MoDL: Model Based Deep Learning Architecture for Inverse Problems (1712.02862v4)

Abstract: We introduce a model-based image reconstruction framework with a convolution neural network (CNN) based regularization prior. The proposed formulation provides a systematic approach for deriving deep architectures for inverse problems with the arbitrary structure. Since the forward model is explicitly accounted for, a smaller network with fewer parameters is sufficient to capture the image information compared to black-box deep learning approaches, thus reducing the demand for training data and training time. Since we rely on end-to-end training, the CNN weights are customized to the forward model, thus offering improved performance over approaches that rely on pre-trained denoisers. The main difference of the framework from existing end-to-end training strategies is the sharing of the network weights across iterations and channels. Our experiments show that the decoupling of the number of iterations from the network complexity offered by this approach provides benefits including lower demand for training data, reduced risk of overfitting, and implementations with significantly reduced memory footprint. We propose to enforce data-consistency by using numerical optimization blocks such as conjugate gradients algorithm within the network; this approach offers faster convergence per iteration, compared to methods that rely on proximal gradients steps to enforce data consistency. Our experiments show that the faster convergence translates to improved performance, especially when the available GPU memory restricts the number of iterations.

• The paper introduces a framework that integrates data consistency enforcement and CNN denoising to improve image reconstructions in inverse problems.
• It leverages weight sharing and end-to-end training to reduce network parameters and reliance on extensive training data, boosting efficiency.
• Experimental results on multi-channel brain MRI data demonstrate superior performance, achieving high PSNR values even at high acceleration factors.

MoDL: Model-Based Deep Learning Architecture for Inverse Problems

Hemant K. Aggarwal, Merry P. Mani, and Mathews Jacob's paper on "MoDL: Model-Based Deep Learning Architecture for Inverse Problems" presents an innovative framework blending model-based image reconstruction with convolutional neural network (CNN) regularization priors. This solution particularly emphasizes addressing inverse problems in imaging by integrating specific forward models, leading to more efficient and effective reconstructions.

The critical advantage of the MoDL framework is its systematic approach to deriving deep architectures tailored for various inverse problems, thus greatly enhancing image reconstruction quality. Unlike direct inversion methods that often require extensive networks with numerous parameters and large datasets, this model-based approach leverages smaller networks due to its explicit incorporation of the forward model, reducing the need for substantial training data and extensive training time.

Framework Overview

The MoDL framework is built on a variational approach combining a data consistency term with a learned CNN-based regularization prior. Specifically, the image reconstruction is formulated as: $$\mathbf{x} = \arg \min_{\mathbf{x}} \|\mathcal{A}(\mathbf{x}) - \mathbf{b}\|_2^2 + \lambda \|\mathbf{x} - \mathcal{D}_{\mathbf{w}}(\mathbf{x})\|^2$$ where $\mathcal{A}$ represents the imaging system operator, $\mathbf{b}$ denotes the measurements, $\lambda$ is a regularization parameter, and $\mathcal{D}_{\mathbf{w}}$ is the CNN denoiser.

In MoDL, the iterative solution alternates between applying a CNN denoiser and enforcing data consistency through numerical optimization, specifically using conjugate gradient (CG) methods for complex forward models. This approach ensures a tighter integration between the forward model and the learned priors, leading to improved reconstruction accuracy.

Methodology

• Network Sharing: Unlike traditional deep learning methods, which train separate networks for each iteration or noise level, MoDL relies on weight sharing across iterations, maintaining the same CNN parameters throughout. This significantly lowers the number of network parameters and reduces overfitting risk, especially beneficial when training data are limited.
• End-to-End Training: The MoDL framework utilizes end-to-end training, optimizing all parameters jointly to enhance performance. This approach outperforms pre-trained strategies by tailoring the CNN weights specifically for the targeted imaging recovery task.
• Data Consistency Enforcement: By using CG optimization within the network, MoDL ensures efficient and accurate enforcement of data consistency. This method achieves faster convergence compared to proximal gradient methods, which are limited by GPU memory constraints.

Experimental Results and Analysis

The paper evaluates MoDL using multi-channel brain MRI data, comparing its performance against various state-of-the-art methods, including compressed sensing techniques and other deep learning frameworks. The experiments demonstrate MoDL's superior performance across different acceleration factors (ranging from 2x to 20x) and noise levels.

Key numeric results highlight:

• At 6x acceleration, MoDL achieves an average PSNR of 39.24 dB, outperforming methods like CSTV (34.50 dB) and CG-PD-NS (36.52 dB).
• Even at high acceleration factors (e.g., 10x), MoDL maintains robust reconstruction quality (average PSNR of 37.35 dB), showcasing its ability to generalize across varying acquisition settings.

Implications and Future Directions

The integration of numerical optimization within the deep network paves the way for more complex forward models and hybrid regularization strategies, potentially revolutionizing application areas like parallel imaging in MRI. Furthermore, the weight-sharing strategy offers a scalable solution adaptable to different imaging tasks without extensive retraining.

Future research could explore extending the framework to other modalities and incorporating additional types of regularization priors. Additionally, investigating the theoretical convergence properties of the MoDL framework could further solidify its foundation and applicability in broader contexts.

In conclusion, the MoDL framework represents a significant advance in model-based deep learning for imaging reconstruction, offering a practical, efficient, and scalable solution poised to impact various domains within medical imaging and beyond.