Modeling techniques for multiscale difference equation signal models

In this paper, we describe novel encoding and decoding methods for multiscale difference equation (MSDE) signal models. Recently, MSDE signal models have been introduced to exploit self-similarities in signals. The encoding process for MSDE models requires an adaptive signal representation from the dictionary of translations and dilations of the signal to be modeled. Here, we propose a new iterative technique that tries to select an 'active' set under the modeling constraints so that the portion of the signal representation due to the 'inactive' set has small or zero norm. Using methods similar to 'line searches and backtracking' used for global non-linear optimization, this method yields the global minima for exact representation. The decoding process involves an eigenvector analysis of a particular matrix. We provide a fast algorithm to obtain the required eigenvector in O(N/sup 2/ log N) operations. We also provide a perturbation analysis of MSDE models that gives a bound on the reconstruction error due to errors in the MSDE coefficients.