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 ℓ
1
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 ℓ
1
and ℓ
2
optimization.