This coherence of curvelets under the wave flow has far-reaching consequences. In image (2), we depict a wave front---where the underlying function is nonsmooth---initially in the shape of an ellipse. If we let it evolve under the wave equation, the ellipse will shrink to the fishtail pattern (the outgoing ellipse, if it exists, is not shown). A caustic has formed, and the wavefront seems to have lost some of its smoothness near the tips of the tail, named cusps. This is not a concern in phase-space, or in the curvelet domain, where the cusp pattern is in fact explained as a superposition of curvelets. Curvelets do not see caustics; keep in mind that the wave equation is linear and that the superposition principle is valid.

We have developed a numerical wave solver based on the same sparsity ideas.