Thomas Blumensath

Research in constrained inverse problems

My research interests are in the development and study of numerical methods for the inversion of ill-posed or underdetermined systems under non-convex constraints. I am particularly interested in the area of sparse and structure inverse problems and the emerging field of compressed sensing. Compressed sensing is an area at the intersection between numerical analysis, high dimensional geometry, convex and non-convex optimisation, computational harmonic analysis and signal processing, and deals with the inversion of underdetermined linear mappings between Euclidean spaces under a sparsity constraint. My recent contributions to this area have been in the development and study of provably efficient numerical algorithms for sparse and structured inverse problems and I have been one of the main drivers behind the extension of many of the ideas developed in compressed sensing to more general constraint sets and to a more general Hilbert space setting.

Research in sampling theory

Signals such as sounds, images and electromagnetic waves are at the heart of almost every scientific and engineering discipline. They are fundamental to modern medical technology as well as most technologies we encounter in our daily lives. The acquisition, transmission, storage, processing and interpretation of signals is therefore of the utmost importance. In our digital age, most storage, transmission and processing is done in the digital domain. This means that information about natural phenomena has to be measured and converted into a format suitable for digital processing.

I am working on novel approaches that exploit signal structure to represent signals using far fewer measurements than would be required by traditional approaches. There are many possible applications for these new techniques, for example, in medical imaging, reducing the number of measurements can significantly reduce the risk to patients.

My research focuses on three main aspects of the problem; 1) building signal models that encapsulate as much prior domain knowledge as possible, 2) designing sampling systems that measure the relevant signal information and 3) developing efficient computational strategies that use the signal models to reconstruct the signals from the measured information. This work extends many ideas recently proposed in the field of compressed sensing to more general signal models and more general Hilbert space settings.

Research in non-convexly constrained optimization for large data-sets

An important part in the development of novel sampling themes and inverse methods is the development of efficient computational optimization algorithms. One particular approach I have been pursuing are greedy methods. Here I have been developing two approaches, Pursuit type algorithms and projection based approaches.

Links to old pages

Thomas Blumensath
Symbiosis Research Fellow and Facilitator Applied Mathematics School of Mathematics Building 58 - Room 2087 University of Southampton SO17 1BJ Tel.: +44 (0) 23 8059 4546 thomas.blumensath@soton.ac.uk
Symbiosis: Mathematics in nano- and bio-engineering - School of Mathematics - Southampton Statistical Sciences Research Institute - School of Engineering Sciences
Research in constrained inverse problems My research interests are in the development and study of numerical methods for the inversion of ill-posed or underdetermined systems under non-convex constraints. I am particularly interested in the area of sparse and structure inverse problems and the emerging field of compressed sensing. Compressed sensing is an area at the intersection between numerical analysis, high dimensional geometry, convex and non-convex optimisation, computational harmonic analysis and signal processing, and deals with the inversion of underdetermined linear mappings between Euclidean spaces under a sparsity constraint. My recent contributions to this area have been in the development and study of provably efficient numerical algorithms for sparse and structured inverse problems and I have been one of the main drivers behind the extension of many of the ideas developed in compressed sensing to more general constraint sets and to a more general Hilbert space setting. Research in sampling theory Signals such as sounds, images and electromagnetic waves are at the heart of almost every scientific and engineering discipline. They are fundamental to modern medical technology as well as most technologies we encounter in our daily lives. The acquisition, transmission, storage, processing and interpretation of signals is therefore of the utmost importance. In our digital age, most storage, transmission and processing is done in the digital domain. This means that information about natural phenomena has to be measured and converted into a format suitable for digital processing. I am working on novel approaches that exploit signal structure to represent signals using far fewer measurements than would be required by traditional approaches. There are many possible applications for these new techniques, for example, in medical imaging, reducing the number of measurements can significantly reduce the risk to patients. My research focuses on three main aspects of the problem; 1) building signal models that encapsulate as much prior domain knowledge as possible, 2) designing sampling systems that measure the relevant signal information and 3) developing efficient computational strategies that use the signal models to reconstruct the signals from the measured information. This work extends many ideas recently proposed in the field of compressed sensing to more general signal models and more general Hilbert space settings. Research in non-convexly constrained optimization for large data-sets An important part in the development of novel sampling themes and inverse methods is the development of efficient computational optimization algorithms. One particular approach I have been pursuing are greedy methods. Here I have been developing two approaches, Pursuit type algorithms and projection based approaches. Links to old pages The School of Engineering at the University of Edinburgh has kindly agreed to host the following webpages for the foreseeable future: Compressed Sensing Sparse Representations for Signal Processing and Coding