LRDUN: A Low-Rank Deep Unfolding Network for Efficient Spectral Compressive Imaging

1. Problem Statement

Spectral Compressive Imaging (SCI) aims to reconstruct high-dimensional Hyperspectral Images (HSI) from compressed 2D measurements (e.g., via Coded Aperture Snapshot Spectral Imaging - CASSI). This is a severely ill-posed inverse problem because the number of unknowns (the full 3D HSI data cube) vastly exceeds the number of measurements.

Limitations of Existing Deep Unfolding Networks (DUNs):

• Computational Redundancy: Current DUNs operate directly on the full high-dimensional HSI data cube at every stage of the unfolding process.
• Ill-posedness: They attempt to map 2D residuals back to the 3D HSI space directly, creating a massive dimensionality gap that exacerbates the ill-posed nature of the problem.
• Inefficiency: Processing the full data cube leads to high computational costs (FLOPs) and limits scalability, despite achieving good reconstruction quality.

2. Methodology

The authors propose LRDUN, a novel framework that fundamentally reformulates the SCI problem by integrating Low-Rank (LR) decomposition directly into the sensing model and the deep unfolding architecture.

A. Novel Imaging Models (Low-Rank Reformulation)

Instead of reconstructing the full HSI tensor $\mathbf{X} \in \mathbb{R}^{H \times W \times B}$, the authors exploit the inherent low-rank property of HSIs. They decompose the HSI into:

1. Spectral Basis ($\mathbf{E}$): A matrix $\in \mathbb{R}^{B \times k}$ capturing global spectral correlations.
2. Subspace Images ($\mathbf{A}$): A tensor $\in \mathbb{R}^{H \times W \times k}$ encoding spatial structures and local spectral sparsity.

• Relation: $\mathbf{X} = \mathbf{A} \times_3 \mathbf{E}$, where $k \ll B$ (physical rank).

By substituting this decomposition into the CASSI sensing equation ($\mathbf{y} = \Phi\mathbf{x} + \mathbf{n}$), they derive two distinct, lower-dimensional imaging models:

• Spectral Basis Model: $\mathbf{y} = \Phi_A \mathbf{e} + \mathbf{n}$
• Subspace Image Model: $\mathbf{y} = \Phi_E \mathbf{a} + \mathbf{n}$

This reformulation drastically reduces the number of unknowns from $H \times W \times B$ to $k \times (H \times W + B)$, significantly mitigating the ill-posedness.

B. Optimization and Unfolding

The reconstruction is formulated as a joint optimization problem minimizing data fidelity and regularization terms for both $\mathbf{A}$ and $\mathbf{E}$. The authors solve this using an Alternating Proximal Gradient Descent (PGD) algorithm, which is then "unfolded" into a deep neural network with $N$ stages.

• E-problem: Updates the spectral basis.
• A-problem: Updates the subspace images.
• Each stage alternates between a Data-Fidelity step (physics-driven gradient descent) and a Prior step (learnable neural network).

C. Generalized Feature Unfolding Mechanism (GFUM)

A key innovation is the GFUM, which decouples the physical rank ($k$) from the feature dimensionality ($C$) of the neural networks.

• The Problem: In standard unfolding, the network input/output is constrained to the physical dimension $k$, limiting the network's ability to learn complex features.
• The Solution: GFUM elevates the $k$-dim variables into a higher-dimensional feature space ($C \ge k$).
  • Physical Component ($k$): Enforces data fidelity.
  • Auxiliary Component ($C-k$): Acts as an information carrier, learning to encode residuals, noise, and mask-aware cues to assist the refinement process.
• This allows the "ProxyNets" (the prior modules) to operate in a richer feature space without violating the physical constraints of the sensing model.

D. Network Architecture

• ProxyNet E (Spectral Basis): A lightweight 1D CNN with GELU activations and residual connections. It enforces orthogonality on the spectral basis using QR Retraction (projecting the result onto the Stiefel manifold) to resolve scale ambiguity.
• ProxyNet A (Subspace Images): A U-Net architecture featuring Spatial Conv Attention Blocks (SCAB). The SCAB uses a large $11 \times 11$ depth-wise convolution to capture long-range spatial dependencies, crucial for recovering fine textures.

3. Key Contributions

1. Novel Imaging Models: Derived two low-dimensional imaging models (spectral basis and subspace) by explicitly integrating LR decomposition with the sensing physics, transforming a high-dimensional ill-posed problem into tractable subproblems.
2. LRDUN Framework: Developed a principled deep unfolding network that jointly solves these subproblems, balancing physical interpretability with data-driven learning.
3. Generalized Feature Unfolding Mechanism (GFUM): Introduced a mechanism to decouple physical rank from network feature dimensions, significantly enhancing representational capacity and flexibility.
4. State-of-the-Art Performance: Achieved superior reconstruction quality with drastically reduced computational complexity compared to existing SOTA methods.

4. Experimental Results

The method was evaluated on simulated datasets (CAVE, KAIST) and real-world CASSI measurements.

• Quantitative Performance:

  • LRDUN-9stg achieved the highest average PSNR (40.96 dB) and SSIM (0.982) on the KAIST dataset, outperforming previous SOTA DUNs like RDLUF, MiJUN, and DAUHST.
  • Efficiency: Despite higher accuracy, LRDUN-9stg required only 30.58 G FLOPs, significantly lower than competitors like BIRNAT (2122 G) or DGSMP (646 G).
  • Lightweight Variant: LRDUN-3stg (0.69M params, 10.26 G FLOPs) outperformed the heavy MST++ (1.33M params) by 3.45 dB in PSNR.

• Qualitative Performance:

  • Visual results show LRDUN recovers intricate textures and sharp edges better than competing methods, which often suffer from over-smoothing or artifacts.
  • The model demonstrated strong generalization on real-world data without fine-tuning, maintaining spectral consistency where other methods degraded.

• Ablation Studies:

  • LR Embedding: Proved that decomposing the sensing model (LRDUN) is superior to applying LR as a post-processing regularizer or using nuclear norm minimization.
  • GFUM: Confirmed that increasing feature dimension $C$ (where $C > k$) improves performance, validating the auxiliary component's role in carrying complementary information.
  • QR Retraction: Essential for stabilizing the optimization of the spectral basis; without it, the model fails to converge.

5. Significance

This paper establishes a new paradigm for SCI reconstruction by shifting the focus from recovering the full high-dimensional data cube to estimating compact, low-dimensional physical components.

• Physics-Informed Efficiency: By embedding the low-rank prior directly into the sensing model rather than just as a regularizer, the method reduces the computational burden and ill-posedness at the source.
• Scalability: The GFUM allows the network to scale its representational power independently of the physical constraints, offering a flexible design for future SCI systems.
• Practical Impact: The significant reduction in computational cost (FLOPs) while maintaining SOTA accuracy makes LRDUN highly suitable for real-time applications and deployment on resource-constrained hardware, bridging the gap between theoretical reconstruction quality and practical efficiency.