180 
1) express A(k x ,k y ,k z ) in spherical coordinates 
tudes can be obtained as follows. Define the cartesian coordi 
nate system 
k P = A* +k y + K 
k.. 
arc tan - 
k a = arccos 
fi 
M 0 
0 < k v <2n 
0 < L <n 
+ k 2 + kt 
y 
(9) 
as A(k p , k 0 , ke); notice that this function can be represented 
only in a hemisphere because of the hermitian symmetry of the 
Fourier transform; 
u 
k X 
u = 
V 
= C T 
by 
w 
k. 
with" 
C = 
- sirup 
cosy 
0 
-COS0 cosy 
- COS0 sirup 
sin& 
sinQ cosy 
sinQ sirup 
cosò 
(12) 
(13) 
2) compute the radial projection of A(k p , k«,, k^) as 
P(K„Ks)= \R(K s ,K„Ks)dK t 
0 
(10) 
3) the angular coordinates of to in spherical representation 
(see Fig. 1) can be found as 
(cp,0) = argmin^ ^ P(K v ,K e ) (11) 
In the new cartesian reference system defeined through(12) and 
(13), equation (5) becomes 
|£ 2 (Cn)|=|£,(/r'Cu)| = 
= |£,(CC "'i? "'Cu)| 
Define for convenience! Lj(u)| = |Lj(Cu)|, i = 1,2, from 
which(14) can be written as 
since, from the inclusion of co within the locus A(k) = 0, 
Pikp ,k$) > 0 with P((p,0) = 0; 
¿2(11) = ^(C-'/T’Cu) = Zi(^w-'(\|/)u) 
(15) 
4) define co the versor of the direction (cp,0). 
The use of radial projection (10) simplifies the 3-D search for a 
line of the locus A(k) = 0 into the minimization of a 2-D fun 
ction which can be solved by standard numerical methods. 
where 
cosy 
R w (y) = C-'RC = 
s/ny 
0 
r(y) 
0 
- sinxp 
cosy 
0 
0~ 
0 
0 1 
(16) 
Matrix R^vy) clearly shows the structure of a rotation by y 
around the co axis as 
p,(u,v)= J Lj(u,v,w)dw i=i,2 
(17) 
Figure 1 Change of coordinate reference systems 
given the structure of R^y), it is easy to prove that the axial 
projections defined in (17) relate as 
3. Estimation of the rotational angle y 
Once the rotational axis has been determined, the estimate of 
the rotational angle y can be conveniently approached in a 
cartesian coordinate system with an axis along co, as shown in 
Fig. 1. The determination of this coordinate system and the 
representation in this system of relationship (5) between magni- 
' u 
= P\ 
f 
r-'(y) 
u ' 
J- v _, 
y\j 
from which y can be obtained resolving a 2-D rotation estimate 
problem. 
J CeSO(3) and then C' 1 = C T