Processing and extraction of information from the data defined on the sphere find applications in cosmology, geophysics, planetary sciences, 3D antenna radiation design, medical imaging, computer vision, virtual reality, wireless communication, acoustics and computer graphics, to name a few. To support spatially localized data analysis in these applications, we present spatial-Slepian transform (SST). We employ well-optimally concentrated Slepian functions, which are obtained as a solution of the Slepian spatial-spectral concentration problem of finding bandlimited and spatially optimally concentrated functions on the sphere, to formulate the proposed transform. Due to the optimal energy concentration of 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 its spatial-Slepian coefficients, formulate an algorithm for fast computation of SST, and carry out computational complexity analysis. We compute the spatial variance of spatial-Slepian coefficients and conduct experiments to show that spatial-Slepian coefficients have better spatial localization than scale-discretized wavelet coefficients. We present the formulation of SST for zonal Slepian functions, which are spatially optimally concentrated in the axisymmetric polar cap region and provide an illustration using a bandlimited Earth topography map.

To demonstrate the utility of the proposed transform, we carry out localized variation analysis, in which we employ SST to detect hidden localized variations in the signal. We illustrate, through a toy example, that spatial-Slepian transform yields a much better estimate of the underlying region of hidden localized variations than scale-discretized wavelet transform.