Nonrigid Structure-from-Motion: Estimating Shape and Motion with Hierarchical Priors

Abstract

This paper describes methods for recovering time-varying shape and motion of non-rigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. Time-varying shape is modeled as a rigid transformation combined with a non-rigid deformation. Reconstruction is ill-posed if arbitrary deformations are allowed, and thus additional assumptions about deformations are required. We first suggest restricting shapes to lie within a low-dimensional subspace, and describe estimation algorithms. However, this restriction alone is insufficient to constrain reconstruction. To address these problems, we propose a reconstruction method using a Probabilistic Principal Components Analysis (PPCA) shape model, and an estimation algorithm that simultaneously estimates 3D shape and motion for each instant, learns the PPCA model parameters, and robustly fills-in missing data points. We then extend the model to model temporal dynamics in object shape, allowing the algorithm to robustly handle severe cases of missing data.

Decomposition and dictionary learning for 3D trajectories

“…In this case, the physical signification of the orthogonal matrix R ∈ R D×D is obviously lost. The extension, which can be called nDRI-MP [37], modifies only the registration (steps [4][5], extending the definition of the inner variable…”

“…These shapes are decomposed on a shape basis, and this is a common way to analyze 3D data [2,3,4,5]. Recently, the duality between shape basis and trajectory basis has been shown by Akhter et al [6].…”

“…As indicated in the Introduction, a 3D object can be described as a linear combination of shape basis vectors [1,4]. Using the dual model (1), the object is now a linear combination of trajectory basis vectors [6].…”

“…As shape basis is dedicated to a studied object, it has to be re-estimated for each new object. Torresani et al [2] use a generalized EM algorithm to learn 3D shapes that were adapted to the data they studied, and this approach was improved by [4]. Akhter et al represents a 3D object with trajectory basis vectors in the dual model (1), independent of the studied object; so the generic basis of DCT can be used [6].…”

Combining Local-Physical and Global-Statistical Models for Sequential Deformable Shape from Motion

“…The most standard approach to solve these ambiguities is using statistical priors to approximate the global deformable structure as a linear combination of low-rank bases of shapes (Brand 2001;Bregler et al 2000;Moreno-Noguer and Porta 2011;Torresani et al 2008), by means of a linear combination of 3D point trajectories (Akhter et al 2008;Park et al 2010;Valmadre and Lucey 2012), or even using a shapetrajectory combination (Gotardo and Martínez 2011b). This is typically used with additional smoothness constraints that further disambiguate the problem (Bartoli et al 2008;Garg et al 2013;Paladini et al 2009).…”