PROBLEM STATEMENT Given partial 2D or 3D trajectories of the motion of a uniformly colored bouncing ball, that is viewed by a single or multi- ple cameras, estimate its full 3D state, over time, i.e. location, orientation, an- gular and linear velocities. MOTIVATION Scene understanding can benefit from exploiting the fact that a dynamic scene and its visual observations are invariably determined by the laws of physics. MAIN IDEA • Model the physics of the scene using physics-based simulation • Acquire visual observations • Define an objective function that con- nects the model to the observations • Produce physically plausible interpre- tations of the scene by performing black-box optimization PHYSICS BASED SIMULATION (A) Dynamics of a bouncing ball The bouncing ball is affected by gravity and air resistance while in flight and fric- tion while in bounce with a surface.

(B) Equations of motion We assume standard equations of mo- tion for the flight phase and add air re- sistance. We derive equations for the bounce phase by extending [1]. (C) Simulation of a bouncing ball We define a parameterized ball throwing simulation process S that: • receives a 21-D vector of scene properties and initial conditions • at each point in time, produces a 12-D vector of location, orientation, linear and angular velocities • is implemented by augmenting the Newton Game Dynamics simulator with our physics modeling • performs at 500fps, but is sub-sampled to real acquisition rate (30fps), in order to account for aliasing effects PHYSICALLY PLAUSIBLE SCENE INTERPRETATION We estimate the physically plausible explanation e of the observed scene by formu- lating an optimization problem, where: • the hypothesis space of x is defined over the domain of simulation process S • the observation data o are trajectories of a bouncing ball (potentially partial, 3D or 2D, from single or multiple cameras) • the objective function quantifies the discrepancy between the result of an invocation to S and the observations • the objective function is optimized by means of Differential Evolution [5] CONTRIBUTIONS • First method to consider attributes of state that can only be estimated through physics-based simulation • Extension to existing work [2–4] in exploiting physics based simulation in vision • Proposal of an effective method that is clear, generic, top-down, simulation based • Incorporation of realistic physics • Selected generic and modular components allow for extension to other broader or different contexts EXPERIMENTAL RESULTS (A) Multiview estimation of 3D trajectories (synthetic/real)

(B) Single view estimation of 3D trajectories Finding ball throwing simulations that optimally repro- duce 2D observations.

(C) Seeing the “invisible” Implicit information, like the state of the ball while occluded (left) and the angular components of its 3D state (right), are computer based on a single camera.

KEY REFERENCES [1] P.J. Aston and R. Shail. The Dynamics of a Bouncing Superball with Spin. Dynamical Systems, 22(3):291– 322, 2007. [2] K. Bhat, S. Seitz, J. Popovi´c, and P. Khosla. Com- puting the Physical Parameters of Rigid-body Motion from Video. In ECCV 2002, pages 551–565. Springer, 2002. [3] D.J. Duff, J. Wyatt, and R. Stolkin. Motion Estimation using Physical Simulation. In IEEE International Con- ference on Robotics and Automation (ICRA), pages 1511–1517. IEEE, 2010. [4] D. Metaxas and D. Terzopoulos. Shape and Nonrigid Motion Estimation through Physics-based Synthesis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(6):580–591, 1993. [5] R. Storn and K. Price. Differential Evolution–A Sim- ple and Efficient Heuristic for Global Optimization over Continuous Spaces. Journal of Global Optimization, 11(4):341–359, 1997. MORE INFORMATION For more information, visit http://www.ics.forth.gr/ kyriazis/?e=1 or contact {kyriazis,oikonom,argyros}@ics.forth.gr This work was partially supported by the IST-FP7-IP-215821 project GRASP