A boostlet transform for wave-based acoustic signal processing in space-time

Sparse representation systems that encode signal architecture have had a profound impact on sampling and compression paradigms. Remarkable examples are multi-scale directional systems, which, similar to our vision system, encode the underlying architecture of natural images with sparse features. Inspired by this philosophy, we introduce a representation system for wave-based acoustic signal processing in 2D space--time, referred to as the \emph{boostlet transform}, which encodes sparse features of natural acoustic fields using the Poincaré group and isotropic dilations. Boostlets are spatiotemporal functions parametrized with dilations, Lorentz boosts, and translations in space--time. Physically speaking, boostlets are supported away from the acoustic radiation cone, i.e., having broadband frequency with phase velocities other than the speed of sound, resulting in a peculiar scaling function. We formulate a discrete boostlet frame using Meyer wavelets and bump functions and examine its sparsity properties. An analysis with experimentally measured fields indicates that discrete boostlet coefficients decay significantly faster and attain superior reconstruction performance than wavelets, curvelets, shearlets, and wave atoms. The results demonstrate that boostlets provide a natural, compact representation system for acoustic waves in space-time.