Abstract
Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).
Highlights
Phase retrieval generally alludes to the non-linear inverse problem of recovering some signal from phaseless measurements
It has been shown that the phase retrieval problem for frames in finite-dimensional Hilbert spaces [7] and a forteriori in finite-dimensional reflexive Banach spaces [2] is always stable, which elicits the question: Why are we concerned with stability estimates for phase retrieval from discrete Gabor measurements at all? The reason is that phase retrieval for frames in infinite-dimensional spaces is always unstable [2,7] and in addition one can construct sequences of finite-dimensional subspaces of infinite-dimensional Hilbert spaces along with frames for which the stability constant of phase retrieval increases exponentially in the dimension of the constructed subspaces [7]
Recent research [1] into the infinite-dimensional phase retrieval problem has led us to believe that the instability of phase retrieval is not an insurmountable obstacle to reconstruction
Summary
Phase retrieval generally alludes to the non-linear inverse problem of recovering some signal (which in this paper will be modelled by x ∈ CL ) from phaseless measurements. It was shown that stability can be restored for examples that exhibit a disconnectedness in the measurements by only reconstructing the phase semi-globally or in an atoll sense. It was shown in [15] that such disconnectedness in the measurements is the only source of instabilities for phase retrieval. The eigenvector-based angular synchronisation approach [17] which relies on a certain weak form of invertibility of the phase retrieval problem to prove a stability result for deterministic measurement systems.
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