In it we explain how recent progress in the theory of compressed sensing allows one to significantly enhance the performance of sparse recovery techniques in imaging applications.
My former undergraduate student Anyi (Casie) Bao was a winner of the SFU Department of Mathematics Undergraduate Research Prize. Her work involved the development and analysis of compressed sensing-based strategies for correcting for corrupted measurements in Uncertainty Quantification. A draft version of the resulting paper can be found here:
Matt King-Roskamp – an undergraduate student in my group co-supervised with Simone Brugiapaglia – was awarded runner-up in two poster competitions this summer:
- The SFU Science Undergraduate Research Journal (SURJ) poster competition
- The SFU Symposium on Mathematics & Computation
His work, entitled Optimal Sampling Strategies for Compressive Imaging, presents new, theoretically optimal sampling techniques for imaging using compressed sensing.
Simone Brugiapaglia, Rick Archibald and I have been investigating the robustness of constrained l1-minimization regularization to unknown measurement errors. The vast majority of existing theory for such problems requires an a priori bound on the noise level. Yet in many, if not most, real-world applications such a bound is unknown. Our work provides the first recovery guarantees for a large class of practical measurement matrices, thereby extending existing results for subgaussian random matrices. Besides this, our work sheds new light on sparse regularization in practice, addressing questions about the relationship between model fidelity, noise parameter estimation and reconstruction error.
In it we demonstrate how the structured sparsity of polynomial coefficients of high-dimensional functions can be exploited via weighted l1 minimization techniques. This yields approximation algorithms whose sample complexities are essentially independent of the dimension d. Hence the curse of dimensionality is mitigated to a significant extent.
I am part of the organizing committee for the 1st Biennial Meeting of the SIAM Pacific Northwest Section to be held from Oct 27 to 29, 2017 at Oregon State University. We look forward to seeing you in Corvallis in the Fall!
Frames of Hilbert spaces are ubiquitous in image and signal processing, coding theory and sampling theory. However, they are far less widely known in numerical analysis.
In a new paper with Daan Huybrechs, we take a look at frames from a numerical analyst’s perspective:
First, we point out that frames can be useful tools in numerical analysis where orthonormal bases may be difficult or impossible to construct. Second, we investigate issues concerning stability and accuracy in frame approximations. Our main result is that frame approximations are stable and accurate, provide the function being approximated has representations in the frame with small-norm coefficients.
One application of this work is meshfree approximation of functions on complex geometries using so-called Fourier extensions. Daan maintains a GitHub page with fast algorithms for computing such approximations.
In it, we generalize Rodrigo’s previous work (with Arno Kuijlaars and Nick Trefethen) from equispaced nodes to arbitrary nonequispaced nodes. In particular, our result quantifies the tradeoff between convergence rates and ill-conditioning for nodes distributed according to modified Jacobi weight functions. We also determine a necessary and sufficient sampling rate for stable approximation with polynomial least-squares fitting.
SFU Science wrote a news article about my latest project on MRI reconstruction:
This project is a joint collaboration between me, Rick Archibald, Anne Gelb, Jan Hesthaven, Rodrigo Platte, Guohui Song and Ed Walsh.
I gave a minitutorial at the 2016 SIAM Conference on Uncertainty Quantification on compressed sensing and its application to UQ. My slides are available here:
I have also written a preliminary bibliography of papers relevant to the tutorial: PolynomialCSBibliography
Please feel free to contact me with comments and questions!