Kronecker-Markov Prior for Dynamic 3D Reconstruction

Recovering dynamic 3D structures from 2D image observations is highly under-constrained because of projection and missing data, motivating the use of strong priors to constrain shape deformation. In this paper, we empirically show that the spatiotemporal covariance of natural deformations is dominated by a Kronecker pattern. We demonstrate that this pattern arises as the limit of a spatiotemporal autoregressive process, and derive a Kronecker Markov Random Field as a prior distribution over dynamic structures. This distribution unifies shape and trajectory models of prior art and has the individual models as its marginals. The key assumption of the Kronecker MRF is that the spatiotemporal covariance is separable into the product of a temporal and a shape covariance, and can therefore be modeled using the matrix normal distribution. Analysis on motion capture data validates that this distribution is an accurate approximation with significantly fewer free parameters. Using the trace-norm, we present a convex method to estimate missing data from a single sequence when the marginal shape distribution is unknown. The Kronecker-Markov distribution, fit to a single sequence, outperforms state-of-the-art methods at inferring missing 3D data, and additionally provides covariance estimates of the uncertainty.