In 1D signal processing local energy and phase can be determined by the analytic signal. Local energy, phase and orientation of 2D signals can be analyzed by the monogenic signal for all i(ntrinsic)1D signals in an rotational invariant way by the generalized Hilbert transform. In order to analyze both i1D and i2D signals in one framework the main idea of this contribution is to lift up 2D signals to the higher dimensional conformal space in which the original signal can be analyzed with more degrees of freedom by the generalized Hilbert transform on the unit sphere. An appropriate embedding of 2D signals on the unit sphere results in an extended feature space spanned by local energy, phase, orientation/direction and curvature. In contrast to classical differential geometry, local curvature can now be determined by the generalized Hilbert transform in monogenic scale space without any derivatives.