Stability estimates for phase retrieval from discrete Gabor measurements
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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 [5] and possibly severely ill-conditioned in finite dimensional Hilbert spaces [5]. Recently, it was also shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable when one is willing to accept a more relaxed semi-global stability regime [1]. We present first evidence that this semi-global stability regime allows one to do phase retrieval from measurements induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales linearly in the space dimension. To this end, we utilise well-known reconstruction formulae which have been used repeatedly in recent years [4], [6-8].
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