Image Processing Reference

In-Depth Information

Figure 6.16:
Reconstruction of a piece of paper being torn apart
Taylor
et al.
[
2010
]. Courtesy of A.

Jepson. © 2010 IEEE.

In
Fayad
et al.
[
2010
], the authors extended this approach to allow for more complex deformations

of the individual patches, given 2D points tracked across a video sequence. Instead of imposing

local planarity, local deformation are regularized using the quadratic deformation model discussed

in Section
6.3.1 Fayad
et al.
[
2009
]. Each overlapping group of tracked 2D points is reconstructed

independently using the quadratic deformation model of Eq.
6.19
. As in the global case, this requires

a rest shape, which can be obtained from the first few initial frames of the sequence using a rigid

structure-from-motion technique. As in
Va ro l
et al.
[
2009
], the scale ambiguity of each patch, is

resolved by exploiting the overlap between the patches. The top two rows of Fig.
6.15
depict results

obtained on real images of a piece of paper undergoing large deformations. The bottom row features

a comparison of this approach with
Va ro l
et al.
[
2009
]. Note that the reconstructions of
Fayad
et al.

[
2009
] are more accurate than those of
Va ro l
et al.
[
2009
]. This seems reasonable both because the

method of
Fayad
et al.
[
2009
] exploits the whole sequence instead of just two images, and because

allowing the patches to deform is a better approximation of the observed phenomenon.

The two methods described above implicitly assume some amount of smoothness in the local

deformations. By contrast, the method of
Taylor
et al.
[
2010
] relies exclusively on the preservation

of local Euclidean distances between feature points found on the surface, much as the
Ecker
et al.

[
2008
],
Perriollat
et al.
[
2010
] methods introduced in Section
4.2.2.1
. Note, however, that in the

NRSFM framework, the true distances between pairs of points are unknown. To overcome this

problem, triplets of neighboring points that move rigidly are identified and the global shape re-

constructed as a soup of triangles whose vertices remain at a fixed distance from each other. More

specifically, under orthographic projection, the 3D length of an edge between points
q
i
1

and
q
i
2

is

related to the length of its projection in the image plane by

2

2

2
=
(d
i
1
−
d
i
2
)
2
,

q
i
1
−

q
i
2

2
−

p
i
1
−

p
i
2

(6.22)

where
p
i
is the 2D projection of point
i
, and
d
i
is its depth. Furthermore, the sum of pairwise depth

differences within a single triangle is always equal to zero, which can be written as

(d
2
−
d
1
)
+
(d
3
−
d
2
)
+
(d
1
−
d
3
)
=

0
.

(6.23)

Search WWH ::

Custom Search