Image Processing Reference
taken into account in the computation of e ( x ) and its Jacobian A . Shrinkage to a trivial solution
can then be prevented by replacing the matrix of Eq. 4.19 by a stiffness matrix chosen so that
the regularization term
x 0 )
approximates the sum of the squares of second derivatives
of the vector ( x
x o ) , such as the one used in Fua and Leclerc [ 1995 ]. This term both penalizes
non-smooth deformations and prevents scaling. Note that such a stiffness matrix also defines local
geometric constraints, since it only encodes links between neighboring mesh vertices.
The constrained least-squares minimization method is effective and fast, but, due to its iterative
nature and to the specific formulation of the regularization term, it requires an initial shape estimate
that is not too different from the desired result. It is therefore well adapted either in a frame-to-
frame tracking context, or to surfaces that deform relatively little so that the reference shape can be
used to initialize the computation. For situation where a single image of a surface undergoing large
deformations is given as input, it was recently shown that an initialization to this problem can be
computed with a discriminative predictor Salzmann and Urtasun [ 2010 ].
The many different shape regularizers and constraints that have been discussed in this chapter
have made it possible to design effective algorithms for monocular non-rigid template-based recon-
struction. In particular, local smoothness used in conjunction with inequality constraints has proved
able to recover the shape of surfaces undergoing complex deformations with folds and creases. The
major drawback of these techniques arises from the fact that they require a reference image in which
the shape of the surface is known. In the next chapters, we will discuss another class of methods that
do not rely on this assumption.