Non-Rigid Structure from Motion: Prior-Free Factorization Method Revisited

A simple prior free factorization algorithm \cite{dai2014simple} is quite often cited work in the field of Non-Rigid Structure from Motion (NRSfM). The benefit of this work lies in its simplicity of implementation, strong theoretical justification to the motion and structure estimation, and its invincible originality. Despite this, the prevailing view is, that it performs exceedingly inferior to other methods on several benchmark datasets \cite{jensen2018benchmark,akhter2009nonrigid}. However, our subtle investigation provides some empirical statistics which made us think against such views. The statistical results we obtained supersedes Dai {\it{et al.}}\cite{dai2014simple} originally reported results on the benchmark datasets by a significant margin under some elementary changes in their core algorithmic idea \cite{dai2014simple}. Now, these results not only exposes some unrevealed areas for research in NRSfM but also give rise to new mathematical challenges for NRSfM researchers. We argue that by \textbf{properly} utilizing the well-established assumptions about a non-rigidly deforming shape i.e, it deforms smoothly over frames \cite{rabaud2008re} and it spans a low-rank space, the simple prior-free idea can provide results which is comparable to the best available algorithms. In this paper, we explore some of the hidden intricacies missed by Dai {\it{et. al.}} work \cite{dai2014simple} and how some elementary measures and modifications can enhance its performance, as high as approx. 18\% on the benchmark dataset. The improved performance is justified and empirically verified by extensive experiments on several datasets. We believe our work has both practical and theoretical importance for the development of better NRSfM algorithms.