Trajectory triangulation: 3D motion reconstruction with ℓ1 optimization
In this paper, we first explain the formulation of the trajectory triangulation: 3D reconstruction of a moving point from a series of 2D projections. The system has to be overconstrained to be solved by least squares techniques. We take advantage of the sparseness of real-world motions in the transformed domain, and borrow the concept of compressive sampling to reformulate the problem with ℓ1 optimization so that it is possible to reconstruct the trajectory even in an underconstrained system. Thus, fewer measurements are needed to reconstruct a 3D trajectory of even larger bandwidth coverage. We conduct experiments on both synthetic and real-world motion data to verify our proposed method, and compare the reconstruction results based on ℓ1 and ℓ2 optimization.