FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging (2008.02683v3)

Abstract: Inverse problems are essential to imaging applications. In this paper, we propose a model-based deep learning network, named FISTA-Net, by combining the merits of interpretability and generality of the model-based Fast Iterative Shrinkage/Thresholding Algorithm (FISTA) and strong regularization and tuning-free advantages of the data-driven neural network. By unfolding the FISTA into a deep network, the architecture of FISTA-Net consists of multiple gradient descent, proximal mapping, and momentum modules in cascade. Different from FISTA, the gradient matrix in FISTA-Net can be updated during iteration and a proximal operator network is developed for nonlinear thresholding which can be learned through end-to-end training. Key parameters of FISTA-Net including the gradient step size, thresholding value and momentum scalar are tuning-free and learned from training data rather than hand-crafted. We further impose positive and monotonous constraints on these parameters to ensure they converge properly. The experimental results, evaluated both visually and quantitatively, show that the FISTA-Net can optimize parameters for different imaging tasks, i.e. Electromagnetic Tomography (EMT) and X-ray Computational Tomography (X-ray CT). It outperforms the state-of-the-art model-based and deep learning methods and exhibits good generalization ability over other competitive learning-based approaches under different noise levels.

• The paper introduces FISTA-Net, a model-based deep learning network that unfolds the FISTA algorithm to solve imaging inverse problems.
• FISTA-Net learns key parameters like gradient step size and thresholding values adaptively from data, offering a tuning-free advantage over traditional methods.
• Comparative analysis shows FISTA-Net achieves superior reconstruction quality on EMT and CT datasets compared to existing model-based and network-based approaches.

FISTA-Net: A Fast Iterative Shrinkage Thresholding Network for Imaging Inverse Problems

The paper presents FISTA-Net, an innovative model-based deep learning approach designed to tackle inverse problems in imaging, particularly focusing on applications such as electromagnetic tomography (EMT) and X-ray computational tomography (CT). FISTA-Net synthesizes the interpretability and robustness of the Fast Iterative Shrinkage/Thresholding Algorithm (FISTA) with the flexibility and adaptability of deep neural networks. The architecture of FISTA-Net is built by unfolding the steps of FISTA into a cascaded deep network incorporating gradient descent, proximal mapping, and momentum modules.

Key Features & Methodology:

1. Model-Based Learning: FISTA-Net is grounded in the classical FISTA framework, ensuring a balance between data fidelity and noise suppression through its iterative process. The paper enhances this by unfolding the algorithm into a network structure allowing internal parameters like gradient step size, thresholding values, and momentum weights to be learned autonomously from training data.
2. Adaptive Gradient & Thresholding: Unlike traditional FISTA, the gradient matrix in FISTA-Net is dynamically updated during iterations, and a novel proximal network facilitates nonlinear thresholding. The adaptive learning of these key parameters offers a tuning-free advantage, enhancing convergence and optimization flexibility across diverse imaging tasks.
3. Convergence Mechanisms: Constraints are imposed on the learned parameters to ensure positive and monotonic convergence behavior through iterations, addressing typical anomalies found in unconstrained optimization scenarios. This ensures the stability and reliability of the reconstruction process.

Comparative Analysis:

In empirical evaluations on both EMT and CT datasets, FISTA-Net consistently outperformed existing model-based and network-based methods, including state-of-the-art approaches such as ISTA-Net and FBPConvNet. It demonstrated superior reconstruction quality, especially in low signal-to-noise scenarios, proving its capacity for robust generalization beyond the training dataset. Quantitative metrics, such as PSNR and SSIM, confirmed the efficacy of FISTA-Net in preserving image details while effectively suppressing noise.

Practical & Theoretical Implications:

The proposed architecture has practical implications for real-time applications where computational efficiency and model accuracy are critical, such as medical diagnostics and industrial imaging. The design leverages convolutional neural networks to represent complex non-linear proximal mappings, thereby expanding its usability across various non-linear imaging tasks without significant reconfiguration.

Theoretically, FISTA-Net offers a compelling framework to integrate deep learning with traditional optimization methods, paving the way for research that deepens the connection between variational models and data-driven architectures. Its unfolding approach provides a clear pathway for future work to explore more sophisticated learning paradigms that tightly couple physical model constraints with empirical data learning.

Future Directions:

Looking ahead, further exploration into adaptive moment estimation, integration with other iterative solvers or variational methods, and extending FISTA-Net capabilities to encompass broader inverse problem categories may reveal novel insights and extensions. The quantitative validation presented in this paper also suggests opportunities for investigating reinforcement learning strategies to advance and refine real-time imaging solutions.

In summary, FISTA-Net is a highly promising approach combining robust theoretical foundations with innovative neural network architectures, offering significant advancements for imaging science.