In computational work, data sets must often be represented on the surface of a sphere or inside a ball, requiring uniform grids. We construct a new volume-preserving projection between a cube and the set of unit quaternions. The projection consists of two steps: an equal-volume mapping from the cube to the unit ball, followed by an inverse generalized Lambert projection to either of the two unit quaternion hemispheres. The new projection provides a one-to-one mapping between a grid in the cube and elements of the special orthogonal group SO(3), i.e., 3D rotations. We provide connections to other rotation representation schemes, including the Rodrigues–Frank vector and the homochoric parameterizations, and illustrate the new mapping through example applications relevant to texture analysis.

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Given a two-dimensional smooth manifold M and a bijective projection p from M on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on M may be generated from a plane WT by the inverse projection p −1 . Examples where the projection maps the whole manifold onto a plane include the two-sphere, the upper sheet of the two-sheeted hyperboloid and the paraboloid. When no such global projection is available, the construction may be performed locally, i.e., around a given point on M. We apply this procedure both to the Continuous WT, already treated in the literature, and to the Discrete WT. Finally, we discuss the case of a WT on a graph, for instance, the graph defined by linking the elements of a discrete set of points on the manifold.

Aims. A method for constructing new uniform grids on the sphere is given. Methods. We define a bijection in R 2 , which maps squares onto discs and preserves areas. Then we use this bijection, combined with Lambert azimuthal projection, for lifting uniform grids from the square to the sphere. Results. We can obtain uniform spherical grids that allow a hierarchical data manipulation and have an isolatitudinal distribution of cells. Compared with HEALPix grids, nowadays the most used in astronomy and astrophysics, our grids have the advantage of allowing easier implementation, and in addition one can move approximating functions from the square to the sphere by a simple technique.

Abstract. In this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations, giving explicit expressions of these wavelets. Our wavelets have small support, a fact which is very important in working with large amounts of data, since the algorithms for decomposition, compression and reconstruction deal with sparse matrices. We also give a quasi-interpolant associated to a given triangulation and study the approximation error. Some numerical examples are given to illustrate the efficiency of our wavelets.