A continuous boostlet transform for acoustic waves in space-time

Abstract: Sparse representation systems that encode the signal architecture have had an exceptional 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, the present study introduces a representation system for acoustic waves in 2D space-time, referred to as the boostlet transform, which encodes sparse features of natural acoustic fields with the Poincar\'e group and isotropic dilations. Continuous boostlets, $\psi_{a,\theta,\tau}(\varsigma) = a^{-1} \psi \left(D_a^{-1} B_\theta^{-1}(\varsigma-\tau)\right) \in L^2(\mathbb{R}^2)$, are spatiotemporal functions parametrized with dilations $a > 0$, Lorentz boosts $\theta \in \mathbb{R}$, and translations $\smash{\tau \in \mathbb{R}^2}$ in space--time. The admissibility condition requires that boostlets are supported away from the acoustic radiation cone, i.e., have phase velocities other than the speed of sound, resulting in a peculiar scaling function. The continuous boostlet transform is an isometry for $L^2(\mathbb{R}^2)$, and a sparsity analysis with experimentally measured fields indicates that boostlet coefficients decay faster than wavelets, curvelets, wave atoms, and shearlets. The uncertainty principles and minimizers associated with the boostlet transform are derived and interpreted physically.