  This paper considers a different aspect of anatomical face modeling:
kinematic modeling of the jaw, i.e., the Temporo-Mandibular Joint (TMJ).
Previous work often relies on simple models of jaw kinematics, even though the
actual physiological behavior of the TMJ is quite complex, allowing not only
for mouth opening, but also for some amount of sideways (lateral) and
front-to-back (protrusion) motions. Fortuitously, the TMJ is the only joint
whose kinematics can be accurately measured with optical methods, because the
bones of the lower and upper jaw are rigidly connected to the lower and upper
teeth. We construct a person-specific jaw kinematic model by asking an actor to
exercise the entire range of motion of the jaw while keeping the lips open so
that the teeth are at least partially visible. This performance is recorded
with three calibrated cameras. We obtain highly accurate 3D models of the teeth
with a standard dental scanner and use these models to reconstruct the rigid
body trajectories of the teeth from the videos (markerless tracking). The
relative rigid transformations samples between the lower and upper teeth are
mapped to the Lie algebra of rigid body motions in order to linearize the
rotational motion. Our main contribution is to fit these samples with a
three-dimensional nonlinear model parameterizing the entire range of motion of
the TMJ. We show that standard Principal Component Analysis (PCA) fails to
capture the nonlinear trajectories of the moving mandible. However, we found
these nonlinearities can be captured with a special modification of autoencoder
neural networks known as Nonlinear PCA. By mapping back to the Lie group of
rigid transformations, we obtain parameterization of the jaw kinematics which
provides an intuitive interface allowing the animators to explore realistic jaw
motions in a user-friendly way.
