Injectivity of Multi-window Gabor Phase Retrieval

TL;DR

This paper investigates the injectivity of phase retrieval using multi-window Gabor frames, providing explicit constructions that require fewer measurements and depend on the signal dimension, with applications in diffraction imaging, ptychography, and audio processing.

Contribution

It introduces explicit constructions of phase retrievable multi-window Gabor frames with fewer measurements, improving efficiency in structured signal recovery.

Findings

Constructed an infinite family of phase retrievable multi-window Gabor frames.

Showed phase retrievability with fewer measurements than previous methods.

Demonstrated measurement count depends on signal dimension, not ambient dimension.

Abstract

In many signal processing problems arising in practical applications, we wish to reconstruct an unknown signal from its phaseless measurements with respect to a frame. This inverse problem is known as the phase retrieval problem. For each particular application, the set of relevant measurement frames is determined by the problem at hand, which motivates the study of phase retrieval for structured, application-relevant frames. In this paper, we focus on one class of such frames that appear naturally in diffraction imaging, ptychography, and audio processing, namely, multi-window Gabor frames. We study the question of injectivity of the phase retrieval problem with these measurement frames in the finite-dimensional setup and propose an explicit construction of an infinite family of phase retrievable multi-window Gabor frames. We show that phase retrievability for the constructed frames can be achieved with a much smaller number of phaseless measurements compared to the previous results for this type of measurement frames. Additionally, we show that the sufficient for reconstruction number of phaseless measurements depends on the dimension of the signal space, and not on the ambient dimension of the problem.