  Wavelets on the sphere are reintroduced and further developed independently
of the original group theoretic formalism, in an equivalent, but more
straightforward approach. These developments are motivated by the interest of
the scale-space analysis of the cosmic microwave background (CMB) anisotropies
on the sky. A new, self-consistent, and practical approach to the wavelet
filtering on the sphere is developed. It is also established that the inverse
stereographic projection of a wavelet on the plane (i.e. Euclidean wavelet)
leads to a wavelet on the sphere (i.e. spherical wavelet). This new
correspondence principle simplifies the construction of wavelets on the sphere
and allows to transfer onto the sphere properties of wavelets on the plane. In
that regard, we define and develop the notions of directionality and
steerability of filters on the sphere. In the context of the CMB analysis,
these notions are important tools for the identification of local directional
features in the wavelet coefficients of the signal, and for their
interpretation as possible signatures of non-gaussianity, statistical
anisotropy, or foreground emission. But the generic results exposed may find
numerous applications beyond cosmology and astrophysics.
