October 26, 2015
Pier Luigi Dragotti (Imperial College London): "Sampling and inverse problems regularized by sparsity" @ Universiteit Antwerpen, Department of Mathematics and Computer Science
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 (e.g., 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. The reconstruction algorithms are all based aroound variations of evergreen Prony's method.
We then consider applications of these ideas to super-resolution imaging, neuroscience and inverse problems. 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 and a method for estimating diffusion fields driven by localized sources using spatio-temporal sensor measurements.