Fast directional spatially localized spherical harmonic transform

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.Comment: 12 pages, 5 figure

Revisiting Slepian concentration problem on the sphere for azimuthally non-symmetric regions

The problems of filtering, spectral analysis and spectral estimation have been investigated on the sphere using azimuthally symmetric functions as kernels which treat all the directions uniformly. In this work, we extend the concentration problem on the sphere for an azimuthally non-symmetric spatial region on the sphere. Our approach is different in a sense that we obtain the family of spatially concentrated bandlimited mutually orthogonal functions by maximizing the contribution of spherical harmonics components of all degrees and orders within the spectral bandwidth. We also provide analysis of the eigenfunctions for different bandwidths and non-symmetric regions and illustrate the concentration of eigenfunctions with the help of examples. Also we formulate the definition of filtering using azimuthally non-symmetric functions. The proposed eigenfunctions can be used to revisit the problems of estimation, localized spectral analysis, smoothing and filter design on the sphere