Ward receives IMA Prize in Mathematics and its Applications

ICES Professor Rachel Ward received the 2016 Institute for Mathematics and its Applications' (IMA) Prize in Mathematics and its Applications.

Ward, an associate professor in mathematics, was recognized for her contributions to the mathematics of machine learning and signal processing.

The prize is awarded annually to a mathematical scientist who is within 10 years of having earned her Ph.D. degree. The award recognizes an individual who has made a transformative impact on the mathematical sciences and their applications.

Ward shared the prize with Deanna Needell, an associate professor in the Department of Mathematics at Claremont McKenna College. The two are frequent collaborators, and according to the IMA, both are lauded by their peers as being among the most talented applied analysts in the country. Their most oft mentioned work in respect to this award was their 2013 paper on “Stable image reconstruction using total variation minimization,” published in the SIAM Journal on Imaging Sciences.

IMA says a particularly striking application of the paper's principles arises in medical imaging, such as magnetic resonance imaging (MRI), where the underlying image to be recovered is, say, a horizontal section of a brain or neck. Like most natural images of interest, this section can be thought of as a two-dimensional function that will be constant or slowly varying over most of the domain, interrupted only by sharp changes across a low-dimensional set of values corresponding to edges.

At the same time, each measurement in an MRI scan corresponds to a Fourier transform component, representing the response of the image to a particular frequency. Each measurement takes time and costs money, and thus it is desirable to obtain high-quality MRI reconstructions using as few measurements as possible.

“My work has answered questions like: what is a good subset of frequencies to take if the total scan time is limited to one hour, or alternatively, if one has a fixed budget of frequencies? And how should one reconstruct the underlying image from these frequencies?” Ward said. “The joint work with Deanna gave theoretical guarantees for a popular reconstruction method used in practice, total variation minimization, and suggested a stochastic sampling strategy for selecting frequencies which achieves these guarantees.”

Needell says that “rather than measuring in every ‘direction’ as in a typical MRI, compressed sensing promotes measuring in a small number of random directions, and it turns out that is enough to still ensure accurate image reconstruction.”

Other applications for this type of data acquisition and analysis include sensor and distributed networks, statistical problems, compression, and image processing problems.

Ward credits her Ph.D. advisor, Ingrid Daubechies, for getting her interested in these types of problems.

“Her construction of compactly supported smooth wavelets. The combination of practicality and mathematical beauty blew my mind,” Ward said.