Skip to content

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

Conference: AAAI 2026 arXiv: 2601.11680 Code: Unavailable Area: Interpretability Keywords: PET reconstruction, frequency domain analysis, ADMM unrolling, amplitude-phase decoupling, low-dose imaging

TL;DR

This work identifies three categories of degradation in low-count PET that are separable in the frequency domain — Poisson noise and photon deficiency induce high-frequency phase perturbations, while attenuation correction errors suppress low-frequency amplitude — and proposes FourierPET: an ADMM-unrolled, frequency-aware reconstruction framework that achieves comprehensive state-of-the-art performance across three datasets with only 0.44M parameters.

Background & Motivation

Triple degradation in low-count PET: (i) Poisson noise reduces signal-to-noise ratio; (ii) photon deficiency causes loss of structural detail; (iii) attenuation correction (AC) errors introduce systematic intensity bias. These three degradations are entangled in the spatial domain and difficult to disentangle.

Common limitations of existing methods: Whether iterative algorithms (OSEM), end-to-end networks (DeepPET), or post-processing methods (RED), all treat degradations indiscriminately in the spatial domain without exploiting their separability.

Core finding (frequency domain analysis): Through amplitude-phase swap experiments and frequency bias profile analysis, the following is quantitatively verified: - Phase variance is concentrated in the high-frequency HH subband → corresponding to noise/photon deficiency - Amplitude bias dominates the low-frequency LL subband → corresponding to AC bias - Separately correcting amplitude and phase yields complementary improvements

Method

Overall Architecture

A frequency-aware regularization term is embedded into an ADMM optimization framework, unrolled over \(K=3\) iterations to form a learnable network:

\[\min_x \underbrace{\mathcal{L}(\mathbf{A}x, y)}_{\text{data fidelity}} + \lambda_a \underbrace{\mathcal{R}_{\text{amp}}(|\mathcal{F}(x)|)}_{\text{low-freq. amplitude correction}} + \lambda_p \underbrace{\mathcal{R}_{\text{phase}}(\angle\mathcal{F}(x))}_{\text{high-freq. phase stabilization}}\]

Each ADMM iteration comprises three modules: x-update (SCM) → z-update (APCM) → u-update (DAM).

Key Designs

  1. Spectral Consistency Module (SCM, x-update):

    • Spatial branch: parallel 3×3 and 5×5 depthwise separable convolutions for multi-scale local feature extraction
    • Frequency branch: State-Space Fourier Neural Operator (SSFNO), which applies SSD processing to the real and imaginary parts of the FFT output and propagates hidden state \(h\) across iterations
    • Measurement consistency is enforced via the back-projection matrix \(\mathbf{A}^\top\)

  2. Amplitude-Phase Correction Module (APCM, z-update):

    • Haar DWT decomposes the input into four subbands: LL/HL/LH/HH
    • Amplitude branch: 1×1 DWConv + BN + GELU, with an additional FFN applied to the LL subband to recover low-frequency components suppressed by AC bias
    • Phase branch: \((cos\Phi, sin\Phi)\) encoding + high-frequency FFN for HH subband phase drift correction + cross-subband fusion
    • Corrected output is reconstructed via iFFT + iDWT

  3. Dual Adjustment Module (DAM, u-update): A learnable scalar \(\mu\) replaces the fixed step size to adaptively control the dual ascent step.

Loss & Training

\[\mathcal{L}_{total} = 0.5 \cdot \mathcal{L}_{Smooth\text{-}L1} + 0.3 \cdot \mathcal{L}_{SSIM} + 0.01 \cdot \mathcal{L}_{freq}\]
  • Frequency loss: \(\mathcal{L}_{freq} = |\mathcal{F}(x_{out}) - \mathcal{F}(x_{gt})|_1\)
  • Optimizer: AdamW, learning rate \(10^{-3} \to 10^{-5}\) (cosine annealing)
  • Unrolling depth \(K=3\), inner iterations \(\mathcal{N}=2\); training performed on a single RTX 4090

Key Experimental Results

Main Results (Three-Dataset Comparison)

