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:
Frames and numerical approximation
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.
Alexei Shadrin, Rodrigo Platte and I have just finished a paper on stability and instability in approximating analytic functions from nonequispaced points:
Optimal sampling rates for approximating analytic functions from pointwise samples
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:
Compressed sensing and application to uncertainty quantification
I have also written a preliminary bibliography of papers relevant to the tutorial: PolynomialCSBibliography
Please feel free to contact me with comments and questions!
My postdoc Il Yong Chun, visiting student Chen Li and I have developed some new and improved recovery guarantees for parallel acquisition systems. These are described in two submitted conference papers:
Optimal sparse recovery for multi-sensor measurements
Sparsity and parallel acquisition: optimal uniform and nonuniform recovery guarantees
These new results build on a previous paper of ours which introduced a compressed sensing framework for multi-sensor systems – see this past news item. The new results explain the practical recovery capabilities of a much broader class of sensing scenarios.
Bernard Deconinck (President), Malgorzata Peszynska (Vice President), Leming Qu (Treasurer) and myself (Secretary) have started a new SIAM section for the Pacific Northwest region (SIAMPNWS)! The aim of the section is to bring together communities in the Pacific Northwest that are active in industrial and applied mathematics. If you are in the Pacific Northwest and would like to be involved, please drop us a line.
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.