Blind compressed sensing: theory

Authors:
Sivan Gleichman;Yonina C. Eldar
Affiliations:
Department of Electrical Engineering, Technion - Israel Institute of Technology;Department of Electrical Engineering, Technion - Israel Institute of Technology
Venue:
LVA/ICA'10 Proceedings of the 9th international conference on Latent variable analysis and signal separation
Year:
2010

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Abstract

Compressed sensing successfully recovers a signal, which is sparse under some basis representation, from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. In this work we define the blind compressed sensing problem, which aims to avoid the need for this prior knowledge, and discuss the uniqueness of its solution. We prove that this problem is ill possed in general unless further constraints are imposed. We then suggest three possible constraints on the sparsity basis that can be added to the problem in order to render its solution unique. This allows a general sampling and reconstruction system that does not require prior knowledge of the sparsity basis.