This paper proposes a new method for reconstruction of star-shaped 3D surfaces from scattered datasets, where such surfaces are considered as signals living in the space of square integrable functions on the unit sphere. We first propose a generalization of the Fourier transform on the sphere. A practical reconstruction method is then presented, which interpolates a spherical signal on an equiangular grid, from non-uniformly sampled dataset representing a 3D point cloud. The experiments show that the proposed interpolation method results in smoother surfaces, and higher reconstruction PSNRs than the nearest neighbor interpolation method.