Early Approaches toNon-Rigid Reconstruction - Deformable Surface 3D Reconstruction from Monocular Images - page 14

basis shapes. An alternative solution to modeling this coupling is to represent the motion of all mesh
vertices as a function of a much smaller number of control points. The fine mesh is then obtained
by interpolating the deformation between these control points.
One way to achieve this is through the use of Free-Form Deformations. Originally in-
troduced for animation purposes Sederberg and Parry [ 1986 ], they were quickly adapted to re-
cover shapes from images Delingette et al. [ 1991 ]. Interpolation can be done through Bézier vol-
umes Coquillart [ 1990 ], polynomial curves Welch and Witkin [ 1994 ], or B-splines Eck and Hoppe
[ 1996 ], Faloutsos et al. [ 1997 ], Krishnamurthy and Levoy [ 1996 ]. A disadvantage of standard free-
form deformations is their lack of ability to model local deformations.This was overcome by introduc-
ing Dirichlet Free-Form Deformations, first to animate a hand Moccozet and Magnenat-Thalmann
[ 1997 ], and then for model-fitting purposes Ilic and Fua [ 2002 , 2006 ]. Similarly, RBFs have shown
good ability at modeling local deformations when fitting a surface to 3D data Carr et al. [ 2001 ]. In
that case, the control points act as the centers of the RBFs. A drawback, however, of the control
points based techniques is that there is no automated way to create the appropriate set of control
points.
An alternative to explicitly relying on control points that define the surface shape is to introduce
a multi-resolution approach Hoppe et al. [ 1994 ]. In this case, the deformation of an initial coarse
mesh is computed, and, following a subdivision strategy Catmull and Clark [ 1978 ], Doo and Sabin
[ 1978 ], the mesh and its deformations are then refined. Several subdivision schemes have been
proposed Dyn et al. [ 1990 ], Kobbelt [ 2000 ], Loop [ 1987 ]. Such multi-resolution approaches were
also used with dynamic vertex connectivity Kobbelt et al. [ 2000 ], and for mesh editing Zorin et al.
[ 1997 ]. In the latter, a limitation arose from the fact that the editable regions were defined only in the
initial coarse mesh. Laplacian surfaces Sorkine et al. [ 2004 ], Zhou et al. [ 2005 ] were thus proposed
to overcome this problem. However, to the best of our knowledge, multi-resolution methods have
not been applied in the context of image-based shape recovery. A potential reason might be that the
surface is interpolated, which tends to yield visually pleasing results but may not correspond to what
is observed in the images.

2.4 LEGACY OFTHE PREVIOUS APPROACHES

In the remainder of this survey, we will focus on approaches that tend to be more recent than those
discussed above and do not belong to any of the three categories introduced in the previous sections
of this chapter. Nevertheless, these newer methods build on some of the components of the earlier
ones.

In particular, many methods described below rely on linear subspace models to regularize the
shape of the reconstructed surface. For example, in Chapter 4 , we will study the use of global and
local learned linear models to constrain shape reconstruction from monocular images. In Chapters 5
and 6 , we will show that linear global models have also been extensively applied in the context of
non-rigid structure-from-motion. In this case, the modes are directly estimated from the 2D tracks
of points throughout a video sequence instead of being learned from training data. Representing