On the construction of low-pass filters on the unit sphere

This paper considers the problem of construction of low-pass filters on the unit sphere, which has wide ranging applications in the processing of signals on the unit sphere. We propose a design criterion for the construction of strictly bandlimited low-pass filters in the spectral domain with optimal concentration in the specified polar cap region in the spatial domain. Our approach uses the weighted sum of the first optimally concentrated eigenfunctions from appropriately formulated Slepian concentration problems on the sphere. Furthermore, in order to reduce the computational complexity of the proposed algorithm, we develop a closed-form expression to accurately model these eigenfunctions. We illustrate the construction of low-pass filters using the proposed approach and demonstrate the advantage of our method approach compared to a diffusion based approach in the literature in terms of control over both bandwidth in the spectral domain and concentration in the spatial domain.