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 Em[DHfu1Q
function z = zernfun(n,m,r,theta,nflag) ;?C#IU
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. O25lLNmO
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gGfoO[B
% and angular frequency M, evaluated at positions (R,THETA) on the hsu{ey p
% unit circle. N is a vector of positive integers (including 0), and oyo(1>
% M is a vector with the same number of elements as N. Each element = k\J<
% k of M must be a positive integer, with possible values M(k) = -N(k) tTd\|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ">?vir^
% and THETA is a vector of angles. R and THETA must have the same KZ~*Nz+H2
% length. The output Z is a matrix with one column for every (N,M) [w ;kkMJAy
% pair, and one row for every (R,THETA) pair. G[jW<'f
% 3Hf0MAt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike g^zs,4pPU<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V|\7')Qq
% with delta(m,0) the Kronecker delta, is chosen so that the integral O|_h_I-2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, HSq}7S&U
% and theta=0 to theta=2*pi) is unity. For the non-normalized r(gXoq_w
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .F+@B\A<
% TX
% The Zernike functions are an orthogonal basis on the unit circle. ]qhPd_$?D'
% They are used in disciplines such as astronomy, optics, and +SJd@y@fR
% optometry to describe functions on a circular domain. ;#Q%j%J
% LR"9D
% The following table lists the first 15 Zernike functions. 4tY ss
% V)}rEX
% n m Zernike function Normalization qWw\_S
% -------------------------------------------------- |JCU<_<
% 0 0 1 1 A_KW(;50
% 1 1 r * cos(theta) 2 I}R0q
% 1 -1 r * sin(theta) 2 bV/jfV"%E
% 2 -2 r^2 * cos(2*theta) sqrt(6) Y3Q9=u*5
% 2 0 (2*r^2 - 1) sqrt(3) o.I6ulY8
% 2 2 r^2 * sin(2*theta) sqrt(6) Yup3^E
w&
% 3 -3 r^3 * cos(3*theta) sqrt(8) y(
y8+ZT
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) s&j-\bOic9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @B}aN@!/
% 3 3 r^3 * sin(3*theta) sqrt(8) >rvQw63\
% 4 -4 r^4 * cos(4*theta) sqrt(10) {T].]7Z
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !>:?rSg*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2#k5+?-c61
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) oY, %Iq
% 4 4 r^4 * sin(4*theta) sqrt(10) i~r l o^
% -------------------------------------------------- fDLG>rXPT
% 5xL~`-IA&v
% Example 1: }NB}"%2
% f5 `g
% % Display the Zernike function Z(n=5,m=1) K$d$m <
% x = -1:0.01:1; cph:y
% [X,Y] = meshgrid(x,x); G}p\8Q}'
% [theta,r] = cart2pol(X,Y); )2M>3C6>f
% idx = r<=1; &\_iOw8
% z = nan(size(X)); m4*@o?Ow
% z(idx) = zernfun(5,1,r(idx),theta(idx)); iTaWu p
% figure =G]@+e
% pcolor(x,x,z), shading interp jmeRrnC}
% axis square, colorbar RD.V'`n"
% title('Zernike function Z_5^1(r,\theta)') c/uNM
% 2PG [7u^
% Example 2: 4f<$4d^md
% jRatm.N
% % Display the first 10 Zernike functions TiH)5
% x = -1:0.01:1; c_>f0i
% [X,Y] = meshgrid(x,x); 8,uB8C9
% [theta,r] = cart2pol(X,Y); 0x!2ihf
% idx = r<=1; P67o{EdK
% z = nan(size(X)); ]~3U
% n = [0 1 1 2 2 2 3 3 3 3]; t]e;;q=L.
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; fj&i63?e
% Nplot = [4 10 12 16 18 20 22 24 26 28]; h;0S%ZC
% y = zernfun(n,m,r(idx),theta(idx)); KI+VXH}Y5{
% figure('Units','normalized') F;>!&[h}G
% for k = 1:10 9VbOQ {8
% z(idx) = y(:,k); zL J/5&
% subplot(4,7,Nplot(k)) XO'l Nb.
