2021/08/13 by Philipp Grohs, Grohs, Philipp, Martin Rathmair +1 · 1 citation
Earth and Planetary Sciences · Engineering · Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Geophysical Methods and Applications #Seismic Imaging and Inversion Techniques
paper · pdf · doi:10.48550/arxiv.2108.06154
openalex publication_date 2021/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of reconstructing the missing phase information from spectrogram data |G f|, with Gf(x,y)=∫_ℝ f(t) e-π(t-x)2e-2πi t ydt, the Gabor transform of a signal f∈ L2(ℝ). More specifically, we are interested in domains Ω⊆ ℝ2, which allow for stable local reconstruction, that is |Gg| ≈ |Gf| in ~Ω \Longrightarrow ∃ τ∈\mathbbT: Gg ≈ τGf in ~Ω. In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Comm. Pure Appl. Math. (2019)] and [P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate functions. J. Eur. Math. Soc. (2021)] we established a characterization of the stability of this phase retrieval problem in terms of the connectedness of the observed measurements. The main downside of the aforementioned results is that the similarity of two spectrograms is measured w.r.t. a first order weighted Sobolev norm. In this article we remove this flaw and essentially show that the Sobolev norm may be replaced by the L2-norm. Using this result allows us to show that it suffices to sample the spectrogram on suitable discrete sampling sets -- a property of crucial importance for practical applications.