Abstract: The generalized phase retrieval problem over compact groups aims to recover a set of matrices, representing an unknown signal, from their associated Gram matrices, leveraging prior structural knowledge about the signal. This framework generalizes the classical phase retrieval problem, which reconstructs a signal from the magnitudes of its Fourier transform, to a richer setting involving non-abelian compact group. Our main result shows that for a suitable class of semi-algebraic priors, the generalized phase retrieval problem not only admits a unique solution (up to inherent group symmetries), but also satisfies a bi-Lipschitz property.  This implies robustness to both noise and model mismatch, an essential requirement for practical use, especially when measurements are severely corrupted by noise. 

Abstract: The fundamental matrix of a pair of pinhole cameras lies at the core of many systems that reconstruct 3D scenes from 2D images. However, for more than two cameras, the relations between the various fundamental matrices of camera pairs are not yet completely understood. In joint work with Viktor Korotynskiy, Anton Leykin, and Tomas Pajdla, we characterize all polynomial constraints that hold vanishing for an arbitrary triple of fundamental matrices. Unlike most constraints in previous works, our constraints hold independently of the relative scaling of of the fundamental matrices, which is unknown in practice. We also provide a partial characterization for essential matrix triples arising from calibrated cameras.

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