GUARNIERI, ALBERTO 
  
IL (u) =| (CR Ca) = Zi (Rw^ (y)u) (31) 
where 
cosy -—siny O0 ry) 
R,(y)=C'RC=|siny cosy 0|= 0 (32) 
0 0 1 0 9 
Matrix R,(y) clearly shows the structure of a rotation by y around the c axis as 
DULV) = Eo dw i=12 (33) 
given the structure of Ra(), it is easy to prove that the axial projections defined in (33) relate as 
A. or (34) 
from which y can be obtained resolving a 2-D rotation estimate problem. From ® and 0 the rotational matrix R(w,6) 
can be reconstructed by means of Rodrigues’ formula as 
R(Ó,y) - e^" 2 I c Ósiny 4 à? (1— cosy) (35) 
5 ESTIMATION OF THE 3-D TRANSLATIONAL VECTOR t 
The translational vector t can be estimated as follows: 
a) De-rotate the image L(x) as 
dy (x) = 1, (Rk) = 4, (x - t) (36) 
b) apply a 3-D cartesian phase correlation algorithm [9] on Zi(k) and D;(k) ^ F[dx(x) | k] = Le *", ie. 
compute the normalized product between transform 
L; (k)D, (k) —g 
Ok) = = (37) 
IZ, (k)D, (k)| 
c) evaluate its inverse Fourier transform 
q(x) = F"[O(k) |x]= 8(x-t) (38) 
The translational vector t can be estimated from the peak of impulsive function (x). The method can be efficiently 
implemented by using MD FFT algorithms [9]. The registration of 3-D views is a natural application of methods for 
estimating 3-D rigid rotations and translations. 
CONCLUSIONS 
This work presents an extension to the rotational case of the well known translational phase correlation Robustness and 
accuracy of the proposed algorithm have been tested and experimental evidence of these characteristics is reported. The 
proposed algorithm retains the good properties typical of the translational phase correlation. The only drawback of the 
2D method is that it requires cartesian to polar coordinates conversion which is a numerically delicate and computa- 
tionally intensive operation. This drawback can be bypassed by resorting to an alternative algorithm for estimating 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 325