181
From co and 0 the rotational matrix R(co,0) can be reconstructed Acknowledgement
by means of Rodrigues’ formula [9] as
b) apply a 3-D cartesian phase correlation algorithm [4] on
¿,(k) and £> 2 (k) = F[d 2 (x) | k] = Z,,(k) e ■ j27tKTt , i.e. [2] S. Weik, “Registration of 3-D Partial Surface Models Using
f(9(k) I xl = 6 (x — t) (12) [4] R. Murray, Z. Li and S. Sastry, A Mathema
to Robotic Manipulation, CRC Press, 1994
D views is a natural application of methods for estimating 3-D
rigid rotations and translations.
The proposed algorithm was applied to a bas-relief in Padua:
Fig. 2 shows two partially overlapping range images while Fig.3
shows the resulting composition of range and intensity data of
these images . This object, whose dimension are about 60 x 100
cm, is very articulated since there are many anatomical details
such as faces, arms, hands, etc., in full 3-D relief. The regions of
the range images associated to the same scene were determined
by a manual procedure and the proposed algorithm was used in
order to determine the rotation and the translation between the
taking positions of the range camera. Fig. 4 shows the compo
sition of the range and intensity images of Fig. 3 .
Conclusions
The available results show that the proposed method is suitable
to give unsupervised estimates within 1° degree precision of the
angular parameters. It can be applied to 3-D views registration
in tasks where this precision is adeguate or it can be used in
order to obtain affective starting points for standard feature-
based methods, which as well-known, can give accurate solu
tions once they are properly initiated. The novelty of presented
procedure relies upon the fact that it is a ferquency domain
approach; therefore it uses the global information of the data
and not sets of features. This is probably one of the causes of its
robustness.
The translational vector t can be estimated as follows:
4. Estimation of the 3-D Translational Vector t
R((ù ,\\i ) = e av =1 +(£>sin\\j + co 2 (1 - cosy)
This study has been developed within the project “Definition of
a quality model for 3D digital cartography carried out by digital
photogrammetry, with suitable features for representing urban
buildings, for planning mobile phone networks”. Partly financed
by MURST (Italian Ministry of University and Research) in
1997 as a project of relevant national interest. National coor
dinator: Riccardo Galetto Head of the research unit Antonio
Vettore.
a) De-rotate the image / 2 (x) as
References
d 2 (x) = / 2 (Æk) = /, (x -1)
(10) [1] P.J. Besl and N.D. McKay, “A Method for Registration of
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compute the normalized product between transform
Luminance also Depth Information”, Proc. of International
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Modeling, Ottawa, Canada, May 1997, pp. 93-100.
c) evaluate its inverse Fourier transform
[3] L. Lucchese, G.M. Cortelazzo, A.Vettore, “Estimating 3-D
Rototranslations from Range Data by a Frequency Domain
Technique”, in Optical 3-D Measurement Techniques IV,
A. Gruen / H. Khamen, Ed. Wichmann, Zuerich, September
1997, pp. 444-453 .
The translational vector t can be estimated from the peak of
impulsive function y(x). The method can be efficiently imple
mented by using MD FFT algorithms [4]. The registration of 3-
[5] Anil K. Jain, Fundamental of Digital Image Processing,
Prentice Hall Information and System Sciences
Series,Thomas Kailath, Series Editor, 1989.
