Automatic Estimation of the Number of Deformation Modes in Non-rigid SfM with Missing Data

This paper proposes a new algorithm to estimate automatically the number of deformation modes needed to describe a non-rigid object with the well-known low-rank shape model, focusing on the missing data case. The 3D shape is assumed to deform as a linear combination of K rigid shape bases according to time varying coefficients. One of the requirements of this formulation is that the number of bases must be known in advance. Most non-rigid structure from motion (NRSfM) approaches based on this model determine the value of K empirically. Our proposed approach is based on the analysis of the frequency spectra of the x and y coordinates corresponding to the individual image trajectories, which are seen as 1D signals. The frequency content of the 2D trajectories is encoded using the modulus of the Discrete Cosine Transform (DCT) of the signals. Our hypothesis is that the value of K that gives the best prediction of the missing data also provides the best 3D reconstruction. Our proposed approach does not assume any prior knowledge and is independent of the 3D reconstruction algorithm used. We validate our approach with experiments on synthetic and real sequences.