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 _gw paAJ
function z = zernfun(n,m,r,theta,nflag) i5gNk)D
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 3sp-0tUE
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N j<)`|?@e(
% and angular frequency M, evaluated at positions (R,THETA) on the 0cq<!{d
% unit circle. N is a vector of positive integers (including 0), and J3$@: S'
% M is a vector with the same number of elements as N. Each element Z9eP(ip
% k of M must be a positive integer, with possible values M(k) = -N(k) -t: U4r(
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, F>eo.|'
% and THETA is a vector of angles. R and THETA must have the same A_crK`3
% length. The output Z is a matrix with one column for every (N,M) .-)kIFMi
% pair, and one row for every (R,THETA) pair. ]KQQdr
% w-3Lw<
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike
k; >Vh'=X
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), CZf38$6 X
% with delta(m,0) the Kronecker delta, is chosen so that the integral @@ cc/S
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~_
u3_d.
% and theta=0 to theta=2*pi) is unity. For the non-normalized jZ''0Lclpc
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !|8"}ZF
% IyAD>Q^
% The Zernike functions are an orthogonal basis on the unit circle. Mbn;~tY>
% They are used in disciplines such as astronomy, optics, and M 0$E_*
% optometry to describe functions on a circular domain. U$;UW3-
% t%StBq(q
% The following table lists the first 15 Zernike functions. dWdD^>8Ef
% rU6A^p\,
% n m Zernike function Normalization !+]KxB
% -------------------------------------------------- +.Xi7x+#O
% 0 0 1 1 u<4bOJn({
% 1 1 r * cos(theta) 2 <v=s:^;C0
% 1 -1 r * sin(theta) 2 ]^,! ;do
% 2 -2 r^2 * cos(2*theta) sqrt(6) Hbn78,~.
% 2 0 (2*r^2 - 1) sqrt(3) e;2A{VsD8
% 2 2 r^2 * sin(2*theta) sqrt(6) s6'=4gM
% 3 -3 r^3 * cos(3*theta) sqrt(8) Qe-PW9C
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @8$z2
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) F x^X(!)~]
% 3 3 r^3 * sin(3*theta) sqrt(8) M6GiohI_"P
% 4 -4 r^4 * cos(4*theta) sqrt(10) -hc8IS
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i[:cG
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) zRbY]dW
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _3.rPS,s
% 4 4 r^4 * sin(4*theta) sqrt(10) cICfV,j
% -------------------------------------------------- UZ#oaD8H6
% x2'pl
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% Example 1: lQEsa45
% Ubgn^+AI
% % Display the Zernike function Z(n=5,m=1) z:Z-2WV2o
% x = -1:0.01:1; ~@(C+ 3,
% [X,Y] = meshgrid(x,x); xP/q[7>#Q
% [theta,r] = cart2pol(X,Y); Y6Ux*vhK
% idx = r<=1; aNA]hl
% z = nan(size(X)); e\O-5hp7
% z(idx) = zernfun(5,1,r(idx),theta(idx)); XMdCQ=
% figure _GrifGU\
% pcolor(x,x,z), shading interp %ZX9YuXQ
% axis square, colorbar 0a bQY
% title('Zernike function Z_5^1(r,\theta)') PQa0m)H@
% OzwJ 52
% Example 2: Hp>L}5 y[
% C!ch
!E#
% % Display the first 10 Zernike functions pb)kN%
% x = -1:0.01:1; '.M4yif\g
% [X,Y] = meshgrid(x,x); %M))Ak4~a
% [theta,r] = cart2pol(X,Y); 3+(lKd
% idx = r<=1; &AWrM{e
% z = nan(size(X)); iQS,@6
% n = [0 1 1 2 2 2 3 3 3 3]; ZhoV,/\+
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; >o O]S]W
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3zu6#3^
% y = zernfun(n,m,r(idx),theta(idx)); P+=m.
% figure('Units','normalized') GdY@$&z{i
% for k = 1:10 LrT EF
j
% z(idx) = y(:,k); szb@2fK
% subplot(4,7,Nplot(k)) >]_^iD]*t
% pcolor(x,x,z), shading interp L`X5\D'X
% set(gca,'XTick',[],'YTick',[]) SOn)'!g
% axis square 3u& ,3:
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) e([>sAx!1
% end 9 M%Gnz
% Pq8oK'z-
% See also ZERNPOL, ZERNFUN2. aKWxL e
>3@3~F%xAX
% Paul Fricker 11/13/2006 J7^UQ
M=lU`Sm
:8hI3]9
% Check and prepare the inputs: GZ,MC?W
% -----------------------------
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4dcm)Xr
error('zernfun:NMvectors','N and M must be vectors.') m#Z&05^
end 2QM{e!9
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if length(n)~=length(m) %ut8/T
error('zernfun:NMlength','N and M must be the same length.') #QIY+muN
end C\~}ySQc.e
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n = n(:); J?yasjjgP
m = m(:);
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if any(mod(n-m,2)) rq4g~e!S
error('zernfun:NMmultiplesof2', ... )#cZ&
O
'All N and M must differ by multiples of 2 (including 0).') u[Kz^ga<
end VsA J2g9L
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if any(m>n) =gD)j&~}_
error('zernfun:MlessthanN', ... Q;w[o
'Each M must be less than or equal to its corresponding N.') \Ta5c31S+
end Z,e|L4&
v/9ZTd
if any( r>1 | r<0 ) KFwuz()7
error('zernfun:Rlessthan1','All R must be between 0 and 1.') T3@2e0u )
end z!O;s
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _,)_(R ,h
error('zernfun:RTHvector','R and THETA must be vectors.') d"06
gp
end iD G&