ERCIM “ Alain Bensoussan ” Fellowship

During my fellowship, I was incorporated into the image group on the “Human Motion Imitation” (HUMIM) project. This project consists in proposing a human motion model implying different mathematical techniques. Consequently, this project implies different skills in computer science, human physiology, mechanics and physics, optimisation’s problems, differential geometry and statistics, and requires a deep team effort. Among the different methods used in the human motion modeling, one can quote “inverse kinematics”, “particle filters tracking and statistical learning” and “manifolds learning”. Inverse kinematics is the problem of estimating the position of joints of the human body from the pose of the end-effectors. End-effectors are the ending of the kinematic chains, so endeffectors can be, for instance, the hands and the feet. Inverse kinematics differ from forward kinematics which gives the position of each joint relative to the position of the corresponding parent joint, and for which we compute exactly the position of the end-effectors. In the inverse kinematics, the position of joints are estimated from the position of end-effectors, but the problem being in general ill-posed, the solution is not unique. Despite of this diffficulty, inverse kinematics were a revolution in computer graphic science. Indeed, for instance, in computer animation, using forward kinematics, we move all joints to get a new pose; with inverse kinematics, the new pose is estimated from the new position of each end-effector, and so it is enough to move only the end-effectors. Regarding the ill-posed character of inverse kinematics, it can be solved by adding prior knowledge about human configurations. In the HUMIM project, we are particularly interested in the tracking of human motion and as a future application: physiotherapeutic reeducation. A tracking of motion consists in recognizing automatically a motion from a video. To achieve this, we have to give the relation between the motion that we aim to estimate (hidden states) and the observed video. This relation is a probabilistic relation represented mathematically by p(I|s), where s is the hidden state that we want to estimate (representing the motion’s parameter in our case) and I is the observation (the video sequence). As in the case of inverse kinematics, such a problem is an ill-posed problem. To solve it, we consider a prior knowledge on the hidden state represented by a probability model p(s) called “prior distribution”. Finally, the hidden states