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 g{%';
function z = zernfun(n,m,r,theta,nflag) 'GFzI:Xr
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. r>Ln*R,9D
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Zx_m?C_2_
% and angular frequency M, evaluated at positions (R,THETA) on the pR"qPSv'
% unit circle. N is a vector of positive integers (including 0), and Y[!a82MTzn
% M is a vector with the same number of elements as N. Each element >=V+X"\Z
% k of M must be a positive integer, with possible values M(k) = -N(k) gy{a+Wbc*
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ~K9U0ypH
% and THETA is a vector of angles. R and THETA must have the same zgqw*)C~
% length. The output Z is a matrix with one column for every (N,M) QP#Wfk(C
% pair, and one row for every (R,THETA) pair. j1ZFsTFMWp
% }$-VI\96
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike BGX@n#:
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), US4Um>j
% with delta(m,0) the Kronecker delta, is chosen so that the integral AJT0)FCpR
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, z7q2+;L
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9zJ`;1
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Iqsk\2W]a3
% K +~v<F
% The Zernike functions are an orthogonal basis on the unit circle. K\b O[J
% They are used in disciplines such as astronomy, optics, and \ax%I)3
% optometry to describe functions on a circular domain. HhvG#Sam!
% GcnY=%L?
% The following table lists the first 15 Zernike functions. @m V C
% h6*`V
% n m Zernike function Normalization j;)6uia*A
% -------------------------------------------------- >| ?T|
% 0 0 1 1 A-5+#
% 1 1 r * cos(theta) 2 Aq!['G
% 1 -1 r * sin(theta) 2 spJ(1F{|V
% 2 -2 r^2 * cos(2*theta) sqrt(6) ??Zmj:8E'
% 2 0 (2*r^2 - 1) sqrt(3) lQBM0|n
% 2 2 r^2 * sin(2*theta) sqrt(6) Rs`a@Fn
% 3 -3 r^3 * cos(3*theta) sqrt(8) &r%*_pX
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) PoJ$%_a}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) F-^HN%
% 3 3 r^3 * sin(3*theta) sqrt(8) +,Az\aT/%
% 4 -4 r^4 * cos(4*theta) sqrt(10) (GG"'bYk
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ug21d42Z4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) h
'[vB^
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
#-T.@a1X
% 4 4 r^4 * sin(4*theta) sqrt(10) "ILWIzf.]
% -------------------------------------------------- `fZD%o3l
% "Vq]|j,B/c
% Example 1: 'c&@~O;^d
% L]d@D0.Z
% % Display the Zernike function Z(n=5,m=1) GYC&P]
% x = -1:0.01:1; 5vft}f
% [X,Y] = meshgrid(x,x); hXm}d\
% [theta,r] = cart2pol(X,Y); y.p6%E_`
% idx = r<=1; D a[C'm=
% z = nan(size(X)); P]"deB|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); N?;o_^C
% figure :(>9u.>l?5
% pcolor(x,x,z), shading interp B#"|5
% axis square, colorbar iIaT1i4t.
% title('Zernike function Z_5^1(r,\theta)') {X<4wxeTo
% ( 'n8=J
% Example 2: o^Yspp
% @b\ S.
% % Display the first 10 Zernike functions 5xDN&su
% x = -1:0.01:1; i ,pN1_-
% [X,Y] = meshgrid(x,x); TE%#$q
% [theta,r] = cart2pol(X,Y); RX5.bVp
eE
% idx = r<=1; i 1I>RK
% z = nan(size(X)); `uh@iD'KI
% n = [0 1 1 2 2 2 3 3 3 3]; Wi[m`#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qQOD
% Nplot = [4 10 12 16 18 20 22 24 26 28]; K;p<f{PE
% y = zernfun(n,m,r(idx),theta(idx)); #we>75l{+R
% figure('Units','normalized') T_?nd T2
% for k = 1:10 K\+}q{
% z(idx) = y(:,k); Jh4&Qh|t
% subplot(4,7,Nplot(k)) M+;P?|a
% pcolor(x,x,z), shading interp sD8m<
% set(gca,'XTick',[],'YTick',[]) tIb21c q
% axis square dAr)%RZ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =HY1l}\
% end [W$Z60?RR
% 1@^Ek8C
% See also ZERNPOL, ZERNFUN2. /%YiZ#
H [Lt%:r
% Paul Fricker 11/13/2006 ZBmXaP[9
a4(?]ND~6
[z% ?MIT
% Check and prepare the inputs: pp]_/46nN
% ----------------------------- wD],{ y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) f{Fe+iPc
error('zernfun:NMvectors','N and M must be vectors.') D!}K)T1~R
end 7~"(+f
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if length(n)~=length(m) 6I>5~?#
error('zernfun:NMlength','N and M must be the same length.') P:(EU s}0
end 6W;?8Z_1
l>D-Aan
n = n(:); -nk#d%a\
m = m(:); px|>v8
if any(mod(n-m,2)) !ml_S)
error('zernfun:NMmultiplesof2', ... 'Z.OF5|eGT
'All N and M must differ by multiples of 2 (including 0).') N
pXgyD
end b>QM~mq3^I
dGsS<@G
if any(m>n) e" Eqi-
error('zernfun:MlessthanN', ... 8nIMZV
'Each M must be less than or equal to its corresponding N.') K2xH'v
O (
end wI!
