MAGPIE: Multilevel-Adaptive-Guided Solver for Ptychographic Phase Retrieval

TL;DR

<3-5 sentence high-level summary> MAGPIE presents a stochastic multilevel solver for ptychographic phase retrieval by marrying a quadratic surrogate-based majorization (as in the PIE family) with a structure-aware MG/OPT coarse-grid surrogate. The method guarantees monotone descent and automatic cross-scale consistency, achieving accelerated convergence and improved reconstruction quality over traditional PIE variants and LBFGS. Comprehensive experiments demonstrate robustness to noise, varying overlap, and realistic objects, with deeper multigrid levels delivering the most significant gains. The approach is scalable, adaptable to real-time feedback contexts, and extendable to blind ptychography and GPU-accelerated minibatching.

Abstract

We introduce MAGPIE (Multilevel-Adaptive-Guided Ptychographic Iterative Engine), a stochastic multigrid solver for the ptychographic phase-retrieval problem. The ptychographic phase-retrieval problem is inherently nonconvex and ill-posed. To address these challenges, we reformulate the original nonlinear and nonconvex inverse problem as the iterative minimization of a quadratic surrogate model that majorizes the original objective. This surrogate not only ensures favorable convergence properties but also generalizes the Ptychographic Iterative Engine (PIE) family of algorithms. By solving the surrogate model using a multigrid method, MAGPIE achieves substantial gains in convergence speed and reconstruction quality over traditional approaches.

Figures (14)

• Figure 1: Experimental setup and data acquisition for ptychography.
• Figure 2: Illustration of the majorization property adapted from SMM_graph.
• Figure 3: Columns (left to right) show the magnitude (top) and phase (bottom) of the three inputs used in our numerical experiments: (1) the simulated Fresnel zone‐plate probe; (2) the test synthetic object (magnitude from the Baboon image; phase from the Cameraman image); and (3) the realistic synthetic object inspired by integrated‐circuit imaging.
• Figure 4: Log-log plots of residuals (top) and errors (bottom) for L-BFGS, rPIE, and MAGPIE_$l$ (levels 1 to $\log(m)$) applied to a synthetic object ($n = 512$) with a probe ($m = 128$) using noise level $\eta = 0.05$, $\texttt{overlap\_ratio} = 0.5$, $\alpha=0.01$, and $\texttt{tol} = 10^{-4}$.
• Figure 5: Reconstructions of L-BFGS, rPIE, and MAGPIE (at level $\log(m)$) applied to a synthetic object ($n = 512$) with a probe ($m = 128$) using noise level $\eta = 0.05$, $\texttt{overlap\_ratio} = 0.5$, $\alpha=0.01$, and $\texttt{tol} = 10^{-4}$.

Theorems & Definitions (27)

• Definition 2.1: Revised Exit Wave
• Proposition 3.1: Majorization
• Proposition 3.2: Convergence and optimality
• Proposition 3.3: Convergence rate
• Proposition 4.1: Well-definedness
• Proposition 4.2: Consistency
• Theorem 4.3: Descent property
• Proposition 4.4: Automatic regularization selection
• Lemma C.1
• proof