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 c54oQ1Q&"
function z = zernfun(n,m,r,theta,nflag) L"P$LEk
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. AK} wSXF
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N y08.R.
l
% and angular frequency M, evaluated at positions (R,THETA) on the 00[Uk'Q*5
% unit circle. N is a vector of positive integers (including 0), and 5O%Q*\(
% M is a vector with the same number of elements as N. Each element D({%FQ"
% k of M must be a positive integer, with possible values M(k) = -N(k) @GK0j"_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pMe'fC~*
% and THETA is a vector of angles. R and THETA must have the same -uHD|
}
% length. The output Z is a matrix with one column for every (N,M) I>B-[QEC
% pair, and one row for every (R,THETA) pair. T'"aStt6
% #;#
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O=?WI
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /Q8E12
% with delta(m,0) the Kronecker delta, is chosen so that the integral xlZ"F
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @bSxT,2
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8vOKm)[%
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. @7e h/|Y,
% !ZJ"lm
% The Zernike functions are an orthogonal basis on the unit circle. :GBWQXb G
% They are used in disciplines such as astronomy, optics, and ;!v2kVuS]
% optometry to describe functions on a circular domain. `lX |yy"
% *$1M=$
% The following table lists the first 15 Zernike functions. 0&mOu #l
% ~Pq1@N>n
% n m Zernike function Normalization
yl0&|Ub
% -------------------------------------------------- w]J9Kv1)-
% 0 0 1 1 wC+_S*M-K
% 1 1 r * cos(theta) 2 cah1'Y
% 1 -1 r * sin(theta) 2 g"Mqh!{
FI
% 2 -2 r^2 * cos(2*theta) sqrt(6) SV0E7qX
% 2 0 (2*r^2 - 1) sqrt(3) `xMmo8u4
% 2 2 r^2 * sin(2*theta) sqrt(6) Ue^2H[zs-
% 3 -3 r^3 * cos(3*theta) sqrt(8) {7.."@Ob<v
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $hZb<Xz
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) pC2ZN
% 3 3 r^3 * sin(3*theta) sqrt(8) u.ubw(vv
% 4 -4 r^4 * cos(4*theta) sqrt(10) G0Q}
1
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W ZdEfY{
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) :vZ8n6J[
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kv{uf$X*ve
% 4 4 r^4 * sin(4*theta) sqrt(10) , 7&`V=C
% -------------------------------------------------- ?f<JwF<
% 5 0uYU[W
% Example 1: +[Cdd{2
% ~47Bbom
% % Display the Zernike function Z(n=5,m=1) (C>FM8$J
% x = -1:0.01:1; Y /$`vgqs
% [X,Y] = meshgrid(x,x); <ZGEmQ
% [theta,r] = cart2pol(X,Y); `@1y|j:m
% idx = r<=1; l$N
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% z = nan(size(X)); Ysbd4rN
% z(idx) = zernfun(5,1,r(idx),theta(idx)); HI)MBrj;r
% figure d$Y3 a^O|
% pcolor(x,x,z), shading interp o8Vtxnkg
% axis square, colorbar 3NAU|//J
% title('Zernike function Z_5^1(r,\theta)') c@;$6WSG^
% g S xK9P
% Example 2: ^L#\z7
% G8b/eWtP
% % Display the first 10 Zernike functions 9c9FC
% x = -1:0.01:1; \i_y(;
% [X,Y] = meshgrid(x,x); f'P}]_3(
% [theta,r] = cart2pol(X,Y); AT Dm$ *
% idx = r<=1; o>*vG
% z = nan(size(X)); j}$dYbf$
% n = [0 1 1 2 2 2 3 3 3 3]; Au3>=x`
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; l,AK
% Nplot = [4 10 12 16 18 20 22 24 26 28]; KzD5>Xf]4$
% y = zernfun(n,m,r(idx),theta(idx)); k.=67L
% figure('Units','normalized') /^ *GoB
% for k = 1:10 e[_W( v
% z(idx) = y(:,k); 7.g)_W{7}
% subplot(4,7,Nplot(k)) #!V
[(/
% pcolor(x,x,z), shading interp NJK?5{H'
% set(gca,'XTick',[],'YTick',[]) Xb +)@Y4h
% axis square DE"KbA0}
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nn><
k"
% end cfI5KLG~#
% pgT XyAP{
% See also ZERNPOL, ZERNFUN2. N'hj
3S='/^l
% Paul Fricker 11/13/2006 u=^0n2ez
Fq3[/'M^
iC- ?F
cA
% Check and prepare the inputs: 8was/^9;
% ----------------------------- 0_b7*\x c
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) p_h)|*W{
error('zernfun:NMvectors','N and M must be vectors.') \%\b*OO
end nTrfbK@
]}z;!D>
if length(n)~=length(m) _|%pe]St
error('zernfun:NMlength','N and M must be the same length.') V#
Mw
end VesW7m*z
iw1((&^)"
n = n(:); 63:0Vt>hZ^
m = m(:); {MX_t/o=f
if any(mod(n-m,2)) ;-84cpfu
error('zernfun:NMmultiplesof2', ... 47I5Y5
'All N and M must differ by multiples of 2 (including 0).') ONQp-$
end 5MY+O\
9D74/3b*
if any(m>n) |F5^mpU
error('zernfun:MlessthanN', ... W}B4^l
'Each M must be less than or equal to its corresponding N.') mY"DYYR>
end pAg;Rib
o?A/
if any( r>1 | r<0 ) xTQV?g
J
error('zernfun:Rlessthan1','All R must be between 0 and 1.') $4.mRS97g
end wqDRFZ1*P
^Q8m)0DP
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) !ZP1?l30
error('zernfun:RTHvector','R and THETA must be vectors.') $t5
0<1
end y$@d%U*rW^
YLk; ^?
