On the conditional equivalence of phase retrieval algorithms

Jakob Schroeder; Andreas Döpp

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arXiv · 2606.07257

On the conditional equivalence of phase retrieval algorithms

Jakob Schroeder·Andreas Döpp

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Abstract

Phase retrieval - recovering a complex-valued field from intensity measurements - is typically solved using variants of the Gerchberg-Saxton (GS) algorithm, understood as alternating projections between measurement planes. Meanwhile, modern computational imaging increasingly relies on gradient-based optimization and automatic differentiation. Here we show that these two approaches are mathematically identical: the GS magnitude replacement step is exactly a unit gradient descent step on an amplitude least-squares loss. This equivalence enables seamless integration of classical phase retrieval with differentiable physics pipelines. We further identify two complementary probabilistic interpretations of this equivalence: globally, the amplitude loss is the negative log-likelihood under Gaussian amplitude noise; locally, each projection step arises as a Bayesian update with the propagated field as prior. The local view provides qualitative guidance for relaxation in iterative phase retrieval.

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Jakob Schroeder, Andreas Döpp. 2026-06-05. On the conditional equivalence of phase retrieval algorithms. https://arxiv.org/abs/2606.07257

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