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 t ]_VG
function z = zernfun(n,m,r,theta,nflag) A .EbXo/
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K%F,='P}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jO1r)hw N>
% and angular frequency M, evaluated at positions (R,THETA) on the FMClSeO7
% unit circle. N is a vector of positive integers (including 0), and OVhE??#
% M is a vector with the same number of elements as N. Each element &'
Ne!o8
% k of M must be a positive integer, with possible values M(k) = -N(k) |>tKq;/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Z`KC%!8K
% and THETA is a vector of angles. R and THETA must have the same -/ g B|J
% length. The output Z is a matrix with one column for every (N,M) G}:lzOlMH
% pair, and one row for every (R,THETA) pair. 5[YDZ7g"~
% znaUB v_
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D-4f >
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), zY('t!u8
% with delta(m,0) the Kronecker delta, is chosen so that the integral LS88.w\=S@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~XWQhIAM4
% and theta=0 to theta=2*pi) is unity. For the non-normalized 1M
781
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. f,:9N 5Z
% Db1pW=66:
% The Zernike functions are an orthogonal basis on the unit circle. /5:bvg+
% They are used in disciplines such as astronomy, optics, and
1][S#H/?
% optometry to describe functions on a circular domain. [`rba'
% b+&%1C
% The following table lists the first 15 Zernike functions. h >s!K9
% \3S8 62B7
% n m Zernike function Normalization <\}KT*Xp
% -------------------------------------------------- C@L$~iG
% 0 0 1 1 f^"N!f a
% 1 1 r * cos(theta) 2 (KF=On;=Y
% 1 -1 r * sin(theta) 2 @)4]b+8Z
% 2 -2 r^2 * cos(2*theta) sqrt(6) MgNU``
% 2 0 (2*r^2 - 1) sqrt(3) }`,t$NV`
% 2 2 r^2 * sin(2*theta) sqrt(6) j&Wl0
% 3 -3 r^3 * cos(3*theta) sqrt(8) (r D_(%o
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) B3
5E8/
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 6DuEL=C
% 3 3 r^3 * sin(3*theta) sqrt(8) %+K<<iyR|
% 4 -4 r^4 * cos(4*theta) sqrt(10) R]btAu;Z
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R5 9S@MsuD
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) \8 h;K>=h
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KjE+QUa
% 4 4 r^4 * sin(4*theta) sqrt(10) J_v$YwE
% -------------------------------------------------- A3S<..g2
% YGPy@-,E
% Example 1: \DD0s8
% ~(IB0=A{v
% % Display the Zernike function Z(n=5,m=1) dOoK Lry
% x = -1:0.01:1; MvWaB
% [X,Y] = meshgrid(x,x); iIq)~e/ Z
% [theta,r] = cart2pol(X,Y); +[tE ^`-F
% idx = r<=1; ?}a;}Q6
% z = nan(size(X)); qh2ON>e;
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ,J{ei7TN
% figure 2m35R&
% pcolor(x,x,z), shading interp ;Mpy#yIU.
% axis square, colorbar x\s|n{
% title('Zernike function Z_5^1(r,\theta)') Gmq/3tw
% ,;Hu=;
% Example 2: D6Goa(!9d
% H+ 0$tHi
% % Display the first 10 Zernike functions W034N[9
% x = -1:0.01:1; [5MJwRM^!;
% [X,Y] = meshgrid(x,x); ZOQTINf
% [theta,r] = cart2pol(X,Y); (v}>tb*#`
% idx = r<=1; PV/77{'
% z = nan(size(X)); r;Gi+Ca5
% n = [0 1 1 2 2 2 3 3 3 3]; (s7;^)}zx
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; R%qGPO5Z\c
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [I$BmGQ
% y = zernfun(n,m,r(idx),theta(idx)); 6u`)QUmItg
% figure('Units','normalized') 72Iy^Y[MX
% for k = 1:10 |*'cF-lp6v
% z(idx) = y(:,k); !>e5z|1
% subplot(4,7,Nplot(k)) ,>eMG=C; g
% pcolor(x,x,z), shading interp 0DmMG
% set(gca,'XTick',[],'YTick',[]) weE/TW\e
% axis square wm$}Pch
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !2'jrJGc
% end l#H#+*F
% ]zQo>W$
% See also ZERNPOL, ZERNFUN2. -xDGH
MV\|e1B}
% Paul Fricker 11/13/2006 3plzHz ,x
p Wt)
A
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% Check and prepare the inputs: _',prZ*
% ----------------------------- Z6_N$Z.A
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) sM?MLB\Za
error('zernfun:NMvectors','N and M must be vectors.') _-9@qe
end I{lT> go
ni6{pK4Wqm
if length(n)~=length(m) %9M~f*
error('zernfun:NMlength','N and M must be the same length.') j^;I3_P
end N#Zhxu,g!
