Phase retrieval from local measurements: Improved robustness via eigenvector-based angular synchronization

Abstract

We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [27]. The improved algorithm admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [27] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that outperforms competing approaches on large problems. Finally, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

Lower Lipschitz bounds for phase retrieval from locally supported measurements

“…In [3], Balan et al showed that if α and β are injective on C d / ∼, then β is bi-Lipschitz with respect to d 1 , and α is bi-Lipschitz with respect to D 2 , where in both cases R N is equipped with the Euclidean norm. Motivated by applications such as (Fourier) ptychography [18,22] and related numerical methods [13,14], we will study frames which are constructed as the shifts of a family of locally supported measurement vectors. Specifically, we assume that {m 1 , m 2 , .…”

On signal reconstruction from FROG measurements

“…Theoretical analysis of the basin of attraction in the related problem of random systems of quadratic equations was performed in [32,33,34,35]. Recently, nonconvex methods for Fourier phase retrieval, accompanied with theoretical analysis, were proposed in [15,14,20]. However, in the FROG setup the problem is quartic rather than quadratic.…”

“…One popular way to overcome the non-uniqueness is by collecting additional information on the sought signal beyond its Fourier magnitude. For instance, this can be done by taking multiple measurements, each one with a different known mask [12,13,14]. An important special case employs shifted versions of a single mask.…”

A direct solver for the phase retrieval problem in ptychographic imaging

“…Moreover, the exponential window shows strong artifacts especially in the phase of the reconstruction as it does not fit the window form given in applications. In our next experiment we compare the proposed tight projector against the pattern projector used in [14]. We already know that both projection spaces coincide if all shifts are taken into account.…”

“…The crucial step of the algorithm is the first, i.e., we have to ensure by a suitable choice of the window w that (10) has a unique solution which can be computed in a stable manner. In [13,14] it has been shown that this is the case for the following choice for w:…”