Stable Camera Motion Estimation Using Convex Programming

We study the inverse problem of estimating n locations $t_1, ..., t_n$ (up to global scale, translation and negation) in $R^d$ from noisy measurements of a subset of the (unsigned) pairwise lines that connect them, that is, from noisy measurements of $\pm (t_i - t_j)/\|t_i - t_j\|$ for some pairs (i,j) (where the signs are unknown). This problem is at the core of the structure from motion (SfM) problem in computer vision, where the $t_i$'s represent camera locations in $R^3$. The noiseless version of the problem, with exact line measurements, has been considered previously under the general title of parallel rigidity theory, mainly in order to characterize the conditions for unique realization of locations. For noisy pairwise line measurements, current methods tend to produce spurious solutions that are clustered around a few locations. This sensitivity of the location estimates is a well-known problem in SfM, especially for large, irregular collections of images. In this paper we introduce a semidefinite programming (SDP) formulation, specially tailored to overcome the clustering phenomenon. We further identify the implications of parallel rigidity theory for the location estimation problem to be well-posed, and prove exact (in the noiseless case) and stable location recovery results. We also formulate an alternating direction method to solve the resulting semidefinite program, and provide a distributed version of our formulation for large numbers of locations. Specifically for the camera location estimation problem, we formulate a pairwise line estimation method based on robust camera orientation and subspace estimation. Lastly, we demonstrate the utility of our algorithm through experiments on real images.Comment: 40 pages, 12 figures, 6 tables; notation and some unclear parts updated, some typos correcte

Shape basis interpretation for monocular deformable 3D reconstruction

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.In this paper, we propose a novel interpretable shape model to encode object non-rigidity. We first use the initial frames of a monocular video to recover a rest shape, used later to compute a dissimilarity measure based on a distance matrix measurement. Spectral analysis is then applied to this matrix to obtain a reduced shape basis, that in contrast to existing approaches, can be physically interpreted. In turn, these pre-computed shape bases are used to linearly span the deformation of a wide variety of objects. We introduce the low-rank basis into a sequential approach to recover both camera motion and non-rigid shape from the monocular video, by simply optimizing the weights of the linear combination using bundle adjustment. Since the number of parameters to optimize per frame is relatively small, specially when physical priors are considered, our approach is fast and can potentially run in real time. Validation is done in a wide variety of real-world objects, undergoing both inextensible and extensible deformations. Our approach achieves remarkable robustness to artifacts such as noisy and missing measurements and shows an improved performance to competing methods.Peer ReviewedPostprint (author's final draft