Deterministic Construction of Sparse Sensing Matrices via Finite Geometry

Abstract

Compressed sensing is a novel sampling technique that provides a fundamentally new approach to data acquisition. Comparing with the traditional method, compressed sensing asserts that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. As a long-standing subject in combinatorial design theory, the packing design gives rise to deterministic sparse matrices with low coherence. With this framework in mind, we investigate a series of packing designs originated from finite geometry. The connection between the sensing matrix and finite geometry is revealed, using the packing design as a bridge. More specifically, we construct a series of $m \times n$ binary sensing matrices with sparsity order $k=\Theta (m^{1/3})$ or $k=\Theta (m^{1/2})$ . Moreover, we use an embedding operation to merge our binary matrices with matrices having low coherence. Comparing with the original binary matrices, this embedding operation generates modified matrices with better recovery performance. Numerical simulations show that our binary and modified matrices outperform several typical sensing matrices. The sparse property of our matrices helps to accelerate the recovery process, which is very preferable in practice.