Efficient Compressive Sensing with Deterministic Guarantees Using Expander Graphs

Compressive sensing is an emerging technology which can recover a sparse signal vector of dimension n via a much smaller number of measurements than n. However, the existing compressive sensing methods may still suffer from relatively high recovery complexity, such as O(n^3), or can only work efficiently when the signal is super sparse, sometimes without deterministic performance guarantees. In this paper, we propose a compressive sensing scheme with deterministic performance guarantees using expander-graphs-based measurement matrices and show that the signal recovery can be achieved with complexity O(n) even if the number of nonzero elements k grows linearly with n. We also investigate compressive sensing for approximately sparse signals using this new method. Moreover, explicit constructions of the considered expander graphs exist. Simulation results are given to show the performance and complexity of the new method

Compressive Source Separation: Theory and Methods for Hyperspectral Imaging

With the development of numbers of high resolution data acquisition systems and the global requirement to lower the energy consumption, the development of efficient sensing techniques becomes critical. Recently, Compressed Sampling (CS) techniques, which exploit the sparsity of signals, have allowed to reconstruct signal and images with less measurements than the traditional Nyquist sensing approach. However, multichannel signals like Hyperspectral images (HSI) have additional structures, like inter-channel correlations, that are not taken into account in the classical CS scheme. In this paper we exploit the linear mixture of sources model, that is the assumption that the multichannel signal is composed of a linear combination of sources, each of them having its own spectral signature, and propose new sampling schemes exploiting this model to considerably decrease the number of measurements needed for the acquisition and source separation. Moreover, we give theoretical lower bounds on the number of measurements required to perform reconstruction of both the multichannel signal and its sources. We also proposed optimization algorithms and extensive experimentation on our target application which is HSI, and show that our approach recovers HSI with far less measurements and computational effort than traditional CS approaches.Comment: 32 page