PerformingTemplate-Based Reconstruction - Deformable Surface 3D Reconstruction from Monocular Images - page 43

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)