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<_Ko+VF
function z = zernfun(n,m,r,theta,nflag) } i)$n(A)K
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. YY4-bNj[p
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N D;F{1[s(
% and angular frequency M, evaluated at positions (R,THETA) on the :PnSQjV:
% unit circle. N is a vector of positive integers (including 0), and )yb+M ez
% M is a vector with the same number of elements as N. Each element c;I, O
% k of M must be a positive integer, with possible values M(k) = -N(k) ;+I4&VieK
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, e}cnX`B
% and THETA is a vector of angles. R and THETA must have the same [ij,RE7,T
% length. The output Z is a matrix with one column for every (N,M) I(n* _bFq
% pair, and one row for every (R,THETA) pair. =ziy`#fm,
% gw3NS8
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _b4fS'[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), D\T!4q'Q
% with delta(m,0) the Kronecker delta, is chosen so that the integral F}rPY:
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, oBlzHBn>0
% and theta=0 to theta=2*pi) is unity. For the non-normalized '3kcD7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~k4W<
% O'}llo
% The Zernike functions are an orthogonal basis on the unit circle. cc>
% They are used in disciplines such as astronomy, optics, and /'>;JF
% optometry to describe functions on a circular domain. }Pg'
vJW
% t&814Uf&\
% The following table lists the first 15 Zernike functions. ? Ekq6uz\)
% .Tm- g#
% n m Zernike function Normalization {.#zHL
;
% -------------------------------------------------- 3BMS_,P
% 0 0 1 1 DB&SOe
% 1 1 r * cos(theta) 2 ,bSVVT-b
% 1 -1 r * sin(theta) 2 Bx X$5u
% 2 -2 r^2 * cos(2*theta) sqrt(6) gf$HuCh|
% 2 0 (2*r^2 - 1) sqrt(3) u5gZxO1J5
% 2 2 r^2 * sin(2*theta) sqrt(6) t58m=4
% 3 -3 r^3 * cos(3*theta) sqrt(8) 4&}\BU*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) coB 6 rW
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) r2G*!qK*1
% 3 3 r^3 * sin(3*theta) sqrt(8) Xn7[n
% 4 -4 r^4 * cos(4*theta) sqrt(10) .9\Cy4_qSd
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D$_8rHc\A
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) q?VVYZXP
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .wFU:y4r
% 4 4 r^4 * sin(4*theta) sqrt(10) ?2~U2Ir]:
% -------------------------------------------------- oa9)Dv
% ?\yB)Nd y
% Example 1: $k(9 U\y-
% ofEqvoi@
% % Display the Zernike function Z(n=5,m=1) pa]
TeH
% x = -1:0.01:1; mvf
_@2^
% [X,Y] = meshgrid(x,x); p6blD-v
% [theta,r] = cart2pol(X,Y); q lY\*{x4
% idx = r<=1; _XN~@5elrC
% z = nan(size(X)); s}b*5@8|tA
% z(idx) = zernfun(5,1,r(idx),theta(idx)); tq E>Zx=X
% figure CSL4P)
% pcolor(x,x,z), shading interp t61'LCEis
% axis square, colorbar H*qD: N
% title('Zernike function Z_5^1(r,\theta)') "=`~iXT{e
% By/bVZks
% Example 2: anZIB
% dt.-C_MO
% % Display the first 10 Zernike functions S1>Z6
% x = -1:0.01:1; 9XN~Ln@}
% [X,Y] = meshgrid(x,x); jg^^\n
% [theta,r] = cart2pol(X,Y); 0O['w<_
% idx = r<=1; |7S:l9;
% z = nan(size(X)); S^g]:Xh&
% n = [0 1 1 2 2 2 3 3 3 3]; :A$wX$H01
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; s @M
% Nplot = [4 10 12 16 18 20 22 24 26 28]; g Np-f
% y = zernfun(n,m,r(idx),theta(idx)); B=x~L
% figure('Units','normalized') ) hPVX()O!
% for k = 1:10 Hrv),Ce
% z(idx) = y(:,k); ;G$)MS'nB
% subplot(4,7,Nplot(k)) vcD'~)G(*
% pcolor(x,x,z), shading interp &1$8q0
% set(gca,'XTick',[],'YTick',[]) AuM:2N2
% axis square '!j(u@&!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) +wjlAqMQ
% end 1'OD3~[R
% h&'J+b
% See also ZERNPOL, ZERNFUN2. Dpp@*xX>
I9s$bRbT
% Paul Fricker 11/13/2006 9e76pP(
S%P3ek>3
k%a?SU<f
% Check and prepare the inputs: $ACe\R/%
% ----------------------------- [EcV\.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6g5]=Q@U:
error('zernfun:NMvectors','N and M must be vectors.') <e^6.!;W
end 0<"tl0p_
YmA) @1@U
if length(n)~=length(m) ees^O{ 8
error('zernfun:NMlength','N and M must be the same length.') A&?WP\_z
end IM2/(N.%
|3W3+Rn!
n = n(:); FRD<0o /`
m = m(:); (T`q++
if any(mod(n-m,2)) j?d!}v
error('zernfun:NMmultiplesof2', ... 'NRN_c9
'All N and M must differ by multiples of 2 (including 0).') 0I6499FQ
end %!W6<ioW
5D>BV*"
if any(m>n) %G^(T%q| m
error('zernfun:MlessthanN', ... N+[}Gb"8q
'Each M must be less than or equal to its corresponding N.') \Z8Y(]6*
end 8:BQHYeJK
O\:;q*]
if any( r>1 | r<0 ) u<J2p?`\&`
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]
+sSg=N7i
end @b>YkJDk
vJzx Py|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 5!2J;.&
error('zernfun:RTHvector','R and THETA must be vectors.') MH2OqiCI
end .Lp Nm'=R
U5 -zB)V
r = r(:); v^57j:sD
theta = theta(:); ``/y=k/au
length_r = length(r); 2M5*bNU_:
if length_r~=length(theta) o4U]lK$
error('zernfun:RTHlength', ... h7)VJY
'The number of R- and THETA-values must be equal.') u_hE7#i
end ,5`."-0}
/"g[Ay
% Check normalization: |A2W8b
{]
% -------------------- &8o :
if nargin==5 && ischar(nflag) ]Sk#a-^~
isnorm = strcmpi(nflag,'norm'); |
3hT {
if ~isnorm ,Uv{dG
error('zernfun:normalization','Unrecognized normalization flag.') KLj 4LOs
end GC,vQ\
else y_;]=hEL
isnorm = false; j
P{:A9T\
end #%9oQ6nO
&T5fH!?4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% e@6RC bj
% Compute the Zernike Polynomials 7/[TE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ktkn2Twa/
[w+yQ7P
% Determine the required powers of r: &zaW"uy3T
% ----------------------------------- K*J4&5?/
m_abs = abs(m); A}
x_zt
rpowers = []; ..v@Q%
for j = 1:length(n) 8T!fGzHx
rpowers = [rpowers m_abs(j):2:n(j)]; 58a)&s[+
end
3J'Bm"
rpowers = unique(rpowers); 'Y~8_+J?
v3=&{}+j.
% Pre-compute the values of r raised to the required powers, 3Qm
t]q
% and compile them in a matrix: *B)Jv9
% ----------------------------- >e5q2U
if rpowers(1)==0 .If"'hMY
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ;C7BoHB9
rpowern = cat(2,rpowern{:}); z&6]vN'
rpowern = [ones(length_r,1) rpowern]; d&$.jk8 2
else `[g#Mxw
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [MSDk"o&