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Task adapted reconstruction for inverse problems
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta.ORCID iD: 0000-0001-9928-3407
Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden ; Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway.
Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

Keywords [en]
Inverse problems, image reconstruction, tomography, deep learning, feature reconstruction, segmentation, classification, regularization
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-262725OAI: oai:DiVA.org:kth-262725DiVA, id: diva2:1362340
Note

QC 20191021

Available from: 2019-10-18 Created: 2019-10-18 Last updated: 2019-10-21Bibliographically approved
In thesis
1. Data-driven Methods in Inverse Problems
Open this publication in new window or tab >>Data-driven Methods in Inverse Problems
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis on data-driven methods in inverse problems we introduce several new methods to solve inverse problems using recent advancements in machine learning and specifically deep learning. The main goal has been to develop practically applicable methods, scalable to medical applications and with the ability to handle all the complexities associated with them.

In total, the thesis contains six papers. Some of them are focused on more theoretical questions such as characterizing the optimal solutions of reconstruction schemes or extending current methods to new domains, while others have focused on practical applicability. A significant portion of the papers also aim to bringing knowledge from the machine learning community into the imaging community, with considerable effort spent on translating many of the concepts. The papers have been published in a range of venues: machine learning, medical imaging and inverse problems.

The first two papers contribute to a class of methods now called learned iterative reconstruction where we introduce two ways of combining classical model driven reconstruction methods with deep neural networks. The next two papers look forward, aiming to address the question of "what do we want?" by proposing two very different but novel loss functions for training neural networks in inverse problems. The final papers dwelve into the statistical side, one gives a generalization of a class of deep generative models to Banach spaces while the next introduces two ways in which such methods can be used to perform Bayesian inversion at scale.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 196
Series
TRITA-SCI-FOU ; 2019;49
Keywords
Inverse Problems, Machine Learning, Tomography
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-262727 (URN)978-91-7873-334-7 (ISBN)
Public defence
2019-10-31, F3, Lindstedtsvägen26, KTH, Stockholm, 14:00 (English)
Opponent
Supervisors
Funder
Swedish Foundation for Strategic Research
Available from: 2019-10-21 Created: 2019-10-18 Last updated: 2019-10-21Bibliographically approved

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