+L&Q
C NfJ:e2
if any( r>1 | r<0 ) (@ fa~?v>@
error('zernfun:Rlessthan1','All R must be between 0 and 1.') lC=N:=Mu
end ^p 2.UW
jQ_dw\
{0
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =!(*5\IM
error('zernfun:RTHvector','R and THETA must be vectors.') f4'El2>-86
end CYt jY~
xN`r4
r = r(:); Dc.n-ipv$
theta = theta(:); d$fvg8^
length_r = length(r); }UKgF.
if length_r~=length(theta) V)0[`zJ
error('zernfun:RTHlength', ... wfBuU>
'The number of R- and THETA-values must be equal.') [J)/Et
end 5=Kq@[(4
s>jr1~~3O_
% Check normalization: q Vm"f,ruo
% -------------------- *$i; o3
if nargin==5 && ischar(nflag) %/l-A
pu
isnorm = strcmpi(nflag,'norm'); VY/|WD~"CW
if ~isnorm s~=KhP~
error('zernfun:normalization','Unrecognized normalization flag.') )o#6-K+b
end EkJVFHfh
else URYZV8=B~
isnorm = false; W/ g|{t[
end tYs8)\{
\G$QNUU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WI1T?.Gc
% Compute the Zernike Polynomials U~uwm/h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fav5e'[$
l`@0zw+
% Determine the required powers of r: 6exI_3A4jh
% ----------------------------------- +I|Rk&
m_abs = abs(m); #^|| ]g/N
rpowers = []; WD15pq l
for j = 1:length(n) "^;#f+0
rpowers = [rpowers m_abs(j):2:n(j)]; CO-Iar
end t< sp%zXZ
rpowers = unique(rpowers); tm(v~L%$>]
?gLR<d_
% Pre-compute the values of r raised to the required powers, }@Xh xZu
% and compile them in a matrix: SQ}S4r
% ----------------------------- A LXUaE.
if rpowers(1)==0 !|:RcH[
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GI4?|@%vD!
rpowern = cat(2,rpowern{:}); gUl1CH&
rpowern = [ones(length_r,1) rpowern]; Iq{o-nq
else w6vLNX
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C<_Urnmn
rpowern = cat(2,rpowern{:}); (O$}(Tn
end 1p8:.1)q
9khjwt
% Compute the values of the polynomials: Le*`r2
% -------------------------------------- gs?8Wzh90*
y = zeros(length_r,length(n)); /@VsqD
for j = 1:length(n) 8tU>DJ}0
s = 0:(n(j)-m_abs(j))/2; d]U`?A,
pows = n(j):-2:m_abs(j); ]k[x9,IU\y
for k = length(s):-1:1 Hi^35
p = (1-2*mod(s(k),2))* ... K[kds`
prod(2:(n(j)-s(k)))/ ... +A@m9
prod(2:s(k))/ ... Nepi|{
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^f9>l;Lb
prod(2:((n(j)+m_abs(j))/2-s(k))); 5J
ySFG3
idx = (pows(k)==rpowers); ton1oq
y(:,j) = y(:,j) + p*rpowern(:,idx); 4S tjj!ew
end ^w.]Hd2
W!t{rI7 2
if isnorm 6
jmrD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); z<!O!wX_aI
end le.anJAr
end a0PE^U
% END: Compute the Zernike Polynomials ymYBm:"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% GQb i$kl
FH.f- ZU
% Compute the Zernike functions: I_ONbJ9]
% ------------------------------ c&E]E(
idx_pos = m>0; /jM_mrpz
idx_neg = m<0; _BbvhWN&+
9TC)
w|
z = y; q]CeD
if any(idx_pos) +~N!9eMc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); uQnT[\k?
end C0QM#"[
if any(idx_neg) HmMO*k<6@
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Or7
mD
end O5zE {#
u"`*DFjo*
% EOF zernfun