Leveraging Phase Information to Boost Unrolled Network Learning for Image Deblurring

Technical Summary: UPADNet – Unrolled Phase and Amplitude Decomposition Network for Image Deblurring

Problem Statement
Image deblurring aims to recover a sharp latent image ($u$) from a blurred and noisy observation ($z$), typically modeled as a linear shift-invariant convolution corrupted by additive noise ($z = h * u + n$). While deep learning approaches have achieved significant success, they often suffer from limitations including a heavy reliance on large annotated datasets, limited generalization across diverse real-world degradations (e.g., varying noise levels, camera motion), and high computational costs. Furthermore, standard deep networks often treat the image as a spatial variable, potentially overlooking the critical role of Fourier phase information, which encodes structural and geometric details essential for sharp image recovery.

Methodology
The authors propose UPADNet (Unrolled Phase and Amplitude Decomposition Network), a model-driven deep architecture that explicitly decomposes the image restoration problem into amplitude and phase components in the Fourier domain. The methodology proceeds in three stages:

1. LMMSE Estimation of Amplitude and Phase:
   Recognizing that naive Fourier domain relations (where amplitude is multiplicative and phase is additive) fail under noise and kernel uncertainty, the authors derive novel Linear Minimum Mean Squared Error (LMMSE) estimators.

   • They formulate generalized linear operators to estimate the amplitude ($\hat{A}_Z$) and phase ($\hat{\theta}_Z$) of the blurred observation.
   • Theoretical analysis (Theorems 1 and 2) proves that these estimators minimize the expected squared error compared to naive multiplicative/additive models by leveraging statistical moments (means, variances, and covariances) of the amplitude and phase distributions.

2. Iterative Optimization via Alternating Direction Method:
   An iterative optimization algorithm is developed to recover the latent sharp image by minimizing a regularized objective function. This objective enforces statistical consistency with the derived LMMSE estimators and includes:

   • Data Fidelity Terms: Enforcing consistency between the estimated and observed amplitude and phase.
   • Regularization: $\ell_1$ regularization on the image amplitude ($M \circ A_U$) to promote sparsity in the Fourier domain, and $\ell_2$ penalties on the blur kernel components to ensure smoothness. An additional $\ell_1$ term on the image phase ($\theta_U$) enforces structural symmetry.
   • Solution Strategy: The non-convex objective is solved using Half-Quadratic Splitting (HQS), introducing auxiliary variables to decouple non-smooth penalties. This results in a sequence of closed-form subproblems for updating amplitude and phase variables.

3. Unrolling into a Deep Network (UPADNet):
   Instead of using fixed statistical parameters, the iterative algorithm is "unrolled" into a trainable deep network.

   • Architecture: Each iteration of the optimization algorithm corresponds to a UPADBlock. A UPADBlock consists of four sub-blocks that sequentially update the amplitude and phase of the latent image ($A_U, \theta_U$) and the blur kernel ($A_H, \theta_H$).
   • Learnable Parameters: The statistical matrices $\{W_i\}$ and the frequency-domain filter $M$, which are theoretically fixed in classical formulations, are learned from data.
   • Weight Generator: A lightweight module dynamically produces iteration-specific weights and filters using mechanisms like SimpleGate (SG) and Simplified Channel Attention (SCA).
   • Activation: To approximate the soft-thresholding proximal operators while ensuring smooth gradient flow, the authors replace ReLU with GELU activations.
   • Multi-scale Design: Multiple UPADBlocks are stacked with intermediate downsampling and upsampling to enlarge the effective receptive field.

Key Contributions

• Phase-Amplitude Decomposition: The paper introduces a principled framework that explicitly models and estimates both amplitude and phase in the Fourier domain, rather than restoring the spatial image directly.
• Novel LMMSE Estimators: The derivation of optimal linear estimators for blurred amplitude and phase that outperform naive Fourier domain relations under noise.
• UPADNet Architecture: The development of an end-to-end trainable network that unrolls the derived optimization algorithm, allowing the network to learn the underlying statistical parameters directly from paired clean and degraded data.
• Robustness: The design specifically targets robustness in high-noise and limited-data regimes by leveraging the interpretability of the unrolled optimization and the structural priors of phase information.

Experimental Results
The authors evaluated UPADNet on three benchmarks: GoPro (synthetic motion blur), RealBlur (real-world camera shake), and a synthetic COCO dataset with controlled noise and motion kernels.

• Performance: UPADNet achieved state-of-the-art results on GoPro (34.63 dB PSNR, 0.975 SSIM) and RealBlur (41.29 dB PSNR on RealBlur-R), outperforming recent transformer-based (e.g., Stripformer, FFTformer) and diffusion-based methods.
• Efficiency: The model uses only 27M parameters, significantly fewer than large-scale competitors like AdaRevD (220M).
• Noise Robustness: In experiments with increasing Gaussian noise ($\sigma = 0.01$ to $0.05$), UPADNet maintained stable performance, whereas other unrolled and deep networks showed significant degradation.
• Data Efficiency: When trained with reduced data ratios (down to 60% of the dataset), UPADNet exhibited minimal performance drop compared to competitors, demonstrating superior generalization in low-data regimes.
• Ablation: Studies confirmed that increasing the number of unrolled blocks (up to 72) and using GELU activations with a learning rate of $1\times10^{-3}$ yielded optimal performance.

Significance and Claims
The paper claims that the explicit modeling of Fourier phase and amplitude is crucial for high-quality image deblurring, particularly in challenging real-world scenarios. By unrolling an optimization algorithm derived from LMMSE theory, UPADNet bridges the gap between the interpretability of traditional iterative methods and the adaptability of deep learning. The authors assert that this approach offers a robust solution that generalizes well across varying blur types, noise levels, and limited training data, highlighting the value of structured frequency-domain modeling within deep unrolling frameworks.