A structural property of the wavelet packet transform method to localise incoherency of a signal

Abstract This paper proves a general structural property of the wavelet tree for a given seminorm in the context of the wavelet packet transform method. This structural property can be used in denoising algorithms of different applications to guarantee the optimality of novel search strategies. The property holds for any input signals using any orthogonal analysing wavelet families. The property holds for any norms, which results to be a convex function through the wavelet tree. Using a defined norms, seminorms or pseudonorms, this property can be used to detect incoherent parts of an input signal by using the minimal depth of the tree. In this sense the proposed denoising procedure works without thresholds for the localisation of different kinds of noise, as well as for a stop criterium for an optimal representation of the incoherency. The proof of this property is performed by mathematical induction and the demonstration is based on orthonormality with the help of the multiresolution framework aspects of the wavelets packet tree. The Theorem is independent of the definition of the adopted norm and of the incoherent part of an input signal. In this sense, the discovered property is a general one, which is related to any norm and any nature of the signal incoherence. It can be used in different applications, in which a minimum of a norm is required to be calculated through the wavelet tree.