Dictionary Adaptation in Sparse Recovery Based on Different Types of Coherence

Abstract: In sparse recovery, the unique sparsest solution to an under-determined system of linear equations is of main interest. This scheme is commonly proposed to be applied to signal acquisition. In most cases, the signals are not sparse themselves, and therefore, they need to be sparsely represented with the help of a so-called dictionary being specific to the corresponding signal family. The dictionaries cannot be used for optimization of the resulting under-determined system because they are fixed by the given signal family. However, the measurement matrix is available for optimization and can be adapted to the dictionary. Multiple properties of the resulting linear system have been proposed which can be used as objective functions for optimization. This paper discusses two of them which are both related to the coherence of vectors. One property aims for having incoherent measurements, while the other aims for insuring the successful reconstruction. In the following, the influences of both criteria are compared with different reconstruction approaches.