Stable Gabor Phase Retrieval and Spectral Clustering

We consider the problem of reconstructing a signal $f$ from its spectrogram, i.e., the magnitudes $|V_varphi f|$ of its Gabor transform
$$V_varphi f (x,y):=int_{mathbb{R}}f(t)e^e{-2pi i y t}dt, quad x,yin mathbb{R}.$$ Such problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification.

While it is well-known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connection between phase retrieval, spectral clustering and spectral geometry. We show that the stability of the Gabor phase reconstruction is bounded by the reciprocal of the emph{Cheeger constant} of the flat metric on $mathbb{R}^2$, conformally multiplied with $|V_varphi f|$. The Cheeger constant, in turn, plays a prominent role in the field of spectral clustering, and it precisely quantifies the `disconnectedness’ of the measurements $V_varphi f$.

It has long been known that a disconnected support of the measurements results in an instability—our result for the first time provides a converse in the sense that there are no other sources of instabilities.

Due to the fundamental importance of Gabor phase retrieval in coherent diffraction imaging, we also provide a new understanding of the stability properties of these imaging techniques: Contrary to most classical problems in imaging science whose regularization requires the promotion of smoothness or sparsity, the correct regularization of the phase retrieval problem promotes the `connectedness’ of the measurements in terms of bounding the Cheeger constant from below. Our work thus, for the first time, opens the door to the development of efficient regularization strategies.

• Speaker: Philipp Grohs (University of Vienna)
• Thursday 05 October 2017, 15:00–16:00
• Venue: MR 14, CMS.
• Series: Applied and Computational Analysis; organiser: Carola-Bibiane Schoenlieb.