  We consider an extension of the continuous shearlet transform which
additionally uses higher order shears. This extension, called the Taylorlet
transform, allows for a detection of the position, the orientation, the
curvature and other higher order geometric information of singularities.
Employing the novel vanishing moment conditions of higher order,
$\int_\mathbb{R} g(t^k)t^m dt=0$, on the analyzing function, we can show that
the Taylorlet transform exhibits different decay rates for decreasing scales
depending on the choice of the higher order shearing variables. This enables a
more robust detection of the geometric information of singularities.
Furthermore, we present a construction that yields analyzing functions which
fulfill vanishing moment conditions of different orders simultaneously.
