  We propose a transform for signals defined on the sphere that reveals their
localized directional content in the spatio-spectral domain when used in
conjunction with an asymmetric window function. We call this transform the
directional spatially localized spherical harmonic transform (directional
SLSHT) which extends the SLSHT from the literature whose usefulness is limited
to symmetric windows. We present an inversion relation to synthesize the
original signal from its directional-SLSHT distribution for an arbitrary window
function. As an example of an asymmetric window, the most concentrated
band-limited eigenfunction in an elliptical region on the sphere is proposed
for directional spatio-spectral analysis and its effectiveness is illustrated
on the synthetic and Mars topographic data-sets. Finally, since such typical
data-sets on the sphere are of considerable size and the directional SLSHT is
intrinsically computationally demanding depending on the band-limits of the
signal and window, a fast algorithm for the efficient computation of the
transform is developed. The floating point precision numerical accuracy of the
fast algorithm is demonstrated and a full numerical complexity analysis is
presented.
