Spherical Deconvolution

This paper proposes nonparametric deconvolution density estimation overS2. Here we would think of theS2elements of interest being corrupted by randomSO(3) elements (rotations). The resulting density on the observations would be a convolution of theSO(3) density with the trueS2density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis onSO(3) andS2, involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expectedS2and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution.