International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Figure 2: First row contains three 2.5D views of the robot.
The second row contains the complete 3D model viewed
from different angles.
the right foot of the robot and left hand of the bunny) were
already present in the individual views (due to the presence
of noise in the acquisition phase), and are not due to error
in our technique. Such parts would be removed during the
reconstruction phase.
5 ANALYSIS
Qualitative analysis of the resultant 3D models was per-
formed by visual inspection of the registered surfaces. The
features on the registered 3D models were compared with
the features visible in their pictures (taken from the view-
ing angles). Silhouettes of the 3D models were compared
with its picture taken from a similar pose to find errors in
geometry. Our qualitative analysis show that the 3D mod-
els resulting from our technique are very accurate. We
could only observe small seams, present between some
parts of the surfaces, whose magnitude was within the mesh
resolution. These small seams are present due to noise and
variations in the surface sampling during the acquisition
phase and are unavoidable. The integration and reconstruc-
tion phases of the 3D modeling removes these seams by
approximating the data by a single smooth surface.
For quantitative analysis, the ground truth data must be
available. In our case, since the ground truth was not avail-
able, we took a different approach to perform quantitative
analysis. We have compared the transformations result-
ing from the pairwise registration with the transformations
resulting from the global registration. Since global regis-
tration distributes the error present in pairwise correspon-
dences evenly over all the views of the 3D model, the dif-
ference between the pairwise and global transformations
should give us an estimate of the error that was present in
the pairwise correspondence/registration.
The comparison was performed as follows. View 1 of the
objects was taken as a reference in each case. The rota-
tion matrices of each view n resulting from pairwise regis-
tration Rp, and global registration Eug, were calculated.
Next the amount of rotational difference 0,, present in the
two rotation matrices was calculated according to Eqn. 3
and Eqn. 4.
Ran = RpRg (3)
race E 18
à eos cs) X 180 (4)
3 T
In Eqn. 3, Ra, is a rotation matrix representing the dif-
ference between Ry, and Rg,. Eqn. 4 is derived from
Rodrigue's formula. 0,, represents the amount of rotation
error (about a single axis) present in the rotation matrices
of pairwise registration and global registration. The dif-
ference t,, between the translation vectors of each view n
resulting from pairwise registration tp, and global regis-
tration tg, is calculated according to Eqn. 5.
[ton TEX tnl ;
HS Ee (5)
mesh resolution
In Eqn. 5, the difference between the translation vectors is
normalized with respect to the mesh resolution in order to
make it scale-independent. In our experiments the mesh
resolution of the bunny was twice the mesh resolution of
the robot.
Fig. 4(a) shows the 0, and Fig. 4(b) shows the £,, for all
the views of the bunny. Similarly Fig. 5 shows the 0, and
t, for all the views of the robot. 0,, and t, for view | of
the bunny and the robot are zero because view 1 is taken
as the reference view. In the case of the bunny (Fig. 4),
view 4 has the maximum difference in rotation (1.2?) and
translation (0.9.mesh resolution). This is because view 4
is at one end of the pairwise correspondence chain (see
Fig. 3(a)). The overlap information used by the pairwise
correspondence and registration algorithm is shown by the
graph of Fig. 3. Each node represents a view and an arc
represents an overlap. The dotted arcs represent overlaps
that were not used by the pairwise registration (since pair-
wise registration requires a spanning tree graph). Note that
our technique considers all possible overlaps for the global
registration and not just the ones given in Fig. 3. The over-
all difference between the rotation and translation result-
ing from our pairwise registration and global registration
is very small (see Fig. 4 and Fig. 5). In the case of the
bunny, the average 0, is equal to 0.34° and the average tn
is equal to 0.24 mesh resolutions. In the case of the robot,
the average 0,, is equal to 0.30° and the average ?,, Is equal
to 0.20 mesh resolutions. In other words since there was
very small error in the pairwise registration, the global reg-
istration algorithm had to distribute a very small amount of
error. Since the pairwise registration was derived from the
pairwise correspondence, the corollary is that the corre-
spondence algorithm is accurate. Had the correspondences
International Archi
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Figure 3: Graphs
of the views of t
pairwise correspc
aa a que
(a) View Num
Figure 4: The dif
ized translation (1
pair wise and glo
been inaccurate tl
tion would have t
accumulated betv
of Fig. 3 and hen
distribute these lz
ferences between
6 CONCLUSI!
We have presente
using our automa
with global regist
and only assumes
the views which i:
acquisition. We |
titative analysis o
performed by visi
els. The quantitati
the results of pait
tration results. In
nique to be able to
set of views.
ACKNOWLEDC
We would like to
Mellon University
in our experimen
grant number DP(
REFERENCES
Ashbrook, A., Fi:
1998. Finding Su
