DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models

TL;DR

This work tackles nonlinear Fourier phase retrieval by reframing it as an inverse problem solvable with a pretrained unconditional diffusion prior. The authors marry Denoising Diffusion Restoration Models (DDRM) with classical alternating projection methods, using DDRM’s posterior sampling to refine reconstructions from noisy Fourier magnitude measurements without task-specific training. The proposed DDRM-PR framework applies a Fourier-phase-specific adaptation of DDRM, incorporating an HIO-based measurement-consistency step and randomized initializations to produce robust reconstructions, demonstrated on simulated CelebA-HQ data and experimental scattering-imaging data with consistent gains in distortion- and perceptual-quality metrics. The approach offers practical advantages in generalization and flexibility across phase retrieval tasks, while outlining avenues for improving noisy-case theory, complex-valued signals, and diffusion-prior conditioning.

Abstract

Diffusion models have demonstrated their utility as learned priors for solving various inverse problems. However, most existing approaches are limited to linear inverse problems. This paper exploits the efficient and unsupervised posterior sampling framework of Denoising Diffusion Restoration Models (DDRM) for the solution of nonlinear phase retrieval problem, which requires reconstructing an image from its noisy intensity-only measurements such as Fourier intensity. The approach combines the model-based alternating-projection methods with the DDRM to utilize pretrained unconditional diffusion priors for phase retrieval. The performance is demonstrated through both simulations and experimental data. Results demonstrate the potential of this approach for improving the alternating-projection methods as well as its limitations.

Figures (5)

• Figure 1: Ground-truth test images (top row), reconstructions using the developed approach (middle row), and HIO initialization results (bottom row) for the case with parameters: $\alpha=0.5$, $N=1$, $\eta=0.15$, $\eta_b=0.20$, $t=15$, and $T_{init}=350$.
• Figure 2: Ground-truth test images (top row), reconstructions using the developed approach (middle row), and HIO initialization results (bottom row) for the case with parameters: $\alpha=1$, $N=1$, $\eta=0.25$, $\eta_b=0.22$, $t=30$, and $T_{init}=400$.
• Figure 3: Ground-truth test images (top row), reconstructions using the developed approach (middle row), and HIO initialization results (bottom row) for the case with parameters: $\alpha=2$, $N=1$, $\eta=0.25$, $\eta_b=0.18$, $t=15$, and $T_{init}=400$.
• Figure 4: Ground-truth test images (top row), reconstructions using the developed approach (middle row), and HIO initialization results (bottom row) for the case with parameters: $\alpha=3$, $N=1$, $\eta=0.78$, $\eta_b=0.17$, $t=30$, and $T_{init}=300$.
• Figure 5: Performance comparison with experimental data. Each row corresponds to a different test image. The first column shows the ground truth representing the target amplitude-only signal. The second column displays the speckle pattern captured on the detector, which is the observed intensity measurement after passing through the scattering medium. The third column presents the HIO result illustrating the initial reconstruction obtained with HIO algorithm. The last column shows the reconstruction of DDRM-PR, which integrates pretrained diffusion models for enhanced phase retrieval.

Theorems & Definitions (11)

• Theorem 4.1: Simplified form of DDRM
• Definition A.1
• Definition A.2
• Lemma A.1
• proof
• Corollary A.1.1
• Lemma A.2
• Corollary A.2.1
• Lemma A.3: Reparametrization trick
• Lemma A.4