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 @-z#vJ5Qe{
function z = zernfun(n,m,r,theta,nflag) XA[GF6W,Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. -DO*,Eecv
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 7k<4/|CQ{
% and angular frequency M, evaluated at positions (R,THETA) on the vT<q zN
% unit circle. N is a vector of positive integers (including 0), and CfMq?.4%E}
% M is a vector with the same number of elements as N. Each element TtL2}Wdd.%
% k of M must be a positive integer, with possible values M(k) = -N(k) xM1>kbo|
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1,
\WM*2&
% and THETA is a vector of angles. R and THETA must have the same :!a9|Fh~
% length. The output Z is a matrix with one column for every (N,M) {&Kq/sRz
% pair, and one row for every (R,THETA) pair. ~Od4(
}/G
% wHW";3w2~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike GHHErXT\a
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), e75k-
% with delta(m,0) the Kronecker delta, is chosen so that the integral U!F~><
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, WUid5e2
% and theta=0 to theta=2*pi) is unity. For the non-normalized U*ZP>Vv
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. p[(VhbN
% mMqT-jT
% The Zernike functions are an orthogonal basis on the unit circle. \TG!M]D:
% They are used in disciplines such as astronomy, optics, and %Fc,$ =
% optometry to describe functions on a circular domain. I/bED~Z:a
% xMsos?5}
% The following table lists the first 15 Zernike functions. ;Ef:mr"Nu
% PXGS5,
% n m Zernike function Normalization S;$@?vF
% -------------------------------------------------- 4z-sR/ d
% 0 0 1 1 P'#m1ntxQ
% 1 1 r * cos(theta) 2 s-eC' )w~E
% 1 -1 r * sin(theta) 2 Vw*;xek?
% 2 -2 r^2 * cos(2*theta) sqrt(6) lrjlkgSN
% 2 0 (2*r^2 - 1) sqrt(3) G7k0P-r,0
% 2 2 r^2 * sin(2*theta) sqrt(6) tb7Wr1$<
% 3 -3 r^3 * cos(3*theta) sqrt(8) <^,w,A
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,ZcW+!
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) W[o~AbU
% 3 3 r^3 * sin(3*theta) sqrt(8) BRP9j
y
% 4 -4 r^4 * cos(4*theta) sqrt(10) 7?K?-Oj
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wVBY^TE
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ?;.j)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?@9kVB*|
% 4 4 r^4 * sin(4*theta) sqrt(10) b .k
J&c
% -------------------------------------------------- KQ3]'2q
% c,KT1me
% Example 1: UYH;15s
% 4-[L^1%S[
% % Display the Zernike function Z(n=5,m=1) KO(+%>^R
% x = -1:0.01:1; 9+.0ZP?
% [X,Y] = meshgrid(x,x); (LPMEQhI:
% [theta,r] = cart2pol(X,Y); -zg,pK$+
% idx = r<=1; 1)u
3
% z = nan(size(X)); 2O {@W +Mt
% z(idx) = zernfun(5,1,r(idx),theta(idx)); KyW6[WA9
% figure FG7}MUu
% pcolor(x,x,z), shading interp ?eT^gWX
% axis square, colorbar /-<S F T`
% title('Zernike function Z_5^1(r,\theta)') fGJPZe
% #NVtZs!V/
% Example 2: M#on-[
% \_FX}1Wc2.
% % Display the first 10 Zernike functions cu|gM[
% x = -1:0.01:1; < pI2}
% [X,Y] = meshgrid(x,x); #M6@{R2_
% [theta,r] = cart2pol(X,Y); ^~(vP:
% idx = r<=1; x^}kG[s
% z = nan(size(X)); (,PO(
% n = [0 1 1 2 2 2 3 3 3 3]; \`o+Le+%
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ^jb55X}
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {zBf *x
% y = zernfun(n,m,r(idx),theta(idx)); DW@PPvfs
% figure('Units','normalized') 3 q
% for k = 1:10 jcQ{,9
H`l
% z(idx) = y(:,k); ;rpjXP
% subplot(4,7,Nplot(k)) T%K(opISc(
% pcolor(x,x,z), shading interp VO>A+vx3M
% set(gca,'XTick',[],'YTick',[]) >e*m8gm#
% axis square blph&[`}I
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @HJ&"72$<
% end ?hvPPEJf
% %(Sy XZ
% See also ZERNPOL, ZERNFUN2. }6.R.*Imz
`?{QCBVj
% Paul Fricker 11/13/2006 -WwFUm
OwV>`BIwns
=C8 t5BZ"
% Check and prepare the inputs: *PE1)bF
% ----------------------------- 33|>u+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) /K2VSj3\
error('zernfun:NMvectors','N and M must be vectors.') cu(2BDfiL
end 314PcSc
%5RY Ea
if length(n)~=length(m) oAe]/ j$
error('zernfun:NMlength','N and M must be the same length.') B#AAG*Ai8
end U 9TEC)
Y8`4K* 58%
n = n(:); 0G1?
