A reconstruction set of a discrete wavelet maxima representation

The structure of a reconstruction set (the set of all signals satisfying a given representation) is studied. Assuming finite data length and using the finite-dimensional linear space approach, a general form of a reconstruction set from a discrete wavelet maxima representation is presented. Necessary and sufficient conditions for uniqueness are described. These conditions can be implemented as a test for uniqueness. Although, in most cases, the discrete wavelet maxima representation is unique, the conjecture about the uniqueness of the wavelet maxima representation is not true. A family of sequences with the same maxima representation is shown. The exact reconstruction set is calculated, and it is shown to consist of similarly shaped sequences