  Curvelets are efficient to represent highly anisotropic signal content, such
as a local linear and curvilinear structure. First-generation curvelets on the
sphere, however, suffered from blocking artefacts. We present a new
second-generation curvelet transform, where scale-discretised curvelets are
constructed directly on the sphere. Scale-discretised curvelets exhibit a
parabolic scaling relation, are well-localised in both spatial and harmonic
domains, support the exact analysis and synthesis of both scalar and spin
signals, and are free of blocking artefacts. We present fast algorithms to
compute the exact curvelet transform, reducing computational complexity from
$\mathcal{O}(L^5)$ to $\mathcal{O}(L^3\log_{2}{L})$ for signals band-limited at
$L$. The implementation of these algorithms is made publicly available.
Finally, we present an illustrative application demonstrating the effectiveness
of curvelets for representing directional curve-like features in natural
spherical images.