m = m(:); |E0>-\6
if any(mod(n-m,2)) v9INZ1# v
error('zernfun:NMmultiplesof2', ... \-N
4G1
'All N and M must differ by multiples of 2 (including 0).') {&8-OoH ~
end _ 0%sYkUc
Jf@M>BT^A
if any(m>n) 6+BR5Nr
error('zernfun:MlessthanN', ... 'YQ"Lf
'Each M must be less than or equal to its corresponding N.') ,i#]&f`c;5
end f:\jPkf'
Ev%4}GwO4
if any( r>1 | r<0 ) 9r@r\-
error('zernfun:Rlessthan1','All R must be between 0 and 1.') LEvdPG$)
end "0 \U>h
&Eg>[gAIlp
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) JLm0[1Lzd
error('zernfun:RTHvector','R and THETA must be vectors.') H7?C>+ay
end 1.!rq,+>1
vE7 L> 7
r = r(:); !OekN,6
theta = theta(:); ^RrufwUA
length_r = length(r); *DObtS_
6
if length_r~=length(theta) B;Ab`UX#t
error('zernfun:RTHlength', ... #>GUfhou)
'The number of R- and THETA-values must be equal.') e*jt(p[Ge
end |[(4h
"AP''XNi
% Check normalization: E.Xfb"]
% -------------------- 1uz9zhG><
if nargin==5 && ischar(nflag) r<c yxR~
isnorm = strcmpi(nflag,'norm'); ZdeRLX
if ~isnorm KG)7hja<6g
error('zernfun:normalization','Unrecognized normalization flag.') 7lY&/-V
end A>(m}P
else 7)S`AQ2:)
isnorm = false; d$8rzd
end \Xc6K!HJM
2;r(?ebw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?~uTbNR
% Compute the Zernike Polynomials }legh:/*?O
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 55MsF}p
_%w-y(Sqn
% Determine the required powers of r: HE(|x1C)j
% ----------------------------------- Y v }G"-=
m_abs = abs(m); frbKi _1
rpowers = []; > xkl7D
for j = 1:length(n) g*F?
rpowers = [rpowers m_abs(j):2:n(j)]; R<{bb'
end 9V`/zq?
rpowers = unique(rpowers); "{105&c\
wX@&Qv
% Pre-compute the values of r raised to the required powers, D oX!P|*
% and compile them in a matrix: /1ooOq]
% ----------------------------- q]YPDdR#
if rpowers(1)==0 N~_GJw@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); !}|n3wQ
rpowern = cat(2,rpowern{:}); `Gzukh
rpowern = [ones(length_r,1) rpowern]; F2]v]]F!
else =:n>yZ3T
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `4__X;
rpowern = cat(2,rpowern{:}); f1(V~{N,+
end 7tMV*{+Z
``j..v,
% Compute the values of the polynomials: *T$o"*}
% -------------------------------------- U:m[*
}+<
y = zeros(length_r,length(n)); T^g2N`w2
for j = 1:length(n) j9u/R01d
s = 0:(n(j)-m_abs(j))/2; ^5j|
pows = n(j):-2:m_abs(j); IlG)=?8XZ
for k = length(s):-1:1 -;&aU;k
p = (1-2*mod(s(k),2))* ... Pj>r(Cv
prod(2:(n(j)-s(k)))/ ... Ls )y.u
prod(2:s(k))/ ... Q(
.d!CQ>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7YsBwo
prod(2:((n(j)+m_abs(j))/2-s(k))); *gfx'$
idx = (pows(k)==rpowers); <DP_`[+C
y(:,j) = y(:,j) + p*rpowern(:,idx); #Mw|h^Wm
end ~Z!!wDHS
|E-/b6G
if isnorm +gqtW86
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >;kCcfS3ct
end YMOy6C
end -jnx0{/
% END: Compute the Zernike Polynomials azR<Y_tw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m d:$OC3
ac"Pn?
q
% Compute the Zernike functions: Og[NRd+
% ------------------------------ {2G9>'
idx_pos = m>0; sE@t$'=
idx_neg = m<0; tgK$}#.*
h~haA8i?{
z = y; ^IGutZov
if any(idx_pos) &S}%)g%Iv9
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); gQ4Q
h;
end 5!u.w
if any(idx_neg) 5_Yl!=
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Wc [@,
end BV,P;T0"D
\PU3{_G]
% EOF zernfun