  We propose a novel scheme for efficient Dirac mixture modeling of
distributions on unit hyperspheres. A so-called hyperspherical localized
cumulative distribution (HLCD) is introduced as a local and smooth
characterization of the underlying continuous density in hyperspherical
domains. Based on HLCD, a manifold-adapted modification of the Cram\'er-von
Mises distance (HCvMD) is established to measure the statistical divergence
between two Dirac mixtures of arbitrary dimensions. Given a (source) Dirac
mixture with many components representing an unknown hyperspherical
distribution, a (target) Dirac mixture with fewer components is obtained via
matching the source in the sense of least HCvMD. As the number of target Dirac
components is configurable, the underlying distributions is represented in a
more efficient and informative way. Based upon this hyperspherical Dirac
mixture reapproximation (HDMR), we derive a density estimation method and a
recursive filter. For density estimation, a maximum likelihood method is
provided to reconstruct the underlying continuous distribution in the form of a
von Mises-Fisher mixture. For recursive filtering, we introduce the
hyperspherical reapproximation discrete filter (HRDF) for nonlinear
hyperspherical estimation of dynamic systems under unknown system noise of
arbitrary form. Simulations show that the HRDF delivers superior tracking
performance over filters using sequential Monte Carlo and parametric modeling.
