  In this paper, we address the problem of exact recovery condition in
retrieving 3D human motion from 2D landmark motion. We use a skeletal kinematic
model to represent the 3D human motion as a vector of angular articulation
motion. We address this problem based on the observation that at high tracking
rate, regardless of the global rigid motion, only few angular articulations
have non-zero motion. We propose a first ideal formulation with $\ell_0$-norm
to minimize the cardinal of non-zero angular articulation motion given an
equality constraint on the time-differentiation of the reprojection error. The
second relaxed formulation relies on an $\ell_1$-norm to minimize the sum of
absolute values of the angular articulation motion. This formulation has the
advantage of being able to provide 3D motion even in the under-determined case
when twice the number of 2D landmarks is smaller than the number of angular
articulations. We define a specific property which is the Projective Kinematic
Space Property (PKSP) that takes into account the reprojection constraint and
the kinematic model. We prove that for the relaxed formulation we are able to
recover the exact 3D human motion from 2D landmarks if and only if the PKSP
property is verified. We further demonstrate that solving the relaxed
formulation provides the same ground-truth solution as the ideal formulation if
and only if the PKSP condition is filled. Results with simulated sparse
skeletal angular motion show the ability of the proposed method to recover
exact location of angular motion. We provide results on publicly available real
data (HUMAN3.6M, PANOPTIC and MPI-I3DHP).
