3D reconstruction of a smooth articulated trajectory from a monocular image sequence

An articulated trajectory is defined as a trajectory that remains at a fixed distance with respect to a parent trajectory. In this paper, we present a method to reconstruct an articulated trajectory in three dimensions given the two dimensional projection of the articulated trajectory, the 3D parent trajectory, and the camera pose at each time instant. This is a core challenge in reconstructing the 3D motion of articulated structures such as the human body because endpoints of each limb form articulated trajectories. We simultaneously apply activity-independent spatial and temporal constraints, in the form of fixed 3D distance to the parent trajectory and smooth 3D motion. There exist two solutions that satisfy each instantaneous 2D projection and articulation constraint (a ray intersects a sphere at up to two locations) and we show that resolving this ambiguity by enforcing smoothness is equivalent to solving a binary quadratic programming problem. A geometric analysis of the reconstruction of articulated trajectories is also presented and a measure of the reconstructibility of an articulated trajectory is proposed.