Compressed Sensing with Deep Image Prior and Learned Regularization (1806.06438v4)

Abstract: We propose a novel method for compressed sensing recovery using untrained deep generative models. Our method is based on the recently proposed Deep Image Prior (DIP), wherein the convolutional weights of the network are optimized to match the observed measurements. We show that this approach can be applied to solve any differentiable linear inverse problem, outperforming previous unlearned methods. Unlike various learned approaches based on generative models, our method does not require pre-training over large datasets. We further introduce a novel learned regularization technique, which incorporates prior information on the network weights. This reduces reconstruction error, especially for noisy measurements. Finally, we prove that, using the DIP optimization approach, moderately overparameterized single-layer networks can perfectly fit any signal despite the non-convex nature of the fitting problem. This theoretical result provides justification for early stopping.

• The paper introduces a novel compressed sensing approach that integrates untrained deep image priors to reconstruct images from limited measurements.
• It leverages untrained deep generative models and a Gaussian-based learned regularization to reduce reconstruction error under noisy conditions.
• Empirical evaluations across diverse datasets demonstrate superior performance compared to traditional methods, highlighting its practical impact in data-constrained environments.

Compressed Sensing with Deep Image Prior and Learned Regularization

The paper "Compressed Sensing with Deep Image Prior and Learned Regularization" introduces an innovative approach to addressing the compressed sensing problem by integrating deep image prior (DIP) with learned regularization. The authors have crafted a methodology that is novel in its reliance on an untrained deep generative model, circumventing the necessity for extensive pre-training on large datasets—an obstacle in medical imaging applications where image data accessibility is limited.

Methodological Framework

The foundational mechanism of the compressed sensing problem tackled herein involves reconstructing an unknown signal from a limited set of measurements, defined as $y = Ax^* + \eta$, where $A$ is the measurement matrix and $\eta$ represents noise. Traditional methods necessitating sparse representations have been expanded to more complex models that incorporate manifold assumptions or generative models. In this paper, the authors leverage the DIP’s capability to operate without pre-training, which broadens its applicability across domains lacking voluminous datasets.

Two central innovations are introduced:

1. Deep Image Prior for Compressed Sensing (CS-DIP): By initializing a DCGAN generator’s weights randomly, optimizing with gradient descent is employed to minimize discrepancies between generated output and observed measurements. The DIP inherently aids in preserving structural image priors.
2. Learned Regularization Technique: Enabling enhanced accuracy by integrating a prior on network weights based on Gaussian distribution assumptions, thereby reducing reconstruction error amidst noisy datasets. This regularization acts as a refined adaptive weight decay, learned from a minimal set of similar images rather than extensive training data.

Theoretical Insights

A pivotal theoretical contribution of this paper lies in demonstrating, via mathematical rigor, that the DIP optimization technique with overparameterized networks can fit any signal perfectly. The authors prove the convergence of gradient descent even within nonconvex optimization spaces, substantiating the necessity of early stopping to forestall overfitting—validating empirical heuristic findings from prior studies.

Empirical Evaluation

The authors conduct a robust series of experiments across varied datasets:

• Grayscale and RGB image datasets, such as MNIST and medical chest X-rays, demonstrate the method’s efficacy in conditions of limited measurements.
• Comparative analyses with baseline methods like BM3D-AMP, TVAL3, and Lasso indicate superior performance, especially notable in scenarios requiring fewer measurements with substantial noise.
• Learned regularization yields quantifiable reductions in mean-squared error (MSE), particularly advantageous amidst noise or compression, thus highlighting its potential for practical data-constrained environments.

Implications and Future Directions

This research propels compressed sensing forward by mitigating the constraints of data availability linked to pre-trained models—a leap with theoretical backing that deepens understanding of neural network optimization dynamics in ill-posed problem spaces. It opens avenues for deployment in medical settings where image acquisition is challenging. Future work might delve into exploring alternative neural architectures within DIP, finessing domain-specific regularization techniques, and broadening its applicability to various non-linear inverse problems.

Ultimately, the paper enriches both theoretical landscapes and practical applications of compressed sensing, offering significant contributions for researchers working with deep learning models, signal processing, and fields requiring infrastructure for handling scant data resources.