Deep Bayesian inversion
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Jonas Adler
Abstract
Characterizing statistical properties of solutions of inverse problems is essential in many applications, and in particular those that involve uncertainty quantification. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for many imaging applications, such as those that arise in medical imaging. We introduce two novel deep learning-based methods for using Bayesian inversion to solve large-scale inverse problems: a sampling-based method that relies on a Wasserstein GAN with a novel mini-discriminator and a direct approach that trains a neural network with a novel loss function. The performance of both methods is demonstrated on clinical ultra low dose 3D helical CT image reconstruction. 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 promising performance, which clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.
Abstract
Characterizing statistical properties of solutions of inverse problems is essential in many applications, and in particular those that involve uncertainty quantification. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for many imaging applications, such as those that arise in medical imaging. We introduce two novel deep learning-based methods for using Bayesian inversion to solve large-scale inverse problems: a sampling-based method that relies on a Wasserstein GAN with a novel mini-discriminator and a direct approach that trains a neural network with a novel loss function. The performance of both methods is demonstrated on clinical ultra low dose 3D helical CT image reconstruction. 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 promising performance, which clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
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Part I: Mathematical aspects of data-driven methods in inverse problems
- On optimal regularization parameters via bilevel learning 1
- Learned regularization for inverse problems 39
- Inverse problems with learned forward operators 73
- Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
- Learned reconstruction methods for inverse problems: sample error estimates 163
- Statistical inverse learning problems with random observations 201
- General regularization in covariate shift adaptation 245
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Part II: Applications of data-driven methods in inverse problems
- Analysis of generalized iteratively regularized Landweber iterations driven by data 273
- Integration of model- and learning-based methods in image restoration 303
- Dynamic computerized tomography using inexact models and motion estimation 331
- Deep Bayesian inversion 359
- Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
- Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
- Classification with neural networks with quadratic decision functions 471
- Index 495
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
-
Part I: Mathematical aspects of data-driven methods in inverse problems
- On optimal regularization parameters via bilevel learning 1
- Learned regularization for inverse problems 39
- Inverse problems with learned forward operators 73
- Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
- Learned reconstruction methods for inverse problems: sample error estimates 163
- Statistical inverse learning problems with random observations 201
- General regularization in covariate shift adaptation 245
-
Part II: Applications of data-driven methods in inverse problems
- Analysis of generalized iteratively regularized Landweber iterations driven by data 273
- Integration of model- and learning-based methods in image restoration 303
- Dynamic computerized tomography using inexact models and motion estimation 331
- Deep Bayesian inversion 359
- Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
- Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
- Classification with neural networks with quadratic decision functions 471
- Index 495
