Reconstructing signals from their blind compressed measurements through consistent extension of autocorrelation sequence

The main challenge in blind compressive sensing is to uniquely reconstruct a sparse signal from its undersampled measurements without prior knowledge of the representing basis. This paper proposes a reconstruction algorithm that estimates a signal from its blind compressed measurements using a linear prediction method of autocorrelation sequence extension. The method extends the lower dimensional autocorrelation sequence of the blind compressed measurement vector to a higher dimensional autocorrelation sequence. The autocorrelation matrix associated with the extended autocorrelation sequence is symmetric and diagonalisable. The matrix that diagonalises the extended autocorrelation matrix exhibits performance close to the Karhunen-Loeve transform. Hence, it is identified as the matrix of sparsifying basis with respect to which the underlying signal exhibits sparsity. This matrix of sparsifying basis is utilised to retrieve the sparse set of representing coefficients using the orthogonal matching pursuit algorithm. The sparse signal is estimated maintaining consistency with the available measurements. The algorithm is formulated as a cascade of three lifting steps, namely, the autocorrelation extension, identification of the sparsifying transform, and the recovery and reconstruction of signals. The signals are reconstructed uniquely with the reconstruction error lower bounded to the order of ${10}^{-3}$.