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 N{Is2Ia
function z = zernfun(n,m,r,theta,nflag) 7sLs+|<"
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. d(v )SS
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [IV8
% and angular frequency M, evaluated at positions (R,THETA) on the )}u.b-Nt.
% unit circle. N is a vector of positive integers (including 0), and vNJ!i\bX
% M is a vector with the same number of elements as N. Each element `86 9XE
% k of M must be a positive integer, with possible values M(k) = -N(k) kTC6fNj[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, GhpH7%s
% and THETA is a vector of angles. R and THETA must have the same ]MB^0:F-
% length. The output Z is a matrix with one column for every (N,M) :Z=A,G
% pair, and one row for every (R,THETA) pair. VnIJ$5Y
% t5eux&C
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~@sx}u
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `7N[rs9|S
% with delta(m,0) the Kronecker delta, is chosen so that the integral 8Cm^#S,+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, VK?,8Y
% and theta=0 to theta=2*pi) is unity. For the non-normalized })"9TfC
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. RqcX_x(p
% @p`#y
% The Zernike functions are an orthogonal basis on the unit circle. fMLm_5 (H
% They are used in disciplines such as astronomy, optics, and :&TOQ<vM
% optometry to describe functions on a circular domain. ]@WJ&e/'@
% 6Ajiz_~U
% The following table lists the first 15 Zernike functions. -?e~S\JH
% ^PWZ1.T
% n m Zernike function Normalization o'D6lkf0
% -------------------------------------------------- Wigm`A=,r
% 0 0 1 1 /{qr~7k,oQ
% 1 1 r * cos(theta) 2 NrL%]dl3/
% 1 -1 r * sin(theta) 2 fNB*o={r|
% 2 -2 r^2 * cos(2*theta) sqrt(6) '-ACNgNn
% 2 0 (2*r^2 - 1) sqrt(3) j4brDlo?@
% 2 2 r^2 * sin(2*theta) sqrt(6) -JUv'fk
% 3 -3 r^3 * cos(3*theta) sqrt(8) dmE-WS
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) WJJ!NoP
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) $9ON3>
% 3 3 r^3 * sin(3*theta) sqrt(8) n|^-qy'w
% 4 -4 r^4 * cos(4*theta) sqrt(10) .GS|H d
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) T8qG9)~3
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) *(r85lEou)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) M)3h 4yQ
% 4 4 r^4 * sin(4*theta) sqrt(10) TWxMexiW
% -------------------------------------------------- 9`c :sop
% v3@)q0@
% Example 1: }b,a*4pN
% l}<s~ip
% % Display the Zernike function Z(n=5,m=1) 9 -TFyZYU
% x = -1:0.01:1; &|9?B!,`
% [X,Y] = meshgrid(x,x); {OQ sGyR?
% [theta,r] = cart2pol(X,Y); ];Z_S`JR
% idx = r<=1; R\X=Vg
% z = nan(size(X)); ,
:kCt=4%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); c?z%z&
% figure GU"MuW`u2
% pcolor(x,x,z), shading interp v8wN2[fC
% axis square, colorbar %*r Pd>*
% title('Zernike function Z_5^1(r,\theta)') @];Xbbw+c
% orL7y&w(v:
% Example 2: iOD9lR`s
% R?]>8o,
% % Display the first 10 Zernike functions LFh(.
}
% x = -1:0.01:1; iAXx`>}m
% [X,Y] = meshgrid(x,x); Dcp,9"yt%
% [theta,r] = cart2pol(X,Y); RNIfw1R
% idx = r<=1; ;N4mR6
% z = nan(size(X)); SZyPl9.b
% n = [0 1 1 2 2 2 3 3 3 3]; Ie+z"&0
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; /=-E`%R}!
% Nplot = [4 10 12 16 18 20 22 24 26 28]; I:Z38xz -[
% y = zernfun(n,m,r(idx),theta(idx)); 4V[+6EV
% figure('Units','normalized') 1zl@$ Nt
% for k = 1:10 @o>2:D1G
% z(idx) = y(:,k); tM!1oWH
% subplot(4,7,Nplot(k))
G%4vZPA
% pcolor(x,x,z), shading interp @Yt[%tOF+
% set(gca,'XTick',[],'YTick',[]) G.(9I~!
