My postdoc Il Yong Chun and I recently completed a new paper titled Compressed sensing and parallel acquisition. In it we present a theoretical framework for the use of compressed sensing in multi-sensor systems. These systems are found numerous applications, ranging from parallel MRI, to multi-view imaging and generalized sampling theory. Our work provides a first mathematical analysis of the gains offered by parallel acquisition and compressed sensing, and presents new insight into questions of optimal sensor design.
My new collaboration (with Rick Archibald, Anne Gelb, Jan Hesthaven, Rodrigo Platte, Guohui Song and Ed Walsh) was featured in the 2015 American Institute of Mathematics Newsletter.
To quote from the article: …our group is proving how other data inherent to the [MRI] scanning process, such as resonant frequencies and signal decay rates, which are currently used only to provide contrast, can be useful in diagnosing a condition or measuring a response to treatment. Much more information can be extrapolated from the same scan: temperature, blood ow, di usion, structure, and physiology, for example. We’re developing nonconventional image reconstruction techniques to get this information faster and better than has ever been possible before.
It is possible to approximate analytic functions with endpoint singularities using nearly the same number of degrees of freedom as is needed for analytic functions. My new paper with Jesus-Martin Vaquero and Mark Richardson introduces a technique to do this.
My new paper considers function interpolation via infinite-dimensional compressed sensing with weighted l1 minimization. In it I derive new recovery guarantees which are provably sharp in a number of important cases, and explain the practical usefulness of including weights in the optimization.
My summer undergraduate student Matthew King-Roskamp won second place in the poster competition at the 2015 SFU Symposium on Mathematics and Computation. Congratulations Matt!
I was awarded the AMMCS Kolmogorov-Wiener Prize for Young Researchers at the 2015 AMMCS-CAIMS Congress.
Rodrigo Platte and I have a new paper in which we propose a mapped polynomial method for high accuracy function approximation on arbitrary grids. We show how the method achieves numerical stability with near-optimal numbers of degrees of freedom – in effect circumventing the so-called impossibility theorem – by sacrificing classical convergence for finite, but arbitrarily high accuracy.
My new paper proposes an infinite-dimensional l1 minimization framework for function interpolation and analzes its approximation properties. In particular, I show that weighted l1 minimization with polynomials is an optimal method for approximating smooth, one-dimensional functions from scattered data.
I was awarded an Alfred P. Sloan Research Fellowship in Mathematics.