Improved success rate and stability for phase retrieval by including randomized overrelaxation in the hybrid input output algorithm

In this paper, we study the success rate of the reconstruction of objects of finite extent given the magnitude of its Fourier transform and its geometrical shape. We demonstrate that the commonly used combination of the hybrid input output and error reduction algorithm is significantly outperformed by an extension of this algorithm based on randomized overrelaxation. In most cases, this extension tremendously enhances the success rate of reconstructions for a fixed number of iterations as compared to reconstructions solely based on the traditional algorithm. The good scaling properties in terms of computational time and memory requirements of the original algorithm are not influenced by this extension.

💡 Research Summary

The paper addresses the long‑standing challenge of phase retrieval, where only the magnitude of a Fourier transform and the support (geometrical shape) of an object are known. In two or more dimensions and with an oversampling ratio of at least two, the problem is theoretically solvable, but practical iterative algorithms often stall in local minima. The authors first review the conventional hybrid input‑output (HIO) algorithm combined with error‑reduction (ER) steps. HIO uses a feedback parameter β (typically 0.5–1.0) to push the estimate outside the support region, while ER simply projects the current estimate onto both the support and the measured amplitude constraints. Although the HIO+ER combination is widely used, it still fails for many realistic objects, especially when the object’s structure is complex or the data contain noise.

To overcome these limitations, the authors introduce randomized over‑relaxation (OR) into the HIO framework, yielding a new scheme they denote (HIO+OR)+ER. Over‑relaxation replaces a projection operator Pμ with Qμ;λ = 1 + λ(Pμ − I), where λ is a relaxation factor. When λ = 1 the operator reduces to the original projection; for λ ≠ 1 the update is “over‑relaxed” (λ > 1) or “under‑relaxed” (λ < 1). Crucially, instead of fixing λ, the authors draw λA from a uniform distribution in the interval