GUARNIERI, ALBERTO 
Fig. 3 shows an example of functions /,(x) and l(x), related to Apollo's head, located at Padua University Collection. 
  
Fig. 3: Example of image data /,(x) and its corresponding rotated version (x) in 3D space. 
4 ESTIMATION OF THE 3-D ROTATIONAL MATRIX R 
Write the 3-D rotational matrix R as follow [6] 
R = R(d,0)= e“° (22) 
where 
0" -0o 4, 
Q | v, 0 -o,| € so(3) (23) 
=, €. 0 
= [a, , 6, , 0, lem , is a unit vector determining the rota-tional axis, © is a skew-symmetric matrix obtained from 
the vector ® and 0€ R is the rotational angle in radians. Define the difference function A(k) between tyransforms as 
AK) = Ak kk CAC [nao AGED (24) 
Le LO] [20 Lo | 
  
It can be proved [6] that R, as rotational matrix, has eigenvalues A; = 1, A, = e”° and s 7 e 7? . Call o the eigenvector 
corresponding to A; = 1. The vector © is a solution of the equation A(k) = 0, indeed A(k) = 0 if R'k =k or equivalent 
Rk = k . In other words the locus A(k) = 0 includes a line through ®. For objects without special symmetries (as natural 
objects typically are) this property of the function A(k) can be exploited in order to determine the versor c and then the 
angle 0 by means of the following procedure: 
1) express A(k,,k,,k;) in spherical coordinates 
k, = Jk} +k +k’ k,20 
k 
k, = arctan— 0sk <27 
k 
Xx 
[ 
Q5) 
k, - arccos—  0<k, <a 
JE +R +R 
as A(kp» ke, kg); notice that this function can be represented only in a hemisphere because of the hermitian symmetry of 
the Fourier transform; 
  
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 323