Various practical applications require effective representation of functions on the sphere or on the ball. Examples include the gravitational field modeling, geomagnetism, helioseismology, astronomy, cosmology, and seismology. In most applications the functions on the sphere have been traditionally represented in terms of spherical harmonics. The nature of spherical harmonics as global functions, however, creates problems. The spherical harmonic representations rely on delicate cancellation/interference of spherical harmonics and are slowly convergent. The existing representations on the ball using orthogonal polynomials or other methods have the same drawbacks or are even worse. The primary objective of this project is to develop innovative multiscale data representations on the sphere and on the ball based on newly created wavelet type systems, called "needlets". The needlet system on the sphere consists of almost exponentially localized radial band-limited functions, which are automatically extendable to harmonic functions in the exterior of the sphere and thereby enabling needlet representations to provide a highly effective framework for representation and analysis of harmonic functions such as the gravitational potential. The needlet compatibility with spherical harmonics permits for fast conversions between needlet and spherical harmonic representations. This makes them easy to integrate into the existing models based on spherical harmonics. Furthermore, the superb localization of needlets at fine scales makes needlet-based models highly amenable to efficient local updates, which is a significant advantage over traditional models based on spherical harmonics. The needlet system on the ball has a similar structure and consists of almost exponentially localized algebraic polynomials. Theoretical results show that the new representations are superior to the existing mono- and multiscale methods used in these areas. An important element of the proposed research is the employment of nonlinear approximation methods for effective representation and approximation of functions from needlets. These are multilevel techniques which allow control of the uniform (or other) norm of the error of approximation. Another step forward will be the development of anisotropic elements on the sphere and ball (e.g. curvlets) for better extraction of curvlinear features of the data.
The targeted applications of this research are mainly in the domain of geodesy. Most geodetic applications rely on the ability to compute accurately the gravitational (disturbing) potential. This project will pursue the implementation of multiscale needlet representations for modeling of the gravitational potential. Other potential applications of the new representations are in geomagnetism, helioseismology, astronomy, cosmology, seismology, where spherical harmonic representations are widely used. An important goal of this project is to promote the broad utilization of the new representations in other diverse disciplines (from geophysics to high-speed videoendoscopy) and to stimulate interest in younger mathematicians to this area. This research project offers an excellent opportunity for graduate and undergraduate students at the University of South Carolina to participate in testing ideas for further development.
