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 76e%&ZG)Q
function z = zernfun(n,m,r,theta,nflag) 9qyA{
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. -$Y@]uf^
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N EPx_xX
% and angular frequency M, evaluated at positions (R,THETA) on the 7WZ).,qxY
% unit circle. N is a vector of positive integers (including 0), and "4W@p'
% M is a vector with the same number of elements as N. Each element Oc\Bu6F
% k of M must be a positive integer, with possible values M(k) = -N(k) :e9}k5kdk
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ^`0^|u=
% and THETA is a vector of angles. R and THETA must have the same FPM@%U
% length. The output Z is a matrix with one column for every (N,M) #"tHT<8 u
% pair, and one row for every (R,THETA) pair. z}I4m
% x!6&)T?!n
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike p3?!}VM!y
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), r!/=Iy@
% with delta(m,0) the Kronecker delta, is chosen so that the integral Rw4"co6
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =`VA_xVu
% and theta=0 to theta=2*pi) is unity. For the non-normalized G$X+g{
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. rn1^6qy)
% .xXe *dm%
% The Zernike functions are an orthogonal basis on the unit circle. 4;G:.k!K
% They are used in disciplines such as astronomy, optics, and u\~dsD2)q
% optometry to describe functions on a circular domain. XXbAn-J
% EL_rh TWw
% The following table lists the first 15 Zernike functions. |&JCf=
% *=]hc@
% n m Zernike function Normalization pJM~'tlHV
% -------------------------------------------------- p-]vf$u
% 0 0 1 1 ]"'$i4I{R
% 1 1 r * cos(theta) 2 lq2Ah=FuN
% 1 -1 r * sin(theta) 2 u,<#z0R|;$
% 2 -2 r^2 * cos(2*theta) sqrt(6) QR'yZ45n4
% 2 0 (2*r^2 - 1) sqrt(3) z[kz[
% 2 2 r^2 * sin(2*theta) sqrt(6) :W'Yt9v)
% 3 -3 r^3 * cos(3*theta) sqrt(8) Z i-)PK^
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Cx>iSx
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) xyGk\= S
% 3 3 r^3 * sin(3*theta) sqrt(8) /jJi`'{U
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4k9O6
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5GD6%{\O
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) YE<_a;yh1
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qTM,'7Rwn
% 4 4 r^4 * sin(4*theta) sqrt(10) $[zy|Y(
% -------------------------------------------------- !acm@"Ea
% 9NCo0!Fb
% Example 1: a]NQlsE}l
% W5a)`%H
% % Display the Zernike function Z(n=5,m=1) J!?hajw7N
% x = -1:0.01:1; IipG?v0z~
% [X,Y] = meshgrid(x,x); YGy.39@31
% [theta,r] = cart2pol(X,Y); :S
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% idx = r<=1; o&2(xI2
% z = nan(size(X)); S{cy|QD
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 6?-vj2,
% figure ?yKW^,q+
% pcolor(x,x,z), shading interp w_-v!s2
% axis square, colorbar 5mNd5IM
% title('Zernike function Z_5^1(r,\theta)') CRy;>UI
% ve|:z
% Example 2: H]@M00C
% /A3tY"Vn
% % Display the first 10 Zernike functions c}9.Or`?
% x = -1:0.01:1; <"I#lib
% [X,Y] = meshgrid(x,x); 0pP;[7k\
% [theta,r] = cart2pol(X,Y); BElVkb
% idx = r<=1; #DMt<1#:
% z = nan(size(X)); HorFQ?8
% n = [0 1 1 2 2 2 3 3 3 3]; =,B44:`r
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; T;(k
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Wi3:;`>G<p
% y = zernfun(n,m,r(idx),theta(idx)); >;Er[Rywr
% figure('Units','normalized') DyiyH%SSD
% for k = 1:10 v]CH
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|
% z(idx) = y(:,k); Y*-#yG9
% subplot(4,7,Nplot(k)) _97A9wHj
% pcolor(x,x,z), shading interp
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% set(gca,'XTick',[],'YTick',[]) I
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% axis square +an.z3?w
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5c?1JH62o8
% end \W5fcxf
% :f?};t+
% See also ZERNPOL, ZERNFUN2. h$`P|#V&
s)HLFdis@
% Paul Fricker 11/13/2006 E"p;
5 rpX"(
z:B4
% Check and prepare the inputs: P !:LAb(
% ----------------------------- b,]QfC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <=;#I_E#E
error('zernfun:NMvectors','N and M must be vectors.') '8+<^%c
end 92|\`\LP%
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if length(n)~=length(m) p8CDFLuV
error('zernfun:NMlength','N and M must be the same length.') I^h^QeBis
end .t\#>Fe
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n = n(:); sTx23RJ9
m = m(:); R"NR-iU
if any(mod(n-m,2)) &s.S)'l4l
error('zernfun:NMmultiplesof2', ... IbFS8 *a\
'All N and M must differ by multiples of 2 (including 0).') ]"Y?
ZS;H
end *3;H6
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if any(m>n) e nsou!l
error('zernfun:MlessthanN', ... 7`113`1
'Each M must be less than or equal to its corresponding N.') iT f]Pd'
end |KF_h^
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if any( r>1 | r<0 ) [
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') WCc7 MK
end .xnJT2uu'
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Gt>*y.]
error('zernfun:RTHvector','R and THETA must be vectors.') cB,O"-
end HE>6A|rgDr
UVND1XV^f
r = r(:); =ELl86=CG
theta = theta(:); -:mT8'.F-
length_r = length(r); WvV!F?uqZ
if length_r~=length(theta) -\ {.]KL
error('zernfun:RTHlength', ... QrmiQ]d*p
'The number of R- and THETA-values must be equal.') v(5zSo
end :Fe}.* t
NGsG4y^g?z
% Check normalization: WX@a2c.'
% -------------------- S6~&g|T,
if nargin==5 && ischar(nflag) i7N|p9O.
isnorm = strcmpi(nflag,'norm'); 8RGU^&
if ~isnorm 6|h~pH
error('zernfun:normalization','Unrecognized normalization flag.') O7&