r = r(:); js
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theta = theta(:); Bu?"b=B*
length_r = length(r); Yjz'lWg
if length_r~=length(theta) 0@a6r=`el
error('zernfun:RTHlength', ... g3{)AX[Uy
'The number of R- and THETA-values must be equal.') M52kau
end ^EU&6M2
cn ,zUG!-h
% Check normalization: N3^pFy`
% -------------------- GEP YSp
if nargin==5 && ischar(nflag) 'qLk"
isnorm = strcmpi(nflag,'norm'); AEkgm^t.{
if ~isnorm |7WzTz
error('zernfun:normalization','Unrecognized normalization flag.') J)(H-xvV
end &B3Eq1A
else ><iE VrpN
isnorm = false; gUDd2T#
end %o<&O(Y
2a*1q#MpAt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G}i\UXFE
% Compute the Zernike Polynomials q|2{W.P5qi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% AF
D/
J
]OY6.m
% Determine the required powers of r: ri~d Wx
% ----------------------------------- xMg&>}5
m_abs = abs(m); aA%$<ItH
rpowers = []; 9\TvX!)h
for j = 1:length(n) _J&u{
rpowers = [rpowers m_abs(j):2:n(j)]; q,d]i/T
end rBs7,h
rpowers = unique(rpowers); Faa:h#
d%9I*Qo0,
% Pre-compute the values of r raised to the required powers, x@.iDP@(
% and compile them in a matrix: /6F 1=O(c>
% ----------------------------- Ed#%F-1sX
if rpowers(1)==0 M4M
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rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); `{I,!to
rpowern = cat(2,rpowern{:}); H_;Dq*
rpowern = [ones(length_r,1) rpowern]; F']Vg31c
else 8s8q`_.)(
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3f's>+,#%
rpowern = cat(2,rpowern{:}); 3leg,qd
end #f.@XIt'
,Z_nV+l_
% Compute the values of the polynomials: v)N6ZOj*C
% -------------------------------------- V]H<:UE
y = zeros(length_r,length(n)); /(n)I
for j = 1:length(n) <t]c'
s = 0:(n(j)-m_abs(j))/2; CCq<y
pows = n(j):-2:m_abs(j); psRm*,*O
for k = length(s):-1:1 K *vNv4
p = (1-2*mod(s(k),2))* ... oiO3]P]P
prod(2:(n(j)-s(k)))/ ... S,AZrgh,"X
prod(2:s(k))/ ... U'-MMwE]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... e_]1e7t
prod(2:((n(j)+m_abs(j))/2-s(k))); !dhZs?/UI
idx = (pows(k)==rpowers); =i%2/kdi0b
y(:,j) = y(:,j) + p*rpowern(:,idx); Fh v)
end qCgP8U/jv
NL&g/4A[a
if isnorm R$,`}@VqZ3
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2!68W
X
end C==tJog[
end 9[T#uh!DC
% END: Compute the Zernike Polynomials Xki/5roCQ|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% eV9:AN }K=
l$m^{6IYc
% Compute the Zernike functions: w?M*n<)
O
% ------------------------------ AaTtYd
idx_pos = m>0; oE)c8rE
idx_neg = m<0; I4|p;\`fK
^fK8~g;rB
z = y; u6r-{[W}
if any(idx_pos) rLL;NTN+/
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }sJ%InL
end "r"]NyM
if any(idx_neg) 3pDZ}{ZZU
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [&3"kb
end w5|@vB/pj
PYz| d
% EOF zernfun