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 ,#0#1k<Dm
function z = zernfun(n,m,r,theta,nflag) K>\v<!%a
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. C9FAX$$^(Y
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N *kj+6`:CPs
% and angular frequency M, evaluated at positions (R,THETA) on the ew c:-2Y^
% unit circle. N is a vector of positive integers (including 0), and 6vU%Y_n=y]
% M is a vector with the same number of elements as N. Each element N!\1O,
% k of M must be a positive integer, with possible values M(k) = -N(k) u2I@ fH/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ?fc<3q"
% and THETA is a vector of angles. R and THETA must have the same N];K
% length. The output Z is a matrix with one column for every (N,M) P/k#([:2
% pair, and one row for every (R,THETA) pair. P.^*K:5@
% DD>n-8M@>
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 4JH^R^O<n
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), u:wf:^
% with delta(m,0) the Kronecker delta, is chosen so that the integral )hVn/*mH
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, o nv0gb/J
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9%MgA ik(
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. DoICf1
% QV#HN"F/K
% The Zernike functions are an orthogonal basis on the unit circle. $HRl:KDdP~
% They are used in disciplines such as astronomy, optics, and yU~wZjw
% optometry to describe functions on a circular domain. e_S,N0
% #.,LWL]
% The following table lists the first 15 Zernike functions. #B_H/9f(
% mK^E@uxN
% n m Zernike function Normalization }%y5<n*v\
% -------------------------------------------------- {t]8#[lo
% 0 0 1 1 >Wd_?NaI
% 1 1 r * cos(theta) 2 5+(Cp3
% 1 -1 r * sin(theta) 2 ,~Lx7 5{
% 2 -2 r^2 * cos(2*theta) sqrt(6) /(%!txSNEt
% 2 0 (2*r^2 - 1) sqrt(3) UdpuQzV<4`
% 2 2 r^2 * sin(2*theta) sqrt(6) f]Rh<N$
% 3 -3 r^3 * cos(3*theta) sqrt(8) diKl}V#u
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) /f=31<+MtF
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .lSoC`HE
% 3 3 r^3 * sin(3*theta) sqrt(8) *A0d0M]cg
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4`+R
|"4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) cCG!X%9
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) lxR]Bh+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) WZviC_
% 4 4 r^4 * sin(4*theta) sqrt(10) 'PTQ
S,E
% -------------------------------------------------- pqohLA
% 1V,DcolRY
% Example 1: Nr*o
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% 0R-W9qP
% % Display the Zernike function Z(n=5,m=1) rWN%j)#+
% x = -1:0.01:1; h5v=h>c
% [X,Y] = meshgrid(x,x); m,rkKhXP
% [theta,r] = cart2pol(X,Y); <Iil*\SC
% idx = r<=1; yy`XtJBWWs
% z = nan(size(X)); dvAz}3p0]
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 5'|W(yR}
% figure 8rLhOA
% pcolor(x,x,z), shading interp u!FF{~5cs
% axis square, colorbar B @8lD\
% title('Zernike function Z_5^1(r,\theta)') E>u U6#v
% q0nIJ(
% Example 2: *}>)E]O@
% Fj`K$K?
% % Display the first 10 Zernike functions >h$Q%w{V
% x = -1:0.01:1; Z dT-
% [X,Y] = meshgrid(x,x); ;O<-4$
% [theta,r] = cart2pol(X,Y); j=u)
z7J
% idx = r<=1; xg'xuz$U
% z = nan(size(X)); IJ7wUZp"
% n = [0 1 1 2 2 2 3 3 3 3]; Y3H5}4QD
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1%";|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; nJwP|P_
% y = zernfun(n,m,r(idx),theta(idx)); G4\|bwh
% figure('Units','normalized') 5>VX]nE3!
% for k = 1:10 {r#uD5NJ/
% z(idx) = y(:,k); Q5Epq
sKyC
% subplot(4,7,Nplot(k)) BxaGBK<k
% pcolor(x,x,z), shading interp qXoq<
|
% set(gca,'XTick',[],'YTick',[]) mp*?GeV?M
% axis square {"|la;*I
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m;ju@5X
% end 5inCAPXz
% )OK"H^}f
% See also ZERNPOL, ZERNFUN2. +&<k}Mz
FRsp?i
K)
% Paul Fricker 11/13/2006 !Yz
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n8i: /ypB
equi26jhr
% Check and prepare the inputs: jPn.w,=)27
% ----------------------------- 02-% B~oP
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vTC{
error('zernfun:NMvectors','N and M must be vectors.') k+hl6$:Qj%
end }-Jo9dNs
t~":'le`zr
if length(n)~=length(m) BQ B<+o'
error('zernfun:NMlength','N and M must be the same length.') ;(Az
end Ydyz-
;s+3#Py
n = n(:); Qm_;o(
m = m(:); .fS{j$
if any(mod(n-m,2)) PO,zP9
error('zernfun:NMmultiplesof2', ... {e0(M*u
'All N and M must differ by multiples of 2 (including 0).') Q(4~r+
end C 1)+^{7ef
E
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if any(m>n) ?6h~P:n.
