Image Processing Reference

In-Depth Information

Figure 4.13:
Repetitive texture.
Top Row
The established correspondences between the reference and

the target image, reconstructed 3D mesh reprojected into the target image, and the same mesh seen from

a different viewpoint for the method of
Salzmann and Fua
[
2011
].
BottomRow
Similar outputs for the

method of
Shaji
et al.
[
2010
]. © 2010 IEEE.

than one point in the input image. This amounts to adding new lines in the matrix
M
of Eq.
4.1
and

to introducing indicator variables that encode which ones of these correspondences are truly active.

The quadratic problem of
Salzmann and Fua
[
2011
] becomes a mixed integer quadratic problem,

which is NP-hard. Nevertheless, a branch-and-bound strategy was shown to yield good approxi-

mate solutions
Shaji
et al.
[
2010
], at the cost of increased computational complexity with respect

to
Salzmann and Fua
[
2011
]. In
Sanchez-Riera
et al.
[
2010
], correspondences are also established

simultaneously as the shape is recovered. In that case, given a shape prior modeled as a mixture of

Gaussians, a strategy based on Kalman filtering is employed to progressively reduce the number of

2D point candidates that can be matched to a 3D point.

4.2.3 DISTANCE CONSTRAINTS

As discussed in Section
4.2.2
, whether enforced locally or globally, smoothness by itself does not

suffice to make the 3D monocular surface reconstruction problem well-posed and to guarantee

a unique solution. Additional constraints are required. Enforcing distances across the deforming

surface to be preserved has proved an effective way of disambiguating shape recovery.

In
Salzmann
et al.
[
2008a
], reconstruction was performed under a global linear subspace

model. The modal weights
c
were expressed as the weighted sum of Eq.
4.9
and the weights
β
i

became the unknowns of the problem. Overcoming the ambiguities left by the smoothness con-

straints was done by choosing the weights
β
i
that result in a surface in which the Euclidean distances

between neighboring vertices remain as similar as possible to their value in the reference configura-

tion. These constraints can be expressed as

2
=
l
i,j
,

∀
(i, j)
∈
E

v
i
−

v
j

,

(4.15)

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