Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging
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Marcello Carioni
Abstract
Unsupervised deep learning approaches have recently become one of the crucial research areas in imaging owing to their ability to learn expressive and powerful reconstruction operators, even when paired high-quality training data is scarcely available. In this chapter, we review theoretically principled unsupervised learning schemes for solving imaging inverse problems, with a particular focus on methods rooted in optimal transport and convex analysis. We begin by reviewing the optimal transport-based unsupervised approaches, such as the cycle-consistency-based models and learned adversarial regularization methods, which have clear probabilistic interpretations. Subsequently, we give an overview of recent works on provably convergent learned optimization algorithms applied to accelerate the solution of imaging inverse problems, alongside their dedicated unsupervised training schemes. We also survey provably convergent plug-and-play algorithms (based on gradient-step deep denoisers), which are among the most important and widely applied unsupervised approaches for imaging problems, along with some learned explicit deep neural network-based regularizers. Together with a detailed survey, we provide an overview of the key mathematical results that underlie the methods reviewed in the chapter to keep the discussion self-contained.
Abstract
Unsupervised deep learning approaches have recently become one of the crucial research areas in imaging owing to their ability to learn expressive and powerful reconstruction operators, even when paired high-quality training data is scarcely available. In this chapter, we review theoretically principled unsupervised learning schemes for solving imaging inverse problems, with a particular focus on methods rooted in optimal transport and convex analysis. We begin by reviewing the optimal transport-based unsupervised approaches, such as the cycle-consistency-based models and learned adversarial regularization methods, which have clear probabilistic interpretations. Subsequently, we give an overview of recent works on provably convergent learned optimization algorithms applied to accelerate the solution of imaging inverse problems, alongside their dedicated unsupervised training schemes. We also survey provably convergent plug-and-play algorithms (based on gradient-step deep denoisers), which are among the most important and widely applied unsupervised approaches for imaging problems, along with some learned explicit deep neural network-based regularizers. Together with a detailed survey, we provide an overview of the key mathematical results that underlie the methods reviewed in the chapter to keep the discussion self-contained.
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
