Spatial-Slepian Transform on the Sphere

TL;DR

The paper introduces the spatial-Slepian transform (SST) for localized signal analysis on the sphere, leveraging optimally concentrated Slepian functions to enable spatially localized signal representation and recovery.

Contribution

It proposes a novel transform based on Slepian functions for localized spherical signal analysis, including an efficient computation algorithm and applications to Earth topography data.

Findings

SST provides localized signal analysis on the sphere.

The transform forms a tight frame with rotated Slepian functions.

Application to Earth topography demonstrates utility in detecting localized variations.

Abstract

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.