Compressive sensing matrices and hash families

Deterministic construction of measurement matrices for compressive sensing can be effected by first constructing a relatively small matrix explicitly, and then inflating it using a column replacement technique to form a large measurement matrix that supports at least the same level of sparsity. In particular, using easily developed null space conditions for ℓ₀- and ℓ₁-recoverability, properties of the pattern matrix used to select columns lead to well-studied matrices, separating and distributing hash families. Two-stage compression and recovery techniques are developed that employ more computationally intensive ℓ₀-recoverability for small matrices and simpler ℓ₁-recoverability for one larger matrix; this can reduce the number of measurements required.