  We present spatial-Slepian transform~(SST) for the representation of signals
on the sphere to support localized signal analysis. We use well-optimally
concentrated Slepian functions, obtained by solving the Slepian
spatial-spectral concentration problem of finding bandlimited and spatially
optimally concentrated functions on the sphere, to formulate the proposed
transform and obtain the joint spatial-Slepian domain representation of the
signal. Due to the optimal energy concentration of the Slepian functions in the
spatial domain, the proposed spatial-Slepian transform allows us to probe
spatially localized content of the signal. Furthermore, we present an inverse
transform to recover the signal from the spatial-Slepian coefficients, and show
that well-optimally concentrated rotated Slepian functions form a tight frame
on the sphere. We develop an algorithm for the fast computation of the
spatial-Slepian transform and carry out computational complexity analysis. We
present the formulation of SST for zonal Slepian functions, which are spatially
optimally concentrated in the polar cap~(axisymmetric) region, and provide an
illustration using the Earth topography map. To demonstrate the utility of the
proposed transform, we carry out localized variation analysis; employing SST
for detecting hidden localized variations in the signal.
