Sampling and Reconstruction driven by Sparsity Models: Theory and Applications

Abstract:

Modelling signals as sparse in a proper domain has proved fruitful in many signal and image processing applications. Recently, the notion of sparsity has lead to new sampling theories that have demonstrated that the prior knowledge that signals can be sparsely described in a basis or in a parametric space can be used to sample and perfectly reconstruct such signals at a significantly reduced rate. The insight that sub-Nyquist sampling can, under some circumstances, allow perfect reconstruction is revolutionizing signal processing, communications and inverse problems.

In this talk we first recall that sampling involves the reconstruction of continuous-time or continuous-space signals from discrete measurements (samples) and show how to relate the discrete measurements to some properties of the original continuous signal (for example its Fourier transform at specific frequencies). This is achieved by using the theory of approximation of exponentials and the so called generalized Strang-Fix conditions. Given this partial knowledge of the original signal, we then reconstruct it by using sparsity priors and in particular we provide exact reconstruction formulas for specific classes of 1-D and 2-D signals.

We then consider applications of these ideas to super-resolution imaging, neuroscience and sensor networks. In particular, we present:

• a method for enhancing image resolution by one order of magnitude
• a new fast algorithm for calcium-transients detection from two-photon calcium imaging
• a method for estimating diffusion fields driven by localized sources using spatio-temporal sensor measurements

This is joint work with T. Blu (CUHK), J. A. Uriguen (ICL), J. Onativia Bravo (ICL) J. Murray-Bruce (ICL), A. Scholefield (ICL) and S. Schultz (ICL).

This work is supported by the European Research Council (ERC) starting investigator award Nr. 277800 (RecoSamp).

Biography

Dr Pier Luigi Dragotti, Professor of Signal Processing, Imperial College of London

Pier Luigi Dragotti is Professor of Signal Processing with the Electrical and Electronic Engineering Department at Imperial College London. He received the Laurea degree (summa cum laude) in electrical and electronic engineering from the University of Naples Federico II, Naples, Italy, in 1997; the master's degree in communications systems from the Swiss Federal Institute of Technology of Lausanne (EPFL), Lausanne, Switzerland, in 1998, and the PhD degree from EPFL in April 2002. He has held several visiting positions at different universities and research centers. In particular, he was a Visiting Student with Stanford University, Stanford, CA, USA, in 1996, a Researcher with the Mathematics of Communications Department, Bell Labs, Lucent Technologies, Murray Hill, NJ, USA, in 2000 and a Visiting Scientist with the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2011. He was Technical Co-Chair for the European Signal Processing Conference in 2012, an Associate Editor of the IEEE Transactions on Image Processing from 2006 to 2009 and an Elected Member of the IEEE Image, Video and Multidimensional Signal Processing Technical Committee (2008-2013). Currently he is an elected member of the IEEE Signal Processing Theory and Method Technical Committee.

He is a recipient of the ERC Starting Investigator Award for the project RecoSamp. His work includes sampling theory, wavelet theory and its applications, image and video compression, image-based rendering, and image super-resolution.