Early Approaches toNon-Rigid Reconstruction - Deformable Surface 3D Reconstruction from Monocular Images - page 8

2.1.3 PHYSICS-BASEDMETHODS FOR COMPUTERVISION
As they became very popular in Computer Graphics for simulation and animation purposes,
physics-based models also gained acceptance in Computer Vision for non-rigid motion analy-
sis Kambhamettu et al. [ 1994 ]. In both fields, their main purpose was to restrict the potential defor-
mations of an object to plausible ones only. However, in Computer Graphics where the simulation
results have to look realistic, physical accuracy, or at least plausibility, is more important than in Com-
puter Vision. There, the main concern is quality of fit to image data and robustness to erroneous
measurements. The role of the model is that of a regularizer that turns the model fitting process into
one that is easier to perform.
The original Snakes Kass et al. [ 1988 ] are a good example of this. The external energy that
serves as a regularizer is written as a quadratic function that approximates the sum of the square of the
curvatures along the surface, which itself is an approximation of the true elastic deformation energy.
The fact that it is not a particularly accurate approximation of the true energy is more than made up
by the fact that it can be expressed in quadratic form, thereby allowing a very effective semi-implicit
optimization scheme. The same formulation was later extended to 2D shape recovery Pilet et al.
[ 2008 ] and 3D surface modeling from stereo Fua and Leclerc [ 1995 ] using triangulated meshes.
Many other variations of the physics-based models have been proposed since to reconstruct
surfaces from images. In the medical imaging domain, balloon forces Cohen and Cohen [ 1993 ]
were introduced to make the surface expand from its initial state so that it could be started
from inside the object to be outlined. Deformable superquadrics Metaxas and Terzopoulos [ 1993 ],
Terzopoulos and Metaxas [ 1991 ] were proposed to reconstruct more complex shapes by modeling
both global and local deformations. Finally, in McInerney and Terzopoulos [ 1993 , 1995 ], the FEM
formulation was followed more closely, and a deformable surface was modeled as a thin-plate un-
der tension. More recently, the use of the Boundary Element Method has also been advocated to
track deformable objects in 2D Greminger and Nelson [ 2003 ] and in 3D Greminger and Nelson
[ 2008 ]. Comparisons of these different FEM formulations are available both specifically for medical
imaging McInerney and Terzopoulos [ 1996 ] and in a broader context Montagnat et al. [ 2001 ].
There has been some interest in more accurate modeling of the true physics of deformable
objects via the nonlinear finite element method in Computer Vision. However, unlike in Computer
Graphics where one can tune the forces and material parameters until satisfactory deformations are
produced, recovering surface shape by fitting a model to the image data requires these parameters to
be fixed during the optimization process. Some approaches that rely on sophisticated models have
nonetheless been proposed for fitting a mesh to 3D range data Huang et al. [ 1995 ], Jojic and Huang
[ 1997 ], Tsap et al. [ 1998 ] and for video-based shape recovery Bhat et al. [ 2003 ], Tsap et al. [ 2000 ].
They involve an analysis-by-synthesis approach and a more-or-less exhaustive search through the
parameter space until those that yield the best fit are found. Recently a nonlinear FEM formula-
tion Ilic and Fua [ 2007 ] has been proposed to recover the deformations of beam structures in the
image plane, where image features act as forces, as in the original Snakes Kass et al. [ 1988 ]. To the