  It is known that the continuous wavelet transform of a function $f$ decays
very rapidly near the points where $f$ is smooth, while it decays slowly near
the irregular points. This property allows one to precisely identify the
singular support of $f$. However, the continuous wavelet transform is unable to
provide additional information about the geometry of the singular points. In
this paper, we introduce a new transform for functions and distributions on
$\R^2$, called the Continuous Shearlet Transform. This is defined by
$\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{\psi_{ast}}$, where the analyzing
elements $\psi_{ast}$ are dilated and translated copies of a single generating
function $\psi$ and, thus, they form an affine system. The resulting continuous
shearlets $\psi_{ast}$ are smooth functions at continuous scales $a >0$,
locations $t \in \R^2$ and oriented along lines of slope $s \in \R$ in the
frequency domain. The Continuous Shearlet Transform transform is able to
identify not only the location of the singular points of a distribution $f$,
but also the orientation of their distributed singularities. As a result, we
can use this transform to exactly characterize the wavefront set of $f$.
