Compressive Sensing

Compressive sensing is a recent field of mathematical signal processing which predicts that accurate signal and image recovery from seemingly incomplete measured data is possible. The key ingredient consists in the empirical observation that many types of signals can be well-represented by a sparse expansion in a suitable basis. Efficient recovery algorithms include l1-minimization and greedy algorithms are available.

Remarkably, all provably (near-)optimal measurement matrices modelling the linear data acquisition process know so far are based on randomness. While Gaussian and Bernoulli random matrices indeed provide optimal guarantees on the minimal amount of measurement required for exact and approximate sparse recovery, they are of limited use for practical purposes due to the lack of structure. Therefore, several structured random matrices have been studied in the context of compressive sensing, including random partial Fourier modelling sparse recovery from randomly selected Fourier coefficients and partial random circulant matrices connected to subsampled random convolutions.

Compressive sensing is motivated by many applications including magnetic resonance imaging, radar, wireless communications, coded aperture imaging, analog to digital conversion, astronomical signal processing, sparse regression problems, numerical solution of (parametric) differential equations, function recovery in high dimensions and more.

The talk will give a basic introduction to compressive sensing with an emphasis on the work of the speaker.