Geometric, Variational Integrators for Computer Animation

We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important computational tool at the core of most physics-based animation techniques. Several features make this particular time integrator highly desirable for computer animation: it numerically preserves important invariants, such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the implementation of the method. These properties are achieved using a discrete form of a general variational principle called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate the applicability of our integrators to the simulation of non-linear elasticity with implementation details

Smooth 2D Coordinate Systems on Discrete Surfaces

International audienceWe introduce a new method to compute conformal param- eterizations using a recent definition of discrete conformity, and estab- lish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.Comment: 26 pages; published online in Foundations of Computational Mathematics (2011

Pose normalization for eye gaze estimation and facial attribute description from still images

Our goal is to obtain an eye gaze estimation and a face description based on attributes (e.g. glasses, beard or thick lips) from still images. An attribute-based face description reflects human vocabulary and is therefore adequate as face description. Head pose and eye gaze play an important role in human interaction and are a key element to extract interaction information from still images. Pose variation is a major challenge when analyzing them. Most current approaches for facial image analysis are not explicitly pose-invariant. To obtain a pose-invariant representation, we have to account the three dimensional nature of a face. A 3D Morphable Model (3DMM) of faces is used to obtain a dense3D reconstruction of the face in the image. This Analysis-by-Synthesis approach provides model parameters which contain an explicit face description and a dense model to image correspondence. However, the fit is restricted to the model space and cannot explain all variations. Our model only contains straight gaze directions and lacks high detail textural features. To overcome this limitations, we use the obtained correspondence in a discriminative approach. The dense correspondence is used to extract a pose-normalized version of the input image. The warped image contains all information from the orignal image and preserves gaze and detailed textural information. On the pose-normalized representation we train a regression function to obtain gaze estimation and attribute description. We provide results for pose-invariant gaze estimation on still images on the UUlm Head Pose and Gaze Database and attribute description on the Multi-PIE database. To the best of our knowledge, this is the first pose-invariant approach to estimate gaze from unconstrained still images