New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle‐shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j, each element has an envelope that is aligned along a “ridge” of length 2−j/2 and width 2−j.