Even/odd decomposition made sparse: A fingerprint to hidden patterns

The very fundamental operation of even/odd decomposition is at the core of some of the simplest information representation and signal processing tasks. So far most of its use has been for rearranging data to provide fast implementations of various types of transforms (Fourier, DCT, …) or for achieving elementary data transformation, such as the Walsh–Hadamard transform. This work proposes to look into the decomposition framework to obtain a richer perspective. In the context of an iterated even/odd decomposition, it is possible to pinpoint intermediate layered levels of symmetries which cannot be easily captured in the original data. In addition this determines a hierarchical fingerprinting for any sort of continuous finite support analog signal or for any discrete-time sequence which may turn out useful in several recognition or categorization tasks. It also may help to achieve sparsity within a natural hierarchical framework, which could be easily extended for many other types of orthogonal transformations. This paper also suggests a global measure of the energy imbalance across the hierarchy of the decomposition to capture the overall fingerprinting of this interpretation.