The Uniqueness Question of Discrete Wavelet Maxima Representation

Abstract

In this paper, we analyze the discrete wavelet maxima representation from the reconstruction point of view. Assuming finite data length and using the finite dimensional linear space approach we present necessary and sufficient conditions for the given representation to be unique. The algorithm which tests for uniqueness is shown. A general form of a solution to the reconstruction problem is described. The above results are valid for any bank of linear filters where the outputs are sampled at extreme values. For illustration we show two wavelet transform examples. The first is the common unique-representation case. The second is an interesting examples of family of sequences which have the same maxima representation. that is, we show an example of a non-unique discrete wavelet maxima representation.