Blind Compressive Sensed Signal Reconstruction Using Householder Transforms

The recovery/reconstruction of blind compressive sensed signal is a highly ill-posed problem that asserts the reconstruction of the signal from its reduced set of measurements without knowing the representing basis. In this paper, we propose a method that makes use of the sparsifying property of Householder transformation. The method identifies the reflection vector, relative to which a Householder transform exists, that gives the sparse representation of the signal. The estimate of the signal is updated using the identified sparse set of coefficients and the Householder transform by applying
$\ell_{1}$
-trend filtering. The proposed method achieves a signal reconstruction comparable with the existing techniques for speech, tidal wave, image and synthetic signals. The unique reconstruction of the signal upto a lower error bound is achieved using the proposed method.