An adaptive basis estimation method for compressed sensing with applications to missing data reconstruction

The subject of compressed sensing, especially, the related concept of sparse representation has been growing into an exciting area with a diverse set of applications in the fields of image sensing and analysis, signal compression, network reconstruction, etc. The efficacy of the associated techniques depends on the ability to discover a suitable basis for a sparse representation of the underlying signal. This paper presents a method for discovering this basis adaptively from the data. Specifically, the method estimates the dictionary of basis functions that maps the sub-sampled signal to the sparse representation of the signal. We present an application of this technique to the reconstruction of missing data, which is an important problem in all data-driven methods. Two case studies, namely, the reconstruction of missing data in a liquid level system and missing pixels of a 2-D signal (image) are presented. Results show that the proposed algorithm outperforms the existing KSVD algorithm in terms of both accuracy and speed of the reconstruction.