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n = n(:); FcRW;e8-
m = m(:); spGB)k,^
if any(mod(n-m,2)) >9q&PEc
error('zernfun:NMmultiplesof2', ... KTn}w:+B\
'All N and M must differ by multiples of 2 (including 0).') <0QH<4
end ewfP G,S
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if any(m>n) niqN{
error('zernfun:MlessthanN', ...
Tjl:|F8
'Each M must be less than or equal to its corresponding N.') BvR-K\rx
end '{J&M|<A
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if any( r>1 | r<0 ) \L(*]:EP
error('zernfun:Rlessthan1','All R must be between 0 and 1.') BwBm[jtP
end GF>'\@Th
( @3\`\X
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) C":o/;,1
error('zernfun:RTHvector','R and THETA must be vectors.') ;Ww7"-=sw
end 05spovO/'
Z7_ zMM
r = r(:); $zyIuJN#
theta = theta(:); VH(S=G5Yb
length_r = length(r); W ]Nv33i
[
if length_r~=length(theta) /,X[k !
error('zernfun:RTHlength', ... E[*Fz1>
'The number of R- and THETA-values must be equal.') c{&*w")J
end 8S<@"v
KM!k$;my
% Check normalization: ']>Mp#j
% -------------------- qqu.EE
if nargin==5 && ischar(nflag) s.x&LG
isnorm = strcmpi(nflag,'norm'); ~,BIf+\XF
if ~isnorm +{/*z
error('zernfun:normalization','Unrecognized normalization flag.') sp
]zbX?
end K,e w >U
else S=nP[s
isnorm = false; \N4
y<
end _^'I
,N
e;kI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GN0`rEh
% Compute the Zernike Polynomials q-`RI*1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9!Ar`Io2@
]ZoD'-,
% Determine the required powers of r: GQl$yZaK{
% ----------------------------------- UH"#2< |b
m_abs = abs(m); ~{D[
>j][
rpowers = []; bg3"W,bv%
for j = 1:length(n) :Pg}Zz <
rpowers = [rpowers m_abs(j):2:n(j)]; 7As|Ns`
end OZIW_'Wm/
rpowers = unique(rpowers); )6w}<W*1E
2{Chu85
% Pre-compute the values of r raised to the required powers, (C\hVy2X?N
% and compile them in a matrix: %rF?dvb;?
% ----------------------------- !p[9{U->o;
if rpowers(1)==0 !j\" w p
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @0eHS+
rpowern = cat(2,rpowern{:}); b.@P%`@a.
rpowern = [ones(length_r,1) rpowern]; ^<:sdv>Y5
else :mS# h@l
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4_UU<GEp
rpowern = cat(2,rpowern{:}); Pf$pt
end W?We6.%
cwuO[^S}
% Compute the values of the polynomials: a3VM'
% -------------------------------------- 3VUWX5K?
y = zeros(length_r,length(n)); #CnHf
for j = 1:length(n) AxZD-|.
s = 0:(n(j)-m_abs(j))/2; #!9S}b$
pows = n(j):-2:m_abs(j); q\q=PB6r
for k = length(s):-1:1 _kdL'x
p = (1-2*mod(s(k),2))* ... DEw8*MN
prod(2:(n(j)-s(k)))/ ... `- \/$M9s=
prod(2:s(k))/ ... &%2*Wu;
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ; h`0ir4[A
prod(2:((n(j)+m_abs(j))/2-s(k))); R.s^o]vT
idx = (pows(k)==rpowers); 2~*Ez!.3
y(:,j) = y(:,j) + p*rpowern(:,idx); k` {@pt.
end S8l1"/?aHE
Y(+^;Y3U
if isnorm x%<
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 2iU7 0(H
end e }*0ghKI
end Lqp8yVO
% END: Compute the Zernike Polynomials Pe_!?:vF
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `$YP<CJeq
|w*R8ro_
% Compute the Zernike functions: 'i8U
% ------------------------------ JI/_ce
idx_pos = m>0; oR~e#<$;
idx_neg = m<0; Ln.ZVMZ;
m$LVCB
z = y; KT.?Xp:z
if any(idx_pos) NJ MJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); @O}7XRJ_8
end /?6gdN
if any(idx_neg) 8*SP~q
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); JTqq0OD}
end EQe5JFR
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% EOF zernfun