University of Cambridge
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.