Householder Transform based Estimation of Signal and Sparsifying Basis from Blind Compressive Measurements

Blind compressive sensing generates a reduced set of measurements of a signal without the knowledge of the sparsifying basis. In this context, the unique recovery and reconstruction of the underlying signal is a challenging problem. One of the properties of the Householder operator is to transform a non-sparse vector in a finite dimensional vector space to a sparse representation of the vector, preserving its ℓ 2 -norm. In this work, we present an algorithm that invokes the sparsifying property of the Householder transforms for the recovery and reconstruction of a signal from its reduced set of blind measurements. The algorithm is an iterative procedure using two objective functions. The first objective function is to estimate the sparse set of representing coefficients. The second objective function jointly estimates the unknown sparsifying transform and the unknown signal. Both the objective functions are entangled to ensure that the estimates meet the fidelity requirement with the available reduced set of measurements. An ℓ 1 -trend filtering is applied to minimize overfitting the estimate of the signal. The signal is uniquely recovered and reconstructed upto an acceptable lower error bound.