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 ]VCVV!G_=n
function z = zernfun(n,m,r,theta,nflag) 5Qh$>R4!"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. :]rb} 1nLB
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N c;13V(Djy
% and angular frequency M, evaluated at positions (R,THETA) on the wqnHaWd*
% unit circle. N is a vector of positive integers (including 0), and xk:=.Qqh
% M is a vector with the same number of elements as N. Each element ;J>upI
% k of M must be a positive integer, with possible values M(k) = -N(k) 0ap_tCY
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3 i;sB
% and THETA is a vector of angles. R and THETA must have the same $1E'0M`
% length. The output Z is a matrix with one column for every (N,M) @,:6wKMc
% pair, and one row for every (R,THETA) pair. s;ivoGe}
% JqmxS*_P
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '-l.2IUyT
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1o8C4?T&
% with delta(m,0) the Kronecker delta, is chosen so that the integral #lY_XV.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 3T= ?!|e
% and theta=0 to theta=2*pi) is unity. For the non-normalized f'oO/0lx
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ct<]('Hm(
% B4Fuvi
% The Zernike functions are an orthogonal basis on the unit circle. (6crWw{3
% They are used in disciplines such as astronomy, optics, and WR<?_X_
% optometry to describe functions on a circular domain. cSD$I^$oq
% 3cA'9
% The following table lists the first 15 Zernike functions. .}c&"L;W
% $ o
}
% n m Zernike function Normalization chE}TK
% -------------------------------------------------- PU2^4h/[`
% 0 0 1 1 KbSE=3
% 1 1 r * cos(theta) 2 )
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% 1 -1 r * sin(theta) 2 q21l{R{Y
% 2 -2 r^2 * cos(2*theta) sqrt(6) qN"Q3mU^h*
% 2 0 (2*r^2 - 1) sqrt(3) WqJrDj~
% 2 2 r^2 * sin(2*theta) sqrt(6) Z_h-5VU-
% 3 -3 r^3 * cos(3*theta) sqrt(8) (U B?UJc
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 8-PHW,1@a3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fpa~~E-
% 3 3 r^3 * sin(3*theta) sqrt(8) h.*v0cq:
% 4 -4 r^4 * cos(4*theta) sqrt(10) ]<*-pRN
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !kS/Ei
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _M)
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% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |kGQ~:k+P
% 4 4 r^4 * sin(4*theta) sqrt(10) dLfB){>S
% -------------------------------------------------- Fy$f`w_H@
% |E9'ii&?B
% Example 1: oMNSQMlI
% {ldt/dl~
% % Display the Zernike function Z(n=5,m=1) DS1{~_>nFu
% x = -1:0.01:1; 8Drz
i!}
% [X,Y] = meshgrid(x,x); agkGUK/
% [theta,r] = cart2pol(X,Y); cSTF$62E
% idx = r<=1; #M)+sK$H%f
% z = nan(size(X)); <Ej`zGhWz
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 2}n7f7[/b
% figure P?ms^
% pcolor(x,x,z), shading interp Rc vp@
% axis square, colorbar RKPX*(i~
% title('Zernike function Z_5^1(r,\theta)') 5HaI$>h6
% 4nrn
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% Example 2: _CMNmmp`e
% bE;c&g
% % Display the first 10 Zernike functions q5G`q&O5
% x = -1:0.01:1; DF>3)oTF
% [X,Y] = meshgrid(x,x); w>o/)TTJL
% [theta,r] = cart2pol(X,Y); mxE<
% idx = r<=1; YsMM$rjP+
% z = nan(size(X)); brX[-
% n = [0 1 1 2 2 2 3 3 3 3]; [w90gp1O[
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Ew JNpecX
% Nplot = [4 10 12 16 18 20 22 24 26 28]; dmWCNeja.
% y = zernfun(n,m,r(idx),theta(idx)); );zLgNx,
% figure('Units','normalized') j5wfqi
% for k = 1:10
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% z(idx) = y(:,k); oOHY+'V
% subplot(4,7,Nplot(k)) )Dp0swJ
% pcolor(x,x,z), shading interp M1icj~Jr
% set(gca,'XTick',[],'YTick',[]) =4$ErwI_dm
% axis square iB|htH'T
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 4f&"1:
% end ,{; *b
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% #aQQd8
% See also ZERNPOL, ZERNFUN2. s"XwO8yhM
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% Paul Fricker 11/13/2006 &1Cs'
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% Check and prepare the inputs: (:I]v_qEYS
% ----------------------------- !S%0#d2
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {a__/I>)
error('zernfun:NMvectors','N and M must be vectors.') <F8e?xy
end PXyv);#Q`
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if length(n)~=length(m) :..WL;gC
error('zernfun:NMlength','N and M must be the same length.') {-lpYD^k3
end ap8q`a{j^
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n = n(:); P2@Z7DhQ
m = m(:); Wb>;L@jB7
if any(mod(n-m,2))
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error('zernfun:NMmultiplesof2', ... L9<\vJ
'All N and M must differ by multiples of 2 (including 0).') \_
end *NG\3%}%|@
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if any(m>n) Vef!5]t5
error('zernfun:MlessthanN', ... 4i
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'Each M must be less than or equal to its corresponding N.') E#\Oe_eq~N
end b8_F2
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if any( r>1 | r<0 ) fZj,Q#}D
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ="5D}%
end <:Mz2Rg
y%X!l(gQ
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) d]Y;rqjue
error('zernfun:RTHvector','R and THETA must be vectors.') 5.*,IedY
end *FktI\tS
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r = r(:); lm
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theta = theta(:); _-lE$
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length_r = length(r); b*.aaOb
if length_r~=length(theta) n0!2-Q5U)h
error('zernfun:RTHlength', ... 3C<G8*4);/
'The number of R- and THETA-values must be equal.') ,~=]3qmbR
end 6Iqy"MQuq
.1q}mw
% Check normalization: vc&v+5Y
% -------------------- EG`6T
if nargin==5 && ischar(nflag) Q#G xo
isnorm = strcmpi(nflag,'norm'); uDP:kM
if ~isnorm aopZ-^
error('zernfun:normalization','Unrecognized normalization flag.') D;NL*4zt
end eb}P/
else Y X^c}t}U
isnorm = false; 6^WNwe\
end yKoZj
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x};~8lGT>t
% Compute the Zernike Polynomials .whi0~i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GTM0Qvf?
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% Determine the required powers of r: =aZ d>{Y
% ----------------------------------- =T,Q7Dh
m_abs = abs(m); AU3Rz&~
rpowers = []; 5XUm} D$
for j = 1:length(n) !9WGZfK+0Y
rpowers = [rpowers m_abs(j):2:n(j)]; OemY'M?ZQ
end 7;o:r$08&}
rpowers = unique(rpowers); NX,m6u
Q{|%kU"
% Pre-compute the values of r raised to the required powers, Yu\$Y0 {]
% and compile them in a matrix: X2@Ef2EkM
% ----------------------------- C[<}eD4bV
if rpowers(1)==0 h/t;ZLUAZP
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 9gcW;
rpowern = cat(2,rpowern{:}); &U7v=a
rpowern = [ones(length_r,1) rpowern]; I09 W=
else Tj#S')s8
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~c35Y9-5
rpowern = cat(2,rpowern{:}); ?!P0UTe~
end ea"X$<s>-
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% Compute the values of the polynomials: L@{'J
% -------------------------------------- &liON1GLM
y = zeros(length_r,length(n)); "D
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for j = 1:length(n) X|T|iB,vT
s = 0:(n(j)-m_abs(j))/2;
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pows = n(j):-2:m_abs(j); (s,&