Block-constrained compressed sensing

Abstract : This PhD. thesis is dedicated to combine compressed sensing with blocks structured acquisition. For several years, Compressed Sensing (CS) has been an appealing framework for signal and image processing applications, in order to reduce acquisition times while perfectly recovering an unknown signal $x\in \mathbb{C}^n$. However, current CS theories lead to sampling patterns, consisting in sets of isolated measurements, that can be hard to implement on real devices. In contrast, in various applications such as Magnetic Resonance Imaging, the samples should lie on a continuous trajectory due to electromagnetic constraints. Therefore, sampling patterns used in practice are very structured and they depart from typical CS sampling schemes. Yet, with no theoretical justification by current CS theories, they lead to satisfactory reconstruction results. In this work, it is proposed to explain the gap between theoretical Compressed Sensing results and successful structured acquisition made in practice. To do so, two perspectives have been considered. In the first part of this work, theoretical CS results are derived with blocks acquisition constraints, for the recovery of \emph{any} $s$-sparse signal and for the recovery of a vector with a given support $S$. We show that structured acquisition can be successfully used in a CS framework, provided that the signal to reconstruct presents an additional structure in its sparsity, adapted to the sampling constraints. In the second part of this work, we propose numerical methods to generate efficient block sampling schemes. Standard CS gives a probability distribution from which isolated measurements should be drawn. The idea is then to project this probability distribution on a set of measures supported on admissible patterns.
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  • HAL Id : tel-01264178, version 1

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Claire Boyer. Block-constrained compressed sensing. Information Theory [math.IT]. Université Paul Sabatier, 2015. English. ⟨tel-01264178⟩

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