Compressive Sensing with Side Information: Analysis, Measurements Design and Applications

Compressive sensing is a breakthrough technology in view of the fact that it enables the acquisition and reconstruction of certain signals with a number of measurements much lower than those dictated by the Shannon-Nyquist paradigm. It has also been recognised in the last few years that it is possible to improve compressive sensing systems by leveraging additional knowledge – so-called side information – that may be available about the signal of interest. The goal of this thesis is to investigate how to improve the acquisition and reconstruction process in compressive sensing systems in the presence of side information. In particular, by assuming that both the signal of interest and the side information obey a joint Gaussian mixture model (GMM), the thesis focuses on the analysis and the design of linear measurements for two different scenarios: i) the scenario where one wishes to design a linear projection matrix to capture the signal of interest; and ii) the scenario where one wishes to design a linear projection matrix to capture the side information. In both cases, we derive sufficient and (occasionally) necessary conditions on the number of measurements needed for the reliable reconstruction in the low-noise regime and we also derive linear measurement designs that are close to optimal. Numerical results are presented with synthetic data from both Gaussian and GMM distributions and with real world imaging data that confirm that analysis is well aligned with practice. We also showcase our measurement design scheme can lead to significant improvement on the application example associated with the reconstruction of high-resolution RGB images from gray scale images using low-resolution, compressive, hyperspectral measurements as side information