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 [%?y( q
function z = zernfun(n,m,r,theta,nflag) c.0]1
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. B=dseeG[To
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "S(yZ6r"
% and angular frequency M, evaluated at positions (R,THETA) on the 5
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% unit circle. N is a vector of positive integers (including 0), and lJ&y&N<O
% M is a vector with the same number of elements as N. Each element ]4o?BkL
% k of M must be a positive integer, with possible values M(k) = -N(k) {xToz]YA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 5VKcV&D
% and THETA is a vector of angles. R and THETA must have the same sUbFRq
% length. The output Z is a matrix with one column for every (N,M) np=kTJ
% pair, and one row for every (R,THETA) pair. `|?]CkP
% 0bSz4<}
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike o:9$UV[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ]F+K|X9-
% with delta(m,0) the Kronecker delta, is chosen so that the integral puF%=i
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, akCIa'>t
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]u0Jd#@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #w*"qn#2Uz
% i-.c=M
% The Zernike functions are an orthogonal basis on the unit circle. qtY
m!g
% They are used in disciplines such as astronomy, optics, and .8(%4ejJ(
% optometry to describe functions on a circular domain. fGTOIi@#
% 8lb-}=
% The following table lists the first 15 Zernike functions. 8gI\zgS
% L/fRF"V
% n m Zernike function Normalization 3e
73l
% -------------------------------------------------- H(&Z:{L
% 0 0 1 1 5r7h=[N
% 1 1 r * cos(theta) 2 [q3+$W \r
% 1 -1 r * sin(theta) 2 Jn#K0(FQ
% 2 -2 r^2 * cos(2*theta) sqrt(6) Hm4bN\%
% 2 0 (2*r^2 - 1) sqrt(3) !M^\f
N1
% 2 2 r^2 * sin(2*theta) sqrt(6) ;{Jb6'K1h
% 3 -3 r^3 * cos(3*theta) sqrt(8) RHI&j~
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) `)tA
YH
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]7vf#1i<
% 3 3 r^3 * sin(3*theta) sqrt(8) xqv[?
?
% 4 -4 r^4 * cos(4*theta) sqrt(10) Ow)R|/e/
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tN2 W8d
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) (3W&AM
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |[LE9Lq/
% 4 4 r^4 * sin(4*theta) sqrt(10) 8[R1A
% -------------------------------------------------- Q.ukY@L.'
% ^Plc}W7h
% Example 1: EY$?^iS
% 61|B]ei/
% % Display the Zernike function Z(n=5,m=1) C0(sAF@
% x = -1:0.01:1; >3P9 i ;W
% [X,Y] = meshgrid(x,x);
tT-=hDw
% [theta,r] = cart2pol(X,Y); enumK\
% idx = r<=1; P^zy; Qs7
% z = nan(size(X)); 7P*Z0%Q
% z(idx) = zernfun(5,1,r(idx),theta(idx)); WK4@:k
m6)
% figure YxyG\J\|,
% pcolor(x,x,z), shading interp wT/6aJoX
% axis square, colorbar }e2F{pQ
% title('Zernike function Z_5^1(r,\theta)') a.,i.2
% 1Is%]6
% Example 2: [pR)@$"k'
% &I)\*Ue2t
% % Display the first 10 Zernike functions b{pg!/N4
% x = -1:0.01:1; [gZDQcU
% [X,Y] = meshgrid(x,x); Abf1"#YImy
% [theta,r] = cart2pol(X,Y); j+Zt.KXjT
% idx = r<=1; 9wMEvX70
% z = nan(size(X)); tW(+xu36
% n = [0 1 1 2 2 2 3 3 3 3]; +?V0:Kz]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; )Mi'(C;
% Nplot = [4 10 12 16 18 20 22 24 26 28]; r<|nwFJ
% y = zernfun(n,m,r(idx),theta(idx)); -[$&s FD
% figure('Units','normalized') F.0d4:A+
% for k = 1:10 N&x:K+Zm.
