Image Processing Reference
Early Approaches to Non-Rigid
Modeling the behavior of non-rigid surfaces has been an active area of research for the past twenty
years. Many approaches have been proposed in the context of both Computer Vision and Computer
Graphics. These two fields are closely related, since Computer Vision aims at solving the inverse
problem of Computer Graphics, that is recovering the shape of real objects as opposed to simulating
the deformations of virtual ones. It is therefore not surprising that similar representations often
appear in both domains.
Throughout the years, approaches to non-rigid surface reconstruction have relied on many dif-
ferent techniques to represent and constrain surface deformations. These techniques can be roughly
classified into those that are physics-based, rely on statistical learning methods, or parameterize the
shape to implicitly regularize its deformations.
Some of them have proved very successful for their intended purposes but not necessarily for
generic monocular 3D surface reconstruction. In this chapter, we briefly review these techniques.
We discuss their strengths, that the more recent methods described in the remainder of this survey
exploit, and their weaknesses that these same methods attempt to correct.
In both the Computer Vision and Computer Graphics fields, most early approaches to modeling
deformations of non-rigid objects were inspired by Mechanical Engineering concepts. The key idea
was to model the behavior of an object by describing the true physical laws that govern it. A seminal
work in this field Kass et al. [ 1988 ] advocated using this approach to delineate 2D image shapes and
was quickly extended to 3D modeling Terzopoulos et al. [ 1987 , 1988 ]. In the proposed formalism, a
global energy, written as the sum of an internal one and an external one, is minimized. The internal
energy derives from physical surface properties and typically acts as a regularizer that enforces global
smoothness. It is often taken to be quadratic to convexify the minimization problem and make its
resolution simpler. The external energy encodes the image information and allows image features to
act as attraction forces that tend to deform the surface to make it conform to these features.
The formulation introduced in Kass et al. [ 1988 ] and many of the subsequent methods Fua
[ 1996 ] are directly inspired by Mechanical Engineering techniques, especially the Finite Element
Method (FEM) Bathe [ 1982 ], Zienkiewicz [ 1989 ]. In the remainder of this section, we briefly
introduce FEM. We then discuss why it is too complex to be used in its complete form in either