Data Compression on the Sphere Using Faber Decomposition

We describe a simple function decomposition given by Faber, and from this derive a fast and easily implemented compression scheme on the sphere, which we apply to a standard set of earth data. x1. Introduction The practical problem of eeciently representing functions deened on the sphere has received considerable attention over the last decade. In addition to the speed advantage that they can give to visualization applications in computer graphics (i.e., illumination and climate models), eecient representations of data deened on the sphere also give rise to fast compression algorithms, desirable for analysis and display of sets of planetary data (i.e., topography and remote sensing), which are often so voluminous as to preclude practical manipulation of the entire data set. Several recent approaches to solving this representation problem exploit powerful wavelet constructions. Constructions of wavelets on the sphere based on tensor product splines were presented by Dahlke et al. in 2] and by Lyche and Schumaker in 5]. Other methods used to generate wavelets on the sphere include the use of so-called spherical basis functions by Narcowich and Ward in 6], the use of spherical harmonics by Freeden and Schreiner in 4], and the \lifting" scheme of Schrr oder and Sweldens, introduced in 7]. In this paper, we do not utilize wavelets to compress data on the sphere, but rather, we use a simple function decomposition given by Faber. The sections herein are organized as follows: we rst deene the Faber function decomposition, and next prove an important stability theorem. In x4 we describe a data compression technique based on Faber's construction, and then port this method to the sphere in x5. Finally, after a section describing our implementation, we discuss future research directions.