FourierPET: Deep Fourier-based Unrolled Network for Low-count PET Reconstruction

TL;DR

This work tackles low-count PET reconstruction by revealing that degradation effects separate spectrally into high-frequency phase perturbations caused by Poisson noise and low-frequency amplitude suppression from AC bias. The authors introduce FourierPET, an ADMM-unrolled framework with three dedicated modules: SCM (spectral consistency in the frequency domain), APCM (amplitude-phase correction in frequency bands), and DAM (learned dual update), enabling targeted, interpretable corrections while preserving data fidelity. The approach leverages SSFNO-based global frequency modeling and band-wise spectral shaping, supervised by a composite loss that blends pixel, structural, and spectral objectives. Across multiple datasets and count regimes, FourierPET achieves state-of-the-art performance with fewer parameters and demonstrates strong generalization, including zero-shot adaptation from human to mouse PET. This frequency-aware strategy improves robustness, interpretability, and potential clinical translation for low-dose PET imaging.

Abstract

Low-count positron emission tomography (PET) reconstruction is a challenging inverse problem due to severe degradations arising from Poisson noise, photon scarcity, and attenuation correction errors. Existing deep learning methods typically address these in the spatial domain with an undifferentiated optimization objective, making it difficult to disentangle overlapping artifacts and limiting correction effectiveness. In this work, we perform a Fourier-domain analysis and reveal that these degradations are spectrally separable: Poisson noise and photon scarcity cause high-frequency phase perturbations, while attenuation errors suppress low-frequency amplitude components. Leveraging this insight, we propose FourierPET, a Fourier-based unrolled reconstruction framework grounded in the Alternating Direction Method of Multipliers. It consists of three tailored modules: a spectral consistency module that enforces global frequency alignment to maintain data fidelity, an amplitude-phase correction module that decouples and compensates for high-frequency phase distortions and low-frequency amplitude suppression, and a dual adjustment module that accelerates convergence during iterative reconstruction. Extensive experiments demonstrate that FourierPET achieves state-of-the-art performance with significantly fewer parameters, while offering enhanced interpretability through frequency-aware correction.

Figures (14)

• Figure 1: Motivation.(a) Qualitative and (b) quantitative analyses demonstrate that low-count degradations exhibit separable spectral patterns: Poisson noise and photon starvation mainly perturb the phase, degrading structural fidelity, while attenuation-correction (AC) errors suppress low-frequency amplitude, inducing global intensity bias. (c) Frequency-deviation profiles reveal that phase errors concentrate in high frequencies, whereas amplitude distortions dominate low-frequency bands.
• Figure 2: Overview of the proposed FourierPET architecture. Given a measured sinogram $y$, FourierPET performs $K$ unrolled ADMM iterations to iteratively refine the primal variable $x$, auxiliary variable $z$, and dual variable $u$. Each iteration comprises three steps: (1) the $x$-update enforces measurement consistency and spectral alignment; (2) the $z$-update applies frequency-aware regularization to mitigate degradation; (3) the $u$-update promotes convergence by reconciling $x$ and $z$. The final reconstruction is obtained after $K$ iterations.
• Figure 3: Structure of a single FourierPET iteration, with three sequential components: (a) the $x$-update via SCM, (b) the $z$-update via APCM, and (c) the $u$-update via DAM.
• Figure 4: APCM Core modules. The Amp Branch (left) restores suppressed low-frequency components, while the Phase Branch (right) corrects high-frequency drifts. Together, these submodules provide targeted, frequency-aware compensation for degradations in low-count PET.
• Figure 5: Qualitative comparison on the In‑House dataset. Top: axial slices and corresponding error maps. Bottom: coronal and sagittal views of the same subjects, with red lines indicating axial slice locations. Orange rectangles highlight localized errors in the tumor region of interest (ROI).