[Paper Review] Conditional Normalizing Flows for Low-Dose Computed Tomography Image Reconstruction

Alexander Denker, Maximilian Schmidt+3 more|arXiv (Cornell University)|Jun 11, 2020

Medical Imaging Techniques and Applications25 references4 citations

TL;DR

This paper proposes a hybrid conditional normalizing flow that integrates the filtered back-projection (FBP) as a conditioner to reconstruct low-dose CT images from noisy measurements. By leveraging flow-based density estimation instead of MSE minimization, the method achieves state-of-the-art structural similarity (SSIM) while maintaining competitive PSNR, demonstrating superior structural fidelity over deep learning baselines despite no explicit MSE training.

ABSTRACT

Image reconstruction from computed tomography (CT) measurement is a challenging statistical inverse problem since a high-dimensional conditional distribution needs to be estimated. Based on training data obtained from high-quality reconstructions, we aim to learn a conditional density of images from noisy low-dose CT measurements. To tackle this problem, we propose a hybrid conditional normalizing flow, which integrates the physical model by using the filtered back-projection as conditioner. We evaluate our approach on a low-dose CT benchmark and demonstrate superior performance in terms of structural similarity of our flow-based method compared to other deep learning based approaches.

Motivation & Objective

• Address the challenge of reconstructing high-quality CT images from low-dose, noisy measurements using a statistical inverse problem framework.
• Overcome limitations of MSE-based deep learning methods that fail to capture complex, non-i.i.d. noise and artifact structures in CT reconstructions.
• Develop a conditional density estimator that models the high-dimensional posterior distribution $ p(x|y^\delta) $ for low-dose CT using flow-based generative modeling.
• Integrate physical knowledge from the CT forward model (via FBP) into the conditioning mechanism to improve sample quality and generalization.
• Demonstrate that flow-based likelihood optimization leads to better structural preservation than MSE-based training, especially in SSIM metrics.

Proposed method

• Propose a conditional normalizing flow (cNF) that models the posterior $ p(x|y^\delta) $ using invertible neural networks with coupling layers.
• Use the filtered back-projection (FBP) of low-dose measurements as the conditioning input to inject physical consistency and reduce distributional complexity.
• Adopt a multi-scale RealNVP-like architecture with 6 coupling layers per scale, followed by Haar-based downsampling and invertible 1x1 convolutions for feature permutation.
• Train the model end-to-end using maximum likelihood estimation on the log-likelihood of the ground truth images given the low-dose measurements.
• Condition the flow on FBP outputs via a shared encoder network, enabling the model to learn residual corrections to the FBP reconstruction.
• Estimate the conditional mean via Monte Carlo sampling from the learned flow distribution to produce final reconstructions.

Experimental results

Research questions

• RQ1 Can conditional normalizing flows effectively model the complex, non-Gaussian posterior distribution in low-dose CT reconstruction?
• RQ2 Does integrating the physical FBP reconstruction as a conditioner improve the quality and stability of flow-based CT reconstruction compared to end-to-end learned models?
• RQ3 How does likelihood-based training in normalizing flows compare to MSE-based optimization in terms of structural image quality metrics like SSIM?
• RQ4 Can flow-based models capture structural dependencies and artifacts in low-dose CT data better than standard deep learning baselines?
• RQ5 Is the conditional mean of the flow-based posterior a robust and high-quality reconstruction strategy when the posterior is uni-modal?

Key findings

• The proposed cNF model achieves a test SSIM of 0.88 on the LoDoPaB-CT dataset, outperforming all other deep learning and classical methods, including FBP + U-Net and learned iterative algorithms.
• Despite not being trained to minimize MSE, the model achieves a PSNR of 35.68, which is comparable to state-of-the-art MSE-optimized models like FBP + U-Net and learned PD.
• Both PSNR and SSIM improve with increasing number of Monte Carlo samples used to compute the conditional mean, indicating that the model’s uncertainty estimates are meaningful and sample quality improves with more draws.
• The model’s superior SSIM performance confirms that likelihood-based training captures structural image properties better than MSE-based optimization, which tends to favor pixel-wise accuracy over perceptual quality.
• The integration of FBP as a conditioner enables the flow to focus on learning residual corrections, leading to faster convergence and improved generalization on the low-dose CT benchmark.
• The results suggest that flow-based models are well-suited for statistical inverse problems where the likelihood is complex and non-Gaussian, particularly when artifacts and noise dependencies are non-i.i.d.