos(theta) 2 j]?0}Z*
% 1 -1 r * sin(theta) 2 aWsKJo>j[#
% 2 -2 r^2 * cos(2*theta) sqrt(6) d a?th
% 2 0 (2*r^2 - 1) sqrt(3) Bbt8fJA~
% 2 2 r^2 * sin(2*theta) sqrt(6) #HnyE+tD
% 3 -3 r^3 * cos(3*theta) sqrt(8) \2<yZCn
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) \(>$mtS:
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) a]wcA
% 3 3 r^3 * sin(3*theta) sqrt(8) k>0cTBY&
% 4 -4 r^4 * cos(4*theta) sqrt(10) rIFC#Jd/
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) DN8pJa
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) V\M!]Nnxr
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V+a%,sI
% 4 4 r^4 * sin(4*theta) sqrt(10) '3u]-GU2_
% -------------------------------------------------- pTX'5
% etK,zEd
% Example 1: ;gW|qb+#)j
% qVRO"/R
% % Display the Zernike function Z(n=5,m=1) +#JhhW
Zj(
% x = -1:0.01:1; SQKY;p
% [X,Y] = meshgrid(x,x); U)/Ul>dY
% [theta,r] = cart2pol(X,Y); T4}?w
% idx = r<=1; $9i5<16
% z = nan(size(X)); tEX~72v
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ^$Io;*N4
% figure 7fzyD
% pcolor(x,x,z), shading interp wY
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% axis square, colorbar !zkEh9G
% title('Zernike function Z_5^1(r,\theta)') pnA]@FW
% +e]b,9.sR
% Example 2: ]ifHA# z`~
% T17LYHIT
% % Display the first 10 Zernike functions 8`~3MsE"
% x = -1:0.01:1; <[5$ {)
% [X,Y] = meshgrid(x,x); MJ"Mn^:/
% [theta,r] = cart2pol(X,Y); }NBJ T4R
% idx = r<=1; !6/IKh`J
% z = nan(size(X)); 4"X>_Nt6
% n = [0 1 1 2 2 2 3 3 3 3]; ,sJfMY
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; =i5:*J
% Nplot = [4 10 12 16 18 20 22 24 26 28]; |AfQ_iT6c
% y = zernfun(n,m,r(idx),theta(idx)); ?{z${ bD
% figure('Units','normalized') z57papo
% for k = 1:10 ^$,kTU'=
% z(idx) = y(:,k); ^oB1 &G
% subplot(4,7,Nplot(k)) x0;}b-f
% pcolor(x,x,z), shading interp pVa|o&,
% set(gca,'XTick',[],'YTick',[]) wG?kcfu
% axis square XXwhs-:o
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Mh.eAM8 _
% end U1|4vd9
% gwz _b
% See also ZERNPOL, ZERNFUN2. xAz4ZXj=q
FC(cXPX}
% Paul Fricker 11/13/2006 3<lhoD
)Qj9kJq
E0Y/N?
% Check and prepare the inputs: h16Nr x
% ----------------------------- H.[&gm}p>
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) [}>6n72gNh
error('zernfun:NMvectors','N and M must be vectors.') +2o|#`)i
end m.a1
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if length(n)~=length(m) gO%i5
error('zernfun:NMlength','N and M must be the same length.') RTY4%6]O
end <T/L.>p4
=l'_*B8
n = n(:); a4.:
i
m = m(:); 'htA! KHF
if any(mod(n-m,2)) wEc5{ b5M
error('zernfun:NMmultiplesof2', ... <0
idG
'All N and M must differ by multiples of 2 (including 0).') YY<?w
end ?N*@o.
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if any(m>n) P]x+Q
error('zernfun:MlessthanN', ... wXGFq3`
'Each M must be less than or equal to its corresponding N.') @VS5Mg8
end a&VJYAB
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if any( r>1 | r<0 ) c] R![sa
error('zernfun:Rlessthan1','All R must be between 0 and 1.') g uWqHVSs
end ujqktrhuLb
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) q;PzB4#
error('zernfun:RTHvector','R and THETA must be vectors.') c qyh#uWe
end ^ED>{UiNI
TC#B^m`'p
r = r(:); <sB45sNbU`
theta = theta(:); '|?r&-5 h
length_r = length(r); 2`U&,,-Mf
if length_r~=length(theta) eSBf;lr=
error('zernfun:RTHlength', ... , tj7'c$0
'The number of R- and THETA-values must be equal.') XJ?z{gXJ
end GZX!iT
@BhAFv,7
% Check normalization: kDa#yN\
% -------------------- )II,HT-LY
if nargin==5 && ischar(nflag) M':.b+xN
isnorm = strcmpi(nflag,'norm'); deY<+!
if ~isnorm becQ5w/~
error('zernfun:normalization','Unrecognized normalization flag.') ClZyQ=UAD
end [E7@W[xr
else .Q)"F /
isnorm = false; @il}0
end O^%ace1
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N<(`+?
% Compute the Zernike Polynomials 8E%*o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :/l
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% Determine the required powers of r: Sd6^%YB
% ----------------------------------- 2Hwf:S'
m_abs = abs(m); 8!>pFVNJf
rpowers = []; R\amcQ
9
for j = 1:length(n) xyz86r ^u
rpowers = [rpowers m_abs(j):2:n(j)]; ^D[;JV
end *60)Vo.=
rpowers = unique(rpowers); dD<kNa}2
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% Pre-compute the values of r raised to the required powers, E/;YhFb[
% and compile them in a matrix: !:{_<