  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.
