Blind Compressed Sensing

TL;DR

This paper introduces blind compressed sensing, a method that reconstructs sparse signals without prior knowledge of the sparsity basis, using constraints to ensure uniqueness and achieve results comparable to traditional methods.

Contribution

It proposes a novel framework for blind compressed sensing with conditions for uniqueness and simple retrieval methods, expanding applicability to all sparse signals.

Findings

Achieves reconstruction comparable to standard compressed sensing without prior basis knowledge

Provides conditions under which the solution is unique for different basis constraints

Demonstrates effectiveness through simulations under sparsity and constraint conditions

Abstract

The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the concept of blind compressed sensing, which avoids the need to know the sparsity basis in both the sampling and the recovery process. We suggest three possible constraints on the sparsity basis that can be added to the problem in order to make its solution unique. For each constraint we prove conditions for uniqueness, and suggest a simple method to retrieve the solution. Under the uniqueness conditions, and as long as the signals are sparse enough, we demonstrate through simulations that without knowing the sparsity basis our methods can achieve results similar to those of standard compressed sensing, which relay on prior knowledge of the sparsity basis. This offers a general sampling and reconstruction system that fits all sparse signals, regardless of the sparsity basis, under the conditions and constraints presented in this work.