PerformingTemplate-Based Reconstruction - Deformable Surface 3D Reconstruction from Monocular Images - page 45

Figure 4.15: Demonstrating why constant Euclidean length constraints are ill-suited for sharp folds.
Left: Two points of the discrete representation of a continuous surface in its rest configuration. Right:
When the surface deforms, while the geodesic distance between the two points is preserved, the Euclidean
one decreases. This suggests that distance inequality constraints should be used rather than equalities.

While preserving Euclidean distances has proved effective, it remains an approximation of the
true physical behavior: What is truly preserved on a deforming inextensible surface is the geodesic ,
as opposed to Euclidean, distance between points. To be most effective, the techniques proposed
in Ecker et al. [ 2008 ], Perriollat et al. [ 2010 ] therefore require relatively evenly placed feature points
that can be detected and whose distance from each other is relatively small so that the Euclidean
distance is a reasonable approximation of the geodesic one. This also true of the Salzmann et al.
[ 2008a ] approach, but this requirement is more readily satisfied since the distances constrained are
those between neighboring mesh vertices, independently of the surface texture. As long as inter-
vertex distances remain reasonably small with respect to the local radius of curvature, the requirement
will be met.
As illustrated by Fig. 4.15 , when creases develop on an inextensible surface, the Euclidean
distance between vertices of the mesh representing it may decrease. It is the geodesic distance that
remains constant and, in effect, bounds the Euclidean one. In Salzmann and Fua [ 2011 ], it was
therefore proposed to replace the constant distance constraints of Ecker et al. [ 2008 ], Perriollat et al.
[ 2010 ], Salzmann et al. [ 2008a ] by inequality constraints that force the distance between neighboring
vertices to remain smaller than their geodesic distance, which can be computed in the reference image.
Because of scale ambiguities, these inequality constraints by themselves do not sufficiently constrain
the solution as they do not prevent the mesh from globally shrinking. This is handled by adding a
balloon force not unlike the one proposed in Cohen and Cohen [ 1993 ] that pushes the mesh away
from the camera as far as possible without violating any of the constraints. All these constraints
and forces can be expressed directly in terms of the mesh vertex coordinates, which results in an
optimization problem of the form

x 0 ) 2 − λ d x T d minimize

X

Mx 2 + ( x − (4.16)

subject to v k − v j ≤ l j,k , ∀ (j, k) ∈ E ,

where M is the matrix of Eq. 4.1 , is a matrix that groups the regularization term of Eq. 4.14 for
all patches, and d is the vector that encodes the balloon forces, which amount to maximizing the
depth of surface points. w d is a weight that controls the influence of the balloon forces relative to
the magnitude of reprojection errors. This is a convex minimization problem that can be efficiently