# Stable Gabor phase retrieval for multivariate functions

## TL;DR

This paper extends the understanding of Gabor phase retrieval stability from one-dimensional to multivariate functions, with significant improvements and implications for applications like ptychography in imaging.

## Contribution

It generalizes and enhances previous stability results of Gabor phase retrieval to higher dimensions, broadening practical applicability.

## Key findings

- Generalized stability classification to multivariate functions.
- Improved stability bounds for Gabor phase retrieval.
- Implications for advanced imaging techniques like ptychography.

## Abstract

In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a function $f$ from its spectrogram $|\mathcal{G}f|$, where $$ \mathcal{G}f(x,y)=\int_{\mathbb{R}^d} f(t) e^{-\pi|t-x|^2} e^{-2\pi i t\cdot y} dt, \quad x,y\in \mathbb{R}^d, $$ have been completely classified in terms of the disconnectedness of the spectrogram. These findings, however, were crucially restricted to the onedimensional case ($d=1$) and therefore not relevant for many practical applications. In the present paper we not only generalize the aforementioned results to the multivariate case but also significantly improve on them. Our new results have comprehensive implications in various applications such as ptychography, a highly popular method in coherent diffraction imaging.