The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
Aalto

Discretization and Bayesian Modeling in Inverse Problems and Imaging

Tapio Helin

Doctoral dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Information and Natural Sciences for public examination and debate in Auditorium K at the Aalto University School of Science and Technology (Espoo, Finland) on the 12th of March 2010 at 12 noon.

Overview in PDF format (ISBN 978-952-60-3041-8)   [271 KB]
Dissertation is also available in print (ISBN 978-952-60-3040-1)

Abstract

In this thesis the Bayesian modeling and discretization are studied in inverse problems related to imaging. The treatise consists of four articles which focus on the phenomena that appear when more detailed data or a priori information become available. Novel Bayesian methods for solving ill-posed signal processing problems in edge-preserving manner are introduced and analysed. Furthermore, modeling photographs in image processing problems is studied and a novel model is presented.

This thesis consists of an overview and of the following 4 publications:

  1. Tapio Helin. 2009. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems and Imaging, volume 3, number 4, pages 567-597.
  2. Tapio Helin and Matti Lassas. 2007. Bayesian signal restoration and Mumford-Shah functional. Proceedings in Applied Mathematics and Mechanics, volume 7, number 1, Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, pages 2080013-2080014.
  3. Tapio Helin and Matti Lassas. 2009. Hierarchical models in statistical inverse problems and the Mumford–Shah functional. arXiv:0908.3396v2 [math.ST].
  4. Tapio Helin, Matti Lassas, and Samuli Siltanen. 2010. Infinite photography: new mathematical model for high-resolution images. Journal of Mathematical Imaging and Vision, volume 36, number 2, pages 140-158.

Keywords: inverse problems, Mumford–Shah functional, Bayesian inversion, hierarchical modeling, discretization invariance, image model, Borel measure, metric space

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Last update 2011-05-26