error('zernfun:MlessthanN', ... 5tEkQ(Ei8
'Each M must be less than or equal to its corresponding N.') LZQG.
end '-3K`[
uG-S$n"7K
if any( r>1 | r<0 ) m[BpV.s
error('zernfun:Rlessthan1','All R must be between 0 and 1.') E%a&6W
end BnaI30-
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) p$` ^A
error('zernfun:RTHvector','R and THETA must be vectors.') :SY,;..3e
end $'yWg_(
3ug~m-_
r = r(:); ;j+*}|!
theta = theta(:); Iz>\qC}
length_r = length(r); s+E4AG1r
if length_r~=length(theta) n(CM)(ozU
error('zernfun:RTHlength', ... Rm~8n;7oOr
'The number of R- and THETA-values must be equal.') WC
b5
end b;NV vc(
_rz\[{)
% Check normalization: 3sDyB-\&
% -------------------- 2-@t,T
if nargin==5 && ischar(nflag) $x#qv1
isnorm = strcmpi(nflag,'norm'); XEN-V-Z%*
if ~isnorm +]0hSpZ"p
error('zernfun:normalization','Unrecognized normalization flag.') \tCK7sBn
end .')^4\
else VFm)!'=I
isnorm = false; ID,os_ T=
end Dj 6^|R$z&
_qh\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =5uhIU0O
% Compute the Zernike Polynomials LLMGs: [
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }G!'SZ$F 5
s!1/Bm|_T
% Determine the required powers of r: ?v'CuWS
% ----------------------------------- `,4YPjk^
m_abs = abs(m); 7Q,<h8N\5
rpowers = []; w7 \vrS>&
for j = 1:length(n) Mgu9m8
`J
rpowers = [rpowers m_abs(j):2:n(j)]; 4ywtE}mp
end l>J%Q^
rpowers = unique(rpowers); -iFFXESVX
Cv
p#=x0
% Pre-compute the values of r raised to the required powers, z80*Ylx
% and compile them in a matrix: $_e{Zv[
% ----------------------------- 8cRc5X
if rpowers(1)==0 ?9?o8!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); m~&>+q ^7
rpowern = cat(2,rpowern{:}); p:ZQ*Ue
rpowern = [ones(length_r,1) rpowern]; :_+U[k(#
else (&, E}{p9
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); OC\cN%qlw
rpowern = cat(2,rpowern{:}); TGjxy1A
end #G\-ftA &
?zVcP=p@
% Compute the values of the polynomials: !#E-p?O.
% -------------------------------------- >4HB~9dKU
y = zeros(length_r,length(n)); ]{I>HA5[
for j = 1:length(n) U@(8)[?nxn
s = 0:(n(j)-m_abs(j))/2; %{me<\(
pows = n(j):-2:m_abs(j); {xP-p"?p
for k = length(s):-1:1 jP<6Q|5F
p = (1-2*mod(s(k),2))* ... E;"VI2F
prod(2:(n(j)-s(k)))/ ... w2^s}NO
prod(2:s(k))/ ... CurU6x1
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... h,K&R8S
prod(2:((n(j)+m_abs(j))/2-s(k))); cvx"XxE,
idx = (pows(k)==rpowers); #kJ8 qN
y(:,j) = y(:,j) + p*rpowern(:,idx); R1.Yx?
end ]n$ v ^
z_8Bl2tl
if isnorm 'uwq^b_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "`'+@KlE
end "'>fTk_
end g1B P
% END: Compute the Zernike Polynomials ]]5(:>l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d Z+7S`{
B E#pHg
% Compute the Zernike functions: m5hu;>gt
% ------------------------------ J>nta?/,X
idx_pos = m>0; h}S2b@e|
idx_neg = m<0; sr~VvciIy
D^{jXNDNO
z = y; h[C XH"
if any(idx_pos) !=+;9Ry$z
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Z(J
1A x
end |6`7kb;p
if any(idx_neg) nYj7r*e[
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 475jmQ{q
end j\.e6&5%SS
~{6}SXp4U
% EOF zernfun