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A METHOD FOR ESTIMATING 3D SHAPES MOTION BY A FREQUENCY DOMAIN TECHNIQUE 
G.M. Cortelazzo A. Guamieri 
Dipartimento di Elettronica ed Informatica Dipartimento del Territorio - TESAF 
Università degli Studi di Padova - Italy Università degli Studi di Padova - Italy 
e-mail: corte@dei.unipd.it 
A. Vettore 
Dipartimento del Territorio - TESAF 
Università degli Studi di Padova - Italy - 
e-mail: vettoan@uxl.unipd.it 
Commission VI, Working group 3 
ABSTRACT 
Free-form 3-D surfaces registration is a fundamental problem in 3-D imaging, tipically approached by extensions or variations of the 
ICP algorithm [1, 2]. This work suggestes an alternative procedure for 3-D motion estimation based on the Fourier transform of the 
3-D intensity function, implicitly described by the registered time-sequences of range data. Similar to the frequency domain techni 
ques for estimating motion parameters, this is a non feature-based method suitable for unsupervised registration of 3-D views. This 
method has several advantages related to the fact that it uses the total avalaible information and not sets of features. With respect to 
memory occupancy, the use of a time-sequence of a 3-D intensity function represents a considerable data reduction with respect to a 
pair of time-sequences of 2-D functions. The proposed technique, wich extends to the 3-D case previous frequency domain estima 
tion algorithms developed for planar case, retain their robustness. 
1. Problem statement 
Let /,(x), xeR J , be a 3-D object and let / 2 (x) be a rigidly rotated 
version of /,(x) (these data can be obtained from registered 
range and intensity data captured at different times). It can be 
shown that /,(x) and / 2 (x) relate as: 
l 2 (x) = /, (R~X -1) where 1 ReS0(3) , teR J (1) 
According to (1) / 2 (x) is first translated by the vector t and then 
rotated by the matrix R. 
Denote as 
and (5) can be used in order to determine R. Therefore in the 
frequency domain the estimation of R and t can be decoupled 
andone can estimate first R from (5) and t from (3), according to 
the follow'ing two passes procedure. 
2. Estimation of the 3-D Rotational Matrix R 
Write the 3-D rotational matrix R as follow [3] 
R = R(co,e)=e" 9 (6) 
i,(k) = F[l, (*)| k]= JJJ ~l,(x)e- J ** T ‘dx (2) 
with k=[k x , k y , k 2 ] T 
the 3-D Fourier transform of /¡(x), i= 1,2; it is straightforward to 
prove that the two transforms are related as 
L 2 (k) = L { (R- [ k)e- J2l * TRl (3) 
From (3) one sees that the translation t affetcs only phases and 
not magnitudes. However, if the 3-D surfaces were modeled by 
impulsive functions supported on them, their Fourier transforms 
would have a lot of spurious high frequency content not suited 
to the frequency domain techniques to be presented. For this 
reason we preliminarily build 3-D solids from the range data 
defined by the 3-D surfaces and we apply the proposed techni 
que to them. 
Let S,(x) and S 2 (x), xeR 3 , be the range data defining the sur 
faces of a rigid body moving in R J , at times t, and t 2 respecti 
vely. From Si(x) and S 2 (x) define /|(x) and / 2 (x) as: 
/ i( x)=jl if x&Int(S i (*)) i=l 2 (4) 
[0 elsewhere 
where Int(5j(x)) denotes the interior of the surface S^x). 
Magnitudes are related as 
|i,(k)| = |l,(«-'k)| (5) 
1 
S0(3) is the group of the 3x3 special orthogonal matrices 
where 
-CO 
CO 
0 
E so(3) 2 
(7) 
co = [co x , co y , co z ] T E R 3 , is a unit vector determining the rota 
tional axis, CO is a skew-symmetric matrix obtained from the 
vector to and 0 E R is the rotational angle in radians. 
Define the difference function A(k) between tyransforms as 
A(k) = A(k r ,k v ,k.) = 
lAOOl 
Kool 
|Mk)| 
1,(0) 
¿2«» 
¿,(0) ¿,(0) 
It can be proved [3] that R, as rotational matrix, has eigenvalues 
X, = 1, X 2 = e jQ and X 3 = e 70 . Call co the eigenvector corre 
sponding to X] = 1 . The vector co is a solution of the equation 
A(k) = 0, indeed A(k) = 0 if R' ] k = k or equivalent /?k = k . In 
other words the locus A(k) = 0 includes a line through co. 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 co and then the angle 0 by means of the 
following procedure: 
2 so(3)={S E R jx3 : S T = - S} is the space over reals of the 3x3 
skew-symmetric matrices.The map so(3) —> S0(3) is suriective.