Stable Camera Motion Estimation Using Convex Programming

We study the inverse problem of estimating $n$ locations $\mathbf{t}_1, \mathbf{t}_2, \ldots, \mathbf{t}_n$ (up to global scale, translation, and negation) in $\mathbb{R}^d$ from noisy measurements of a subset of the (unsigned) pairwise lines that connect them, that is, from noisy measurements of $\pm \frac{\mathbf{t}_i - \mathbf{t}_j}{\|\mathbf{t}_i - \mathbf{t}_j \|_2}$ 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 $\mathbf{t}_i$ represent camera locations in $\mathbb{R}^3$. 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 pr...

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