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Research Fellows Directory

Anders Hansen

Dr Anders Hansen

Research Fellow

Organisation

University of Cambridge

Research summary

Applied harmonic analysis is a rapidly growing area of mathematics due to its vast

applications in engineering and physics. Arguably one of the most important

applications is medical imaging such as Magnetic Resonance Imaging (MRI) or X-

ray Tomography (CT).

The way an MRI machine works is surprisingly mathematical. The object of

interest (the image), say the brain, can be modeled as a mathematical function. At

every point of the image there is a number describing the color density (this is

exactly a mathematical function), and if one knows this function, one knows the

image completely. The problem with an MRI machine is that it does not measure

or sample the function itself, but rather a transformation of the function, namely

the Fourier transform (this is due to the physics behind the machine). The task is

therefore to reconstruct the function from this transformation. Due to fundamental

results in sampling theory, it is known that if one was able to sample an infinite

number of measurements one would get a perfect reconstruction of the function.

This is of course impossible in practice, and therefore the task is the following:

given a finite number of samples, find the best possible reconstruction. It is this

task that is the very core of sampling theory.

Impact: Sampling theory is the core in medical imaging, but has many other

applications such as in geophysics (seismic), radar surveillance and wireless

communications. The potential impact of my specific research is the design of

reconstruction algorithms for the next generation of MRI machines. Directly

translated to layman terms this means much better quality of MRI images that

again would hopefully lead to better medical diagnosis.

Grants awarded

From sampling to computation - Applied harmonic analysis meets foundations

Scheme: University Research Fellowship

Dates: Oct 2017 - Sep 2020

Value: £304,023.13

From Spectra to Sampling - Functional Analysis meets Applied Harmonic Analysis

Scheme: University Research Fellowship

Dates: Oct 2012 - Sep 2017

Value: £389,931.69

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