Sparse decompositions in "incoherent" dictionaries

The purpose of this paper is to generalize a result by Donoho, Huo, Elad and Bruckstein on sparse representations of signals/images in a union of two orthonormal bases. We consider general (redundant) dictionaries in finite dimension, and derive sufficient conditions on a signal/image for having a unique sparse representation in such a dictionary. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of a combinatorial optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant. The special case where the dictionary is given by a union of several orthonormal bases is studied in more detail and some examples are given.

• Type: Conference
• Year: 2003
• Where: ICIP
• Authors: Morten Nielsen, Rémi Gribonval
• Keywords: Combinatorial Optimization Problem, ICIP 2003, Image Processing, Linear Programming Problem, Sparse Representations, Unique Sparse Representation