  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.
