Signal overcomplete representation and sparse decomposition based on redundant dictionaries

Decomposing a signal based upon redundant dictionaries is a new method for data representation on signal processing. It approximates a signal with an overcomplete system instead of an orthonormal basis to provide a sufficient choice for adaptive sparse decompositions. Replacing the original data with a sparse approximation can result in not only a higher compression ratio, but also greater flexibility in capturing the inherent structure of the natural signals with the redundancy of dictionaries. This paper gives an overview of a series of recent results in this field, and deals with the relationship between sparsity of signal decomposition and incoherence of dictionaries with BP and MP algorithms. The current and future challenges of the dictionary construction are discussed.