Method Params BrainWeb SSIM↑ BrainWeb PSNR↑ In-House SSIM↑ In-House PSNR↑ UDPET SSIM↑ UDPET PSNR↑
OSEM - 0.9078 28.35 0.7456 23.59 0.7607 19.87
FBPnet 21.35M 0.9327 33.62 0.9592 34.19 0.8907 27.36
RED 28.93M 0.9664 34.45 0.9472 34.15 0.8890 26.51
LCPR-Net 75.93M 0.9769 33.75 0.9222 34.95 0.8919 27.77
FourierPET 0.44M 0.9859 35.36 0.9740 35.19 0.9083 27.98

Ablation Study (In-House Dataset)

Configuration SSIM↑ PSNR↑ RMSE↓ Notes
Baseline (conv blocks) 0.940 33.15 0.0237 No frequency modules
+ SCM (replacing x-update) 0.971 34.62 0.0200 +1.47 PSNR
+ APCM (replacing z-update) 0.967 34.05 0.0210 +0.90 PSNR
Full FourierPET 0.974 35.19 0.0190 SCM + APCM are complementary
SCM Sub-module Ablation SSIM PSNR Notes
w/o \(\mathbf{A}^\top\) 0.8328 22.55 Catastrophic degradation; measurement consistency is critical
w/o SSFNO 0.9530 33.69 Global spectral modeling is beneficial
w/o spatial module 0.9681 34.43 Local features are complementary
Full SCM 0.9740 35.19 All three components are indispensable

Key Findings

  • Remarkable parameter efficiency: With only 0.44M parameters — 65× fewer than RED (28.93M) and 172× fewer than LCPR-Net (75.93M) — FourierPET achieves superior performance across all metrics.
  • \(\mathbf{A}^\top\) is a critical constraint: Removing it causes SSIM to drop sharply from 0.974 to 0.833, underscoring the necessity of physical measurement consistency.
  • Complementarity of amplitude and phase branches: The phase branch alone improves SSIM (structural fidelity), while the amplitude branch alone reduces RMSE (global bias); their combination yields the best overall performance.
  • Zero-shot cross-domain generalization: A model trained on human PET can be directly applied to mouse PET while maintaining high reconstruction quality.

Highlights & Insights

  1. The frequency-domain degradation separability hypothesis is highly insightful and is quantitatively validated through amplitude-phase swap experiments and DWT frequency bias profiling.
  2. The combination of ADMM unrolling and frequency-aware priors preserves physical interpretability while retaining the flexibility of data-driven learning.
  3. The extreme parameter efficiency of 0.44M parameters is of substantial value for resource-constrained clinical deployment.
  4. Zero-shot cross-species generalization suggests that frequency-domain degradation patterns are universal.

Limitations & Future Work

  • Validation is limited to brain and whole-body PET; other modalities such as cardiac PET remain unexplored.
  • The unrolling depth \(K=3\) is fixed; adaptive depth could potentially yield further improvements.
  • The frequency-domain analysis assumes strictly separable degradations, whereas real-world scenarios may involve coupling effects.
  • No comparison is made against recent generative reconstruction methods such as diffusion models.
  • Traditional iterative methods (OSEM, MAP): Accurate physical modeling but slow computation and difficult prior design.
  • End-to-end methods (DeepPET, CNNBPnet): Purely data-driven, lacking physical constraints.
  • Unrolled networks (ADMM-Net, ISTA-Net): This work belongs to this category but is the first to introduce frequency-aware amplitude-phase decoupling.
  • Fourier Neural Operator: This work integrates SSM and FNO into SSFNO for global spectral modeling.

Rating

  • Novelty: ⭐⭐⭐⭐⭐ The frequency-domain degradation separability hypothesis is novel and thoroughly validated.
  • Experimental Thoroughness: ⭐⭐⭐⭐⭐ Three datasets + detailed ablations + cross-domain generalization + frequency-domain visualization.
  • Writing Quality: ⭐⭐⭐⭐ Physical motivation is clearly articulated; module design logic is rigorous.
  • Value: ⭐⭐⭐⭐ The frequency-domain decoupling paradigm is transferable to other inverse problems.