% z(idx) = y(:,k); ]QS](BbD:
% subplot(4,7,Nplot(k)) q^]tyU!w
% pcolor(x,x,z), shading interp
,CKvTxz0
% set(gca,'XTick',[],'YTick',[]) D$hQyhz'
% axis square ~6sE an3p
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 9P0yv3
% end ^#w{/C/n
% rhoeZ
% See also ZERNPOL, ZERNFUN2. $?$9y^\
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% Paul Fricker 11/13/2006 ZpWu,1
nsl*Dm"*F
#TATqzA
% Check and prepare the inputs: e?=elN
% ----------------------------- v
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Pw
xIz
error('zernfun:NMvectors','N and M must be vectors.') ]#5^&w)'
end -#%X3F7/w
|*E"G5WZM
if length(n)~=length(m) 8}z3CuM
error('zernfun:NMlength','N and M must be the same length.') lM+ xU;
end PY-+ Bf
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n = n(:); =)*JbwQ
m = m(:); %YCd%lAe,
if any(mod(n-m,2)) uS-3\$
error('zernfun:NMmultiplesof2', ... I+~bCcgPi
'All N and M must differ by multiples of 2 (including 0).') AsAFUuI
end H/`G
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if any(m>n) J 8q
error('zernfun:MlessthanN', ... agW9Go_F[
'Each M must be less than or equal to its corresponding N.') `#U ]iwW!
end HL8(lPgS
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if any( r>1 | r<0 ) C#$6O8O
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ^]7,1dH}M
end (Y )!"_|
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ~v\hIm3=m
error('zernfun:RTHvector','R and THETA must be vectors.') 48k7/w\
end RJ*F>2
^Xa*lR 3
r = r(:); OM{Dq|
theta = theta(:); O4N-_Kfp/
length_r = length(r); 0 {,h.:
if length_r~=length(theta) ~?-qZ<9/
error('zernfun:RTHlength', ... Pxk0(oBX
'The number of R- and THETA-values must be equal.') S\b K+
end tIp{},bQ^
,{+6$h3
% Check normalization: %ZuLl(
% -------------------- Ge0Lb+<G
if nargin==5 && ischar(nflag) 8H_l[/
isnorm = strcmpi(nflag,'norm'); [,GU5,o
if ~isnorm 6W:1>,xS
error('zernfun:normalization','Unrecognized normalization flag.') Ju4.@
end w49{-Pp[
else qPUA!-'
isnorm = false; (M8hy4Ex
end *(p7NYf1
!3?yG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *:[b'D!A
% Compute the Zernike Polynomials Vq U|kv
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X?R
|x[
Hh@2 m\HA
% Determine the required powers of r: ?CFoe$M
% ----------------------------------- H@4/#V|Uy
m_abs = abs(m); i3d y
rpowers = []; PK}vh%
for j = 1:length(n) N;g$)zCV1
rpowers = [rpowers m_abs(j):2:n(j)]; 9 R
end ?lyltAxs'
rpowers = unique(rpowers); ^ `je
I5Q~T5Ar
% Pre-compute the values of r raised to the required powers, ZBC@xM&-
% and compile them in a matrix: ([tG y
% ----------------------------- E$R_rX4x
if rpowers(1)==0 DUhT>,~]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); p&uCp7]U
rpowern = cat(2,rpowern{:}); q#|r
rpowern = [ones(length_r,1) rpowern]; M_; w%FV
else hRLKb}
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9ClF<5?M
rpowern = cat(2,rpowern{:}); ,$ mLL
end ^9s"FdB]24
uD[^K1Ag]^
% Compute the values of the polynomials: YLigP"*~^
% -------------------------------------- 3r`<(%\
y = zeros(length_r,length(n)); .X^43
q
for j = 1:length(n) Wkww&Y
s = 0:(n(j)-m_abs(j))/2; G_0)oC@Jl:
pows = n(j):-2:m_abs(j); !YIb
for k = length(s):-1:1 Stt* 1gT
p = (1-2*mod(s(k),2))* ... )6g&v'dq
prod(2:(n(j)-s(k)))/ ... ff[C'
prod(2:s(k))/ ... YY\Rua/nG
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9[Y*k^.!
prod(2:((n(j)+m_abs(j))/2-s(k))); cT I,1U
idx = (pows(k)==rpowers); (]}XLMi,|!
y(:,j) = y(:,j) + p*rpowern(:,idx); =:;YTie
end T*8_FR <
&62`Wr 0C
if isnorm [C2kK *JZ
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7Y)s#FJ
end {vjqy&?y
end o3fR3P%$
% END: Compute the Zernike Polynomials Ae.]F)w_\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6Z$b?A3zM
o;%n,S8J|^
% Compute the Zernike functions: EtJD'&
% ------------------------------ uO6c3|Zjs
idx_pos = m>0; \ x:_*`fU
idx_neg = m<0; )S#j.8P'B
yTP[,bM
z = y; 2=Jmi?k
if any(idx_pos) 9W$mDw6f
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6OMb`A@/2
end FDl,Ey^r/
if any(idx_neg) ^971<B(v
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :C>J-zY
end EmF]W+!z%
n|J.)E.
% EOF zernfun