% axis square {qh`8
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LWIU7dw
% end EcP"GO5
% tb_}w@:kU
% See also ZERNPOL, ZERNFUN2. 0ED(e1K#B
c.d*DM}W
% Paul Fricker 11/13/2006 mWka!lT
b},OCVT?
f)gA.Rz
% Check and prepare the inputs: qKWkgackP
% ----------------------------- 7]
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 2M|jWy _
error('zernfun:NMvectors','N and M must be vectors.') #>!!#e!*
end I-+D+DhRx
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if length(n)~=length(m) Bv/v4(G5g
error('zernfun:NMlength','N and M must be the same length.') #<l;YT8
end dyu~T{
z+wBZn{0I
n = n(:); 33
N5> }
m = m(:); 3pl.<;9r
if any(mod(n-m,2)) -<CBxyZa&
error('zernfun:NMmultiplesof2', ... !f"@pR6
'All N and M must differ by multiples of 2 (including 0).') t1Cyyb
end -vhgBru
V_Y SYG9f
if any(m>n) =FdS'<GM
error('zernfun:MlessthanN', ... `bivAL
'Each M must be less than or equal to its corresponding N.') 03{e[#6
end !o>/gI`
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if any( r>1 | r<0 ) KS%xo6k.
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5w{_WR6,
end o2Z#
5-
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _L&C4 <e'
error('zernfun:RTHvector','R and THETA must be vectors.') !9V_U
end x+^iEj`gk
@'~v~3
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r = r(:); V=1Y&y
theta = theta(:); O (wt[AEA
length_r = length(r); +vZ-o{}.jO
if length_r~=length(theta) e'g-mRh
error('zernfun:RTHlength', ... v')T^b
F@
'The number of R- and THETA-values must be equal.') wYNh0QlBH
end W!+5}\?
}0qgvw
% Check normalization: MheP@ [w|@
% -------------------- [
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if nargin==5 && ischar(nflag) "TG}aS
isnorm = strcmpi(nflag,'norm'); "EHwv2Hm>
if ~isnorm Z\`uI+`
error('zernfun:normalization','Unrecognized normalization flag.') 7pr@aA"vgj
end =j}]-!
else dt;R
isnorm = false; hb[K.`g
end Z>M0[DJ_
@K2q*d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FRX'"gIR0
% Compute the Zernike Polynomials M0n@?S
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vvdC.4O
:Q!U;33aG
% Determine the required powers of r: \%rX~UhZ=
% ----------------------------------- D0tI
m_abs = abs(m); =][[TH
rpowers = []; +>37'PD
for j = 1:length(n) &5c)qap;n
rpowers = [rpowers m_abs(j):2:n(j)]; XeJx/'9o{
end 6YYZ S2
rpowers = unique(rpowers);
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@
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% Pre-compute the values of r raised to the required powers, |a/"7B|?\
% and compile them in a matrix: m[(2
% ----------------------------- I`zn#U'
if rpowers(1)==0 !V#(g ./W
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c?j /H$
rpowern = cat(2,rpowern{:}); +-K-CXt
rpowern = [ones(length_r,1) rpowern]; lc#su$xR>
else M)(
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x4R[Q&:M
rpowern = cat(2,rpowern{:}); ^Jsx^?
end 3Sf<oYF
Z[Uz~W6M]
% Compute the values of the polynomials: R\
<HR9 r
% -------------------------------------- mGwBbY+5n
y = zeros(length_r,length(n)); 3|l+&LF!IC
for j = 1:length(n) 45q-x_
s = 0:(n(j)-m_abs(j))/2; @aWvN;v
pows = n(j):-2:m_abs(j); Ry r2
for k = length(s):-1:1 VuPa'2
p = (1-2*mod(s(k),2))* ... YN.rj-;^+
prod(2:(n(j)-s(k)))/ ... [f&ja[m q
prod(2:s(k))/ ... 0,E*9y}
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 349W0>eOT
prod(2:((n(j)+m_abs(j))/2-s(k))); pa4zSl
idx = (pows(k)==rpowers); Ae;>
@k/|=
y(:,j) = y(:,j) + p*rpowern(:,idx); /87?U; |V
end %N=-i]+Id
yiWBIJ2Wu9
if isnorm <TC\Nb$~
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); OpW4@le_r
end G;>b}\Ng
end Myg
&H(~
% END: Compute the Zernike Polynomials [;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q`{crY30
,n-M!y
% Compute the Zernike functions: -1DQO|q#
% ------------------------------
'n6D3Vse
idx_pos = m>0; -}AAA*P
idx_neg = m<0; dpx P
\U\ W Q
z = y; ~C\R!DN,
if any(idx_pos) Q~MV0<{
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ZQlja
end jhr:QS/9
if any(idx_neg) WA\
P`'lg
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); jO &sS?
end DZ<q)EpC
&"p7X>bd
% EOF zernfun