Speeding Medical Images

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MRI scans are one of the most important diagnostic techniques
that doctors can use, but it is very expensive technology, partly due
to the time it takes to produce and analyse such complex images.
Mathematics provides the central algorithms for analysing and
reconstructing the images in real time.

The two common algorithms, Sequential Backward Selection and
Sequential Forward Selection, force a compromise between analytical
speed and image quality.

Tappenden and supervisor, Associate Professor Ian Coope at the University of
Canterbury, chose this issue for her PhD as “the most interesting that uses the skill base
I have – linear algebra and optimisation,” she says.

The algorithms rely on Least Squares approximations and the L2 norm, but other
approximation criteria, like the L1 norm, may be useful. The im
ages are compressed before
processing using Fourier methods, but recent techniques such as compressed sensing are
also being explored as an alternative. MRI data comes in the form of a matrix
but it is not feasible to look at every combination of rows. “An existing criterion
uses the trace of the matrix,” says Tappenden, “whereas we’ve created an algorithm
which uses the determinant criterion to choose an optimal subset of rows to give an
accurate image.” Her paper was published in September in the IEEE journal Transactions
on Image Processing. The algorithms will be implemented by the engineers who
programme the machines.

Tappenden has also written some algorithms to reconstruct MRI scans from sparse
data sets. “If you have some conditions on the image – lots of zeros and few nonzeroes
– then you can do a really good reconstruction with only a tiny bit of data.
These are optimisation problems with some really nice properties.”