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Deep Bayesian Inversion

Jonas Adler, Ozan Öktem

TL;DR

The paper addresses uncertainty quantification for large-scale inverse problems by introducing two deep learning-based Bayesian inversion frameworks: Deep Posterior Sampling using a conditional Wasserstein GAN with a novel mini-batch discriminator to sample from the posterior $p(x|y)$, and Deep Direct Estimation to learn Bayes estimators directly from data. It demonstrates the approach on ultra-low-dose $3$D CT imaging, computing the posterior mean and pointwise standard deviation and performing a hypothesis test for a liver lesion. The methods are computationally feasible (approximately 40 seconds per slice for 1000 posterior samples and about 0.08 seconds per slice for direct estimation) and yield consistent uncertainty quantification, supporting Bayesian inversion for time-critical imaging. The work highlights the potential to integrate reconstruction with clinical decision-making under uncertainty in large-scale medical imaging.

Abstract

Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

Deep Bayesian Inversion

TL;DR

The paper addresses uncertainty quantification for large-scale inverse problems by introducing two deep learning-based Bayesian inversion frameworks: Deep Posterior Sampling using a conditional Wasserstein GAN with a novel mini-batch discriminator to sample from the posterior , and Deep Direct Estimation to learn Bayes estimators directly from data. It demonstrates the approach on ultra-low-dose D CT imaging, computing the posterior mean and pointwise standard deviation and performing a hypothesis test for a liver lesion. The methods are computationally feasible (approximately 40 seconds per slice for 1000 posterior samples and about 0.08 seconds per slice for direct estimation) and yield consistent uncertainty quantification, supporting Bayesian inversion for time-critical imaging. The work highlights the potential to integrate reconstruction with clinical decision-making under uncertainty in large-scale medical imaging.

Abstract

Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

Paper Structure

This paper contains 35 sections, 2 theorems, 57 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Bayesian inversion
  3. Statistical Approach to Inverse Problems
  4. Choosing a prior model
  5. Computational feasibility
  6. Contribution
  7. Deep Bayesian Inversion
  8. Deep Posterior Sampling
  9. Deep Direct Estimation
  10. Numerical Experiments
  11. Set-up
  12. Training data
  13. Network architecture and training
  14. Results
  15. Estimators
  16. ...and 20 more sections

Key Result

Proposition 2

Assume that $Y$ be a measurable space, $W$ is a measurable Hilbert space, and $\mathsf{y}$ and $\mathsf{w}$ are $Y$- and $W$-valued random variables, respectively. Then, the conditional expectation $h^*(y) := \mathop{\mathrm{\mathbb{E}}}\nolimits \bigl[ \mathsf{w} \mid \mathsf{y} = y \bigr]$ solves The minimization above is taken over all $W$-valued measurable functions on $Y$.

Figures (9)

  • Figure 1: Top row shows a single random sample generated by a Gibbs type of roughness priors that are common in inverse problems in imaging (\ref{['sec:appendix_analytical_priors']}). Such a prior is proportional to $e^{-S(x)}$ and images show samples for different choices of $S$. Bottom row shows typical samples of normal dose CT images of humans. Ideally, a prior generates samples similar to those in the bottom row.
  • Figure 2: Test data: Normal dose image (left), subset of CT data from a ultra-low dose 3D helical scan (middle), and corresponding FBP reconstruction (right). Images are shown using a display window set to $[-150, 200]$.
  • Figure 3: Conditional mean and point-wise standard deviation (pStd) computed from test data (\ref{['fig:input_data']}) using posterior sampling (\ref{['sec:WGAN']}) and direct estimation (\ref{['sec:DirectUC']}).
  • Figure 4: The suspected tumor (red) and the reference region (blue) shown in the sample posterior mean image. Right plot shows average contrast differences between the tumor and reference region. The histogram is computed by posterior sampling applied to test data (\ref{['fig:input_data']}), the yellow curve is from direct estimation, and the true value is the red threshold.
  • Figure 5: Deep posterior samples (\ref{['sec:WGAN']}) on test data (\ref{['fig:input_data']}) shown using a display window set to [-150, 200] .
  • ...and 4 more figures

Theorems & Definitions (5)

  • Claim 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3