% pcolor(x,x,z), shading interp )YqXRm
% set(gca,'XTick',[],'YTick',[]) >
%KuNy{
% axis square !Ta>U^7
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .c$316
% end y.q(vzg\_
% v~Y^r2
% See also ZERNPOL, ZERNFUN2. !Xph_SQ!B=
l(Q?rwI8Y
% Paul Fricker 11/13/2006 5+wAzVA
28=O03q
F_4n^@M
% Check and prepare the inputs: of@#:Qs
% ----------------------------- _(KbiEB{
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~#/hzS
error('zernfun:NMvectors','N and M must be vectors.') ,tg0L$qC
end &%/7E_j7
b?'yAXk
if length(n)~=length(m) +U3m#Y )k
error('zernfun:NMlength','N and M must be the same length.') mbueP.q[?
end SZXY/~=h
)sT> i
n = n(:); L~KM=[cn
m = m(:); =3v]gOcO
if any(mod(n-m,2)) jfqopiSi
error('zernfun:NMmultiplesof2', ... P$-X)c$&
'All N and M must differ by multiples of 2 (including 0).') z+>}RT]
end \0gM o&
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if any(m>n) 0MdDXG-7
error('zernfun:MlessthanN', ... ^) s2$A:L
'Each M must be less than or equal to its corresponding N.') NW&b&o
end Ho
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a 7,C>%I
if any( r>1 | r<0 ) FJ6u.u
error('zernfun:Rlessthan1','All R must be between 0 and 1.') pLzk
end Kc^;vT>3
*VZ5B<Ic
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,1"KHv
error('zernfun:RTHvector','R and THETA must be vectors.') 2m2;t0
end w4d--[Q
1N>|yQz
r = r(:); J":,Vd!*-
theta = theta(:); !U~WK$BP
length_r = length(r); J>bJ
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if length_r~=length(theta) c?,i3s+2Y
error('zernfun:RTHlength', ... QhK#Y{xY
'The number of R- and THETA-values must be equal.') ok4@N @
end '>rw(3
X.e7A/ClEo
% Check normalization: qm8&*UuKJ
% -------------------- .?Gd'Lp
if nargin==5 && ischar(nflag) X<%Q"2hW
isnorm = strcmpi(nflag,'norm'); h^o{@/2
if ~isnorm _Iv6pNd/
error('zernfun:normalization','Unrecognized normalization flag.') _\GC(
end n= u&uqA*
else 9b*nLyYVz
isnorm = false; ut I"\1hQ
end y7i*s^ys{
Os1>kwC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BFOq8}fX2
% Compute the Zernike Polynomials w2'f/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6 jn3`D
3z&Fi;<+j
% Determine the required powers of r: @ >U-t{W
% ----------------------------------- ixT:)|'i
m_abs = abs(m); B,=H@[Fj
rpowers = []; Ch3jxgQY
for j = 1:length(n) /Bm( `T
rpowers = [rpowers m_abs(j):2:n(j)]; KW^7H
end &E=>Hj(dTG
rpowers = unique(rpowers); ]3l 9:|
q*7VqB
% Pre-compute the values of r raised to the required powers, 9B7^lR
% and compile them in a matrix: hs$GN]
% ----------------------------- D
'Zt
if rpowers(1)==0 Gnq?"</
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); X'qU*Eo
rpowern = cat(2,rpowern{:}); #Ibp(
rpowern = [ones(length_r,1) rpowern]; ?pB>0b~3-
else F
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 1jF`5k
rpowern = cat(2,rpowern{:}); VQS~\:1
end Q{5kxw1ZF
~"kb7Fxp
% Compute the values of the polynomials: h9G RI
% -------------------------------------- 57&b:0`p
y = zeros(length_r,length(n)); DRi<6Ob
for j = 1:length(n) 65aK2MS@
s = 0:(n(j)-m_abs(j))/2; c:o]d )S
pows = n(j):-2:m_abs(j); G%W8S
\
for k = length(s):-1:1 [.uG5%fa
p = (1-2*mod(s(k),2))* ... sv&;Y\2c
prod(2:(n(j)-s(k)))/ ... -RvQB
prod(2:s(k))/ ... >^*+iEe
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... #T=LR@y
prod(2:((n(j)+m_abs(j))/2-s(k))); &RnTzqv
idx = (pows(k)==rpowers); 2-Ej4I~
y(:,j) = y(:,j) + p*rpowern(:,idx); k@3Q|na
end .G#8a1#
< F.hZGss7
if isnorm 9#MBaO8_"
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); L'0B$6
end P<a)25be/
end sEGO2xeI
% END: Compute the Zernike Polynomials Xy$3VU*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% li}1S
[k;\S XDZo
% Compute the Zernike functions: +
|#O@k
% ------------------------------ 9vGu0Um
idx_pos = m>0; U$WxHYo
idx_neg = m<0; G2Qlt@.T
yEhTNBa*h{
z = y; O\"3J(y,
if any(idx_pos) {_ i\f ]L
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); v{ 0=
end \b?" b
if any(idx_neg) ECrex>zr%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); zGAq-<
end 7G}2,ueI
3 I@}my1
% EOF zernfun