General design algorithm for sparse frame expansions

Signal expansions using frames may be considered as generalizations of signal representations based on transforms and filter banks. Frames, or dictionaries, for sparse signal representations may be designed using an iterative algorithm with two main steps: (1) Frame vector selection and expansion coefficient determination for signals in a training set, selected to be representative of the signals for which compact representations are desired, using the frame designed in the previous iteration. (2) Update of frame vectors with the objective of improving the representation of step (1). This method for frame design was used by [Engan et al., Signal Processing 80 (2000) 2121–2140] for block-oriented signal expansions, i.e. generalizations of block-oriented transforms and by [Aase et al., IEEE Trans. Signal Process. 49(5) (2001) 1087–1096] for non-block-oriented frames—for short overlapping frames, that may be viewed as generalizations of critically sampled filter banks. Here we give the solution to the general frame design problem using the compact notation of linear algebra. This makes the solution both conceptually and computationally easier, especially for the overlapping frame case. Also, the solution is more general than those presented earlier, facilitating the imposition of constraints, such as symmetry, on the designed frame vectors.