[Paper Review] Adversarial Regularizers in Inverse Problems

Sebastian Lunz, Ozan Öktem+1 more|arXiv (Cornell University)|May 29, 2018

Medical Imaging Techniques and Applications24 references79 citations

TL;DR

The paper introduces a neural-network-based regularization functional trained as a critic to distinguish ground-truth images from unregularized reconstructions, enabling unsupervised training for inverse problems and improving denoising and CT reconstruction.

ABSTRACT

Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying data-driven approaches to inverse problems, using a neural network as a regularization functional. The network learns to discriminate between the distribution of ground truth images and the distribution of unregularized reconstructions. Once trained, the network is applied to the inverse problem by solving the corresponding variational problem. Unlike other data-based approaches for inverse problems, the algorithm can be applied even if only unsupervised training data is available. Experiments demonstrate the potential of the framework for denoising on the BSDS dataset and for computed tomography reconstruction on the LIDC dataset.

Motivation & Objective

• Introduce a neural network regularizer within a variational framework for inverse problems.
• Develop a scalable training method that uses distribution discrimination between ground-truth and pseudo-inverse reconstructions.
• Establish theoretical properties of the learned regularizers and analyze distributional behavior.
• Demonstrate effectiveness on denoising and CT reconstruction without supervised data.

Proposed method

• Replace hand-crafted regularization with a neural network Psi_Theta as the regularizer in the variational problem.
• Train Psi_Theta as a critic to distinguish ground-truth samples from pseudo-inverse reconstructions, using a Wasserstein-GAN style loss with a gradient penalty.
• Solve the inverse problem by minimizing ||Ax−y||^2 + λΨ_Θ(x) via gradient descent.
• Provide theoretical results on Wasserstein distance decay and manifold-distance regularization under a weak data-manifold assumption.
• Estimate the regularization parameter λ from noise level using λ = 2 E_{e~p_n} ||A^* e||_2.
• Discuss stability and convergence properties of the approach.

Experimental results

Research questions

• RQ1 Can a neural network learned as a regularization functional improve inverse problem reconstructions under unsupervised training data?
• RQ2 What are the theoretical properties (e.g., Wasserstein distance decay, manifold-aligned regularization) of the learned regularizer?
• RQ3 How does the approach compare to classical variational methods and supervised learning in denoising and CT reconstruction?
• RQ4 How can λ be efficiently estimated from noise characteristics without retraining?

Key findings

Dataset	Method	PSNR (dB)	SSIM
BSDS500 denoising	Total Variation	26.3	0.836
BSDS500 denoising	Unsupervised Adversarial Regularizer	28.2	0.892
BSDS500 denoising	Supervised Denoising NN	28.8	0.908
LIDC CT high noise	Filtered Backprojection	14.9	0.227
LIDC CT high noise	Total Variation	27.7	0.890
LIDC CT high noise	Post-Processing (supervised)	31.2	0.936
LIDC CT high noise	RED	29.9	0.904
LIDC CT high noise	Unsupervised Adversarial Reg.	30.5	0.927
LIDC CT low noise	Filtered Backprojection	23.3	0.604
LIDC CT low noise	Total Variation	30.0	0.924
LIDC CT low noise	Post-Processing (supervised)	33.6	0.955
LIDC CT low noise	RED	32.8	0.947
LIDC CT low noise	Unsupervised Adversarial Reg.	32.5	0.946

• The adversarial regularizer outperforms total variation in denoising on BSDS500 under unsupervised training.
• On CT reconstruction (LIDC/IDRI), the unsupervised adversarial regularizer yields PSNR/SSIM superior to many baselines and competitive with supervised post-processing.
• The method achieves comparable or superior visual quality to supervised methods while using unsupervised data only.
• Theoretical results show the gradient-flow under the learned regularizer reduces the Wasserstein distance to the ground-truth distribution and aligns reconstructed images with the data manifold.
• The framework provides stability guarantees for the data-term minimization under mild assumptions.